2014 Book TopologyOptimizationInStructur PDF
2014 Book TopologyOptimizationInStructur PDF
2014 Book TopologyOptimizationInStructur PDF
George I. N. Rozvany
Tomasz Lewiński
Editors
Topology
Optimization
in Structural
and Continuum
Mechanics
International Centre
for Mechanical Sciences
CISM Courses and Lectures
Series Editors:
The Rectors
Friedrich Pfeiffer - Munich
Franz G. Rammerstorfer - Wien
Elisabeth Guazzelli - Marseille
Executive Editor
Paolo Serafini - Udine
Topology Optimization in
Structural and
Continuum Mechanics
Editors
George I. N. Rozvany
Budapest University of Technology and Economics, Hungary
Tomasz Lewiński
Warsaw University of Technology, Poland
ISSN 0254-1971
ISBN 978-3-7091-1642-5 ISBN 978-3-7091-1643-2 (eBook)
DOI 10.1007/ 978-3-7091-1643-2
Springer Wien Heidelberg New York Dordrecht London
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Structural Shape and Topology Optimization
François Jouve
University Paris Diderot (Paris 7) and Laboratoire J.L.Lions
where Uad is the set of admissible shapes among all the subdomains included
into Ω. It may include some constraints, like, for example, an upper bound
over the volume of ω. J depends on ω through the solution of a partial
differential equation (or a system of PDE’s) posed on ω.
In the sequel we will focus on structural shape optimization. The govern-
ing equations will be, most of the time, the linearized elasticity system. Of
course the choice of the objective function, together with the set of admis-
sible solutions (including constraints) is very important. Popular objective
functions will be described later with examples of applications.
Mathematicians usually choose classical objective functions (sometimes
because of theirs good properties rather than their relevance for applica-
tions), while real applications have sometimes “fuzzy” needs. An example
of a popular objective function is the compliance for elastic solids, that leads
to the following problem: “find the most rigid structure for a given set of
external forces and a given volume of constitutive material”.
But real problems coming from industry are usually more complex, in-
volving multiple constraints and loading cases. Examples of such industrial
constraints are: maximum local admissible stress, molding constraints, min-
imal size of small parts, minimal radii of curvature everywhere, cooling of
the molded piece in less than a given time, etc... Some of them are even
difficult to express mathematically in an explicit way (for example the con-
straints related to the molding process), and may involve the solution of
additional complex problems.
We can then consider two interesting problems: find the best configura-
tion maximizing the mean temperature over Ω or minimizing it. Without
further constraints, these two problems are trivial and admit respectively
χ ≡ 0 and χ ≡ 1 as solutions (domain full of respectively α and β material).
If the most conducting material β is supposed to be more expensive
than the other one, we don’t want to use too much of it. The control of
132 F. Jouve
the balance between the performance we want to optimize and the cost (i.e.
the total volume of material β used) is done introducing a given positive
parameter . The two problems above can now be written:
1. Find the cheapest configuration maximizing the mean temperature
over Ω:
max u(x)dx − χ(x)dx , for a given > 0
χ Ω Ω
grey zones from the solution. The resulting structure is strongly mesh de-
pendent, the size of the peaks depending on the spatial resolution allowed
by the mesh size.
There are three possible ways to change the initial ill-posed problem into a
well posed one:
• change the problem and enlarge the set of admissible solutions to allow
microstructures;
• add constraints (e.g. smoothness of the boundaries, topology con-
straints) or regularizing terms (e.g. a perimeter term) to narrow the set of
admissible solutions in such a way that there is existence of solutions;
• work on finite dimensional sets and do numerical computations.
134 F. Jouve
Although the above classes are not very strict, since a given method could
belong to more than one category, it is possible to give popular examples
belonging to each of then: the first approach leads to topology methods
based on homogenization (homogenization method, SIMP); the second one
includes classical boundary methods and the level set method; parametric
methods, topological gradient and genetic algorithm, for example, belong
to the third one
The remaining of this chapter will be restricted to the first two classes,
and more precisely the homogenization method and the level set method.
Compliance:
J(ω) = χ(x)Ae(u) : e(u)dx = A−1 σ : σdx = f · u ds = c(ω) (1)
Ω Ω ∂Ω
It is a global measure of the rigidity, and the most popular objective func-
tion in topology optimization of structures (for practical and mathematical
reasons).
136 F. Jouve
1/α
J(ω) = k(x)|u(x) − u0 (x)| dx
α
(2)
ω
= Ci A−1 σi : σi dx (3)
i Ω
= Ci fi · ui ds
i ∂Ω
≥ ≥ ≥ ?
Figure 3. Uniaxial traction: fine layers are more effective than large ones
for the same volume of material.
2.4 Homogenization
Homogenization is the rigorous way of computing the effective (macroscopic)
properties of heterogeneous and composite media.
For an heterogeneous medium where ε is the typical size of small details,
the direct computation is too expensive. We look for an equivalent medium,
at the macroscopic scale, that will be obtained as the limit material when
ε → 0.
1d scalar example
Consider a > 0, a 1-periodic function and the following problem, with ε > 0
140 F. Jouve
a small parameter:
⎧
⎨ d x du
− a = f in ]α, β[
dx ε dx
⎩
u(α) = u(β) = 0
where
1
a∗ = β
1 dy
β−α α a(y)
The homogenized conductivity is the harmonic mean value and not the
arithmetic one!
Let Y = (0, 1)d and consider a Y −periodic Hooke’s tensor A(y) for the
following elasticity problem with oscillating coefficients:
x
−div A e(uε ) = f in Ω
ε
uε = 0 on ∂Ω,
We assume (formally) the following ansazt for the two-scales asymptotic
expansion:
+∞ x
uε (x) = ui x, ,
i=0
ε
where ui (x, y) is a function of both variables x and y, periodic in y with
period Y . The following derivation rule is used
x x
∇ ui x, = ε−1 ∇y ui + ∇x ui x, ,
ε ε
where ∇y and ∇x denote respectively the partial derivative with respect to
the first and second variable of ui (x, y). This series is then plugged into the
equation
x
f (x) = −ε−2 [divy (Aey (u0 ))] x,
ε
x
−ε−1 [divy (A(ex (u0 ) + ey (u1 ))) + divx (Aey (u0 ))] x,
ε
+∞ x
− εi [divx (A(ex (ui ) + ey (ui+1 ))) + divy (A(ex (ui+1 ) + ey (ui+2 )))] x,
i=0
ε
Identifying each coefficient as an individual equation yields a cascade of
equations that leads to the homogenized equation
−divx (A∗ ex (u(x))) = f (x) in Ω,
and the homogenized Hooke’s tensor A∗ is given by
A∗ijkl = (A(y)ey (wij )kl + Aijkl (y)) dy.
Y
For each (i, j) ∈ {1, . . . , d}2 , the fields wij are solutions of the cell problem:
−divy (A(y) (eij + ey (wij (y)))) = 0 in Y
y → wij (y) Y -periodic,
where eij are the elements of the canonical basis of the space of symmet-
ric matrices in dimension d. Thus, d(d + 1)/2 different cell problems are
necessary to solve, in order to compute A∗ .
142 F. Jouve
some bounds of the energy over Gθ can be computed and are achieved,
for example, by a particular and explicit class of periodic materials: the
sequential laminates.
Sequential laminates
It is a particular class of composites, built sequentially. Consider two
given materials. In a first step they are stacked in successive layers along
one given direction, with a given proportion of each of them. It leads to the
rank 1 laminate, whose mechanical properties can be computed explicitly,
function of the two given materials and the lamination parameters. Then
this rank 1 laminate can be stacked again with one of the initial pure ma-
terials, with a different stacking direction and different proportions, to get
a rank 2 laminate. Mechanical properties of the rank 2 material can be ex-
plicitly computed as well, and the same occurs for any rank of lamination.
Sequential laminates are useful in two ways:
• They are described by a small number of parameters: the rank of
lamination, the global proportion of each constitutive material, the
lamination directions and proportions along each direction.
• There are explicit formulas for the homogenized properties, functions
of these characterizing parameters.
(κ + μ)(1 − θ)
min A∗−1 τ : τ = A−1 τ : τ + (|τ1 | + |τ2 |)2
A∗ ∈Gθ 4κμθ
Structural Shape and Topology Optimization 143
where τ is solution of
−div τ = 0 in Ω
τn = f on ∂Ω,
and CD denotes the set of the generalized shapes that includes composite
materials made by homogenization. The new set of admissible shapes is
CD = θ ∈ L∞ (Ω; [0, 1]) , A∗ (x) ∈ Gθ(x) , ∀x ∈ Ω ,
Structural Shape and Topology Optimization 145
where for each given θ(x) ∈ [0, 1], Gθ is the set of all homogenized Hooke’s
tensors obtained by mixing A and void in respective proportions θ and
(1 − θ).
Unfortunately Gθ is unknown for elasticity (although it can be fully
characterized by laminates for conductivity problems). But it can be proved
that Gθ is the set of all the Hooke’s tensors obtained by mixing A and void
in a periodic way with proportions θ and (1 − θ).
Relaxation does not change (too much) the problem:
inf J(χ) = min J ∗ (θ, A∗ )
χ∈L∞ (Ω;{0,1}) (θ,A∗ )∈CD
1.5
Relaxation
1.4 Convexification
Penalizing from the start
1.2
1.1
1
0 50 100 150
Iterations
1 2 3 4 5
Solution for loading 1
1/α
J(ω) = k(x)|u(x) − u0 (x)|α dx
ω
α1
∗ ∗
J (θ, A ) = θ(x)k(x)|u(x) − u0 (x)| dx α
+ θ(x) dx,
Ω Ω
where u is solution of
−div (A∗ e(u)) = 0 in Ω
A∗ e(u) · n = f on ∂Ω,
But now – and it is the case for any objective function different from the
compliance – the optimal composites cannot be characterized explicitly.
We do not even know if optimal homogenized materials can be reached
by sequential laminates. Thus we have to use a numerical procedure to
optimize, in an approximate way, the local composites.
Partial relaxation
min J ∗ (θ, A∗ ),
(θ,A∗ )∈LD
with
LD = θ ∈ L∞ (Ω; [0, 1]) , A∗ (x) ∈ Lθ(x) , ∀x ∈ Ω ,
where
set of all the Hooke’s tensors obtained by sequential
Lθ(x) = .
lamination of A and void in proportions θ and (1 − θ)
Remarks:
• In the compliance case (or multiple loads case optimizing a weighted
sum of compliances), the partial relaxation is equivalent to the genuine
relaxation.
• For all other objective functions, it is not true (it may be true but not
proved yet...)
• An ill posed problem is replaced by another, a priori, ill posed problem.
We expect that it is at least “less ill posed”...
• In practical computations, the set of admissible composites is further
restricted to rank q laminates (q is fixed) with fixed lamination directions,
and the rest of the algorithm is similar to the multiple loads case.
152 F. Jouve
Adjoint state
Since we derive, with respect to shape parameters, a function depending
on the solution of a partial differential equation posed on this domain, the
computation of the adjoint state is needed.
Remark: the compliance case is self-adjoint. That is why no adjoint state
is needed for the compliance problem, and why it is so popular.
For u solution of the elasticity problem, the adjoint state p is solution of
−div (A∗ e(p)) = cα θk|u − u0 |α−2 (u − u0 ) on Ω
+boundary conditions
1−α
α
where cα = θ(x)k(x)|u(x) − u0 (x)| dx
α
.
Ω
∗
p allows to compute the derivatives of J with respect to the shape pa-
rameters θ and (mi ), and obtain numerically, through a classical descent
algorithm, the optimal parameters for the class of laminates chosen.
* The constraint i mi,n = 1 is adjusted by a Lagrange multiplier.
* Descent steps tn and t n , adjusted using a line search method, are chosen
such that J ∗ (θn+1 , mi,n+1 ) < J ∗ (θn , mi,n )
Structural Shape and Topology Optimization 153
Topology optimization
The topology optimization methods can be classified in three distinct
families:
• Methods based on homogenization, like the one described in the Section
2. These techniques involve the relaxation of the problem (or its partial
relaxation for the SIMP method) to enlarge the set of admissible solutions,
and look for a density of material at each point of the admissible domain,
discretized on a fixed grid. The main advantages of this approach is to
avoid local minima, mesh and initialization dependency, and topological
restrictions.
Their main weaknesses lie in the difficulty to deal with complex direct
problems, objective functions and geometrical constraints (e.g. thickness
constraints).
• Evolutionary algorithms: purely numeric approach, treating the dis-
cretized problem.
154 F. Jouve
Uad = ω ⊂ Ω, vol(ω) = V0 , ΓN ⊂ ∂ω, meas(ΓD ∩ ∂ω) > 0
Remark: the cone condition does not imply regularity (corners allowed),
but forbids cups and rapidly oscillating boundaries.
Theorem [Chenais 75 (35)]:
If
Uad1 = {ω ∈ Uad , ω verifies the uniform cone condition}
then
inf J(ω)
ω∈Uad1
then
inf J(Ω)
ω∈Uad2
˜
inf J(ω)
ω∈Uad
then
inf J(ω)
ω∈Uad4
Lemma: For any θ ∈ W 1,∞ (Rd ; Rd ) such that
θ
W 1,∞ (Rd ;Rd ) < 1, ( Id + θ)
is a diffeomorphism in Rd .
Definition: The shape derivative
of ω → J(ω) at ω0 is the Fréchet deriva-
tive of θ → J ( Id + θ)ω0 at 0.
then
∂(f · u)
J (ω0 )(θ) = 2 + Hf · u − Ae(u) : e(u) θ · n ds, (4)
∂ω0 ∂n
then
∂(g · p) C0
J (ω0 )(θ) = + Hg · p − Ae(p) : e(u) + k|u − u0 |α θ·n ds,
∂ω0 ∂n α
where the state u is solution of the elasticity system and the adjoint state
p is solution of the following adjoint problem:
⎧
⎨ −div (A e(p)) = C0 k(x)|u − u0 |α−2 (u − u0 ) in ω0
p = 0 on ΓD (5)
⎩
Ae(p)n = 0 on ΓN ∪ ∂ω0 ,
1/α−1
with C0 = k(x)|u(x) − u0 (x)|α dx .
ω0
Then by varying φ:
• φ with compact support in ω leads to the state equation.
• Vary the trace of φ on ΓN leads to the Neumann boundary conditions
on u.
• Vary Ae(φ) n on ΓD leads to the Dirichlet boundary conditions on
u.
Then the Lagrangian is derivated with respect to its second variable v:
∂L
(ω, u, p), φ = 0 = j (u) · φ dx + l (u) · φ ds
∂v ω ∂ω
+ Ae(φ) · e(p) dx
ω
− p · Ae(φ)n + φ · Ae(p)n ds.
ΓD
Again, varying φ gives the adjoint problem and its boundary conditions:
• Taking φ with compact support in ω leads to the adjoint equation:
p=0 on ΓD .
We have found a well-posed boundary value problem for the adjoint state
p.
Structural Shape and Topology Optimization 161
Figure 9. Topology changes are easily handled by the level set representa-
tion.
θ = −v n. (6)
Structural Shape and Topology Optimization 163
that ensures that the objective function is decreasing as soon as the descent
parameter is small enough.
Suppose that the shape ωt evolves with a pseudo-time t and a normal
velocity V (t, x). Then, as the boundary of the shape is characterized by the
level set 0 of ψ, we have
ψ t, x(t) = 0 for all x(t) ∈ ∂ωt .
π(λ + 2μ)
TG = {4μσ(u) : e(u) + (λ − μ) tr(σ(u))tre(u)}
2μ(λ + μ)
• Example 2 (Minimization of k(x)|u − u0 |α ). p is the adjoint state,
Ω
solution of the problem (5)
π
TG = C0 k(x)|u(x) − u0 (x)|α
α
π(λ + 2μ)
+ {4μσ(u) : e(p) + (λ − μ) tr(σ(u)) tr(e(p))}
2μ(λ + μ)
A new hole may be dig at point x if T G is smaller than a given value.
∂ψ
+ V |∇ψ| = 0 in Ω
∂t
ψin+1 − ψin
+ min(Vin , 0) g + (Dx+ ψin , Dx− ψin )
Δt
+ max(Vin , 0) g − (Dx+ ψin , Dx− ψin ) = 0
n
ψi+1 − ψin ψ n − ψi−1
n
with Dx+ ψin = , Dx− ψin = i , and
Δx Δx
g − (d+ , d− ) = min(d+ , 0)2 + max(d− , 0)2 ,
g + (d+ , d− ) = max(d+ , 0)2 + min(d− , 0)2 .
∂ψ
+ sign(ψ) |∇ψ| − 1 = 0 in Ω,
∂t
whose stationary solution is the signed distance to the interface.
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Structural Shape and Topology Optimization 173
1 Introduction
The paper concerns classical topic of topology optimization: given two elas-
tic isotropic materials whose amounts are fixed, fill up an a priori assumed
design domain, in this way forming a least compliant structure which trans-
mits a given load to a given part of the boundary of this domain.
This subject is discussed as the plane elasticity problem, or the problem
of thin elastic plate loaded in plane, which falls into broader category of
vectorial problems in topology optimization. Its achievements have been
preceded by solutions to simpler, scalar elliptic case specified as e.g. sta-
tionary heat conduction, torsion of bars, or pre-stressed membranes. De-
velopment of this branch of optimization has been extensively outlined in
(Tartar, 1980), see also (Allaire, 2002, Sec. 3) and (Cherkaev, 2000, Part II)
for main results.
Main aim of the lecture is to derive the compliance minimization problem
in its relaxed version. Exposition of the topic given in the sequel is alter-
native to the one in (Allaire, 2002, Sec. 4.2.1) where the Hashin-Shtrikman
bound, see (2.142) ibidem, is used for estimating the stress energy density.
It is shown that the bound may be found in a different way, namely by ap-
plying the translation method, see relevant articles in (Cherkaev and Kohn,
1997) or (Cherkaev, 2000, Sec. 8, 15) for details. This lecture can be treated
as an addendum to the results mentioned above.
Key issues of the relaxed minimum compliance problem in two-dimensio-
nal setting are presented in Section 2.3. At this point, it is worth pointing
out that the passage from (32) to (57) is relatively short. Its counterpart
for Kirchhoff plates has been explained in (Lewiński and Telega, 2000, Sec.
23.6). The extension of both formulations to the shallow shell case is shown
in (Dzierżanowski, 2012a). Main steps of the derivation in plane elasticity
setting are recalled in the sequel for making the lecture easier to follow. They
also serve as a basis for approximate treatment of the shape optimization
problem in which case the relaxed potential (57) appears to be non-smooth.
The relaxed formulation of shape optimization admits a direct neglect-
ing of stiffness of the weaker material. It allows for certain simplifications
leading to the analytical solution of (34) but it is a source of essential nu-
merical difficulties at the same time. Various approaches are proposed in
the context of shape optimization to overcome the above-mentioned obsta-
cles. They are thoroughly discussed in (Allaire et al., 1997) and (Bendsøe
and Sigmund, 2003), see also (Stolpe and Svanberg, 2001), (Bruggi, 2008)
or (Talischi et al., 2009). In the present article, the novel approach, called
GRAMP, is put forward. It is based directly on the exact result (57), see
(Dzierżanowski, 2012b). The main advantage of GRAMP is its uniform ap-
plicability to the problems of isotropic materials with various possible ratios
between shear and bulk moduli.
Compliance Minimization of Two-Material Elastic Structures 177
I2 = E1 , I4 = E1 ⊗ E1 + E2 ⊗ E2 + E3 ⊗ E3 . (4)
where k and μ respectively denote the Kelvin and Kirchhoff moduli which
are related to Young’s modulus E and Poisson ratio ν through
Eh Eh
k= , μ= . (15)
2(1 − ν) 2(1 + ν)
and
1 1
A−1 = KA I2 ⊗ I2 + LA (I4 − I2 ⊗ I2 ),
2 2
(19)
1 1
B −1 = KB I2 ⊗ I2 + LB (I4 − I2 ⊗ I2 )
2 2
with K = k −1 , L = μ−1 . From (16) it follows that KA < KB , LA < LB . A
search for the stiffest plate loaded in-plane is realized by
∞
Jmin = min J(χ) χ ∈ L Ω; {0, 1} (20)
where
J(χ) = Υ(χ) + χ(x) dx, (21)
Ω
satisfies
uχ
u in H01 (Ω),
(24)
Aχ ∇uχ
A∗ ∇u in L2 (Ω, E2s ),
where u solves the local homogenized problem
div (A∗ ∇u) = −q in Ω,
(25)
u ∈ H01 (Ω).
It follows that stress and strain energy sequences are convergent in the sense
of (26).
Important property of homogenization in RN , N > 1 stems from the fol-
lowing fact: different {χm }, {χn } ∈ L∞ (Ω; {0, 1}) may converge to the same
limit θ ∈ L∞ (Ω, [0, 1]), but sequences {Aχm }, {Aχn } can tend to different
limits A∗1 , A∗2 ∈ L∞ (Ω; E4s ). Hence, taking all sequences {χn } weakly-∗ con-
verging to a given limit θ results in forming a set of all possible homogenized
182 G. Dzierżanowski and T. Lewiński
where
∗ ∗ ∗
J (θ, A ) = Υ (θ) + θ(x) dx, (29)
Ω
and
Υ∗ (θ) = min σ : A∗ −1 σ dx σ ∈ Σ(Ω) . (30)
Ω
where
2W ∗ (σ, θ) = min σ : A∗ −1 σ A∗ ∈ Gθ (32)
stands for the (doubled) stress energy density of a homogenized plate, and
A∗ = A∗ (x), Gθ = Gθ(x) .
Further modification of the problem can be achieved by swapping the
minimization over θ ∈ L∞ (Ω; {0, 1} with integration in (31). This is pos-
sible due to the Rockafellar Theorem, see (Rockafellar, 1976). For detailed
justification of the application of Rockafellar’s Theorem in the context of a
problem tackled in this paper, the reader is referred to (Allaire, 2002, Sec.
4.1.4), see also (Dzierżanowski, 2012b) for its brief recall.
In this way, the relaxed material layout optimization problem for mini-
mum compliance takes the form
∗
Jmin = min F ∗ (σ) dx σ ∈ Σ(Ω) (33)
Ω
where
F ∗ (σ) = min 2W ∗ (σ, θ) + θ θ ∈ [0, 1] . (34)
for the set of periodic stress resultants that are statically admissible in
Y . Here, n represents a unit vector outward normal to ∂Y . Furthermore,
184 G. Dzierżanowski and T. Lewiński
expand the periodicity mesh to the whole space R2 . Given materials A and
B are distributed in Y according to the function χ ∈ L∞ (Y, {0, 1}) such
that χ = θ. The constitutive tensor field Aχ in Y is expressed similarly
to (9). Mean stress energy in Y is measured by
2 W (τ, χ) = τ : A−1 χ τ . (36)
det τ = −τ : (T τ ). (40)
Proof of the latter relies on the periodicity of τ . Despite this fact, there
is no loss in generality of further discussion. Full justification of (41) can
be found in the above-mentioned references, see also (Lewiński and Telega,
2000), hence it is omitted here.
Compliance Minimization of Two-Material Elastic Structures 185
The next step of the method consists in rewritting (36) in the form
2 W (τ, χ) = 2W (τ, χ) + α det τ − α det τ (42)
where
−1
−1 −1
C(α, θ) = θ A−1 − α T + (1 − θ) B −1 − α T + α T. (47)
and
σI − σII
ζ= (49)
σI + σII
186 G. Dzierżanowski and T. Lewiński
where σI , σII denote the principal values of σ; |σI | ≥ |σII | by convention and
σI + σII = 0. Next, rewrite (47) in the form
⎛ ⎞
C11 0 0
1⎜⎜
⎟
⎟
C= ⎜ 0 C22 0 ⎟ (50)
2⎝ ⎠
0 0 C33
where
(KA + α)(KB + α) (LA − α)(LB − α)
C11 = − α, C22 = C33 = + α.
[K]θ + α [L]θ − α
(51)
By taking the eigenbasis of vector σ as the physical basis in (1) it is possible
to write ⎛ ⎞
σI + σII
1 ⎜⎜
⎟
⎟
σ = √ ⎜σI − σII ⎟ , (52)
2⎝ ⎠
0
and consequently
σ : C(α, θ) σ = (σI )2 C11 (α, θ) + C22 (α, θ) ζ 2 . (53)
where
⎧
⎪ KA KB + LA Kθ
⎪
⎪ 0 ≤ ζ ≤ ζ2 ,
⎪
⎪ LA + [K]θ
⎨
K∗ (θ) = KA KB + [L]θ Kθ − (1 − θ)θΔKΔL ζ (58)
⎪
⎪ ζ2 ≤ ζ ≤ ζ1 ,
⎪
⎪ [K + L]θ
⎪
⎩
KA ζ1 ≤ ζ,
and
⎧
⎪
⎪ LA 0 ≤ ζ ≤ ζ2 ,
⎪
⎪
⎪
⎪
⎨ LA LB + [K]θ Lθ − (1 − θ)θΔKΔL ζ −1
L∗ (θ) = ζ2 ≤ ζ ≤ ζ1 , (59)
⎪
⎪
[K + L]θ
⎪
⎪
⎪ LA LB + KA Lθ
⎪
⎩ ζ1 ≤ ζ,
KA + [L]θ
denote the reciprocals of optimal effective moduli. One may observe that
the case of σI + σII = 0 falls into region ζ1 ≤ ζ.
Values of K∗ and L∗ depend continuously on ζ hence QW is also contin-
uous in this variable. Moreover, QW is smooth hence differentiable in the
whole interval ζ ∈ (0, +∞). Consequently, formula
∂
ε= QW (60)
∂σ
⎧
⎪ kA kB + μA kθ
⎪
⎪
⎪ 0 ≤ ζ! ≤ ζ!2 ,
⎪
⎪ μA + [k]θ
⎨
k∗ (θ) = kA kB + [μ]θ kθ + (1 − θ)θΔkΔμ ζ! (61)
⎪
⎪ ζ!2 ≤ ζ! ≤ ζ!1 ,
⎪
⎪ [k + μ]θ
⎪
⎪
⎩
kA !
ζ!1 ≤ ζ,
188 G. Dzierżanowski and T. Lewiński
and
⎧
⎪
⎪
⎪ μA 0 ≤ ζ! ≤ ζ!2 ,
⎪
⎪
⎪
⎪
⎨ μ μ + [k] μ + (1 − θ)θΔkΔμ ζ!−1
ζ!2 ≤ ζ! ≤ ζ!1 ,
A B θ θ
μ∗ (θ) = (62)
⎪
⎪ [k + μ]θ
⎪
⎪
⎪
⎪ μ μ + k μ
⎪ ζ!1 ≤ ζ!
A B A θ
⎩
kA + [μ]θ
where
εI − εII
!
ζ = (63)
εI + εII
depends on principal values εI , εII , |εI | ≥ |εII |, of deformation tensor ε and
kA + [μ]θ θΔk
ζ!1 = , ζ!2 = . (64)
θΔμ μA + [k]θ
t3
t1
n1
A t2 1
q1 n3
*
n2 1 A A3
A 1*
q1
*
1- q A A2 q3
1-
B
1- q 3
1
2
q2
constructed at the previous step of lamination are allowed in the next step,
i.e.
– materials A and B are available for the first lamination,
– materials A, B and L(A, B) are available for the second one,
– materials A, B, L(A, B), L(A, L(A, B)), L(B, L(A, B))
and L(L(A, B), L(A, B)) are available for the third one,
– etc.
It is shown in (Cherkaev, 2000, Sec. 10) that in this way one may characterize
the whole set Gθ .
t*,ktt t Att
t Ant
t*,knn t Ann 1
*,k
t nt tk+1
nk+1
Gk+1
*
Ak A
1- qk+1 qk+1
τ = Pc qc + Pd qd (65)
where −1 T
fAc (ψ) = Pd (ψ) PdT (ψ)A−1 Pd (ψ) Pd (ψ) (70)
is defined through standard matrix operations in the basis (1). Derivation
of the formula for A∗−1
p involves straightforward but lengthy algebraic cal-
culations hence they are omitted here. Different, albeit equivalent form of
(70) is reported in e.g. (Allaire, 1997).
Compliance Minimization of Two-Material Elastic Structures 191
Lamination parameters
θ1 θ2 (1 − θ1 )
m1 = , m2 = , ...
θ θ
(71)
θp (1 − θp−1 )(1 − θp−2 ) . . . (1 − θ1 )
mp = ,
θ
p
such that k=1 mk = 1, measure the relative fraction of material A in
subsequent laminations. The overall amount of A in a rank-p composite is
thus given by
%p
1−θ = (1 − θk ). (72)
k=1
with
−1 T
fA (ψ) = Pc (ψ) PcT (ψ)APc (ψ) Pc (ψ). (74)
It turns out that rank-2 sequential strong laminates with orthogonal
directions of layerings (so-called T-structures, see Figure 3) are sufficient to
prove the attainability QW = W ∗ , see (38). Effective constitutive tensor
A∗2 associated with this composite possesses a property
2 QW (σ, θ) = σ : (A∗−1
2 σ), A∗2 ∈ Gθ . (75)
sII
1- q1
sI sI
B
q1
A A
K + α(1 − θ) L − α(1 − θ)
K∗ (θ) = , L∗ (θ) = . (77)
θ θ
Let us now set ρ = σσIII with |σI | ≥ |σII |. Hence ρ ∈ [−1, 0) if σI σII < 0
and ρ ∈ (0, 1] if σI σII > 0. We also assume that h = 1. One may check that
(57) is now given by
⎧
⎪ 1 1−ν
⎪
⎨ (1 − ρ)2 + 2 ρ for ρ < 0,
θE E
2 QW (σ, θ) = (σI )2 (78)
⎪
⎪
⎩ 1 (1 + ρ)2 − 2 1 + ν ρ for ρ > 0,
θE E
where E ≡ EA and ν ≡ νA . Equivalently
& '
1 1 + sgn(ρ) ν
2 QW (σ, θ) = (σI )2
1 + |ρ|) − 2
2
|ρ| . (79)
θE E
From (79) it follows that QW is continuous for any σ but it is not smooth
for det σ = 0. Thus, QW is not differentiable in the whole range of σ. This
fact poses a particular obstacle in developing the FEM-based numerical
scheme for shape optimization problems. Possible remedy is proposed in
the subsequent Section.
Conditions for attainability of QW on certain microstructures are dis-
cussed in Sec. 2.3 and 2.4. Hence in further considerations we assume that
QW = W ∗ . Consequently, in case of shape optimization, solution to (34)
can be explicitly found. It reads
|σI | + |σII |
θopt (σ) = min 1, √ (80)
E
hence the integrand in (33) takes the form:
⎧ 2 (√
)
⎪
⎪ E |σI | + |σII | − 1 + sgn(σI σII )ν |σI σII | if θ < 1,
⎨E
F ∗ (σ) = & '
⎪
⎪ 2 1 1 1
⎩ (σI ) + (σII ) − νσI σII + E
2 2
if θ = 1.
E 2 2 2
(81)
Formula (69) becomes
" p $−1
1−θ #
A∗−1
p =A −1
+ c
mk fA (ϕ(nk )) , (82)
θ
k=1
194 G. Dzierżanowski and T. Lewiński
one may define the optimal solid-void interpolation scheme of material prop-
erties as
θE bopt (ρ, θ)
Eopt (ρ, θ) = , νopt (ρ, θ) = − . (86)
aopt (ρ, θ) aopt (ρ, θ)
Numerical implementation of the above results is hampered by the non-
smoothness of stress energy functional Uopt (ρ, θ) for ρ = 0. Namely, if this
is the case, then the constitutive formula is not unique, hence it cannot be
directly inverted into the stress-displacement form which serves as a nat-
ural basis for a numerical implementation by FEM. As a possible remedy,
one may either make use of the solution to the problem of optimal lay-
out formulated for a two-material structure, see (61) and (62), or seek the
smooth, thus invertible, approximation Uapp of Uopt . In this Section, the
latter option is investigated.
Function Uopt is taken as a reference expression in determining aapp (θ)
and bapp (θ). Both coefficients are required to be independent of the sign of
ρ and such that resulting Uapp is isotropic, i.e. it takes the form
θ (93)
fq (θ) =
1 + q(1 − θ)
196 G. Dzierżanowski and T. Lewiński
where fq (θ) denotes the material properties interpolation function. The set
of plots in Figure 4 shows fq for different values of parameter q.
fq(q)
hence
ERAMP (θ) = EGRAMP (θ), νRAMP (θ) = ν = const. (98)
Functions USIMP and URAMP have to obey (88) as any other approximate
stress energy function Uapp . Hence, the ranges of SIMP or RAMP parame-
ters (p and q respectively) are limited. Even more restrictive is the require-
ment of macrostructural isotropy of the solid-void composite. Bounds on
the effective isotropic properties of a two-material mixture were introduced
by Hashin and Shtrikman (1963) and improved by Cherkaev and Gibiansky
(1993). It is worth pointing out, however, that both results coincide for
the mixture of material and void hence, in the sequel, the requirement of
effective isotropy is denoted by UHS ≤ Uapp . Formula determining UHS can
be expressed by (87) with
hence
θE θν + (1 − θ)
EHS (θ) = , νHS (θ) = . (100)
3−2θ 3−2θ
The latter can be independently derived from the Hashin-Shtrikman upper
bounds on Kelvin’s and Kirchhoff’s moduli of a mixture of two isotropic
materials given by
−1
1 θ
kHS = kA − (1 − θ) − ,
kA − kB kA + μA
−1 (101)
1 θ(kA + 2μA )
μHS = μA − (1 − θ) − .
μ A − μB 2μA (kA + μA )
The first step in justifying this fact is to apply kB = 0, μB = 0 to (101).
Next, thus obtained result is rewritten in terms of EA , νA , and EHS , νHS , see
198 G. Dzierżanowski and T. Lewiński
(15) for h = 1, as both material A and the effective composite material are
isotropic. In this way, (100) follows with subscript A dropped for simplicity
of notation.
Table 1 presents lower bounds imposed on the parameters of GRAMP,
RAMP and SIMP approximations. They stem from two requirements which
has to be fulfilled by any approximate stress energy functional Uapp . In-
equality Uapp − Uopt ≥ 0, see (88), restricts the values of parameters to
those for which the approximate energy takes the form of an isotropic func-
tion (87) while Uapp − UHS ≥ 0 imposes the effective isotropy of a composite
material as an additional constraint. Details of the results are reported in
e.g. (Dzierżanowski, 2012b) and (Bendsøe and Sigmund, 1999; Stolpe and
Svanberg, 2001). It is worth pointing out that in the GRAMP scheme pa-
rameters are independent of the basic material’s Poisson ratio value ν and
density θ. On the contrary, in the course of calculations related to RAMP
and SIMP schemes one has to set θ = 1 in order to assure the validity of
expressions in Table 1 for any θ ∈ [0, 1].
The comparison of Uopt , UHS and UGRAMP , URAMP , USIMP with param-
eters of lowest possible values are shown for ν = 1/3 (Figure 5(a)) and
ν = −1/3 (Figure 5(b)). In the latter case of an auxetic material, values of
parameters in the RAMP and SIMP schemes are significantly higher than
for the solid with positive Poisson’s ratio. This in turn, reflects in higher
values of the predicted stress energy and may influence the structural com-
pliance and optimal solid-void layout.
1+ν 1+ν
Uapp = URAMP2 q ≥ max 1−ν
1−ν , 1+ν q ≥ max 3−ν
1−ν , 1+ν
Uapp = USIMP2 p ≥ max 2 2
1−ν , 1+ν p ≥ max 2 4
1−ν , 1+ν
1
independent of the basic material’s Poisson ratio ν and density θ,
2
calculated for θ = 1 in order to enforce validity for any θ ∈ [0, 1].
Compliance Minimization of Two-Material Elastic Structures 199
U (r,q)
r
(a)
-1 1
U (r,q)
r
(b)
-1 1
Figure 5. Plots of functions Uopt (lower solid line), UHS (upper solid line),
UGRAMP with q = 1 (lower dotted line), UGRAMP with q = 3 (upper dotted
line) and:
(a) USIMP with q = 3 (dashdotted line), URAMP with q = 2. Plots match
up with ν = 1/3 (in this unique case URAMP = UHS ) and θ = 0.7;
(b) URAMP with q = 5 (dashed line), USIMP with q = 6 (dashdotted line).
Plots match up with ν = −1/3 and θ = 0.7.
200 G. Dzierżanowski and T. Lewiński
h
P
where I denotes the vector whose all components are equal to 1. Note
2
that W ∗ in (102) should be replaced by certain Wapp = (σθE I)
Uapp
reported in Section 4 if one seeks the approximate solution.
Next,
J (m)
Δθ() = − (m) θ , (103)
J
is calculated with θ denoting the parameter ensuring stability of the
material density updating scheme. The negative sign in (103) stems
from the assumption that the compliance of a plate (or, equivalently,
its stress energy) decreases for increasing ratio of stronger material in
a composite. The loop is rerun until reaches the value for which
(a) (b)
h(m,n) θ(i+m,j+n)
m,n
mod
θ(i,j) = ,
h(m,n)
m,n (106)
m ∈ {−i + 1, −i + 2, . . . , −1, 0, 1, . . . , k − i},
n ∈ {−j + 1, −j + 2, . . . , −1, 0, 1, . . . , l − j}
204 G. Dzierżanowski and T. Lewiński
where
h(m,n) = max{0, rmin − d(m,n) } (107)
and d(m,n) denotes the distance between the centers of given element (i, j)
and the one located at (i+m, j +n) while rmin stands for the filtering radius.
The comparison of unfiltered and filtered design obtained by GRAMP
and SIMP schemes is shown in Figure 9.
where
⎧2 (√
)
⎪
⎪ E |σI | + |σII | − 1 + sgn(σI σII )ν |σI σII | if θ < 1,
⎨E
∗
F (σ) = & '
⎪
⎪ 2 1
⎩ (σI )2 + (σII )2 − 2νσI σII + E if θ = 1.
E 2
(109)
Here denotes the Lagrangian multiplier for the isoperimetric condition
1
θ(x) dx = m (110)
|Ω|
Ω
The function: σ1 = |σI | + |σII | satisfies all conditions for a norm in E2s .
Define a norm dual to · 1 by
|ε : τ |
ε−1 = sup τ ∈ E2
s , ε ∈ E2s . (114)
τ 1
The next step is to prove that
where
f (v) = p · v ds, (120)
∂Ωσ
is equivalent to
J0 = max J1 (v) + f (v) (121)
v∈V
206 G. Dzierżanowski and T. Lewiński
with
J1 (v) = min σ1 − σ : ε(v) dx. (122)
σ∈L2 (Ω,E2s )
div σ∈L2 (Ω,R2 ) Ω
where
B−1 = ε ∈ E2s ε−1 ≤ 1 (124)
or
B−1 = ε ∈ E2s |εI | ≤ 1, |εII | ≤ 1 . (125)
hence
σ : ε ≤ σ1 for ε ∈ B−1 (127)
and
σ: ε
g(σ) = 1 − ≥0 (128)
σ1
for any ε ∈ B−1 . It follows that
hence
min (σ1 − σ : ε) = 0 (130)
σ∈E2s
for ε ∈ B−1 .
Assume now that ε ∈ / B−1 and ε1 = 1 + δ > 1, δ > 0. Then, for
such ε one obtains σ : ε ≤ (1 + δ)σ1 and g(σ) ≥ −δ. Since there exists σ
for which the last relation becomes equality, the minimum in (130) is not
bounded from below. Consequently, (123) is confirmed.
Now turn back to (121), (122) and find
J0 = max f (v) v ∈ V, ε(v(x)) ∈ B−1 for a.e. x ∈ Ω (131)
Compliance Minimization of Two-Material Elastic Structures 207
or, equivalently,
J0 = max f (v) v ∈ V, |εI (v(x))| ≤ 1, |εII (v(x))| ≤ 1
for a.e. x ∈ Ω . (132)
|σI | + |σII |
θopt = √ (133)
E
and θopt < 1. Consequently the isoperimetric condition (110) gives
*
J0
= (134)
E E m |Ω|
∗ 2
Jmin = (J0 )2 . (135)
E m |Ω|
∗ 2
Jmin = (f (v ∗ ))2 . (136)
E m |Ω|
Formulae (135) and (136) provide the following relationship between the
∗
minimum compliance Jmin and the minimum volume V0
∗ 2
Jmin = (σ0 V0 )2 . (138)
E m |Ω|
The equality m |Ω| = V0 is not used here, since the structure of minimal
compliance and the one corresponding to (113) are different. Indeed, struc-
ture solving the problem set by (108) and (109) is constructed of a mate-
rial with rank-2 microstructure, while the microstructure corresponding to
(113) is rank-1, forming the curvilinear strips along trajectories of σ being
the minimizer of (113).
A specific form of the problem (132) where the conditions |εI | ≤ 1,
|εII | ≤ 1 are expressed pointwise for a.e. x ∈ Ω stems from the fact that
integrand in (113) has a linear growth. Both problems (113) and (132) can
be interpreted as equilibrium problems of an effective body made from a
locking material. The mechanics of locking materials has been developed
in the works by Čyras (1972), Borkowski (2004a,b), Demengel and Suquet
(1986) or Telega and Jemio lo (1998). According to this theory, field σ in
(113) should be interpreted as a stress rate and not as a stress. The Michell
problem (113) has a similar mathematical structure as the problem
J1 = min |||σ||| dx σ ∈ Σ(Ω (139)
Ω
1991). This topic has been discussed in (Krog and Olhoff, 1997), (Dı́az et
al., 1995), (Czarnecki et al., 2008). Due to the presence of two stiffness ten-
sors the problem can be set in different ways, hence the relaxed formulation
is expected to be non-unique.
The theory of relaxation of the minimum compliance problem serves
as a pattern for more complicated problems concerning optimum design
of structures made of composites. Recalled results are the inspiration for
contemporary research on numerical homogenization and hierarchical mod-
eling.
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212 G. Dzierżanowski and T. Lewiński
1 Introduction
The physical properties of the elastic bodies are determined by the distribu-
tion of the elasticity moduli Cijkl ; i, j, k, l = 1, 2, 3. Due to known symmetry
properties the number of independent moduli equals : 6+5+4+3+2+1 = 21.
The contemporary technology provides means for controlling the distribu-
tion of these moduli within the design domain. A prerequisite for the opti-
mum design of the layouts of the moduli within the body is a proper insight
into the algebraic structure of the Hooke tensor C. The explicit algebraic
representation of this tensor is delivered in (Rychlewski, 1984; Mehrabadi
and Cowin, 1990; Sutcliffe, 1992; Moakher and Norris, 1996). The two-
dimensional case is discussed in (Blinowski et al., 1996). The papers men-
tioned indicate that the Hooke tensor can be decomposed as follows
C = λ1 ω1 ⊗ ω 1 + . . . + λm ω m ⊗ ω m (1.1)
a common feature: they are based on the approach in which the displace-
ment field is the main state (or behavioral) variable. Consequently, in each
case the FMD is led to a saddle point problem. On the other hand, the
theory of minimum compliance of two-material structures, along with its
relaxation theory, see Cherkaev (2000) and the article (Dzierżanowski and
Lewiński, 2013) in this book, teaches us that the stress-based approach is
the most convenient point of departure, since the compliance is expressed
as minimum of the complementary energy over the stress states being stat-
ically admissible. Since two minima: over the design variables and over
the state variables can be switched, the minimum operation over the de-
sign variables becomes an internal operation for which the stress states are
viewed as fixed. This paves the way to an analytical solution of the nested
optimization problem. The theorem behind this rearrangement is due to
Rockafellar’s paper (1976) which teaches us that the minimization of the
integral functionals over the fields can be replaced by integration of the
function defined through a pointwise minimization.
The present paper is organized as follows. Sec. 2 recalls the equilibrium
problem of a linear elastic body. The algebraic structure of Hooke tensors
is recalled in Sec. 3. Why the compliance can be expressed by the comple-
mentary energy is cleared up in Sec. 4. Further part of the lecture concerns
the FMD problem: for the single load case (Sec. 5–Sec. 8), for the two load
case in 2D (Sec. 9, Sec. 10) and for the multi-load cases in Secs 11–13.
The results of the FMD in the one load case in 2D look similar to
Michell’s solution. The reason of this similarity is explained in Sec. 6; prob-
lem (6.28) is similar to Michell’s kinematic formulation; the difference is in
the shape of the locking locus B. In (6.28) this is a ball with respect to the
Euclidean norm || · ||2 while in Michell’s formulation the ball is defined by
∞-norm. Thus the lecture shows why the FMD problem with a single load
conditions is expressed by both primal and dual problems of an effective
medium with locking, see (Demengel and Suquet, 1986).
The numerical methods used are based on new concepts of interpolating
statically admissible stresses; the underdeterminate algebraic systems are
solved by using the SVD decompositions.
It is remarkable, that the singular values of matrices appear twice: a)
as a tool to express the integrand of the effective locking complementary
potential in (13.4) and b) in the mentioned SVD based numerical method.
Therefore, prior to study this article it is recommended to study the related
chapters of the textbooks on linear algebra which concern the singular val-
ues.
The summation convention for repeated indices is adopted. The eigen-
values of a square symmetric matrix A are denoted by μi (A). The singular
216 S. Czarnecki and T. Lewiński
2 Equilibrium
2.1 Plane stress case
Consider an elastic plate of unit thickness, loaded in-plane, whose middle
plane Ω is a symmetry plane of the boundary loading (tractions) of intensity
T (s), acted along the segment Γ1 of the contour Γ of the domain Ω; s is the
natural parameter of the contour. Let us assume that along the segment
Γ2 of the contour Γ the plate is fixed, hence the in-plane displacement
vector vanishes along the contour Γ2 . The plane Ω is parameterized by
a Cartesian system (x1 , x2 ) with the orthonormal basis (e1 , e2 ). Points
in Ω are denoted by x and are identified with the pair (x1 , x2 ). In the
plane stress state the deformation state is determined by the displacement
field u = (u1 , u2 ) referred to the middle plane. We shall not recall here
the standard arguments leading to the plane stress model. The state of
deformation is characterized by the components εij = (ui,j + uj,i )/2 where
∂
( ),j = ; i, j = 1, 2. The right-hand side of this definition will be
∂xj
denoted by εij (u) hence ε(u) is a symmetric part of the gradient of u. The
virtual displacement field will be denoted by v = (v1 , v2 ); these fields are
assumed to satisfy the same kinematic conditions as the unknown fields u1 ,
u2 : they vanish along Γ2 . By writing v ∈ V (Ω) we understand that v is
sufficiently regular and vanish along Γ2 . Usually, the regularity assumptions
should assure that εij ∈ L2 (Ω) or V (Ω) ⊂ H 1 (Ω, R2 ), see (Duvaut and
Lions, 1976), (Nečas and Hlavaček, 1981), (Ciarlet, 1988). One can show
that if εij (v) = 0, i, j = 1, 2, then v is composed of translations and
infinitesimal rigid rotations around the axis normal to the plane Ω. The
The Free Material Design in Linear Elasticity 217
τij,j = 0 (2.1)
τij nj = Ti on Γ1 (2.2)
where F (x) is an Airy function. This function should satisfy the boundary
conditions due to (2.2). Let us write (2.4) in the brief manner
where φrs = φsr and ijk is the permutation symbol. Thus six independent
stress functions determine the stresses satisfying the equilibrium conditions
within the body. On the surface Γ1 the following conditions should be
fulfilled:
imr jps φrs,mp nj = Ti (2.8)
Then the assumption σ33 ≈ 0 makes it possible to eliminate ε33 which leads
to the equation of the form (3.1). The moduli Cijkl are called reduced
moduli of the generalized plane stress state. In the case of isotropy these
moduli can be written with using the effective Kelvin modulus k and the
Kirchhoff modulus μ as follows
while the other Cijkl components vanish; k > 0, μ > 0. If the Young
modulus equals E and the Poisson ratio equals ν, then
E E
k= , μ= (3.5)
2(1 − ν) 2(1 + ν)
Let us note that & '−1
1 1 1
E= + (3.6)
2 2k 2μ
or the Young modulus is a harmonic mean of the moduli 2k and 2μ.
Let us introduce the auxiliary tensors:
1
E 1 = √ (e1 ⊗ e1 + e2 ⊗ e2 )
2
1
E 2 = √ (e1 ⊗ e1 − e2 ⊗ e2 ) (3.7)
2
1
E 3 = √ (e1 ⊗ e2 + e2 ⊗ e1 )
2
They form an orthonormal basis. Instead of the conventional decomposition:
C = Cijkl ei ⊗ ej ⊗ ek ⊗ el (3.8)
C = CKL E K ⊗ E L , K, L = 1, 2, 3 (3.9)
and the dot means the scalar product in E2s . The tensors ω K are called,
after (Rychlewski, 1984, Blinowski et al. 1996), the eigenstates. The moduli
λK will be called Kelvin moduli; we shall assume that λ1 ≥ λ2 ≥ λ3 .
In the case of isotropy, two Kelvin moduli coincide, see (3.11). The cor-
responding eigenstates describe two possible modes of pure shear. Isotropy
implies that the shear modulus is the same for both the shear modes.
The tensor inverse to C has the same projectors, or
1 1 1
C −1 = P1 + P2 + P3 (3.16)
λ1 λ2 λ3
The optimization process can lead to a degeneration of (3.13), to the forms
C = λ1 P 1 , C = λ1 P 1 + λ2 P 2 (3.17)
Then the condition (3.3) is not satisfied; the question of correctness of the
boundary value problem must be reconsidered, as will be seen later.
Remark 3.1
Algebra of tensors of E2s class can be replaced by the algebra of vectors in
R3 with using the basis (3.7). A tensor a ∈ Es2 can be decomposed as:
a = ãK E K , K = 1, 2, 3. Thus the tensor a is treated as a vector in R3 of
components ãK .
The basis (3.7) is used rather rarely. Usually it is sufficient to use the
basis
B 1 = e1 ⊗ e1 , B 2 = e2 ⊗ e2 , B3 = E3 (3.18)
The Free Material Design in Linear Elasticity 221
which obeys the known rules of rotations. In particular the fields ωK can
be treated as vectors in R3 of components which form an orthonormal basis.
where
σ1 = σ11 , σ2 = σ22 , σ3 = σ33 ,
√ √ √ (3.25)
σ4 = 2σ23 , σ5 = 2σ13 , σ6 = 2σ12
and the equations for the strain components are similar. The equation
σ = Cε can now be interpreted as the equation linking two vectors in R6 ,
or: σK = CKL εL . The eigenvalues of tensor C are the eigenvalues of the
matrix [CKL ].
The isotropic material is characterized by the bulk and shear moduli are
expressed by
E E
k= , μ= (3.26)
3(1 − 2ν) 2(1 + ν)
and E, ν are Young modulus and Poisson’s ratio. The moduli Cijkl read
4
C1111 = C2222 = C3333 = k + μ
3
2
C1122 = C1133 = C2233 = k − μ (3.27)
3
C1212 = C1313 = C2323 = μ
and the other moduli follow from symmetry rules (3.2) or vanish. The
components CKL in the basis B K ⊗ B L can be found by (3.23). The rules
of transformation due to rotations are given in (Mehrabadi and Cowin,
1990), see (Czarnecki and Lewiński, 2006). We note that the moduli
λ1 = 3k, λ2 = 2μ, λ3 = 2μ, λ4 = 2μ, λ5 = 2μ, λ6 = 2μ (3.28)
are eigenvalues of the tensor C.
In case of an arbitrary anisotropy the spectral decomposition of the
tensor C has the form
#
m
C= λK ωK ⊗ ωK , m=6 (3.29)
K=1
where the dot means the scalar product in Es2 . The scalar product (4.3)
defines the norm +
||τ ||C −1 = (τ , τ )C −1 (4.4)
in L2 (Ω, E2s ).
According to the Castigliano theorem the compliance (4.2) is expressed
by
Υ = min ||τ ||2C −1 τ ∈ Σ(Ω) (4.5)
The proof of (4.5) can be found in (Duvaut and Lions, 1976).
#
m
1
τ · (C −1 τ ) = (ω K · τ )2 (5.1)
λK
K=1
#
m
I4 = ωK ⊗ ωK (5.2)
K=1
1 # n
−1
τ · (C )τ = ||τ ||2 + νL (ω L · τ )2 (5.3)
λ1
L=2
imposes
ω L (x) · τ (x) = 0, L = 2, . . . , m (5.7)
since the terms underlined in (5.5) are non-negative. Thus the choice
1
ω1 = τ, ω L · ω 1 = δL1 (5.8)
||τ ||
The Free Material Design in Linear Elasticity 225
1
I = min ||τ ||2 dx τ ∈ Σ(Ω) (5.9)
Ω λ1
d(d + 1)
where λ = (λ1 , . . . , λm ), m = ; the constant E0 is a referential
2
elastic modulus. The condition (6.1) can be interpreted as a condition of
bounding the cost of the material to be used. To arrive at a relatively simple
final result we neglect the conditions: λK ≥ λmin > 0 and λK ≤ λmax . We
admit zero values of the Kelvin moduli λK , K = 2, . . . , m. Thus we
consider the FMD problem with the condition (6.1) or
Yp = min { Jp (τ ) | τ ∈ Σ(Ω)} (6.4)
with
1
Jp (τ ) = min ||τ || dx λ1 > 0,
2
λK 0, K = 2, . . . , m;
Ω λ1
||λ||p dx = Λ (6.5)
Ω
226 S. Czarnecki and T. Lewiński
1
J(τ ) = min ||τ ||2 dx λ1 > 0, λ1 dx = Λ (6.6)
Ω λ1 Ω
We assume now that the results (6.7), (6.8) can be accepted even if τ (x) = 0
for some x ∈ Ω. Let τ = π be the minimizer of the problem:
Z = min ||τ ||dx τ ∈ Σ(Ω) (6.10)
Ω
which proves that σ and π are collinear and form common trajectories of
principal stresses. The optimal tensor (6.12) satisfies the symmetry condi-
tions (3.2).
Note that the functional in (6.10) has a convex integrand (of linear
growth) and the field runs over a convex set in L2 (Ω, E2s ). Thus the prob-
lem (6.10) is well posed. The existence question lies outside the present
paper. We mention only that keeping the field τ within Σel (Ω) would be
artificial, since the minimizing sequencies for (6.10) tend to limits which are
not solutions to any well posed problems of linear elasticity.
Some aspects of the problem (6.10) will be discussed in Sec. 8. Now let us
stress that due to linear growth of the integrand of (6.10) there can appear
subdomains in Ω where π = 0. Then λ1 = 0 in these subdomains, which
goes beyond the assumptions in the formulation (6.6). We note, however,
||π||2
that in the domains where π = 0, the quotient is an indeterminate
λ1
expression (0/0). Therefore, the appearance of the domains where π = 0 is
not an argument to reject the formulation (6.10).
The solution to problem (6.10) determines not only the λ1 distribution,
but, which is the most important, determines the domain, where λ1 = 0 or
the domain where the material should be removed. One can alternatively
say that the shape of the optimal body is defined by the effective domain
of ||π||−1 . To detect the holes in Ω we do not need special techniques
of expanding small holes. We should only be equipped with a numerical
method by means of which the problem (6.10) can be solved with high
accuracy, to determine the boundary line of the effective domain of ||π||−1 .
In this manner we arrive at a new, optimized shape of the plate. Let us
name it Ωef f . For x ∈ Ωef f we have ||π(x)|| > 0. Even for x ∈ Ωef f the
condition (3.3) is not satisfied. Indeed, let us compute
||π(x)||
Cijkl (x)ξij ξkl = Eo (π̂(x) · ξ)2 (6.14)
||π||
For ξ ⊥ π̂ this expression vanishes. Although the material does not satisfy
(3.3), its moduli are distributed in such a way that the optimal anisotropic
body can transmit the given load T to the given support Γ2 . Indeed, the
solution of problem (6.10) determines a new domain Ωef f cut out from
Ω in which ||π(x)|| > 0 where the optimum design method applies. This
new plate is capable of transmitting the load T to the support Γ2 . We
substitute (6.13) into (6.10) to arrive at the elasticity problem which governs
the optimal plate behaviour:
with
Eo
a(u, v) = ||π(x)|| (π̂ · ε(u)) (π̂ · ε(v)) dx (6.16)
||π|| Ωef f
f (v) = 0 ∀ v ∈ Vo (6.19)
We note that this condition holds since f (v) = 0 means that π ⊥ ε(v) and
just such v constitute the set V0 .
Assume that u and u solve (6.15). Then a(u − u , v) = 0 for v ∈
V (Ωef f ). We take v = u − u as kinematically admissible to arrive at the
condition u − u ∈ V0 . The solution u is determined up to terms from V0
which vanish on Γ2 . Let us look more closely on the fields in V0 . If v ∈ V0
then ε(v) · π = 0 or
πij vi,j = 0 (6.20)
Noting that div π = 0 we rearrange (6.20) to the form
The criterion of the correctness of the statics solution concerning the geo-
metrically variable structures is that the virtual work of the loading on all
zero-energy modes (rigid body modes, zero strain modes) vanishes. This
condition here assumed the form (6.19).
Let us come back to the problem (6.10), of crucial importance for the
FMD. Let us disclose the condition (2.4) by writing:
Z= min
2 2
max ||τ ||dx + f (v) − τ · ε(v)dx (6.23)
τ ∈L (Ω, Es ) v∈V (Ω) Ω Ω
The operations max and min can be switched (by arguments similar to those
in Strang and Kohn, 1983):
with
R(v) = min 2 2
[||τ || − τ · ε(v)] dx τ ∈ L (Ω, Es ) (6.25)
Ω
where
B = ε ∈ E2s | ||ε|| 1 (6.27)
Thus we find
Remark 6.1
The problems (6.10) and (6.28) are similar to Michell’s problem formula-
tions, discussed in the article by Lewiński and Sokól (2013) in this book.
230 S. Czarnecki and T. Lewiński
Due to this similarity the optimal Michell layouts and the FMD layouts
have much in common.
A simple benchmark
Consider a plane domain shown in Fig. 1, of the sides (2c + b) by a, divided
b
into the subdomains Ω1 , Ω2 , Ω3 by the vertical lines x1 = ± . The load
2
is selfequilibrated and no kinematic conditions are imposed.
We start from considering (6.28). We take the field v of components
v1 = 0, v2 = x2 .
We note that ||ε(v)|| = 1 in the whole domain Ω. Let us compute
a a
f (v) = qbv2 x1 , − qbv2 x1 , −
2 2
f (v) = qba.
Let us consider now the problem (6.10). We take τ = 0 in Ω1 , Ω3 and
τ11 = 0, τ12 = 0, τ22 = q in Ω2 .
We compute ||τ || = q in Ω2 and then
||τ ||dx = abq
Ω
which concides with f (v). This proves that the duality gap is zero and
both the problems (6.10) and (6.28) are solved exactly; τ = π. The Kelvin
modulus λ1 is
λ1 = 0 in Ω1 , Ω3 ,
The Free Material Design in Linear Elasticity 231
2c + b
λ1 = E0 in Ω2
b
By (5.8)
(ω 1 )11 = 0, (ω 1 )22 = 1, (ω 1 )12 = 0.
The only nonzero modulus Cijkl is C2222 = λ1 . The optimal material
in Ω2 can be viewed as fibrous along x2 of no lateral and shear stiffnesses.
The domains Ω1 , Ω3 appear to be empty.
The plate rests on two non-sliding supports on the lower edge (left and
right node) and is subject to a vertical or/and horizontal load at the lower
or/and upper edge, see Fig. 2. The load traction T = T (s) = Ti (x)ei ,
s ∈ Γ1 is modeled by the weight function
s−s0 2
Ti = Ti (s)Tmax e−( w ) (i = 1, 2)
where
Lx Lx 2Lx
s0 = , ,
2 3 3
depending on the load position.
The values Tmax , w are assumed to be equal Tmax ≈ 3.761, w = 0.15
(emulation of the unit forces). We have assumed that E0 = 1.0 [N/m2 ] –
see (6.1).
λ1 (π)
The distribution of optimal for four variants of loading T are
E0
shown in Fig. 3.
The Free Material Design in Linear Elasticity 233
λ1 (π)
Figure 3. The distribution of optimal (scatter, contours and height
E0
field plots in TieDie color map in the first, second and third column, re-
spectively) for the various variants of loading T (a, b, c and d variant in the
first, second, third and fourth row, respectively)
or
||τ || = ||∇2 F || (8.2)
2
where ∇ F = [F,ij ] i, j = 1, 2.
234 S. Czarnecki and T. Lewiński
Thus the right hand side of (8.2) is the Frobenius norm of the matrix
∇2 F . The condition τ ∈ Σ(Ω) will be replaced by (2.7). Hence
Z = min ||∇2 F ||dx F ∈ S(Ω), ik3 jl3 F,kl nj = Ti on Γ1 (8.3)
Ω
Remark 8.1
The problem (6.10) and its reformulation (8.3) resembles the minimal sur-
face problem in which the minimized functional has the form
+
f→ 1 + ||∇f ||2 dx (8.8)
Ω
where
||∇f ||2 = (f,1 )2 + (f,2 )2 (8.9)
The Euler equation for (8.8) reads
∇f
div =0 (8.10)
||∇f ||
The minimal surface problem has become an important part of the calculus
of variations, see (Nitsche, 1989). Note that the equation (8.10) is elliptic,
since it is equivalent to
Δf + (f,2 )2 f,11 + (f,1 )2 f,22 − 2f,1 f,2 f,12 = 0 (8.11)
and the first term proves ellipticity of the equation.
The Free Material Design in Linear Elasticity 235
with
(α)
f (v) = T α ·v ds (9.2)
Γ1
The field τ (α) satisfying the above equation is statically admissible; the set
of such stress fields is denoted by Σα (Ω) Let C be tensor of the reduced
moduli, cf. Sec. 3. The compliance corresponding to the load indexed by α
is given by (4.5) or
236 S. Czarnecki and T. Lewiński
Υ (α)
= min τ· C −1
τ dx τ ∈ Σα (Ω) (9.3)
Ω
If the field u(α) is the displacement field referring to the load of number α
then the compliance is denoted by Υ(α) = f (α) (u(α) ).
Tensor C −1 is decomposed as in (3.16). We assume that the moduli
are ordered: λ1 ≥ . . . ≥ λ3 ≥ λ0 > 0 and fixed. The design variables are
the eigenstates ω 1 , ω 2 , ω 3 satisfying the orthonormality conditions (3.15).
Since ω 1 is determined by ω2 and ω 3 the unknowns will be only two last
tensors. The set of pairs (ω 2 , ω 3 ) defined in Ω, satisfying (3.15) is denoted
by Q(Ω). The merit function is
α
Υ (ω 2 , ω3 ) = min k(τ , ω2 , ω 3 )dx τ ∈ Σα (Ω), α = 1, 2 (9.5)
Ω
and
1
k(τ , ω 2 , ω 3 ) = ||τ ||2 + ν2 (ω 2 · τ )2 + ν3 (ω 3 · τ )2 (9.6)
λ1
the constants ν2 , ν3 are given by (5.4). The aim is to solve the family of
problems
Iη = min { Fη (ω 2 , ω 3 ) | (ω 2 , ω3 ) ∈ Q(Ω)} (9.7)
indexed by η ∈ [0, 1]. Combining (9.4)–(9.7) we arrive at
Iη = min Wη τ 1 (x), τ 2 (x) dx τ 1 ∈ Σ1 (Ω), τ 2 ∈ Σ2 (Ω) (9.8)
Ω
where
where
1 1 1
Let us stress here, this is not a Gram matrix, since it is defined by two
vectors, and the vectors belong to R3 . The Gram matrix will appear in the
case of three independent loads, see Sec. 11. Note that λ3 does not enter
(9.11). The analytical solution of (9.9) determines the optimal eigenstates
ω 2 , ω3 depending on the stress fields τ 1 , τ 2 which solve (9.8). Assume now
that for the fixed η the problem (9.8) is solved and the fields τ 1 , τ 2 are
known. We introduce
√ +
σ = ητ 1 , τ = 1 − ητ 2 (9.14)
with
||ε||2 − ||κ||2 2(ε · κ)
L(ε, κ) = ε+ κ (9.21)
G(ε, κ) G(ε, κ)
The Free Material Design in Linear Elasticity 239
and 7
2
G(ε, κ) = (||ε||2 − ||κ||2 ) + 4(ε · κ)2 (9.22)
One can prove that the strain potential Wη∗ (ε1 , ε2 ) exists such that
α ∂Wη∗ (ε1 , ε2 )
σ ij = i, j = 1, 2 (9.23)
∂εαij
λ1 λ2
Wλ∗ (ε, κ) = μ1 (g(ε, κ)) + μ2 (g(ε, κ)) (9.30)
4 4
where g(ε, κ) is the matrix of the form (9.13) of arguments ε and κ while
μα are its eigenvalues.
Note that
||λ|| = |λ1 | + |λ2 | + |λ3 | and λK ≥ λ03 > 0.
The subject of consideration is minimization of the functional (9.7) over the
Kelvin moduli
where
Sη (ρ) = min Tη (τ 1 (x), τ 2 (x), ρ(x))dx τ α ∈ Σα (Ω), α = 1, 2
Ω
(10.6)
and Tη is given as the solution to the local minimization problem
Tη (τ 1 , τ 2 , ρ) = min Wη (τ 1 , τ 2 ) λ1 + λ2 = ρ, λ1 λ2 , λ1 , λ2 ∈ R+
(10.7)
We make use now of the result (9.12): we express Wη explicitly by λ1 , λ2
1 1
Wη (τ 1 , τ 2 ) = a1 + a2 , (10.8)
λ1 λ2
where √ +
aα = μα g( ητ 1 , 1 − ητ 2 ) (10.9)
and g(·, ·) is the matrix (9.13).
One should solve the problem for given a1 > 0, a2 > 0:
a1 a2
T (a1 , a2 , ρ) = min + λ + λ = ρ λ λ > 0 (10.10)
λ2
1 2 1 2
λ1
The solution of the above problem reads:
√
aα
λ∗α = ρ √ √ (10.11)
a1 + a2
1 √ √ 2
T (a1 , a2 , ρ) = ( a1 + a2 ) (10.12)
ρ
We insert now (10.12) into (10.6) and write (10.5) in the form:
&7
1 √ +
Jη = α min min μ1 (g( ητ 1 , 1 − ητ 2 )+
τ ∈Σα (Ω) Ω ρ(x)
7 '2
√ 1 +
+ μ2 (g( ητ , 1 − ητ 2 ) dx ρ ∈ L(Ω, R+ ),
ρdx = Λ (10.13)
Ω
242 S. Czarnecki and T. Lewiński
√ 1 + α
Zη = min ητ (x), 2
1 − ητ (x) 1 dx τ ∈ Σα (Ω)
Ω 2
(10.15)
where
+ +
|||(σ, τ )||| 1 = μ1 (g(σ, τ )) + μ2 (g(σ, τ )) (10.16a)
2
or
7 +
|||(σ, τ )||| 1 = ||σ||2 + ||τ ||2 + 2 ||σ||2 ||τ ||2 − (σ · τ )2 (10.16b)
2
1 2
where
B− 12 = (ε, κ) ∈ E2s × E2s |||(ε, κ)|||− 1 1 (10.19)
2
and
σ·ε+τ ·κ
|||(ε, κ)|||− 1 = max 2 (10.20)
2 σ, τ ∈Es |||(σ, τ )||| 1
2
∗
optimal moduli Cijkl (x) can be found on the ground of Pareto optimal
∗α
stress fields τ (α = 1, 2) see Figs 5, 6, 7 and 8.
1
Iη = min Wη τ (x), τ (x), τ (x) dx τ K ∈ ΣK (Ω)
2 3
(11.2)
Ω
where
#3 3
1 # √
1 2 3
Wη τ , τ , τ = min ( η L τ L · ω K )2 ω K ∈ R3 ,
λK
K=1 L=1
8
ωK · ω L = δKL (11.3)
We shall prove that the solution to the above problem has the form
√ √ √
Wη τ 1 , τ 2 , τ 3 = Wλ η1 τ 1 , η 2 τ 2 , η 3 τ 3 (11.4)
3
# 1
Wλ σ 1 , σ 2 , σ 3 = μK g σ 1 , σ 2 , σ 3 (11.5)
λK
K=1
Note that
3
# 3
1 #
(ω K · σ L )2 = tr (AT A) (11.10)
λK
K=1 L=1
√
where σ L = ηL τ L and
3
#
tr (AT A) = xK · (ŜxK ) (11.11)
K=1
√
where Ŝ is expressed by σ L = ηL τ L and xK is the Kth column of the
1
matrix QΛ− 2 . Let eK be the orthonormal basis in R3 . Then
1
xK = √ QeK (11.12a)
λK
or
1
xK = √ ω K (11.12b)
λK
Note that g given by (11.6) is equal to S T S. We know that the eigenvalues
of S T S and SS T are identical. Thus the quantities μK are eigenvalues of
of the matrix Ŝ; we sort them as previously: μ1 ≥ μ2 ≥ μ3 .
The problem (11.3) can now be written as below
3
1 2 3
#
Wη τ , τ , τ = min xK · (ŜxK ) xK · xL = δKL ,
K=1
8
1
for K = L and ||xK || = √ (11.13)
λK
In the first step we minimize the term in which the norm ||xK || is the
biggest, or for K = 3. We compute
1
J3 = min x3 · (Ŝx3 ) ||x3 || = √ = x∗3 · (Ŝx∗3 ) (11.14)
λ3
Thus
μ3
J3 = μ3 ||x∗3 ||2 = (11.16)
λ3
248 S. Czarnecki and T. Lewiński
1
J2 = min x2 · (Ŝx2 ) x2 ⊥ x∗3 , ||x2 || = √ = x∗2 · (Ŝx∗2 ) (11.17)
λ2
where x∗2 is the eigenvector corresponding to the eigenvalue μ2
or
μ2
J2 = μ2 ||x∗2 ||2 = (11.19)
λ2
The last term corresponds to the vector x1 which is orthogonal to both the
vectors x∗2 and x∗3 , or this is the eigenvector corresponding to μ1 : Ŝx∗1 =
μ1 x∗1 ; we compute
μ1
J1 = x1 · (Ŝx1 ) = (11.20)
λ1
The sum J1 + J2 + J3 gives the result (11.5). In the proof we have tacitly
made use of the r earrangement inequality, which is used in the form (cf.
Hardy et al.,1999):
#
n
aK # n
aσ (K )
(11.21)
bK bK
K=1 K=1
∂Wλ σ 1 , σ 2 , σ 3
εK = (11.25)
∂σ K
or performing the maximum operation:
Wλ∗ ε1 , ε2 , ε3 =
(11.26)
max σ 1 · ε1 + σ 2 · ε2 + σ 3 · ε3 − Wλ σ 1 , σ 2 , σ 3 σ K ∈ E2s
yet both of these operations seem difficult. The formulation (11.2) is im-
portant as a starting point for minimization over the moduli λK , which is
the subject of the subsequent section.
K
Zη = min μK g η 1 τ 1 , η 2 τ 2 , η 3 τ 3 dx τ ∈ ΣK (Ω)
Ω K=1
(11.27)
250 S. Czarnecki and T. Lewiński
where m=3. Upon solving this problem one can find the layouts of the
Kelvin moduli:
sK (S(x))
λ∗K = Λ 3 (11.30)
#
sK (S(x))dx
Ω K=1
or of the matrix Ŝ. The optimal Hooke tensor has the form (9.19), where
λ∗K are given by (11.30) and ω ∗K are determined as above. In contrast to
the case of two loads (Sec. 10.1) the modulus λ∗K is in general non-zero. It is
however possible, that in some subdomains the matrix S vanishes and there
the material can be removed. The function of three arguments from E2s .
3
1
#
Iη = min Wη τ 1 (x), . . . , τ n (x) dx τ i ∈ Σi (Ω) (12.1)
Ω
where
#
m
1 # √
n
2
1
Wη τ , . . . , τ n
= min ηiτ i · ωK (12.2)
λK i=1
K=1
ωL · ω K = δKL , K, L = 1, 2, . . . , m; ω K ∈ Rm }
and d = 2, m = 3, n ≥ 3. We shall prove that
√ √
Wη τ 1 , . . . , τ n = Wλ η1 τ 1 , . . . , ηn τ n (12.3)
#m
1
Wλ σ 1 , σ 2 , . . . , σ n = μK Ŝ σ 1 , . . . , σ n (12.4)
λK
K=1
Thus the problem (11.13) holds with a new matrix Ŝ, which confirms
(12.4). The eigenstates minimizing (12.2) are the eigevectors of the ma-
trix Ŝ. The optimal tensor C is given by (9.19), where ω K = ω ∗K . The
integrand of the functional in (12.1) has a quadratic growth, which makes it
possible to make use of the non-linear FEM, as in the theory of hyperelastic
bodies.
Upon solving (11.29) one can find the Kelvin moduli by (11.30). The optimal
eigenstates ω ∗K are the eigenvectors of the matrix Ŝ. All the optimal Kelvin
moduli will be, in general, non-zero. Since the integrand in (11.29) is of
linear growth, there can appear the subdomains where S = 0. There the
material is not necessary. Thus the algorithm given determines not only the
layout of Cijkl but also predicts cutting the domain Ω to the domain where
the material is necessary, due to the load applied.
In case of n = 2 the result (12.4) still holds, since the matrix Ŝ has then
two positive eigenvalues μ1 , μ2 , the same as the matrix S T S. The equation
(12.4) reduces to (9.12), found in a different way. Similarly, Eq. (11.29)
reduces to (10.15).
#
n
Fη (ω 1 , . . . , ω6 ) = ηi Υ(i) (ω 1 , . . . , ω 6 ) (13.1)
i=1
a) Case of n ≥ 6
If σ i are nonzero, then μK (Ŝ) > 0, K = 1, . . . , 6. The matrix
(12.6) has positive eigenvalues, also equal to μK (Ŝ) and n − 6 zero
eigenvalues. One should solve (12.1), where Wη is given by (12.3),
(12.4) where m = 6.
b) Case of n = 5
Now the matrix S has dimensions 6 by 5 while the matrix Ŝ = SS T
is 6 by 6 and has one zero eigenvalues; according to the convention:
μ6 = 0. Thus Wλ is given by (12.4) with summation up to m =
5. Thus the eigenvalue λ6 ceases to have an effect on the solution
of (12.1). Upon solving (12.1) for m = 5 we find τ ∗1 , . . . , τ ∗5 .
Having these fields we solve 5 eigenvalue problems to find (ω K , μK ),
K = 1, . . . , 5, for the matrix Ŝ. The vector ω 6 is orthogonal to ω K ,
K = 1, . . . , 5. Tensor C is given by (3.29) for m = 6 and is defined
in a unique way.
c) Case of n = 4
The matrix S has dimensions 6 by 4 while Ŝ has two zero eigen-
values; μ5 = 0, μ6 = 0. Thus the summation in (12.4) is up to m = 4.
Upon solving (12.1) for m = 4 we find the fields τ ∗1 , . . . , τ ∗4 . We
compute the positive eigenvalues μK and eigenvectors μK of the ma-
trix Ŝ, K = 1, . . . , 4. Additionally we determine the vectors ω 5 , ω6
as orthogonal to ω 1 , . . . , ω 4 but this choice is not unique. Thus the
tensor C given by (3.29) for m = 6 will not have uniquely defined
projectors P 5 and P 6 .
d) Case of n = 1
e) The matrix S has dimensions 6 by 1 while the matrix Ŝ has 5 zero
eigenvalues:
μ2 = μ3 = μ4 = μ5 = μ6 = 0; μ1 > 0; η1 = 1
Now
μ1 (Ŝ) = ||σ 1 ||2
and
1
Wλ (σ 1 ) = ||σ 1 ||2 (13.2)
λ1
The vector ω1 is an eigenvector of the matrix Ŝ = σ 1 (σ 1 )T .
One can check that
1
ω1 = σ1 (13.3)
||σ 1 ||
We have arrived at the results analogous to (5.8), (5.9), where n = 1, d = 2.
Thus in the representation (3.29) only the first term is uniquely determined.
The cases of n = 2, 3 are left to the reader.
254 S. Czarnecki and T. Lewiński
14 Final remarks
1. The main conclusion is: the minimal number of the load conditions
which is indispensable to fix correctly (such that (3.3) holds) the com-
ponents of the tensor C equals m, or equals m = 3 in 2D and m = 6
in 3D. Thus this minimal number of loads is equal to the number of
components of strain (or stress). If the number of the load conditions
is smaller, then, in the whole design domain some of the Kelvin moduli
are zero. In the extreme case of a single load only one Kelvin modulus
is non-zero.
2. The FMD procedure is two-stage. To find the optimal moduli at
each point of the design domain one should: a) solve an auxiliary
The Free Material Design in Linear Elasticity 255
problem, which, in general, has the form (13.4). There could appear
subdomains where all the minimizing fields τ ∗1 , . . . , τ ∗n vanish. The
remaining domain, denoted by Ωef f will be filled up by the optimal
non-homogeneous and anisotropic material of moduli Cijkl . b) we
solve the elasticity problem in the effective domain of given elastic
moduli. The state of stress will be determined uniquely.
3. The FMD concept can be applied not only to the minimization of the
compliance with the isoperimetric condition imposed on the integral
of the p-norm ||λ||p . Other admissible isoperimetric conditions are
discussed by (Barbarosie and Lopes, 2008). The possible FMD exten-
sions to other functionals are overviewed in (Haslinger et al. 2010).
15 Bibliography
C. Barbarosie, S. Lopes, Study of the cost functional for free material design
problems. Numer. Funct. Anal. Optimiz. 29: 115-125, 2008.
M.P. Bendsøe, J.M. Guedes, R.B. Haber, P. Pedersen and J.E. Taylor, An
Analytical Model to Predict Optimal Material Properties in the Context
of Optimal Structural Design, J. Appl. Mech. Trans. ASME 61, 930-937,
1994.
M.P. Bendsøe, A.R. Diaz, R. Lipton, J.E. Taylor, Optimal design of material
properties and material distribution for multiple loading conditions. Int. J.
Numer. Meth. Eng. 38: 1149–1170, 1995.
A. Blinowski, J. Ostrowska-Maciejewska, J. Rychlewski, Two-Dimensional
Hooke’s Tensors-Isotropic Decomposition, Effective Symmetry Criteria. Arch.
Mech., 48, 325-345, 1996.
A.V. Cherkaev, Variational Methods for Structural Optimization, Springer,
New York, 2000.
Ph. G. Ciarlet, Mathematical Elasticity, vol. I, Three dimensional elasticity.
North Holland, Amsterdam 1988.
256 S. Czarnecki and T. Lewiński
1 Introduction
obtained subject to all external excitation frequencies within the large range from
zero and up to the fundamental eigenfrequency. Optimization with respect to a
higher order eigenfrequency was found to produce a large gap between the subject
eigenfrequency and the adjacent lower eigenfrequency, and offered even more
competitive designs for avoidance of resonance in problems where external
excitation frequencies are confined within a large interval with finite lower and
upper limits. In subsequent beam shape optimization papers by Olhoff and Parbery
(1984), Bendsøe and Olhoff (1985), and Olhoff et al. (2012), the design objective
was directly formulated as maximization of the separation (gap) between two
consecutive eigenfrequencies of prescribed orders. The study in Olhoff et al. (2012)
yields the interesting result that, except for beam segments adjacent to the beam
ends (whose designs are characteristic for the specific boundary conditions
considered), the entire inner parts of the optimized beam designs exhibit a
significant periodicity in terms of repeated inner beam segments, the number of
which increases rapidly with increasing values of the orders of the consecutive
upper and lower frequencies of the maximized gaps.
It should be noted that the separation of adjacent eigenfrequencies as considered
by Olhoff et al. (2012) and Jensen and Pedersen (2006) is closely related to the
existence of so-called phononic (or acoustic) band gaps, i.e., gaps in the wave band
structure for periodic materials implying that elastic waves cannot propagate in
certain frequency ranges. Sigmund (2001) applied topology optimization to
maximize phononic band gaps in periodic materials (see also Diaz et al., 2005, and
Halkjær et al., 2006). Moreover, Sigmund and Jensen (2003), Jensen (2003), and
Jensen and Sigmund (2005) performed minimization of the response of band gap
structures (wave damping).
In recent papers, Bruggi and Taliercio (2012) and Niu et al. (2009) performed
topology optimization for maximum fundamental eigenfrequency of structures
composed of micropolar solids and cellular material, respectively, and Yoon (2010a)
applied a parameterisation based on element connectivity. However, topology
optimization with respect to eigenfrequencies of structural vibration was first
considered by Dias and Kikuchi (1992), who dealt with single frequency design of
plane disks. Subsequently, Ma et al. (1994), Dias et al. (1994), and Kosaka and
Swan (1999) presented different formulations for simultaneous maximization of
several frequencies of free vibration of disk and plate structures, defining the
objective function as a scalar weighted function of the eigenfrequencies. The paper
(Pedersen, 2000) dealt with maximum fundamental eigenfrequency design of plates,
and included a technique to avoid spurious localized modes. In contrast to the earlier
work, Krog and Olhoff (1999), Jensen and Pedersen (2006), and Du and Olhoff
(2007b) applied a variable bound formulation (see Bendsøe et al., 1983) which
facilitates proper treatment of multiple eigenfrequencies that very often result from
the optimization. The first of these papers treats optimization of fundamental and
higher order eigenfrequencies of disk and plate structures, while Jensen and
262 N. Olhoff and J. Du
Pedersen (2006), and Du and Olhoff (2007b), also deal with maximization of the
separation of adjacent eigenfrequencies for single- and bi-material plates.
Topology optimization with the objective of maximizing the dynamic stiffness
(minimizing the dynamic compliance) of elastic structures subjected to time-
harmonic external loading of given frequency and amplitude are, e.g., studied by
Ma et al. (1995), Min et al. (1999), Jog (2002), Jensen and Sigmund (2005), Olhoff
and Du (2005, 2013), Kang et al. (2012), and Yang and Li (2013). Similar work on
structural topology optimization for minimum vibration amplitude response over a
range of excitation frequencies has been carried out by Calvel and Mongeau (2005)
and Jensen (2007). Recent papers on topology optimization for minimum frequency
response have been published by Yoon (2010b) and Shu et al. (2011).
Optimization of structural-acoustic systems against sound and noise emission
has benefited from textbooks by Koopmann and Fahnline (1997) and Kollmann
(2000), and proceedings by Munjal (2002) and Bendsøe et al. (2006) from two
IUTAM Symposia. During recent years, topology optimization based acoustic
design of elastic structures subjected to time-harmonic external mechanical
loading of given excitation frequency or frequency range, amplitude, and spatial
distribution has attracted significant attention, and minimization (or
maximization) of the acoustic power radiated from the structural surface(s) into a
surrounding or interior acoustic medium like air have been frequent design
objectives, see, e.g., Christensen et al. (1998), Luo and Gea (2003), Wadbro and
Berggren (2006), Sorokin et al. (2006), Bös (2006), Olhoff and Du (2006), Yoon
et al. (2007), Du and Olhoff (2007a, 2010), Dühring et al. (2008), Yamamoto et
al. (2009), Niu et al. (2010), Nandy and Jog (2011), Du et al. (2011), Kook et al.
(2012), and Yang and Du (2013).
In terms of optimization of composite structures with respect to acoustic
criteria, we may refer the reader to the review article Denli and Sun (2007) and
the bibliography Mackerle (2003), and a large number of papers cited therein. As
examples of various types of problems of optimum structural-acoustic design
with composite materials, we may refer to Hufenbach et al. (2001), Thamburaj
and Sun (2002), Chen et al. (2005); Yamamoto et al. (2008), Jensen (2009) and
Niu et al. (2010).
In Niu et al. (2010), the novel topology optimization based method termed
the Discrete Material Optimization (DMO) method (see Stegmann and Lund
2005, Lund and Stegmann 2005) is applied to furnish the simultaneous design
optimization of fiber angles, stacking sequence and selection of material for
vibrating laminated composite plates with minimum sound emission.
Introductory Notes on Topological Design Optimization… 263
each of the design variables tend to attain one of their limiting values as
explained below, thereby forming a design with aggregations of finite elements
with solid material and void, respectively, see Fig. 1b. The result is a rough
description of outer as well as inner boundaries of the design that represents the
overall optimum topology. This topological design may subsequently be used as
a basis for refined shape optimization, see Olhoff et al. (1991).
where E*e is the elasticity matrix of a corresponding element with the fully solid
elastic material the structure is to be made of. The power p in (1), which is
termed the penalization power, is introduced with a view to yield distinctive “0-
1” designs, and is normally assigned values increasing from 1 to 3 during the
optimization process. Such values of p have the desired effect of penalizing
intermediate densities 0 < Ue < 1 since the element material volume is
proportional to Ue while the interpolation (1) implies that the element stiffness is
less than proportional. Note also that the interpolation (1) satisfies E e (0) = 0 and
E e (1) = E*e , implying that if a final design has density 0 and 1 in all elements,
this is a design for which the structural response has been evaluated with a
correct physical model.
By analogy with (1), for a vibrating structure the finite element mass matrix
may be expressed as
M e ( U e ) U eq M *e (2)
where M *e represents the element mass matrix corresponding to fully solid
Introductory Notes on Topological Design Optimization… 265
material, and the power q 1. Apart from exceptions briefly discussed in the
following section, normally q = 1 is chosen.
The global stiffness matrix K and mass matrix M for the finite element based
structural response analyses behind the optimization, can now be calculated by
NE NE
p
K ¦U e K *e , M ¦U q
e M *e (3)
e 1 e 1
Here, K *e is the stiffness matrix of a finite element with the fully solid material
for the structure, and NE denotes the total number of finite elements in the
admissible design domain.
In the problem formulations in Chapters 3, 5 and 7, Ve, e = 1,…,NE, denotes
the volumes of the finite elements, V0 is the total volume of the admissible
design domain, and for single material design, V* denotes the total volume
NE
by generating a continuous interpolation model for the mass with respect to any
value of the material density between 0 and 1. For example, to achieve continuity
of the interpolation model, we may introduce the following revised form of Eq.
(4),
U M* , U e ! 0.1
M e (Ue ) ® e 6 e * . (4a)
¯c0 U e M e , U e d 0.1
where the coefficient c0 105 enforces the C0 continuity at the value Ue = 0.1 of
the material density. In several of the examples presented later in this paper, for
comparison, we have applied each of the interpolation models (4) and (4a) in the
numerical solution scheme and only found negligible differences in the final
results. The reason is that in both models, the region with lower density has a
very small contribution to the first several eigenfrequencies of the structure.
Furthermore, all intermediate values of the material density will approach zero or
one during the design process, which implies that the change of the interpolation
model in regions with lower density as shown in (4a) must have very limited
influence on the final zero-one design.
*1 *2
where E and E denote the element elasticity matrices corresponding to the
e e
two given solid, elastic materials *1 and *2. Here, material *1 is assumed to be
the stiffer one. The penalization power p in (5) has generally been assigned the
value 3 which resulted in distinctive optimum topology designs in the examples
of bi-material design considered in the sequential papers Olhoff and Du
(2013B,E). It follows from (5) that for a given element, Ue = 1 implies that the
element fully consists of the solid material *1, while Ue = 0 means that the
element fully consists of the solid material *2.
The element mass matrix of the bi-material model may be stated as the
simple linear interpolation
M e ( U e ) U e M *e1 (1 U e )M *e2 (6)
where M *e1 and M *e2 are the element mass matrices corresponding to the two
different, given solid elastic materials *1 and *2.
Introductory Notes on Topological Design Optimization… 267
The SIMP model formulated by (1) and (2) (or (5) and (6)) may be regarded
as an interpolation scheme for the structural stiffness and mass with respect to
material volume density. Recently, a generalized material model based on a
polynomial interpolation was proposed by Jensen and Pedersen (2006), and it
was shown how proper polynomials corresponding to different design objectives
can be easily obtained.
When bi-material design is treated via the problem formulations in the
sequential papers Olhoff and Du (2013B,E), then V* denotes the total volume
NE
¦U V
e 1
e e of the stiffer material *1 available for the structure, while the total
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Introductory Notes on Topological Design Optimization… 271
Abstract A frequent goal of the design of vibrating structures is to avoid resonance of the
structure in a given interval for external excitation frequencies. This can be achieved by,
e.g., maximizing the fundamental eigenfrequency, an eigenfrequency of higher order, or
the gap between two consecutive eigenfrequencies of given order, subject to a given
amount of structural material and prescribed boundary conditions. Mathematical
formulations and methods of numerical solution of these topology optimization problems
are presented for linearly elastic structures without damping in this paper, and several
illustrative results are shown.
1 Introduction
Methods for optimization of simple (unimodal) eigenvalues/eigenfrequencies in
shape and sizing design problems are well established and can be implemented
directly in topology optimization. The formulation for topology optimization
with respect to a simple, fundamental eigenfrequency is presented in Sub-section
2.1, and the sensitivity analysis of a simple eigenfrequency subject to change of a
design variable Ue is outlined in Sub-section 2.2. However, particularly in
topology optimization it is often found that, although an eigenfrequency is
simple during the initial stage of the iterative design procedure, later it may
become multiple due to coincidence with one or more of its adjacent
eigenfrequencies. In order to capture this behaviour, it is necessary to apply a
more general solution procedure that allows for multiplicity of the
eigenfrequency because a multiple eigenfrequency does not possess usual
differentiability properties.
In Sub-section 2.3, the abovementioned eigenfrequency optimization
problems are conveniently formulated by a so-called bound formulation
(Bendsøe et al. 1983, Taylor and Bendsøe 1984, Olhoff 1989). Sub-section 2.4
then presents design sensitivity results for multiple eigenvalues derived in
(Seyranian et al. 1994, and Du and Olhoff 2007), and by usage of these results,
the problems can be solved efficiently by mathematical programming (see, e.g.,
Overton 1988, and Olhoff 1989) or by the MMA method (Svanberg 1987).
Moreover, the procedure of treating the multiple eigenvalues can be greatly
simplified by using the increments of the design variables as unknowns (see
Krog and Olhoff 1999, and Du and Olhoff 2007). Sub-section 2.5 presents the
iterative, numerical solution procedure which is developed such that it is
applicable independently of whether the subject eigenfrequencies are uni- or
multimodal.
Finally, Section 3 presents several numerical examples of topology
optimization of single- and bi-material beam- and plate-like structures, and
Section 4 concludes this paper, which lends itself to (Du and Olhoff 2007).
max { min {Z 2j }}
U1,, UN E j 1,J (1a)
Subject to :
Kij j Ȧ 2j Mij j , j 1, , J , (1b)
ij Tj Mij k G jk , j t k, k , j 1, , J , (1c)
NE
¦U V e e V * d 0 , V * DV0 , (1d)
e 1
0 U d Ue d 1 , e 1, , N E . (1e)
the constraint (1b). The J candidate eigenfrequencies considered will all be real
and can be numbered such that
0 Z1 d Z2 d d Z J , (2)
and it will be assumed that the corresponding eigenvectors are M-
orthonormalized, cf. (1c) where Gjk is Kronecker’s delta. In problem (1a-e), the
symbol NE denotes the total number of finite elements in the admissible design
domain. The design variables Ue, e = 1,…,NE, represent the volumetric material
densities of the finite elements, and (1e) specify lower and upper limits U and 1
for Ue. To avoid singularity of the stiffness matrix, U is not zero, but taken to be
a small positive value like U = 10-3. In (1d), the symbol D defines the volume
fraction V * /V0 , where V0 is the volume of the admissible design domain, and V *
the given available volume of solid material and of solid material *1,
respectively, for a single-material and a bi-material design problem, cf. Sub-
sections 2.1 and 2.3 in the preceding paper Olhoff and Du (2013A).
O j ^ij T
j (K cU1 O j M cU1 )ij j ,, ij Tj (K cU N O j M cU N )ij j
E E
`T
(8)
max {E }
E , U1,, U N E (9a)
Subject to :
E Z 2j d 0 , j n, n 1,, J , (9b)
Constraints:1(b-e) (9c)
Here, as well as in Eqs. (10) below, J is assumed to be larger than the highest
order of an eigenfrequency to be considered a candidate to exchange its order
with the n-th eigenfrequency or to coalesce with this eigenfrequency during the
design process.
The problem of maximizing the distance (gap) between two consecutive
eigenfrequencies of given orders n and n – 1 with n > 1, (see Olhoff 1976, Olhoff
and Parbery 1984, Bendsøe and Olhoff 1985, Jensen and Pedersen 2005, Olhoff
et al. 2012) may be written in the following extended bound formulation, where
two bound parameters are used:
max {E 2 E1} (10a)
E 1, E 2, U1,, U N E
Subject to :
E 2 Z 2j d 0 , j n, n 1, , J , (10b)
Z 2j E1 d 0 , j 1, , n 1, (10c)
Constraints: 1(b-e). (10d)
Note that if in (10) we remove the bound variable E1 and the corresponding set of
constraints (10c) from the formulation, then the eigenfrequency gap
maximization problem (10) reduces to the n-th eigenfrequency maximization
problem (9), and in particular, for n = 1, to the problem of maximizing the
fundamental eigenfrequency in (1).
In problem (9) the eigenfrequency Zn , and in problem (10) both the
eigenfrequencies Zn and Zn1 of the optimum solution may very well be
multiple, and the bound formulations in (9) and (10) are tailored to facilitate
handling of such difficulties.
It is also worth noting that the introduction of the scalar bound variables E
in (9) and E1 and E 2 in (10) implies that even if multiple eigenfrequencies are
present, the optimization problems (9) and (10) are both differentiable if they are
considered as problems in all variables, i.e. the bound parameter(s) E (or
E1 , E 2 ), design variables U e , e 1,, N E , as well as the eigenfrequencies Z j
280 N. Olhoff and J. Du
and eigenvectors ij j , j 1,, J , (implying that all these variables should have
been included under the ‘max’ signs in (9a) and (10a)). This type of problem is
referred to as one of “Simultaneous analysis and design” (SAND), and is a very
large problem in the present context. Therefore, we refrain from solving the
current topology optimization problems in this form.
In the form written above, where only the design variables U e , e 1,, N E ,
and the bound parameters E and E1 , E 2 are included under the ‘max’ signs in
(9a) and (10a), the topology optimization problems (9) and (10) are non-
differentiable because the eigenfrequencies Z j , j 1,, J , are considered as
functions of the design variables U e , e 1,, N E . This is a ‘nested’ formulation
which provides the basis for numerical solution by a scheme of successive
iterations where, in each iteration, the eigenfrequencies Z j and eigenvectors ij j ,
j 1,, J , are established for known design, U e , e 1,, N E , by solution of the
generalized eigenvalue problem (1b) and implementation of the orthonormality
conditions (1c).
To accommodate for occurrence of multiple eigenfrequencies, we in the
subsequent Sub-section 2.4 consider some important sensitivity results for such
eigenfrequencies. In Sub-section 2.5, we make use of these results in the
development of incremental forms of problems (9) and (10) which provide the
basis for construction of a highly efficient scheme for numerical solution of the
topology optimization problems under study.
*1
Similarly, the eigenvalue problem (1b) contained in problem (10) may yield
another R-fold eigenvalue Oˆ O Z 2 , j n R,, n 1 , which corresponds to
j j
the R largest eigenfrequencies Z j in (10c). This case (for which we assume that
1 d n R ), is completely analogous to (11).
282 N. Olhoff and J. Du
~
Assuming that we know the multiple eigenvalue O , the associated sub-set of
orthonormalized eigenmodes, and have computed the derivatives of the matrices
K and M, we can construct the generalized gradient vectors fsk, s, k = n, …, n+N-
1, from (13). Solving the algebraic sub-eigenvalue problem in (12) for 'O then
yields the increments 'O 'O j , j n,, n N 1 , of the multiple eigenvalue
~
^ `
O subject to a given vector ǻȡ 'U1 , , 'U N E of increments of the design
variables.
The N increments 'O j , j n,, n N 1, constitute the eigenvalues of the
sub-eigenvalue problem (12), and represent the directional derivatives of the
~
multiple eigenvalue O O j Z 2j , j n,, n N 1 , with respect to change
'U e of the design variables U e , e 1, , N E . Attention should be drawn to the
fact that the increments 'O j , j n,, n N 1 of the multiple eigenvalue are
generally non-linear functions of the direction of the design increment vector ǻȡ .
Thus, unlike simple eigenvalues, multiple eigenvalues do not admit a usual
linearization in terms of the design variables.
Finally, two important special cases should be observed.
Case of vanishing off-diagonal terms. For the case of multiple eigenvalues, cf.
(17) with N > 1, a very important observation can be made. If in (12) all off-
diagonal scalar products are zero, i.e. if
f skT ǻȡ 0 , s z k , s, k n, , n N 1, (16)
then the increment 'O j of an eigenvalue O j Z 2j becomes determined as
'O j f jjT 'ȡ , j n,, n N 1, (17)
Structural Topology Optimization… 283
Hence, if the design increment vector ǻȡ fulfils (16), then f jj has precisely the
same form as the gradient vector O j in (8) for a simple eigenvalue, and the
eigenvalue increments 'O j in (17) are uniquely determined on the basis of the
eigenmodes ij j , j n,, n N 1 . The formulas for design sensitivity analysis
of multiple eigenvalues then become precisely the same as those for simple
eigenvalues.
0. Problem initialization.
Define value of n and
initialize design variables U e
Main
Inner loop loop
Increments No
'U e converged ?
Yes
U e converged ? No
i.e., 'ȡ H ?
Yes
Stop
max {E } (19a)
E , 'U1 , , 'U N E
Subject to :
E ª¬Z 2j f jjT ǻȡ º¼ d 0 , for j J n N, (19b)
(b) Maximization of the gap (distance) between the n-th and (n-1)-th
eigenfrequencies:
max {E 2 E1}
E 1, E 2, 'U1,, 'U N E (20a)
Subject to :
>
E 2 Z 2j f Tjj ǻȡ d 0 , @ for j J n N, (20b)
E2 >Z 2
j @
' (Z 2j ) d 0 , j n, ..., n N 1, (20c)
>Z 2
j @
'(Z 2j ) E1 d 0 , j n R, ..., n 1, ( R d n 1) (20d)
>Z 2
j @
f Tjj ǻȡ E1 d 0 , for j n R 1, (if R d n 2) (20e)
>
det f skT ǻȡ G sk ǻ(Z 2 ) @ 0, s, k n, ..., n N 1, (20f)
det >f T
sk ǻȡ G sk ǻ (Z
2
)@ 0, s, k n R, ..., n 1, (20g)
NE
Note that in the sub-optimization problems (19) and (20), the only unknowns
are the bound variables E and E1 , E 2 and the increments of the design
286 N. Olhoff and J. Du
a
b Admissible design domain
(a)
(b)
(c)
The optimized topologies are shown in Figs. 3(a-c), and the corresponding
optimum fundamental eigenfrequencies are all found to be bimodal with values
given in the caption of the figure. Fig. 4 shows the iteration history for the first 3
eigenfrequencies of the optimum bimodal design with simply supported ends in
Fig. 3(a). The iteration histories for the optimum designs with the two other
cases of boundary conditions in Figs. 3(b,c) are qualitatively similar. Figs. 5(a-c)
depict the first 3 eigenmodes of the optimized beam-like structure with simply
supported ends in Fig. 3(a), and the results show that the first 2 eigenmodes
(corresponding to the bimodal fundamental eigenfrequency) of the structure are
typical simply supported beam-type vibration modes, while the 3rd one is a more
general 2D vibration mode.
288 N. Olhoff and J. Du
(a)
(b)
(c)
Figure 3(a-c). Optimized single-material topologies (50% volume fraction) for the three
different sets of boundary conditions defined in Figs. 2(a-c). The optimum fundamental
eigenfrequencies are all found to be bimodal and have the values (a) Z1opt
a 174.7 , (b)
Z1opt
b 288.7 , and (c) Z1opt
c 456.4 , implying that they are increased by (a) 154%, (b)
177% and (c) 212% relative to the initial designs.
600
500
Eigenfrequencies
400
Z3
300
200 Z2
(Maximized) Z1
100
0
0 20 40 60 80
Iteration number
Structural Topology Optimization… 289
Figure 4. Iteration history of the first 3 eigenfrequencies associated with the design
process leading to the optimum simply supported beam-like structure in Fig. 3(a). It is
seen that the fundamental eigenfrequency is simple for the initial design, but soon
coalesces with the second eigenfrequency, and the maximum fundamental eigenfrequency
is bimodal.
(a) Z1opt
a 174.7
(b) Z 2 a Z1opt
a 174.7
(a)
(b)
(c)
Figure 6(a-c). Optimized single-material topologies (50% volume fraction) for the three
different sets of boundary conditions in Figs. 2(a-c). The optimum second
eigenfrequencies are found to be (a) Z2opta 598.3 , (b) Z2optb 732.8 , and (c) Z2optc 849.0 ,
and are all bimodal.
up with a maximized gap between the 2nd and the 3rd eigenfrequencies that is
equal to 810, which is 548 % higher than the difference between the
corresponding eigenfrequencies of the initial design. Note that in Fig. 7(c) the
3rd, 4th and 5th eigenfrequencies form a tri-modal eigenfrequency of the final
optimized design.
(a)
(b)
1200
Z5
1000
Z4
Eigenfrequencies
Z3
800
600
Z2
200
Z1
0
0 20 40 60 80 100
Iteration number
(c)
Figure 7. (a) Clamped beam-like 2D structure with a concentrated mass attached at the
mid-point of the lower edge. (b) Optimized topology of the beam-like structure. The gap
between the 2nd and the 3rd eigenfrequencies is maximized. (c) Iteration history for the
first five eigenfrequencies associated with the process leading to the optimized topology.
Notice that the 3rd, 4th and 5th eigenvalues have coalesced to a tri-modal eigenfrequency
for the optimized topology.
292 N. Olhoff and J. Du
Figure 10. Plate-like 3D structure (a=20, b= 20 and t=1) with simple supports at its four
corners and center. (a) Admissible design domain. (b-c) The eigenmodes of the initial
design associated with the bimodal fundamental eigenfrequency Z10 Z20 24.56 .
The optimized topology is shown in Fig. 11(a) (50% volume fraction), and
the corresponding optimum fundamental eigenfrequency is also bimodal.
294 N. Olhoff and J. Du
90
Z3
80
Eigenfrequency
70
Z2
60
Z1
50 (Maximized)
40
30 Multiple eigenfrequency
20
0 10 20 30 40 50
Iteration number
(a) (b)
Figure 11. (a) Optimized topology (50% volume fraction, single-material design)
associated with the maximum fundmental eigenfrequency Z1opt 60.32 , which is
bimodal. (b) Iteration history for the first three eigenfrequencies.
60 250 120
Z5 Z4 Z5
50 Z4 100
200 Z4
Eigenfrequencies
Eigenfrequencies
Eigenfrequencies
Z3
40 (Maximized) Z2 Z3 80
150 Z2
30 (Maximized) 60
100 Z3
20 40
Z1 Z2
Z1 (Maximized)
50
10 20
Z1
0 0 0
0 20 40 60 80 0 20 40 60 80 0 20 40 60 80
Iteration number Iteration number Iteration number
Z3
60
50
(Maximized) Gap: ZZ
3 2
40
30 Z2
20
Z1
10
0 10 20 30 40 50
Iteration number
Figure 15. Optimized topology of the plate-like structure with simple supports at four
corners and a concentrated mass at the center, cf. Fig. 8(a). The gap between the 2nd and
the 3rd eigenfrequencies is maximized. (b) Iteration histories for the first five
eigenfrequencies associated with the process leading to the design (a). It shows that the
second and the third eigenfrequencies form a double eigenfrequency for the initial design,
but that they split as the design process proceeds, and the 3rd and the 4th eigenfrequencies
end up being a double eigenfrequency of the final design. (c) Optimized topology of the
plate-like structure with the upper horizontal edge clamped, other edges free, and a
concentrated mass attached at the mid-point of the lower horizontal edge, cf. Fig. 8(c).
The gap between the 2nd and the 3rd eigenfrequencies is maximized.
4 Conclusions
Problems of topology optimization with respect to structural eigenfrequencies of
free vibrations were studied and presented in Sections 2 and 3 of this paper. The
design objectives were maximization of eigenfrequencies of given order, and
distances (gaps) between two consecutive eigenfrequencies of the structures.
It was necessary to develop and apply special iterative numerical procedures to
handle topology optimization problems associated with both simple and multiple
Structural Topology Optimization… 297
References
Bendsøe, M.P., Olhoff, N., Taylor, J.E., 1983. A variational formulation for multicriteria
structural optimization. J Struct Mech 11: 523-544.
Bendsøe, M.P., Olhoff, N., 1985. A method of design against vibration resonance of
beams and shafts. Optim Control Appl Meth 6: 191-200.
Bratus, A.S., Seyranian, A.P., 1983. Bimodal solutions in eigenvalue optimization
problems. Appl Math Mech 47: 451-457.
Cheng, G., Olhoff, N., 1982. Regularized formulation for optimal design of
axisymmetric plates. Int J Solids Struct 18: 153-169.
Diaz, A.R., Kikuchi, N., 1992. Solutions to shape and topology eigenvalue optimization
problems using a homogenization method. Int J Num Meth Engng 35: 1487-1502.
Diaz, A.R., Lipton, R., Soto, C.A. (1994). A new formulation of the problem of optimum
reinforcement of Reissner-Midlin plates. Comp Meth Appl Mechs Eng 123: 121-139.
Du, J., Olhoff, N., 2007. Topological design of freely vibrating continuum structures for
maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct
Multidisc Optim 34:91-110. See also Publisher’s Erratum in Struct Multidisc Optim
(2007) 34:545.
Du J., Olhoff N., 2010. Topological design of vibrating structures with respect to
optimum sound pressure characteristics in a surrounding acoustic medium. Struct
Multidisc Optim 42, 43-54.
Haftka, R.T., Gurdal, Z., Kamat, M.P., 1990. Elements of Structural Optimization.
Dordrecht: Kluwer.
Haug, E.J., Choi, K.K., Komkov, V. 1986. Design Sensitivity Analysis of Structural
Systems. New York: Academic Press.
On Optimum Design and Periodicity
of Band-gap Structures
1 Introduction
A band-gap structure can quench vibrations and significantly suppress the
propagation of waves for a certain range of frequencies. Such a frequency range
(ii) much larger values of the orders n and n-1 of the adjacent upper and lower
eigenfrequencies of maximized frequency gaps, and for (iii) different values of a
positive minimum cross-sectional area constraint. The new results are obtained
by the approach of finite element and gradient based optimization using
analytical sensitivity analysis, as described in the preceding paper Olhoff and Du
(2013B). The new solutions are compared with corresponding limiting optimum
solutions obtained without minimum cross-sectional area constraint by usage of
the aforementioned method of “Scaled Optimum Beam Elements” developed in
Olhoff (1976).
^
max ' Z 2 Zn2 Zn21
De
` (a)
subject to
KI j Z 2j MIj , j 1, , J , (b)
ITj MIk G jk , j t k, j, k 1, , J , (c ) (2)
NE
¦D l
e 1
e e 1 d 0, (d )
0 D min d D e , e 1, , N E . ( e)
Here, Z j and Ij are the dimensionless j-th eigenfrequency and corresponding
eigenvector, respectively, and ' Z 2 is the difference between the squares of
two consecutive eigenfrequencies of given orders n and n-1 (n = 2, 3, …). In
Eq. (2b), K and M are symmetric positive definite global stiffness and mass
matrices (with corresponding beam element matrices available in (Petyt 2010))
for the generalized structural eigenvalue problem for the vibrating beam structure.
Thus, the J candidate eigenfrequencies ( J ! n ) considered in the optimization
problem will all be real and can be ordered as follows by magnitude:
0 Z1 d Z2 d d Z J (3)
Eq. (2c) imposes the conditions of M -orthonormalization of the corresponding
eigenvectors, where G jk denotes Kronecker’s delta.
The dimensionless optimization problem (2) is discretized by subdividing the
beam into N E finite elements of equal lengths le = 1/ N E with individual cross-
sectional areas D e e 1, , N E , which play the role as design variables of the
discretized problem. Hence, Eq. (2d) expresses the non-dimensional (unit)
volume constraint for the problem, and in Eq. (2e) a positive minimum cross-
sectional area constraint value D min is prescribed for the design variables
D e e 1, , N E . The value of D min is to be chosen less than the mean (unit)
value of the cross-sectional area of the dimensionless beam, and larger than zero
to avoid singularity of the stiffness matrix.
Using an extended bound formulation (Bendsøe et al. 1983, Olhoff 1989,
Jensen and Pedersen 2006), the optimization problem in Eq. (2) can be
reformulated as in Eq. (4) where two scalar variables E1 and E 2 are introduced
in order to facilitate handling of possible multiple eigenfrequencies Zn and Zn 1 .
Note that E1 and E 2 are upper and lower bound parameters in the constraint
equations (4b) and (4c), respectively, and that the difference between them in the
objective function will be maximized. At the same time, E1 and E 2 serve as
design variables together with the cross-sectional areas.
304 N. Olhoff and B. Niu
max
E1 , E 2 ,D1 ,,D N E
^E 2 E1` (a)
subject to
E 2 Z 2j d 0, j n, n 1, , J , (b)
Z 2j E1 d 0, j 1, , n 1, (c )
KI j 2
Z MIj , j 1, , J , (d ) (4)
j
ITj MIk G jk , j t k, j, k 1, , J , ( e)
NE
¦D l
e 1
e e 1 d 0, (f)
0 D min d D e , e 1, , N E . (g)
3 Numerical Examples
3.1 Cantilever beams
Here, examples of cantilever beam designs for maximized higher order
eigenfrequency gaps 'Z3 Z3 Z2 , 'Z4 Z4 Z3 , 'Z9 Z9 Z8 ,
On Optimum Design and Periodicity… 305
Figure 1. Cantilever with maximized frequency gap 'Z3 = 129.72. 'Z3u =39.66 for the
comparison design
Figure 2. Cantilever with maximized frequency gap 'Z4 =195.15. 'Z4u =59.20 for the
comparison design
(a)
(b)
Figure 3. Cantilever with maximized frequency gap 'Z9 =1144.81. 'Z9u =157.91 for the
comparison design. (a) Mode shapes, (b) Optimized design
3(b) is the “best” optimum solution to the problem considered. Fig. 3(a) shows to
suitable scale the free vibration modes I 9 x and I 8 x corresponding to the
normalized and mutually orthogonal mode shape vectors I 9 and I 8 associated
with the eigenfrequencies Z9 and Z8 that define the maximized frequency gap
308 N. Olhoff and B. Niu
Figure 4. Cantilevers with maximized frequency gaps 'Z10 . 'Z10u =177.65 for the
comparison design. (a), (b) and (c) Local optimum solutions. (d) Presumed global
optimum solution
Figure 5. Local optimum cantilever associated with frequency gap 'Z10 =901.09
them that was very similar to one of four alternative, presumed global optimum
designs available in Olhoff (1976) for the problem of maximizing the 10-th
natural frequency when inner points of vanishing cross-sectional area are
allowed.
Fig. 5 shows the design that resulted from applying the uniform comparison
beam as an unbiased initial design when attempting to maximize the frequency
gap 'Z10 . As is seen from the caption of Fig. 5, the value of the frequency gap
310 N. Olhoff and B. Niu
(a)
(b)
Figure 6. Cantilever with maximized frequency gap 'Z19 = 5141.39. 'Z19u =355.31 for
the comparison design. (a) Mode shapes, (b) Optimized design
'Z10 for this distinctly different design is much lower than that of the design in
Fig. 4(d), so the design in Fig. 5 is only a local optimum solution.
Next, we present examples of cantilever beams with maximized gaps
between adjacent frequencies of higher orders, i.e., 'Z19 Z19 Z18 and
'Z20 Z 20 Z19 . The optimized designs are shown in Figs. 6(b) and 7(b).
Both beam designs are distinct, and the eigenfrequencies defining the maximized
frequency gaps of the designs are both found to be unimodal, albeit very close to
neighbouring eigenfrequencies.
Figs. 6(b) and 7(b) clearly show the important result that except for beam
segments adjacent to the beam ends (whose designs are characteristic for the
specific boundary conditions considered), the entire inner part of each of the
optimum beam designs exhibit a significant periodicity in terms of repeated
beam segments of the same type. By comparing the optimized designs in Figs.
6(b) and 3(b) and in Figs. 7(b) and 4(d), respectively, it may be concluded that
the degree of this inner periodicity increases with increasing values of the orders
n and n-1 of the natural frequencies that define the frequency gap subjected to
maximization. The free vibration modes I19 and I18 are drawn on the basis of the
mode shape vectors I19 and I18 corresponding to the natural frequencies Z19 and
On Optimum Design and Periodicity… 311
(a)
(b)
u
Figure 7. Cantilever with maximized frequency gap 'Z20 = 5542.40. 'Z20 =375.04 for
the comparison design. (a) Mode shapes, (b) Optimized design
Z18 that define the frequency gap 'Z19 Z 19 Z18 , and are shown in Fig. 6(a).
Similarly, the free vibration modes I20 and I19 corresponding to the natural
frequencies Z20 and Z19 defining the gap 'Z20 Z 20 Z19 are shown in Fig.
7(a). Here, it is interesting to study the influence on the modes of the inner dip in
the (new) beam segment at the free end of the design in Fig. 7(b). Note finally
that it is obvious from Figs. 6 and 7 that the physical frequency gap mechanism
described in connection with Fig. 3, also manifests itself in the current examples.
Figure 10. Clamped-clamped beam with maximized frequency gap 'Z10 =1617.91.
'Z10u =197.39 for the comparison design
than the corresponding values 'Z4u =78.96, 'Z9u =177.65, and 'Z10u =197.39 for
the uniform, clamped-clamped comparison beam.
The optimized beam designs are shown in Figs. 8, 9 and 10, and it is
interesting to compare the design in Fig. 8 in the present paper with that in the
bottom of Fig. 13 in Olhoff (1976). We note that – as in Figs. 3(b) and 4(d) – it is
seen in Figs. 9 and 10 that periodicity, i.e., repetition of segments of the same
type, already appears in the inner part of the beam designs with maximized
frequency gaps that correspond to relatively low orders of the respective natural
frequencies. However, the segments adjacent to the beam ends are generally
different due to different characteristics of the specific boundary conditions
considered.
Figure 11. Clamped-simply supported beam with maximized frequency gap 'Z4 =274.72.
'Z4u =74.02 for the comparison design
Figure 13. Clamped-simply supported beam with maximized frequency gap 'Z10 =
1541.79. 'Z10u = 192.46 for the comparison design
simply supported at the other end will be optimized. The same lower limit
D min =0.05 as above is prescribed. The frequency gaps considered are 'Z4 , 'Z9
and 'Z10 , and their maximized values are found to be 'Z4 =274.72,
'Z9 =1226.18 and 'Z10 =1541.79, i.e., they are significantly larger than the
corresponding values 'Z4u =74.02, 'Z9u =172.72, and 'Z10u = 192.46 for the
uniform, clamped-simply supported comparison beam.
The optimized beam designs are shown in Figs. 11, 12 and 13. Periodicity
can be observed in the inner part of the beam designs with maximized band-gaps
'Z9 and 'Z10 , see Figs. 12 and 13. The segments adjacent to the simply
supported end are generally different from those adjacent to the free or clamped
end. Two different kinds of segments adjacent to the simple end support are seen
in Figs. 12 and 13 that depict the beams maximizing the frequency gaps 'Z9
and 'Z10 , respectively. This point is discussed in Section 4.
314 N. Olhoff and B. Niu
Figure 14. Cantilever with maximized frequency gap 'Z9 =377.91 subject to a
minimum cross-sectional area constraint value 0.5. 'Z9u =157.91 for the comparison
design
Figure 15. Clamped-clamped beam with maximized frequency gap 'Z9 =460.64 subject
to a minimum cross-sectional area constraint value 0.5. 'Z9u =177.65 for the comparison
design
Figure 16. Clamped-simply supported beam with maximized frequency gap 'Z9 =
431.06 subject to a minimum cross-sectional area constraint value 0.5. 'Z9u = 172.72 for
the comparison design
and 12 are obtained in the beams in Figs. 14, 15 and 16, although the latter are
optimized with the considerably larger value D min =0.5 of the minimum cross-
sectional area constraint.
By the comparison of the above-mentioned figures, we also verify that due to
their larger design freedom, the beams optimized with the small value D min =0.05
of the lower cross-sectional area limit, are associated with significantly larger
increases of the maximized frequency gaps.
4 Discussion
Up to now, we have considered optimum design of Bernoulli-Euler beams with
the objective of maximizing, for a specified value of n, the separation (gap)
between the (higher-order) n-th and (n-1)-th natural frequencies, subject to a
prescribed positive value of a non-dimensional minimum allowable cross-
sectional area D min which has been chosen as D min =0.05 in Sub-sections 3.1, 3.2
and 3.3, and D min =0.5 in Sub-section 3.4.
In this section, we shall briefly discuss the characteristics of this natural
frequency gap optimization problem in the limiting case where the cross-
sectional area function is geometrically unconstrained (except for the given
volume). This means that no minimum constraint is specified for the cross-
sectional area of the beam, i.e., the cross-sectional area is allowed to attain zero
value in discrete points on the beam axis. In this special case (that corresponds
to D min = 0 in the context of this paper), the solutions to our problem of
maximizing the gap between the natural frequencies Zn and Zn 1 are the same
as the solutions to the problem of maximizing a single, higher order natural
frequency Zn of given order n for specified volume, length, and boundary
conditions of the beam. The latter problem is treated in (Olhoff, 1976) where a
large number of optimum designs are available.
The reason why the two different beam optimization problems have identical
solutions, can be explained as follows. When a single natural frequency Zn of
given higher order n is maximized without specification of a minimum constraint
on the cross-sectional area, the optimized beam turns out to possess n-1 degrees
of kinematic freedom to perform rigid motions, since the cross-section vanishes
at inner singular points of the beam. At these points, either inner hinges of zero
bending moment and finite shear force, or, predominantly, inner separations
with both zero bending moment and zero shear force, are created by the
optimization of the n-th natural frequency. This has the effect that
simultaneously with the maximization of the n-th natural frequency, the n-1
degrees of kinematic freedom of the beam turn all the n-1 modes associated with
316 N. Olhoff and B. Niu
the lower order natural frequencies into rigid body motions, and all these
frequencies (including that of order n-1) are therefore equal to zero.
Thus, besides maximizing the n-th natural frequency of the beam, the
problem formulation in Olhoff (1976) covers the current problem of maximizing
the difference (gap) between the n-th and the (n-1)-th natural frequency of the
beam, if D min = 0.
The optimized beams associated with the small minimum cross-sectional area
constraint value D min =0.05 in Sub-sections 3.1, 3.2, and 3.3 strongly indicate the
locations of formation of inner hinges and inner separations in the limiting case
of D min = 0. In Fig. 1, for example, the comparatively large inner beam segment
with active minimum cross-sectional area constraint will shrink to a single point
with the formation of an inner separation between the two parts of the beam in
the limiting case of D min = 0. In Fig. 2, the narrow “dip” in the cross-sectional
area function indicates a point where an inner hinge with zero bending moment
will be created in the case of D min = 0. A large number of similar points with
formation of inner beam separations and hinges in the case of D min = 0 are easily
identified in the figures. It is worth noting that in each of the optimum designs
shown, there is no more than a single inner point with a narrow “dip” in the area
function indicating formation of an inner hinge in the case of D min = 0, but an
increasing number of points that indicates creation of inner separations, when the
given order of the frequency gap is increased. An exception is the design in Fig.
5 which contains several narrow “dips”, but this is a local, and not global
optimum solution.
In the optimum designs with one end clamped and the other end simply
supported, see Figs. 12 and 13, we observe two kinds of segments adjacent to the
simply supported end. Quite surprisingly, for the design shown in Fig. 12, the
beam segment at the simply supported end will shrink to a separation in the
limiting case of D min = 0, where both the bending moment and shear force are
zero at the end point of the beam. Hence, as is discussed in Olhoff (1976), the
beam disconnects from the support, i.e., the separation makes the simple support
superfluous. This implies that the optimum solution in the limiting the case of
D min = 0 will be the same as that with a free end, instead of the original simply
supported end. Contrary to this, the beam segment adjacent to the simply
supported end in Fig. 13 will not shrink to a separation in the limiting case of
D min = 0, but remain connected to the hinge at the end point. Thus, in this case,
the optimum solution behaves as expected and takes advantage of the simple
support.
In Olhoff (1976), the dimensionless beam optimization problem is first
solved for small values of n for various combinations of classical boundary
conditions. This is done by successive iterations based on a numerical integration
On Optimum Design and Periodicity… 317
of the governing equations, where singularities in the n-th eigenmode and its
derivatives are isolated at points of zero beam cross-sectional area for reasons of
accuracy and convergence. Hereby a small number of different types of
optimally designed, non-dimensional beam elements are produced. By proper
scaling, these beam elements can be used as building blocks for optimum beams
associated with much larger values of the prescribed order of their n-th
eigenfrequency.
Since, as mentioned above, at most one inner hinge will appear in a global
optimum beam with D min = 0, and an inner hinge can be included in optimized
non-dimensional beam elements mentioned above, then all other inner points of
vanishing cross-sectional area in an optimum beam associated with a sufficiently
large value of n, will be inner separations between optimized beam elements.
This is very important because the inner separations provide the means to solve
very easily the optimization problem for a sufficiently large value of n by an
optimum scaling of the optimized non-dimensional beam elements by means of
very simple analytical formulas derived in Olhoff (1976).
As an example, let us consider the approach of determining the design of a
vibrating cantilever beam that maximizes, say, the 9-th eigenfrequency Z9 of the
beam, when no minimum cross-sectional area constraint is prescribed, i.e.,
D min =0. (Note that the optimized beam with D min =0.05 is shown in Fig. 3.)
The first step of the method consists in applying Table 2 and Eq. (30) in
Olhoff (1976) which easily yields that the optimum cantilever design associated
with n=9 will have four inner separations and be composed of (or assembled as)
five dimensionless, optimized elements (or segments) along the length of the
beam: an element “a” consisting of a cantilever optimized for n=1 (see Fig. 1 in
the paper), followed by four elements “c” (see Fig. 11 in the paper), each
consisting of a free-free beam optimized for n=3 (the order of the lowest non-
vanishing natural frequency for such a beam). The four (identical) “c” elements
will endow the resulting optimum beam design with periodicity. The beam
elements “a” and “c”, together with no more than four other elements, are
necessary for the optimization of non-dimensional Bernoulli-Euler beams for any
value of n and any combination of classical boundary conditions (clamped,
simply supported and free ends). These elements are all optimized with their
designs shown in the first part of Olhoff (1976), and the elements are listed
together with their optimum characteristics in Table 1 of the paper. Finally, very
simple explicit algebraic expressions [(30), (57) (63) and (64)] are derived and
presented in the paper, for computation of the maximum value of the n-th
eigenfrequency (in the current example Z9 ), of the optimized, assembled beam,
and for the proper scaling of the lengths and volumes of the individual,
optimized beam elements, such that each of these elements will vibrate at the
same frequency as the assembled, optimum beam.
318 N. Olhoff and B. Niu
with O z 1 , indicated by three grey domains in Fig. 17. The 8th and 9th non-
dimensional circular eigenfrequencies of the uniform, comparison beam and the
optimized beam are also shown in Fig. 17. As is well known, no stop band exists
in the infinite uniform beam in absence of damping. However, a relatively large
stop band for the infinite periodic beam is observed, where bending waves
cannot propagate. Similarly, stop bands can be seen in Figs. 18 and 19 for two
other examples. These figures demonstrate that there is almost perfect correlation
between the band-gap size/location of the emerging band structure and the
size/location of the corresponding maximized natural frequency gap in the finite
structure.
Figure 19. Variation of O versus excitation frequency by employing Floquet theory for
an inner repeated segment in Fig. 9, where the frequency gap 'Z9 Z9 Z8 of the
clamped-clamped beam is maximized. The grey domains indicate Floquet-predicted stop
bands. The 8th and 9th eigenfrequencies Z8u and Z9u of the comparison beam, and Z8 and
Z9 of the optimized beam are shown in the figure.
On Optimum Design and Periodicity… 321
It has been demonstrated in many papers, see e.g., Jensen (2003), Søe-
Knudsen and Sorokin (2010), that a structure with a finite number of repeated
unit cells may significantly suppress propagation of waves with frequencies in
the stop band. Fig. 20 shows the displacement response at the right hand end of
the optimized beam shown in Fig. 6(b), when the beam is subjected to a time-
harmonic base excitation in the transverse direction at the left hand end. The base
motion is prescribed with a given displacement amplitude u0 relative to the
fixed reference axis. The transverse displacement u at the right hand end is
u2
indicated in Fig. 20 in the form 10 log10 2 dB.
u0
Figure 20. Displacement response at the right hand end from the flexural vibration of the
optimized beam in Fig. 6(b) when the beam is subjected to time-harmonic base excitation
in the transverse direction at the left hand end. No damping is assumed. The 18th and 19th
eigenfrequencies Z18 and Z19 of the optimized beam are indicated in the figure.
It is seen from Fig. 20 that there is a large drop in the response in the stop
band frequency range. The stop band calculated from the corresponding unit cell
is given in Fig. 18. It demonstrates that the stop band may exist in the optimized
beam obtained by maximization of a frequency gap. It is observed from Figs. 18
and 20 that there is a correlation between the value of O representing the
strength of attenuation in a band gap, and the level of attenuation in the
frequency response function for a finite structure composed of the same periodic
unit cell. The many resonance peaks observed in Fig. 20 are due to the fact that
no damping is included. The resonance peaks can be removed or reduced by
including some damping, and we also found that there is no significant change of
322 N. Olhoff and B. Niu
the band-gap behavior for relatively small damping. The effect of smoothing by
including damping is often used in the topology optimization of band-gap
structures (Sigmund and Jensen, 2003).
6 Conclusions
Maximizing gaps between two adjacent natural frequencies (eigenfrequencies) of
free transverse vibrations of prescribed order is investigated in this paper which
lends itself to Du and Olhoff (2007a,b) and Olhoff et al. (2012). The results are
obtained by finite element and gradient based optimization using analytical
sensitivity analysis. An incremental optimization formulation with consideration
of multiple eigenvalues is applied, which can be used for problems with any mix
of multiple and simple eigenfrequencies. Non-dimensional optimum solutions
are presented for different classical boundary conditions, different orders of the
upper and lower eigenfrequencies of maximized gaps, and values of a minimum
cross-sectional area constraint. However, geometrically unconstrained optimum
solutions obtained in Olhoff (1976) are also discussed and utilized in this paper.
The results show that, except for beam segments adjacent to the beam ends
whose designs are characteristic for the specific boundary conditions considered,
all the inner part of the optimum beam designs exhibits a significant periodicity
in terms of repeated beam segments, the number of which increases with
increasing orders of the upper and lower frequencies of the maximized gaps.
When small values of the minimum cross-sectional area are prescribed,
solutions to the current problems are very close to corresponding solutions
obtainable by simple non-dimensional analytical expressions for limiting
optimum solutions that were derived earlier by a “method of scaled beam
elements” (Olhoff, 1976) in which inner points of vanishing cross-sectional area
in the beams were allowed and exploited.
In wave propagation problems, band-gap is found in an infinite beam
structure constructed by repeated translation of an inner beam segment obtained
by eigenfrequency gap optimization. The band-gap size/location of the emerging
band structure is matching very well with the size/location of the corresponding
maximized natural frequency gap in the finite structure. For the optimized
structures composed of a finite number of repeated segments in the inner part,
the motion transmitted from one end will be significantly suppressed by the
periodic segments. For the beam structures studied here, it can be concluded that
the optimum design maximizing the gap between two adjacent eigenfrequencies
of free transverse vibration of given higher order is periodic. It is also
demonstrated that the approach tailors a band-gap which is matching very well
the maximized frequency gap in the periodic structure characterizing elastic or
acoustic wave propagation.
On Optimum Design and Periodicity… 323
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multicriteria structural optimization. J Struct Mech, 11 (4): 523-544.
Bendsøe, M.P., Olhoff, N., 1985. A method of design against vibration resonance of
beams and shafts. Optim Control Applications & Methods, 6 (3): 191-200.
Bendsøe, M.P., Sigmund, O., 2003. Topology Optimization: Theory, Methods and
Applications. Berlin: Springer-Verlag, 2003.
Brillouin, L., 1953. Wave Propagation in Periodic Structures, 2nd edition. Dover
Publication, New York.
Diaz, A.R., Kikuchi, N., 1992. Solutions to shape and topology eigenvalue optimization
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Methods in Engineering, 35 (7): 1487-1502.
Diaz, A.R., Haddow, A.G., Ma, L., 2005. Design of band-gap grid structures. Structural
and Multidisciplinary Optimization, 29 (6): 418-431.
Du, J., Olhoff, N., 2007a. Topological design of freely vibrating continuum structures
for maximum values of simple and multiple eigenfrequencies and frequency gaps.
Structural and Multidisciplinary Optimization, 34 (2): 91-110.
Du, J., Olhoff, N., 2007b. Topological design of freely vibrating continuum structures
for maximum values of simple and multiple eigenfrequencies and frequency gaps
(V.ol 34, pg 91, 2007). Editors’s Erratum, Structural and Multidisciplinary
Optimization, 34 (6), 545.
Halkjær, S., Sigmund, O., 2004. Optimization of beam properties with respect to
maximum band-gap. Mechanics of the 21st Century, Procedings of 21st International
Congress of Theoretical and Applied Mechanics. Gutkowski, W., Kowalewski, T.A.
(Eds.), IUTAM, Warsaw, Poland (ISBN: 978-1-4020-3456-5).
Halkjaer, S., Sigmund, O., Jensen, J.S., 2006. Maximizing band gaps in plate structures.
Structural and Multidisciplinary Optimization, 32 (4): 263-275.
Hladky-Hennion, A.C., Allan, G., deBilly, M., 2005. Localized modes in a one-
dimensional diatomic chain of coupled spheres. J Applied Physics, 98 (5): 054909.
Hussein, M.I., Hulbert, G.M. and Scott, R.A., 2006a. Dispersive elastodynamics of 1D
banded materials and structures: Analysis. Journal of Sound and Vibration, 289 (4-5):
779-806.
Hussein, M.I., Hamza, K., Hulbert, G.M., Scott, R.A., Saitou, K., 2006b. Multiobjective
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characteristics. Structural and Multidisciplinary Optimization, 31 (1): 60-75.
Hussein, M.I., Hulbert, G.M., Scott, R.A., 2007. Dispersive elastodynamics of 1D
banded materials and structures: Design. Journal of Sound and Vibration, 307 (3-5):
865-893.
Jensen, J.S., 2003. Phononic band gaps and vibrations in one- and two-dimensional
mass-spring structures. Journal of Sound and Vibration, 266 (5): 1053-1078.
Jensen J.S., Sigmund O., Thomsen J.J., Bendsøe M.P., 2002. Design of multi-phase
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Jensen, J.S., Pedersen, N.L., 2006. On maximal eigenfrequency separation in two-
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Topological Design for Minimum Dynamic
Compliance of Structures under Forced Vibration
1 Introduction
Subject to :
Cd | PTU | ,
(1)
(K Z p2 M )U P,
NE
* (V *
¦ U eVe V d 0 , DV0 ) ,
e 1
0 U d Ue d 1 , (e 1 , , N E ) .
In (1), the symbol Cd stands for the dynamic compliance defined as
Cd | P T U | . Here, P denotes the vector of amplitudes of a given external time-
iZ p t
harmonic mechanical surface loading vector p(t ) Pe with the prescribed
excitation frequency Z p , and U represents the vector of magnitudes of the
Topological Design for Minimum Dynamic Compliance… 327
iZ t
corresponding structural displacement response vector a(t ) Ue p . Thus, U
and P satisfy the dynamic equilibrium equation included in (1) for the steady-
state vibration at the prescribed frequency Z p , with K and M representing the N
dimensional global structural stiffness and mass matrices, where N is the number
of DOFs. We note that the above expression for the dynamic compliance Cd
represents the numerical mean value of the magnitudes of the surface
displacements weighted by the values of the amplitudes of the corresponding
time-harmonic surface loading. For the case of static loading ( Z p = 0), the
expression directly reduces to the traditional definition of static compliance, i.e.,
the work done by the external forces against corresponding displacements at
equilibrium.
In (1), NE denotes the total number of finite elements in the admissible design
domain for the topology optimization problem. The symbols Ue, e = 1,…,NE, play
the role of design variables of the problem and represent the volumetric material
densities of the finite elements, with lower and upper limits U and 1 specified
for Ue. To avoid singularity of the stiffness matrix, U is not zero, but taken to be
a small positive value like U = 10-3. In the second but last constraint in (1), the
symbol D defines the volume fraction V * /V0 , where V0 is the volume of the
admissible design domain, and V * is the given available volume, respectively, of
the solid material for a single-material design problem and of the solid material
*1 for a bi-material design problem, cf. Sub-sections 2.1 and 2.3 in the paper
Olhoff and Du (2013A).
It is noted from (1) that the global dynamic stiffness matrix K d defined as
Kd K Z p2 M may be negative definite when the prescribed external excitation
frequency Z p has a high value, e.g. higher than the fundamental eigenfrequency
of the structure. In this case, the scalar product of the transpose of the vector of
amplitudes of the external surface loading and the vector of amplitudes of the
structural displacement response may become negative, and in order to include
this possibility in our problem formulation, we apply the absolute value of this
scalar product as the dynamic compliance Cd, see (1). Moreover, to render the
problem differentiable, we choose the objective function F as the square of the
dynamic compliance. The dynamic equilibrium equation in (1) is solved in a
direct way by Gauss elimination in this paper.
328 N. Olhoff and J. Du
where prime denotes partial derivative with respect to Ue. The sensitivity Pc of
the load vector will be zero if it is design-independent, otherwise it can be
handled using the method described by Hammer and Olhoff (1999, 2000), and
also by Du and Olhoff (2004a,b). The sensitivity Uc of the displacement vector
is given by
(K Z p2 M ) Uc f { Pc (K c Z p2 Mc) U , (3)
where the sensitivities of the stiffness and mass matrices can be directly obtained
from the SIMP material model in Eqs. (4) of the paper Olhoff and Du (2013A). The
vector f is known as the pseudo load and is defined by the term on the right-hand
side of Eq. (3). Instead of solving Eq. (3), the adjoint method (see e.g. Tortorelli and
Michaleris 1994) may be used to calculate the sensitivity of the objective function in
a more efficient manner, which gives the following result
(4)
>
F c 2(P T U) 2U T Pc U T (K c Z p2 M c) U . @
Accordingly, the optimality condition for problem (1) can be expressed in the
following form by means of the method of Lagrange multipliers,
(5)
> @
2(P T U) 2U T Pc U T (K c Z 2p Mc) U /Ve 0 ,
where / is the Lagrange multiplier corresponding to the material volume constraint,
and the side constraints for Ue have been ignored. The optimization problem (1) can
be solved by using the well-known MMA method (Svanberg 1987) or an optimality
criterion method, e.g. the fixed point method, as devised by Cheng and Olhoff
(1982).
Cd Initial design D1
Path 2
Cs2 D2 D1
Cd1
Cs1
Cd2
Z=0 :2 :1 Zp Z
(a)
Initial design D1
Cd
Path 1
D1 D3
Cd1
Cs1 Cd3
Cs3
Z=0 Zp :1 :3 Z
(b)
3 Numerical Examples
Frequencies
60 First eigenfrequency :1
40
20
0
0 5 10 15 20 25 30
Iteration number
(a) (b)
-6
x 10
2
Dynamic and static compliance
0
0 5 10 15 20 25 30
Iteration number
(c) (d)
Figure 2. (a) Admissible design domain (a = 3, b = 2 and c = 0.1) with loading and
support conditions. (b) Iteration history for the first eigenfrequency of the plate (:1 < Zp =
80). (c) Iteration histories for the dynamic and static structural compliance (the latter
corresponds to the same loading amplitude but frequency Z = 0). (d) Material distribution
at iteration step 30.
50% for the given solid material. (SI-units are used throughout.) The
fundamental eigenfrequency of the plate in the initial design (see Fig. 2(a)) is :1
= 61.6, i.e., less than the given loading frequency. Minimization of the dynamic
compliance drives the design away from the resonance point which implies a
continual decrease of :1 as shown in Fig. 2(b). As a result, the static compliance
of the structure increases very quickly (Fig. 2(c)). Fig. 2(d) shows that at
iteration step 30, the plate has become very weak at the two fixed supports. This
indicates creation of a rigid body vibration mode in association with the first
eigenfrequency, and that the structure cannot effectively sustain the static load
associated with Z = 0.
-7
130 x 10
5
120
First eigenfrequency :1 Dynamic compliance Cd for
Structural compliance
100
3
90
Prescribed loading frequency 2 Static compliance Cs Z
80
Z Zp
70
1
60
Initial loading frequency Z 0
50 0
0 5 10 15 20 25 30 5 10 15 20 25 30
Iteration number Iteration number
(a) (b)
Figure 3. (a), (b) Iteration histories for the first eigenfrequency of the plate, the loading
frequency, and the dynamic and static compliances. (c), (d) Optimum topologies (50%
volume fraction, single-material design) for Zp=80 and Zp=150 (with an upper bound on
the static compliance, i.e. C S d C S ).
334 N. Olhoff and J. Du
-7 -7
x 10 x 10
4
Dynamic compliance of structure
6
Dynamic compliance of structure
3 Z Z
p
Z Z
p
Z Z
p
2.5
and different upper bounds on Cs 4
Z Z
p
Cs = 0.5x10-7 with constraint
2 3
Cs = 1x10-7 Cs <= Cs = 0.5x10 -7
1.5 Cs = 3x10-7
2
Cs = 4x10-7
1
1
0.5
0 0
0 50 100 150 200 0 50 100 150 200
Iteration number Iteration number
(a) (b)
Figure 4. (a) Iteration histories of the dynamic compliances of the plate subject to a high
loading frequency (Z = Zp = 150 > :opt = 127.6) and four different upper bound
constraints on the static compliance Cs, i.e. C s d C s . (b) Iteration histories of the dynamic
compliances of the plate subject to a given upper bound constraint on Cs
( C s d C s 0.5 u 10 7 ), for four different loading frequencies Zp = 130, Zp = 150, Zp = 180
and Zp = 200, all of which are higher than the optimum value of the fundamental
eigenfrequency :opt .
Topological Design for Minimum Dynamic Compliance… 335
Fig. 4(a) shows the iteration histories of the dynamic compliance of the plate
subject to the higher loading frequency ( Z = Zp = 150) and four different upper
bound constraints on the static compliance Cs (associated with the same loading
amplitude but zero frequency). The graphs show that the optimum dynamic
compliance decreases as the upper bound constraint on Cs is increased. In Fig.
4(b), iteration histories are shown for minimum compliance topology design of
the plate subject to a given upper bound constraint on the static compliance
( C s d C s 0.5 u 10 7 ) for four different higher loading frequencies. These
graphs show that for the higher loading frequency designs, the dynamic
compliance of the structure decreases as the prescribed loading frequency is
increased. This feature is opposite to that obtained by minimum compliance
topology design subject to prescribed lower or medium loading frequencies. As a
conclusion, variations of the minimum dynamic compliance with respect to
different loading frequencies are depicted in Fig. 5(a), and Fig. 5(b) presents the
static compliances associated with the minimum dynamic compliance designs
subject to different prescribed loading frequencies.
-7 -7
x 10 x 10
1.5 1.5
Minimum dynamic compliance Cd
Dynamic
Design with Dynamic design without
minimum dynamic compliance designs
-7
Cs = 0.5x10
0.5 0.5
Dynamic design without
constraint on Cs
0
:opt 0
:opt
0 50 100 150 200 0 50 100 150 200
Loading frequency Zp Loading frequency Zp
(a) (b)
Figure 5. (a) Minimum dynamic compliances Cd vs. different loading frequencies. (b)
Static compliances Cs (correspond to the same loading amplitude but zero frequency)
associated with the designs in Fig. 5(a) vs. different loading frequencies. Note that if the
prescribed loading frequency is close to or higher than the optimum value :opt = 127.6 of
the fundamental eigenfrequency for the corresponding problem of free vibrations of the
plate, an upper bound constraint C s d C s is prescribed for the static compliance in order to
avoid obtaining a statically too weak structure from the dynamic design.
336 N. Olhoff and J. Du
Young’s modulus E = 107, Poisson’s ratio X = 0.3 and the specific mass Jm = 1.
The design objective is to minimize the dynamic compliance of the inlet. Fig. 6(a)
shows the admissible design domain and the initial loading boundary. Figs. 6(b-d)
show optimized topologies and the associated loading boundaries for three given
loading frequencies Z p 0 (static loading), Z p 800 and Z p 1000 . These
loading frequencies are all lower than the maximum fundamental eigenfrequency
of free vibrations of the inlet, which was found to be :opt = 1328. The
corresponding optimum topology of the inlet (with the same material volume
fraction as before) is shown in Fig. 7.
Figure 7. Optimum topology of the 2D inlet (for 40% volume fraction) obtained by
maximizing the fundamental eigenfrequency of free vibrations of the inlet. The maximum
fundamental eigenfrequency is :opt = 1328.
4 Conclusions
References
Calvel, S., Mongeau, M., 2005. Topology optimization of a mechanical component
subject to dynamic constraints. Technical Report LAAS N0 05374. Available from
http://mip.ups-tlse.fr/perso/mongeau.
Cheng, G., Olhoff, N., 1982. Regularized formulation for optimal design of
axisymmetric plates. Int J Solids Struct 18: 153-169.
Du, J., Olhoff, N., 2004a. Topological optimization of continuum structures with design-
dependent surface loading - Part I: New computational approach for 2D problems.
Struct Multidisc Optim 27: 151-165.
Du, J., Olhoff, N., 2004b. Topological optimization of continuum structures with design-
dependent surface loading - Part II: Algorithm and examples for 3D problems. Struct
Topological Design for Minimum Dynamic Compliance… 339
Abstract This paper is devoted to topology optimization problems formulated with the
design objective of minimizing the sound power radiated from the structural surface(s)
into a surrounding acoustic medium. Bi-material elastic continuum structures without
material damping are considered. The structural vibrations are excited by time-harmonic
external mechanical loading with prescribed excitation frequency, amplitude, and spatial
distribution. Several numerical results are presented and discussed for bi-material plate-
like structures with different sets of boundary and loading conditions.
1 Introduction
where the sound power radiated from the structural surface can be estimated by
using a simplified approach instead of solving the Helmholtz integral equation.
This reduces the computational cost of the structural-acoustical analysis
considerably.
¦ U eVe V *1 d 0 , (V *1 DV0 ) ,
e 1
0 d Ue d 1 , (e 1,, N E ) .
Here, the symbols pf and vn* in the expression for represent the acoustic
pressure and the complex conjugate of the normal velocity of the structural
surface, and Pf denotes the corresponding vector of amplitudes of the acoustic
pressure on the structural surface S. The symbol L represents the fluid-structural
coupling matrix and the symbols K and M denote the N dimensional structural
stiffness and mass matrices, where N is the number of DOFs. The expression
K Z p2 M in (1) represents the dynamic stiffness matrix which we may denote
by KD. The matrices G, H and CD can be generated by the discretized Helmholtz
integral and calculation of the spatial angle along the structural surface (see, e.g.,
Christensen et al. 1998). We consider a bi-material design problem (see Sub-
section 2.3 in Olhoff and Du 2013A) where NE is the total number of finite
elements and the symbol Ue denotes the volumetric density of the stiffer material
Topological Design for Minimum Dynamic Compliance… 343
in element e and plays the role of the design variable in the problem. The symbol
D denotes the fraction of the given volume V *1 of the stiffer material (material *1)
and is given by V *1 /V0 , where V0 is the volume of the admissible design domain.
The remaining part of the total volume V0 is occupied by a softer material
(material *2) as explained in Sub-section 2.3 of Olhoff and Du 2013A.
where c is the sound speed and J f is the specific mass (mass density) of the
acoustic medium. Tests performed by Sorokin (2005) for simple beam and
sphere examples show that the accuracy of (2) depends on not only the frequency
level but also on the size of the structure and the shape of the vibration mode of
the structure. Generally speaking, the accuracy of the approximation increases
with increasing values of the frequency, but may decrease with a change of the
vibration mode. Nevertheless, the tests also show that even for lower frequencies,
(2) may still yield a good approximation of the distribution (up to a multiplying
factor) of the sound pressure along the structural surface. This is useful for our
problem of optimizing the global sound radiation, because even a scaled
distribution of the sound pressure along the structural surface can yield a
topology design which is close to the optimum one.
If we further assume weak coupling, i.e., ignore the acoustic pressure in the
structural equation, the first constraint in (1) will be simplified to the equation of
a vibrating structure subjected only to the external mechanical loading P (see the
third equation of Eqs. (1) and the paper Olhoff and Du, 2006),
(Z p2 M K )U P (3)
With the above simplification, the first two constraint equations in problem
(1) may be replaced by Eqs. (2) and (3), and the sound power flow from the
344 N. Olhoff and J. Du
3 c J f cZ p2 U T S n Uc (6)
where prime denotes partial derivative with respect to Ue. Using Eq. (3) and
applying the adjoint method, see Tortorelli and Michaleris (1994), the sensitivity
(6) of the sound power flow may be calculated in a more efficient manner, which
gives the following result
>
3 c J f cZ p2 U Ts Pc U Ts (K c Z p2 M c) U @ (7)
Based on the above sensitivity results, the optimization problem (1) may be
solved by using the well-known MMA method (see Svanberg 1987) or an
optimality criterion method, e.g. the fixed point method (see Cheng and Olhoff
1982).
3 Numerical Examples
(a) (b)
Figure 1. Plate-like structure (a=20, b=20, t=1) subjected to uniformly distributed
harmonic pressure loading on its upper surface. All edges of the plate are clamped.
Figure 3. The first three eigenmodes of free vibration of the optimum topology design
subject to the given loading frequency Zp = 100. The second and the third
eigenfrequencies constitute a bi-modal eigenfrequency due to the symmetry of the plate.
348 N. Olhoff and J. Du
emitted from the structural surface, we use the method described in Section 2 of
the paper Olhoff and Du 2013D) to minimize the dynamic compliance
C d | P T U | of the plate-like structure in vacuum (and assuming vanishing
structural damping). Optimum topologies for the latter problem corresponding to
the six prescribed loading frequencies Zp = 10, 100, 200, 300, 500 and 1000 are
shown in Figs. 4(a-f).
It is seen that for the low excitation frequencies Zp = 10 and Zp = 100, the
optimum topologies shown in Figs. 4(a) and 4(b) are almost indistinguishable
from the corresponding ones in Figs. 2(a) and 2(b). However, as the value of the
excitation frequency is increased, the differences between the topologies become
more pronounced (see Figs. 4 and 2 for Zp = 200, 300, 500 and 1000).
For further comparison we have calculated the values of both the sound
power and the dynamic compliance of the initial structure and the optimum
structures corresponding to the two different design objectives behind Figs. 2 and
4 (assuming the same loading and boundary conditions), and the results are
presented in Table 1.
These results provide numerical evidence that topology optimization with
respect to either of the two design objectives has a strongly improving effect on
350 N. Olhoff and J. Du
It is seen that the optimum topologies associated with the same loading
frequencies as in Figs. 2 and 6 are quite different, which implies that the
boundary and loading conditions may have large influence on the resulting
topology of the plate-like structure.
It is noteworthy that in the optimum topologies subject to the concentrated
harmonic load (see Fig. 6), the central part of the plate is always filled out with
the stiffer material *1 which also has a higher mass density. This local layout is
very efficient in counteracting the concentrated load which has not only a given
frequency but also a prescribed amplitude. The mass assembly surrounding the
point of action of the force yields a large local inertia, which effectively reduces
the displacement amplitude at the point of action of the force, and thereby
reduces the vibration amplitudes and the density of sound power emission all
over the plate.
Figs. 7 and 8 show comparisons of the power flow distribution between the
initial designs (subject to the two different boundary and loading conditions in
Figs. 1 and 5) and the corresponding optimum designs for the prescribed
excitation frequency Zp = 1000. Similar comparisons for corresponding designs
subject to the prescribed frequency Zp = 500 are given in Figs. 9 and 10. It is
seen that the distribution of the sound power in the optimum designs subject to
the concentrated load show features that are very different from those in the
optimum designs subjected to the uniform load.
(a) (b)
Figure 7. Distribution of the power flow from the structural surface of the initial design.
(a) and (b) correspond to the two different sets of boundary and loading conditions shown
in Figs. 1 and 5, respectively. The loading frequency has the prescribed value Zp = 1000.
Topological Design for Minimum Sound Emission… 353
(a) (b)
Figure 8. Distribution of the power flow from the structural surface of the optimum
design. (a) and (b) correspond to the two different sets of boundary and loading conditions
shown in Figs. 1 and 5, respectively. The loading frequency has the prescribed value Zp =
1000.
Thus, we find that for the higher excitation frequencies, the main part of the
sound power emitted from the optimum designs subjected to the concentrated
load (see Figures 8(b) and 19(b)) is limited within a small annular-like area
surrounding the mass assembly in the vicinity of the concentrated load. This
implies that the optimum designs (see Figures 6(d) and 6(f)) create an efficient
isolation of vibration and sound power radiation that to a large extent terminates
the transmission of bending waves to the boundary of the plate-like structures at
the prescribed frequencies. The features discussed here reveal very strong
similarities between the present optimum designs and so-called band-gap
structural designs (see, e.g., Halkjær et al., 2006).
(a) (b)
Figure 9. Distribution of the power flow from the structural surface of the initial design.
(a) and (b) correspond to the two different sets of boundary and loading conditions shown
in Figs. 1 and 5, respectively. The loading frequency has the prescribed value Zp = 500.
354 N. Olhoff and J. Du
(a) (b)
Figure 10. Distribution of the power flow from the structural surface of the optimum
design. (a) and (b) correspond to the two different sets of boundary and loading conditions
shown in Figs. 1 and 5, respectively. The loading frequency has the prescribed value Zp =
500.
4 Conclusions
References
Cheng, G., Olhoff, N., 1982. Regularized formulation for optimal design of
axisymmetric plates. Int J Solids Struct 18: 153-169.
Christensen, S.T., Sorokin, S.V., Olhoff, N., 1998. On analysis and optimization in
356 N. Olhoff and J. Du
Olhoff, N., Du, J., 2013A. Introductory notes on topological design optimization of
vibrating continuum structures. In: Rozvany, G., Lewinski, T., Eds., Topology
Optimization in Structural and Continuum Mechanics, Int. Centre for Mechanical
Sciences, Udine, Italy, June 18-22, 2012. Springer-Verlag, Vienna, 2013, 15 pp.
Olhoff, N., Du, J., 2013D. Topological design for minimum dynamic compliance of
structures under forced vibration. In: Rozvany, G., Lewinski, T., Eds., Topology
Optimization in Structural and Continuum Mechanics, Int. Centre for Mechanical
Sciences, Udine, Italy, June 18-22, 2012. Springer-Verlag, Vienna, 2013, 15 pp.
Sorokin, S.V., 2005. Private communication.
Svanberg, K., 1987. The method of moving asymptotes a new method for structural
optimization. Int J Numer Meth Engng 24: 359-373.
Tortorelli, D., Michaleris, P., 1994. Design sensitivity analysis: overview and review.
Inverse Problems in Engineering 1: 71-105.
Wadbro E., Berggren M., 2006. Topology optimization of an acoustic horn. Comput
Methods Appl Mech Eng 196: 420-436.
Yamamoto, T., Maruyama, S., Nishiwaki, S., Yoshimura, M. 2009. Topology design of
multi-material soundproof structures including poroelastic media to minimize sound
pressure levels. Computer Methods in Applied Mechanics and Engineering 198(17-
20): 1439-1455.
Yang, R., Du, J., 2013. Microstructural topology optimization with respect to sound
power radiation. Struct Multidisc Optim 47(2): 191-206.
Discrete Material Optimization of Vibrating
Laminated Composite Plates for Minimum Sound
Emission
1 Introduction
Composite materials like fiber reinforced polymers (FRPs) are being used
increasingly in aerospace vehicles, maritime carriers, wind turbine blades, and
various mechanical equipment where high strength, high stiffness and low
weight are important properties. In such applications, the FRPs are usually
stacked in a number of layers, each consisting of strong fibers bonded together
by a resin, to form a laminate. In addition, laminated sandwich structures may
also consist of layers made of foam material. When these composite structures
are used in dynamic environments, vibration control and noise reduction become
minimizing the sound power radiated from the vibrating structural surface into
the acoustic medium. Then Rayleigh’s integral is introduced to calculate the
sound power flow from the structural surface, which leads to a simplified
optimization formulation of the current problem. Section 3 introduces the
parameterization for discrete material optimization (DMO) and discusses penalty
functions for DMO. Section 4 presents a DMO convergence measure. Section 5
deals with the sensitivity analysis required for the numerical optimization
algorithm. Section 6 presents several numerical examples with different
excitation frequencies in order to validate the proposed method, including
single-layer, multi-layer laminated composite plates, and laminated sandwich
plates consisting of layers made of FRPs and foam material. Finally, a section
with conclusions closes the paper.
and vn* represent the acoustic pressure and the complex conjugate of the normal
velocity of the structural surface, and P f denotes the corresponding vector of
amplitudes of the acoustic pressure on the structural surface S. The symbol L
represents the fluid-structural coupling matrix. The matrices G , H and CD
can be generated by the discretized Helmholtz integral and calculation of the
spatial angle along the structural surface (see, e.g., Christensen et al. 1998 a, b),
and P denotes the vector of amplitudes of a given external time-harmonic
iZ t
mechanical loading vector p t = Pe p with the prescribed excitation
frequency Z p . The symbols K and M represent the global structural
stiffness and mass matrices, U denotes the vector of magnitudes of the
iZ p t
corresponding structural displacement response vector a t = Ue , and
Kd K Z M
2
p is defined as the global dynamic stiffness matrix. Here, M
depends on the mass density J i of each of the candidate materials, and it is
assumed that damping can be neglected.
Since laminated composite plates are considered, more design parameters
need to be introduced as compared to single material solid plates. Thus, l
denotes ‘layer’, e denotes ‘element’, N l is the number of layers, N e is the
number of elements, and nl is the number of candidate materials (design
variables) per layer for each element; the number of element design variables
n e for multi-layered elements (with N l layers) is the sum of the number of
Nl
design variables per layer, nl , over all N l layers, such that n e ¦ k 1
nkl , and
e
N
the total number of design variables in the problem is therefore N dv ¦ e
i 1 i
n .
It is emphasized that the ‘classical’ topology optimization formulation has one
design variable per element by setting n e =1 for single material 0/1 design. For
single-layer plates, the number of candidate materials is also the number of
element design variables. Here we consider a more general condition with
multiple layers, where the number of design variables must be summed over all
layers for this element. The symbol tl and Ae represent the thickness of the
l-th layer and area of the e-th element, respectively. In the DMO approach, the
variable xi (i 1, 2, , nl ) per layer for each element can be seen as a local
density variable that indicates possible selection of the i-th material, i.e., with
xi xmax meaning that the i-th material is chosen and xi xmin meaning that
the i-th material is not chosen. The design variables x for all layers in all
elements are denoted as
Discrete Material Optimization of Vibrating Laminated… 363
laminated plate, such that the product kr is much larger than 1, where k= Z p /c is
the wave number, c the speed of sound in the acoustic medium, and r is the
distance between a source point on the structural surface and an observation
point in the acoustic domain, cf. Herrin et al. (2003). Following (Du and Olhoff
2007a), assuming a sufficiently high value of the structural vibration frequency
Z p , the radiation impedance pf /vn at the structural surface will be approximately
equal to the characteristic impedance J f /c of the acoustic medium (Lax and
Feshbach 1947). Thus, the acoustic pressure pf and the normal velocity vn of the
structural surface are approximatively related by the simplified equation
p f J f cvn (3)
where J f is the mass density and c the sound speed in the acoustic medium. The
accuracy of the approximation is discussed in the papers (Du and Olhoff 2007a,
Herrin et al. 2003).
The normal velocity of the surface in (3) can be obtained as
vn n u iZ p (4)
where n is the unit normal and u the amplitude of the displacements of the
structural surface after interpolation based on finite element analysis, i.e., using
the finite element interpolation u = NU e , where N is the shape function and
U e is the nodal displacement vector of element e.
Thus, substituting (3) with vn given by (4) into the expression for the sound
power (objective function) 3 in (1), we after simple algebra obtain the
simplified expression
1
3 ³ J f cZ p2 n u n u dS (5)
S
2
which upon use of the discretized formulations can be written in the matrix form
1
3 J f cZ p2 UT S n U (6)
2
where superscript T stands for transpose, U denotes the global displacement
vector of the structure, and S n defined as
Ne Ne § ·
T
Sn ¦S ne ¦ ¨¨ ³ N nnT NdS ¸ (7)
e 1 e 1 © Se
¸
¹
is termed the surface normal matrix of the structure.
Finally, in this paper we shall consider the acoustic medium to be light, i.e.,
air with the mass density J f =1.2kg/m3 and sound speed c = 343.4m/s. This
means that we may assume weak coupling, i.e., ignore the acoustic pressure on
the structure, and this implies a further simplification of the current optimization
Discrete Material Optimization of Vibrating Laminated… 365
problem. All in all, with the simplifications made, the original formulation (1) of
our optimization problem can be re-written in the very convenient form
1
min 3 J f cZ p2 UT S n U
x 2
S .t. : K Z p2 M U P
N e § N l nl
(8)
·
¦ ¨ ¦¦ [ eli ci tl ¸Ae d R
e 1© l 1 i 1 ¹
ǂǂ0 xmin d x j d xmax 1, j 1, N dv
When comparing (8) with (1), let us first note that the assumption of weak
coupling implies that the feedback acoustic pressure LP f on the structure in (1)
vanishes such that the first constraint in (8) is simply the standard equation for a
vibrating structure without damping, that is subjected to the given external
dynamic loading P, only. At the same time, in (8) the Rayleigh approximation (3)
has dismissed the discretized Helmholtz integral equation in the second
constraint in (1) and instead taken over itself the delivery of acoustic surface
pressures to 3 (cf. the expression for 3 in (1)), and finally substituted these
pressures by surface normal velocities converted into global structural
displacements U in the expression for 3 in (8). We notice that these
displacements U are simply obtained by solution of the standard equation for
forced vibration of the structure in the first constraint of (8).
We may conclude that the application of Rayleigh’s approximation and the
assumption of weak coupling have furnished a formulation (8) of our
optimization that is much simpler than the formulation in (1) based on
Helmholtz’ integral equation and full structural-acoustic coupling. Thus, problem
(8) does not require a system of coupled equations (the first and second
constraint equations in (1)) to be assembled and solved, and is therefore much
easier to implement, much faster to solve numerically, and hence requires
considerably less computer resources than problem (1). Similar advantages must
be expected for the sensitivity analysis of problem (8).
where
nl
p
[i x
l ª1 x l p º (11)
i «¬ j »¼
j 1; j z i
too much to penalize intermediate values of the design variables. Fig. 3(a), (b)
and (c) depict the weighting functions [1 , [ 2 and their sum if the penalty
factor is further increased up to the value p=10. The differences between Fig. 1
and Fig. 3 are seen to be more significant. We note that the two plateau domains
in Figs. 3(a) and (b) are not favourable for penalization of design variables with
values located in these two domains. Due to this, the power p is typically
increased gradually from 1 to 3 only, and not up to a higher value during the
continuation process.
Similarly, the element mass density and element cost per layer, i.e. for the
l-th layer, are also expressed as a weighted sum for the candidate materials,
respectively,
nl nl
[i
Jl ¦ [i J i ¦ nl
Ji (12)
i 1 i 1
¦ k
[
1 k
nl nl
[i
cl ¦[ c ¦ i i nl
ci (13)
i 1 i 1
¦ k
[
1 k
where J i denotes the mass density and ci the unit cost of the i-th candidate
material. The weighting functions [i use the same interpolation formulae in
Eq. (10).
For the resource constraint, linear interpolation is used, which means that
the penalty power p=1. However, for the stiffness and mass matrices, nonlinear
interpolation is used, and the penalty power p is typically increased gradually
from 1 up to 3 during the iterations.
Furthermore, the element stiffness and mass matrices can be obtained on the
basis of first-order laminated composite plate theory, and the global stiffness and
mass matrices can be obtained by assembling the element matrices.
4 Evaluation of Convergence
A convergence measure given in Stegmann and Lund (2005) is adopted to
describe whether the optimization has converged to a satisfactory result. This
convergence measure is described briefly here, while the reader is referred to the
original paper (Stegmann and Lund 2005) for a detailed discussion. For each
layer in each element the inequality is evaluated for all weighting functions, [i
[i t H [12 [ 22 [ n2 (14)
where H is a tolerance level, typically 95-99.5% suggested by Stegmann and
Lund (2005). If the inequality is satisfied for one of the weighting functions in
the layer, it is flagged as converged. The DMO convergence measure hH is
defined as the ratio between the number of converged layers N cl ,tot in all
elements and the total number of layers in all elements N l ,tot Nl Ne
N cl ,tot
hH (15)
N l ,tot
The DMO convergence measure is denoted as h95 if the tolerance level is
95% and full convergence, h95 1 , means that all layers in all elements have a
single weighting function contributing more than 95% to the Euclidian norm of
the weighting functions.
Discrete Material Optimization of Vibrating Laminated… 369
It should be reiterated that linear interpolation with the penalty factor p=1 is
adopted for the resource constraint. With these sensitivity results, the design
problem (8) may be solved by a mathematical programming method, e.g., MMA
developed by Svanberg (1987). A broad account of finite element based design
sensitivity analysis and optimization is available in Lund (1994).
370 N. Olhoff and B. Niu
Uniform
harmonic
pressure load
l=Nl
l=Nl-1
l=Nl-2
FRPs(Tl) l=Nl-3
Foam
l=2
l=1
Poisson’s ratio vxy 0.25 and mass density J 1900 kg m3 . The fiber angles
are taken to be ª¬90o , r45o , 0o º¼ .
For all the examples in this paper, unbiased initial values of design variables
were set to the value corresponding to uniform distribution of all candidate
materials (in this sub-section 0.25). The corresponding initial design provides a
convenient reference for evaluation and discussion of the vibration and sound
power radiation characteristics of the designs optimized by usage of DMO in the
following. In this section the initial design corresponds to a quasi-isotropic layup
of the glass/epoxy material. Here, we start out computing a lower spectrum of
eigenfrequencies Zi of the initial design and find the results given in Table 1,
where Z=Z=884. is a bimodal (double) eigenfrequency.
The optimum designs in Fig. 5 are observed to be similar to each other for
the excitation frequencies 10, 100, 200 and 300, which are all below the first
resonance frequency :1 of the initial design as well as the resulting optimized
design. However, different topologies are found when higher excitation
frequencies are considered. The sound power radiation has generally been
decreased quite substantially for the optimized design relative to the initial
design in all the examples, see Table 3.
Table 3 Single-layer plate: Comparison of the total sound power radiation from the initial
design and the designs optimized for different excitation frequencies Zp
Excitation Initial design Optimized design Relative decrease
frequency Zp Sound power 3 Sound power 3 Fig. |33|/ 3
10 15.61 7.69 5(a) 50.77%
100 1738. 830. 5(b) 52.24%
200 10023. 4263. 5(c) 57.47%
300 51001. 15846. 5(d) 68.93%
500 375728. 91434. 5(e) 75.66%
1000 9310. 7844. 5(f) 15.77%
As can be seen from Table 3, the largest decrease (75.66%) of the sound
power emission 3 relative to that of the initial design, is obtained for the
optimized design in Fig. 5(e) with the prescribed external excitation frequency
Zp =500. The reason for this large decrease of 3 is that Zp =500 is quite close
to (slightly larger than) the first resonance frequency Z1=434. of the initial
design (as is also reflected by the high value of the sound power radiation 3 0
from the initial design at Zp =500 in Table 3). Thus, taking the optimized design
with Zp =500 in Fig. 5(e) to be given, we have computed its six lowest
eigenfrequencies and obtained the results shown in Table 2.
By comparison of results in Table 1 and Table 2, we see that the
optimization with the excitation frequency prescribed as Zp =500 has decreased
quite significantly the nearest (first) resonance frequency :1 from the value 434.
for the initial design to the value 354. for the optimized design. This implies that
large displacement amplitudes have been considerably reduced at the excitation
frequency Zp =500 by the discrete material optimization, and explains the
374 N. Olhoff and B. Niu
significant reduction of the sound radiation by 75.66%. At the same time, the
second resonance frequency :2 is slightly decreased by the optimization for the
case of Zp =500, see Table 1 and Table 2, but it is so much larger than the
excitation frequency Zp =500 that it has only marginally affected the sound
power radiation at Zp.
Hence, for the case of Zp =500, the mechanical cause of the substantial
reduction of the sound power radiation achieved by the optimization is that the
optimization has driven the nearest (first) resonance frequency as far away as
possible (i.e., downward) from the prescribed excitation frequency Zp.
Let us then consider the case of optimization for Zp =1000, where the
excitation frequency Zp is roughly located in the middle of the interval between
the first and second resonance frequencies :1=434. and :6=1591. of the initial
design, see Table 1. This is generally a favourable location of the excitation
frequency for a given design, and is also seen to lead to relatively low sound
radiation with a comparatively small difference between the values of the initial
and the optimized design (cf. Table 3). With Zp=1000, we find that the
optimization yields a design (shown in Fig. 5(f)), where the first resonance
frequency is decreased from 434. to 398. and the second resonance frequency is
increased from 1591. to 1756.
Hence, in this case with Zp =1000, the mechanical cause of the reduction of
the sound radiation obtained by the optimization (see Table 3) is that both of the
neighbouring resonance frequencies have been driven away from the excitation
frequency Zp, and thereby created an enlarged gap between these two resonance
frequencies with a reduced level of sound radiation at Zp and in its vicinity.
and foam material that enable creation of a sandwich structure. The same
orthotropic glass/epoxy composite with permissible fiber angles ª¬90o , r45o , 0o º¼
is used as before, but in addition an isotropic polymeric foam material with
Young’s modulus E 125 MPa , Poisson’s ratio v 0.3 and mass density
J 100 kg m3 is assumed to be available for the structure.
In this and the next Sub-section 6.4, we choose the given global resource
constraint value R to represent the total mass of the structural material.
Accordingly, we define the unit cost factors u1 and u2 (see Section 2) of the
unidirectional fiber composite material and the isotropic foam material,
respectively, as the mass densities J i and J i of these two materials, and take
them to be c1 =1900 and c2 =100 in (8). The allowable total material mass
resource constraint value is taken to be R =10.0. In view of the data given, this
means that the foam must constitute at least 50% of the total volume.
In the current sub-section, we consider the case where the upper and the
lower layers are not allowed to choose the polymeric foam, while the inner 6
layers can locally consist of either the foam or glass/epoxy composite material.
This implies 38 design variables per element, distributed as [4, 5, 5, 5, 5, 5, 5, 4],
and brings the total number of design variables for the whole plate up to 15200,
when a 20u20 finite element mesh is used.
The results of the optimization for Zp=100 and 1000 are presented in Fig. 7
and Fig. 8, respectively, where layer elements with fibers indicate that the
glass/epoxy material is selected with the fiber orientations shown, and elements
in white indicate selection of isotropic foam material. The DMO convergence
measures for the designs with Zp =100 and 1000 are h95 0.93 and h95 0.98 ,
respectively. It is clearly seen from the figures that sandwich-like plates have
resulted from both cases of optimization, in particular for the design with
Zp=100 shown in Fig. 7, where almost all the available foam material is found in
the 4 innermost layers (core) of the 8-layer plate. In the design obtained for
Zp=1000 in Fig, 8, the available foam material is almost entirely placed in the 6
innermost layers, where it surrounds a short, approximately circular cylinder
consisting of the composite material. This will be discussed below. It is also
interesting to note that the upper and the lower layer of the optimized designs in
both of the two cases considered here, are very similar to the optimized designs
of the single-layer plates at the same excitation frequencies, as is seen by
comparing the designs of layers 1 and 8 in Fig. 7 and Fig. 8 with the design in
Fig. 5(b) forZp=100 and the design in Fig. 5(f) for Zp=1000, respectively.
Discrete Material Optimization of Vibrating Laminated… 377
Layer 7 Layer 8
Figure 7. Sandwich plate: Optimized design for the excitation frequencyZp=100 when no
foam material is allowed in the lower layer 1 and upper layer 8
378 N. Olhoff and B. Niu
Layer 7 Layer 8
Figure 8. Sandwich plate: Optimized design for the excitation frequency Zp=1000 when
no foam material is allowed in the lower layer 1 and upper layer 8
Discrete Material Optimization of Vibrating Laminated… 379
Table 4 Results for the initial and optimized designs of sandwich plates when no foam
material is allowed in the surface layers
1st 2nd
Sound Sound
resonance resonance
radiation for radiation for
freq. freq.
Zp=100 Zp=1000
: :
Initial design 449. 1656. 2059. 12808.
Design
optimized with 666. 2114. 967.
Zp=100, Fig. 7
Design
optimized with 369. 1984. 7969.
Zp=1000, Fig. 8
outermost layers 1, 2 and 7, 8, while almost all the weaker foam material is
found in the inner layers 3-6.
Contrary to Fig. 7, the design in Fig. 8 with Zp =1000 has been significantly
driven by dynamics, essentially because inertia forces are proportional to the
square of the frequency of harmonic vibration of mass. Thus, as is seen, the
central part of the plate in Fig. 8 is filled-out by an approximately circular
through-the-thickness cylinder consisting of the stiffer composite material that
has a much higher mass density than the foam material. This local design of the
plate is very efficient in counteracting the time-harmonic external loading that
has not only a given frequency (Zp =1000) but also a prescribed amplitude, and
is found to be in anti-phase with the central part of the forced vibration mode,
whose shape is as depicted in Fig. 11(b). Clearly, the mass assembly in the
central part of the plate yields a large inertia that very effectively reduces the
displacement amplitudes over the central part of the plate, and, thereby, reduces
the vibration amplitudes and the density of sound power emission all over the
plate.
Fig. 9 and Fig. 10 are the same as described for the designs in Figs. 7 and 8,
respectively, in Sub-section 6.3.
Layer 7 Layer 8
Figure 9. Sandwich plate: Optimized design for the excitation frequency Zp=100 and
without restriction for the surface layers
382 N. Olhoff and B. Niu
Layer 7 Layer 8
Figure 10. Sandwich plate: Optimized design for the excitation frequency Zp=1000 and
without restriction for the surface layers
corners. The inner layers of this design also only exhibit very small changes. The
forced vibration mode of the optimized design in Fig. 9 excited at Zp=100 has
the same phase as the uniformly distributed dynamic loading, and is shown in
Fig. 11(a).
Table 5 Results for the initial and optimized designs of sandwich plates without
restriction for the surface layers
1st 2nd Sound Sound
resonance resonance radiation radiation
freq. freq. for for
: : Zp=100 Zp=1000
Initial design 432. 1594. 2713. 14152.
Design optimized with
623. 2029. 952.
Zp=100, Fig. 9
Design optimized with
281. 1963. 6952.
Zp=1000, Fig. 10
The surface layers of the design optimized with Zp=1000 in Fig. 10 also
predominantly consist of composite material, and the inner layers of the plate are
seen to exhibit only minor changes relative to the corresponding layers in Fig. 8.
However, it is noteworthy that in the surface layers of the plate in Fig. 10, a thin,
ring-like shaped region consisting of foam material is found between the large
central part of each of the surface layers and the edges of the plate. Moreover, a
close inspection of all the layers of the plate in Fig. 10 reveals that (with some
small, unimportant exceptions in the innermost layers 4 and 5), the entire plate is
equipped with an inner, through-the thickness zone of foam material that
emanates from the thin, ring-shaped regions with foam in the surface layers, and
follows these regions all the way around the large central part of the plate. A
mechanical explanation of this feature of the plate design in Fig. 10, which is
optimized for and excited by Zp=1000, may be given by considering the
corresponding forced vibration mode in Fig. 11(b), where the outer part of the
plate along the clamped edges is found to be in-phase, and the central part of the
plate to be in anti-phase with the uniformly distributed harmonic dynamic
loading. Guided by Fig. 11(b), we have found that within the thin, ring-like plate
region with foam material in the surface layers, there exists a closed curve along
which the two principal bending curvatures are zero and very small, respectively.
This means that from the point of view of optimization, the
through-the-thickness application of the isotropic, low-stiffness foam material is
“optimal” in this region of the plate, and that it saves composite material here for
other regions where high stiffnesses are useful.
384 N. Olhoff and B. Niu
7 Conclusions
Minimum sound emission from vibrating laminated composite plates without
damping is considered in this paper. The plates are subjected to uniformly
distributed time-harmonic pressure loading with prescribed frequency and
amplitude. Since the plate surfaces are flat, instead of solving the Helmholtz
equation, Rayleigh’s integral approximation is used for computing the total
sound power radiated from the plate into a light acoustic medium such as air.
This substantially reduces the computational cost of the structural-acoustic
analysis and design optimization. Optimization of fiber orientations, stacking
sequence and material selection is performed by Discrete Material Optimization
(DMO) for quadratic, clamped single-layer, multi-layer and sandwich plates.
Interesting features of the optimized designs are observed in numerical
examples.
Numerical results for single-layer and multi-layer plates show that the fiber
orientations of the layers in each of the optimized multi-layer plates are found to
be the same, and to correspond precisely to those in the optimized design of the
corresponding single-layer plate at the same excitation frequency. In the design
of laminated plates with polymeric foam and glass/epoxy composite material as
candidate materials, sandwich-like plates have been obtained for the excitation
frequencies considered. The influence of a restriction on the selection of
candidate materials for the surface layers on the optimum topologies is discussed.
For the same type of plate structure and excitation frequency, the design
optimized without restriction on candidate materials for the surface layers is
consistently better than the design optimized subject to the restriction.
The total sound power radiation from the vibrating laminated composite
plates is generally significantly decreased by the discrete material optimization.
To minimize the sound radiation, the optimization has either driven the nearest
resonance frequency as far away as possible from the prescribed excitation
frequency, or has increased the gap between two neighbouring resonance
Discrete Material Optimization of Vibrating Laminated… 385
References
Bendsøe, M.P., Olhoff, N., 1985. A method of design against vibration resonance of
beams and shafts. Optim Control Appl Methods 6: 191-200.
Bendsøe, M.P., Kikuchi, N., 1988. Generating optimal topologies in structural design
using a homogenization method. Computer Methods in Applied Mechanics and
Engineering 71: 197-224.
Bendsøe, M.P., Sigmund, O., 2003. Topology Optimization: Theory, Methods and
Applications. Springer-Verlag, Berlin.
Bendsøe, M.P., Olhoff, N., Sigmund, O., (Eds.), 2006. Proc. IUTAM Symp. Topological
Design Optimization of Structures, Machines and Materials—Status and Perspectives,
Copenhagen, Denmark, October 26–29, 2005. Springer, Berlin.
Bös, J., 2006. Numerical optimization of the thickness distribution of three-dimensional
structures with respect to their structural acoustic properties. Struct Optim 32: 12-30.
Chen, S.H., Song, X.W., Yang, Z.J., 2005. Modal optimal control of cabin noise of
vehicles. Smart Mater Struct 14: 257-264.
Christensen, S.T., Sorokin, S.V., Olhoff, N., 1998a. On analysis and optimization in
structural acoustics - Part I: Problem formulation and solution techniques. Structural
Optimization 16: 83-95.
Christensen, S.T., Sorokin, S.V., Olhoff, N., 1998b. On analysis and optimization in
structural acoustics - Part II: Exemplifications for axisymmetric structures. Structural
Optimization 16: 96-107.
Denli, H., Sun, J.Q., 2007. Structural-acoustic optimization of composite sandwich
structures: a review. The Shock and Vibration Digest 39 (3): 189-200.
Du, J., Olhoff, N., 2007a. Minimization of sound radiation from vibrating bi-material
structures using topology optimization. Struct Multidisc Optim 33: 305-321.
Du, J., Olhoff, N., 2007b. Topological design of freely vibrating continuum structures
for maximum values of simple and multiple eigenfrequencies and frequency gaps.
Struct Multidisc Optim 34:91-110. See also Publisher’s Erratum in Struct Multidisc
Optim (2007) 34:545.
Du, J., Olhoff, N., 2010. Topological design of vibrating structures with respect to
optimum sound pressure characteristics in a surrounding acoustic medium. Struct
Multidisc Optim 42: 43-54.
Dühring, M.B., Jensen, J.S., Sigmund, O., 2008. Acoustic design by topology
386 N. Olhoff and B. Niu
*
Kurt Maute
*
Department of Aerospace Engineering Sciences,
University of Colorado Boulder, Boulder, CO, USA
1 Introduction
Besides convection and radiation, diffusive transport problems play an im-
portant role in many engineering applications. For example, the transport of
species, electrons, and thermal energy in solids is often modeled by Fickian
diffusion. Therefore, topology optimization of diffusive transport problems
has been studied frequently; see for example the work by Li et al. (1999,
2004); Gersborg-Hansen et al. (2006); Bruns (2007); Kim et al. (2009); Iga
et al. (2009); Zhuang et al. (2010). In comparison to topology optimiza-
tion in elasticity, diffusive transport problems are in general easier to treat
numerically as they only involve a scalar field (e.g. concentration, electric
potential, temperature). However, the physical phenomena, which arise as
the design is changed, are often less intuitive and need careful considera-
tion. In addition, numerical issues particular to steady-state and transient
diffusive problems need to be addressed.
The common goal of topology optimization methods is to find the ge-
ometry and material layout of a body such that an objective is minimized
or maximized, subject to a set of equality and inequality constraints. In
general, the geometry and the material layout can be defined by the dis-
tribution of two or more material phases. In the simplest case of the body
being made of one homogeneous material, this definition leads to a two-
phase material layout problem where one phase is the bulk material and
the other phase represents void. The material phase at a point in a given
design domain can be defined via an integer-valued indicator function or by
either explicitly or implicitly describing the geometry of the material inter-
faces along with a binary variable that defines whether the point is inside
or outside a domain of a particular phase. The well-known density meth-
ods in topology optimization use a relaxed, continuous formulation of the
indicator function, leading to the concept of a continuously varying density
function of a real or fictitious material. The reader is referred to the excel-
lent book by Bendsøe and Sigmund (2003) for a comprehensive overview of
density methods. Level set methods describe implicitly the geometry of the
interfaces; the reader is referred to the book by Sethian (1999) and a recent
review article by van Dijk et al. (2013). The particular features of both, the
density and level set approaches, will be discussed in other chapters of this
book.
Here we will discuss topology optimization of diffusive transport prob-
lems following a density method. Further we will limit our discussion to a
two-phase, void-solid design problem. Thus, we will describe the geometry
of the body via density distribution and interpolate the material properties
as smooth continuous functions of the local density. Assuming an appropri-
ate parameterization of the density distribution and a suitable discretization
of the physical field (e.g. concentration, temperature, etc.) the objective
and constraints will be smooth functions of the optimization variables and
the resulting parameter optimization problem can be solved by standard
nonlinear programming techniques. For simple problems, such as problems
with just one linear constraint and monotonous objectives, optimality cri-
teria methods may be applicable as well.
2 Model Problem
To discuss the basic features of topology optimization of diffusive transport
problems, consider the following configuration, which can be thought of as
Topology Optimization of Diffusive Transport Problems 391
(1)
(2) 1
1
(12)
11
(2)
2
(12)
12
Note: In the above equation the indicator function is defined given a decom-
position of the design domain Ω into Ω(1) and Ω(2) . However, as the relation
is invertible one can use the indicator function to define the decomposition:
In addition, at the interface Γ(12) we enforce that the solution and the
fluxes are continuous:
T (1) − T (2) = 0 (6)
and
(1) (2)
(1) ∂T (12) (2) ∂T (21)
dij nj + dij nj =0 (7)
∂xi ∂xi
(kl)
where nj is the normal pointing from phase k to phase l. The physical
variable T (k) along Γ(12) is defined as follows:
(kl)
T (k) (xi ) = T (xi − nj ) →0 (8)
(12) (21)
Note: Using ni = ni one can write the interface flux condition (7)
(l)
for an isotropic material, i.e. dij = k (l) δij , as follows:
(12)
k (1) ∂T (2) /∂xi nj δij
= (9)
k (2) (12)
∂T (1) /∂xi nj δij
For a constant diffusivity k (2) and as k (1) → 0, the gradient of T (1) along
(12) (21)
nj also needs to decrease and/or the gradient of T (2) along nj needs
(1)
to increase. The physically correct solution in the extreme case of k = 0,
(21)
i.e. phase 1 is void, T (2) along nj vanishes, which represents an adiabatic
boundary condition.
For solving the diffusive transport problem defined above, it is convenient
to rewrite the governing equation in the weak form as follows, with RT being
Topology Optimization of Diffusive Transport Problems 393
the residual:
dT ∂T ∂T
RT = δT c dΩ + δ dij dΩ + δT (aT − q) dΩ
Ω dt Ω ∂xj ∂xi Ω
∂T
− δT kij nj dΓ
Γ ∂x
i (10)
Q−bT
(1) (2)
(1) ∂T (12) (2) ∂T (21)
− δT dij nj + dij nj dΓ = 0
Γ(12) ∂xi ∂xi
0
where we assume that T satisfies the boundary and interface conditions, i.e.
T − T̂ on ΓT and T (1) = T (2) on Γ(12) , and δT is an admissible test function,
i.e. it vanishes on ΓT .
Given the two-phase description discussed above, we are interested in
optimizing the material layout which corresponds to optimizing the geome-
try of phase (2) with phase (1) being the void phase. Regardless of choosing
the indicator function or the subdomain boundaries as primary variables,
the optimization problem reads:
minΓ(12) |I J(T )
s.t. Gi (T ) ≤ 0 i = 1, . . . , NG (11)
RT (T ) = 0
3 Density Methods
Treating directly the indicator function I(xi ) as the primary variable leads
to theoretical and practical issues. The indicator function may alternate
394 K. Maute
adiabatic
TL TR
adiabatic
from point to point between zero and one. The associate functional space is
I(xi ) ∈ L∞ (Ω). The L∞ space contains all bounded but otherwise arbitrar-
ily varying functions. For such functions, the optimization problem (11) is
ill-posed and does not converge as the discretization of I(xi ) and the physi-
cal field T (xi ) is refined. This manifests itself in a strong dependency of the
optimization results on the discretization of the design domain. From an
engineering perspective, one may argue that such a dependency is accept-
able as one is just interested in the solution of a particular discretization.
However, due the ill-posedness of the problem there is little meaning to this
solution, except that it may represent an improvement over some other de-
sign. Furthermore, since often the size of the geometric features decrease as
the mesh is refined, the mesh resolution is likely to be inadequate to pre-
dict the physical solution with sufficient accuracy. Finally, from a practical
point of view, directly parametrizing the indicator function and using these
parameters as optimization variables leads to significant numerical costs as
it requires solving potentially large integer optimization problems. While
the efficiency of integer programming methods, such as branch-and-bound
methods, has increased, solving large integer optimization problems is still
computationally challenging and in most cases impractical.
Resolving the ill-posedness for diffusion problems follows both, from
a mathematical and physical perspective, the same ideas as in elasticity
Bendsøe and Sigmund (2003). Therefore we will not further dive into this
topic, but rather outline the basics. From an engineering view one can ex-
plain the ill-posedness of the topology optimization problem with the role
of the indicator function in the governing equations. Consider the following
heat conduction example depicted in Figure 2.
At the left edge ΓT L of a thin sheet, the temperature is prescribed to
T̂L and at the right edge ΓT R to T̂R . Adiabatic boundary conditions are
imposed on the upper and lower edges. For the sake of simplicity we ignore
volumetric fluxes and surface convection. The objective is to maximize the
Topology Optimization of Diffusive Transport Problems 395
total heat flux across the sheet by varying the distribution of the material
phases (1) and (2) in the design domain with the heat conductivities k (1) <
k (2) . The design domain is the entire sheet. The objective can be determined
as follows:
∂T ∂T
J= dij nj dΩ = dij nj dΩ (12)
ΩL ∂xi ΩR ∂xi
Without additional constraints, the solution of the problem is trivial: the
design domain is occupied by phase (2) as it has the higher conductivity.
When limiting the volume fraction of phase (2), f2 = |Ω2 |/|Ω|, the problem
has a non-trivial solution.
Intuitively, the heat flux at a point is maximum if the constitutive tensor,
di j, is optimum for a given temperature gradient and local volume fraction.
The heat flux across the sheet can then be maximized by finding the op-
timum distribution of local volume fraction and the resulting temperature
gradient field. In general, the local problem is finding the optimum diffu-
sivity tensor over all tensors of heterogeneous materials that are composed
of the phases (1) and (2) with a given volume fraction, f2 . For diffusive
problems, this optimization problem has attracted a significant amount of
research. One can show that the optimum diffusivity tensor can be derived
via homogenization from an optimum micro-structure that consists only of
the (1) and (2) and satisfies the volume fraction constraint. Such a micro-
structure is shown in Fig. 3. Note the orientation of the micro-structure
with respect to a macroscopic reference frame is part of the solution of the
local problem. Interestingly, optimum micro-structures of diffusive prob-
lems have a minimum interface area.
The ill-posedness of the original formulation of the topology optimiza-
tion problem is rooted in the dilemma that the indicator function I, defined
at the macro-scale, can never converge to the optimum continuum solu-
396 K. Maute
adiabatic
adiabatic
4 Physical Phenomena
In general, formulating optimization problems whose solution provide guid-
ance to engineers in the design process, requires deep insight into the physi-
cal phenomena that may dominate the response of both the converged design
as well as intermediate configurations during the design process. The latter
is of great importance for density based topology optimization, in particu-
lar as intermediate designs may involve non-conventional material layouts
and, depending on the material interpolation used, fictitious material prop-
erties. As the following simple study will show this may impact the proper
formulation of objective and constraints and may require the inclusion of
real or fictitious phenomena that otherwise would play no role in the design
problem.
Here we consider a thin sheet subdivided into three subdomains as shown
in Figure 4. The temperature being prescribed at the left edge ΓT L to T̂L
and adiabatic boundary conditions are imposed on the upper and lower
398 K. Maute
TL TL TL
(2) (1) (2) (2) (1) (2) (2) (1) (2)
1 1 2 1 1 2 1 1 2
TL TL TL
(2) (1) (2) (2) (1) (2) (2) (1) (2)
1 1 2 1 1 2 1 1 2
TL TL TL
(2) (1) (2) (2) (1) (2) (2) (1) (2)
1 1 2 1 1 2 1 1 2
40 m
20
m
qA
Figure 6. Convective cooling of rectangular plate.
5 Example
As mentioned above, considering convective boundary conditions leads to
interesting topology optimization problems. Since density methods require
interpolating physical properties as a function of the density, one can con-
sider different interpolation schemes for the convection coefficient. To il-
lustrate the profound effect of the interpolation approach on the topol-
ogy optimization result, we revisit the problem of Bruns (2007), depicted
in Figure 6. A two-dimensional design domain is subject to a heat flux
qA = 1 pW/μm at point A. The plate is cooled via natural convection; the
convection coefficient of the bulk material and the surrounding media is
b0 = 10−3 W/μm2 K; the ambient temperature is set to T∞ = 0 K. Assum-
ing an isotropic material behavior, the conduction coefficient of the bulk
material is k0 = 1 pW/μmK. The design problem is to find the optimum
material layout that minimizes the temperature at point A; the volume of
the bulk material may not exceed 30 % of the volume of the design space.
Here we apply a standard SIMP model to interpolate the heat conduction
coefficient:
k = k0 f β (15)
with β = 3 where f denotes the volume fraction, or density. The nonlin-
ear interpolation of the conduction coefficient along with volume constraint
leads to an implicit penalization of intermediate densities. We consider two
Topology Optimization of Diffusive Transport Problems 401
Figure 7. Material layouts for different mesh refinement levels and convec-
tion interpolations.
8K
0K
Figure 8. Temperature contours for optimized designs.
h 0.5P m h 0.1P m
∂f
P =
dΩ (16)
Ω ∂xi
6 Numerical Issues
Despite the simplicity of the partial differential equations describing diffu-
sive transport problems, their numerical treatment is not trivial. In the
transient case, instability phenomena may occur, independent of whether
implicit or explicit time integrations schemes are used, if the time step and
404 K. Maute
elements Temperature
2 1.035 101
3 −1.22 100
4 5.97 10−2
5 −5.55 10−4
6 3.60 10−8
7 2.52 10−9
8 2.28 10−7
9 1.14 10−6
Table 2. Temperature versus mesh discretization for a simple thermal bar
subject to convection.
The oscillation of the temperature above and below the ambient value
is evident for low mesh densities. Because the convection influence is pro-
portional to the convecting area while the conduction influence is inversely
proportional to the element size, mesh refinement will mitigate the oscil-
lations. This can be seen by examining the eigenvalues of the conduction
matrix. For the symmetric mode to remain the lower eigenvalue, the fol-
lowing condition needs to be satisfied:
which is equivalent to
b S L2
≤1 (24)
6kA
This stability condition shows that oscillations do not occur if the mesh is
sufficiently refined.
Following Bruns (2007), the stability issue for convection dominated
problems can be circumvented if the convection matrix is lumped and ap-
proximated by a diagonal matrix:
bS 1 0
C̃ = . (25)
2 0 1
7 Acknowledgment
The author acknowledges the support of the National Science Foundation
under grants EFRI–1038305 and EFRI–1240374. The opinions and conclu-
sions presented in this paper are those of the authors and do not necessarily
reflect the views of the sponsoring organization. The author would also like
to thank Mr. Peter Coffin for his help in preparing the numerical results.
Bibliography
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50(15-16):2859 – 2873, 2007.
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thermal conductors considering design-dependent effects, including heat
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tion of nonlinear heat conduction problems using topological derivatives.
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Topology Optimization of Diffusive Transport Problems 407
*
Kurt Maute
*
Department of Aerospace Engineering Sciences,
University of Colorado, Boulder, USA
1 Introduction
Finding the geometry of systems to optimize the performance of internal
and external flows is of great importance across a wide spectrum of applica-
tions. For example, fluids apply forces on bodies, such as lift and drag, and
transport species and thermal energy. Manipulating these phenomena is a
central issue for a large number of engineering systems, including aircraft
aerodynamics, injection molding, and liquid cooling. Due to the complexity
of modeling and predicting flows, historically the design of flow problems
was driven by experimental studies. With the advent of computational fluid
dynamics (CFD) in the late 1960s, the flow about more complex geometries
could be simulated and analyzed numerically. Improved numerical schemes
2 Flow Modeling
Assuming that compressibility effects can be neglected, the incompressible
Navier-Stokes equations are valid across a broad range of flow regimes. For
Topology Optimization of Flows: Stokes and Navier-Stokes Models 411
where ρ̂ and vˆi describe the density and velocity, respectively. The sub-
scripts i, j define the spatial directions, ˆ indicates dimensional quantities
ˆ
and a prescribed value. The external body forces are denoted by fˆiB and
¯
the external traction forces by F̂i . The stress tensor σ̂ij is defined as:
∂v̂i ∂v̂j
σ̂ij = −p̂δij + μ̂ + , (5)
∂ x̂j ∂ x̂i
where p̂ and μ̂ describe the pressure and dynamic viscosity, respectively. In
the following, we assume that the dynamic viscosity is constant, i.e. it does
not depend of the flow variables.
Alternatively, the incompressible Navier-Stokes equations (1)-(2) at steady
state can be written in weak form with non-dimensional variables as follows:
∂vi 1 ∂wi ∂wj
R= wi ρ vj dΩ − wi fi dΩ −
B
+ pδij dΩ
Ω ∂xj Ω Ω 2 ∂xj ∂xi
1 ∂wi ∂wj ∂vi ∂vj ∂vi
+ + μ + dΩ + q dΩ
Ω 2 ∂xj ∂xi ∂xj ∂xi Ω ∂xi
∂vi ∂vj
− wi nj −pδij + μ + dΓ = 0, (6)
Γ ∂xj ∂xi
where R is a residual, wi an admissible velocity test function and q an
admissible pressure test function. Note, the divergence of the stress tensor,
∂ σ̂ij /∂ x̂j in Eq. (1), has been integrated by parts. The non-dimensional
dynamic viscosity is defined as:
μ̂ 1
μ= = , (7)
L̂ref v̂ ρ̂ Re
where L̂ref , v̂, ρ̂ are the dimensional reference length, reference velocity,
reference density, and Re is the Reynolds number.
For small Reynolds numbers (Re << 1) the advective and inertial forces
are small compared with viscous forces and can be neglected. In such flow
412 K. Maute
regimes the fluid velocities are small, the viscosity is large, or the length-
scales of the flow are small. In this case the strong form of the flow equations
is:
∂ σ̂ij
Momentum equation: + fˆiB = 0, (8)
∂ x̂j
∂v̂i
Incompressibility condition: = 0, (9)
∂ x̂i
Surface traction condition: σ̂ij nj = F̂i on Γσ , (10)
Dirichlet condition: v̂i = v̄ˆi on Γv and p̂ = p̄ˆ on Γp . (11)
The above equations are the Stokes flow model. Their weak form using
non-dimensional variables is given by:
1 ∂wi ∂wj 1 ∂vi ∂vj
R=+ + + dΩ
Ω 2 ∂xj ∂xi Re ∂xj ∂xi
∂vi 1 ∂wi ∂wj
+ q dΩ − wi fi dΩ −
B
+ pδij dΩ
Ω ∂xi Ω Ω 2 ∂xj ∂xi
1 ∂vi ∂vj
− wi nj −pδij + + dΓ = 0, (12)
Γ μ ∂x j ∂xi
From a mathematical perspective, neglecting the advection term in the
Navier-Stokes equation simplifies the problem significantly for three reasons:
(a) the Stokes equation are linear, (b) there are no numerical instability is-
sues caused by the advection term, and (c) the resolution requirements of
the flow field, i.e. mesh density, are reduced.
3 Density Methods
As discussed for example in the chapter on topology optimization of diffusive
transport problems, one can manipulate the geometry by directly varying
the shape of the interface between fluid and solid sub-domains. In the con-
text of flow optimization, the solid can be considered the “void phase.”
Such approaches are typically restricted to shape variations and topological
changes are not possible unless special interface descriptions are used, e.g.
level set methods. Level set methods for flow topology optimization were
presented by Cunha (2004); Pingen et al. (2007b); Mohammadi and Piron-
neau (2008); Duan et al. (2008a,b,c); Challis and Guest (2009); Kreissl and
Maute (2012).
The most common approach in flow topology optimization follows a den-
sity concept. The integer-valued indicator function, which defines whether
Topology Optimization of Flows: Stokes and Navier-Stokes Models 413
equations. Again ignoring external volume forces, the strong form of the
momentum equations reads:
∂v̂i ∂ σ̂ij
ρ̂ v̂j = + α̂v̂i (16)
∂ x̂j ∂ x̂j
a. b.
D
c. d. 0 1 U
D̂ D
decreasing q.
0 1 U 0 1 U
the drag rapidly increases to the value of the solid cylinder. Note these
results are qualitative, but represent a wide variety of flow regimes and
boundary conditions.
When imposing a constraint on the volume that can be occupied by fluid,
the strong non-linearity observed above leads to an inherent penalization
and favors “0-1” solutions. However, the large gradients of the dissipated
energy (and the flow field in general) with respect to the density may cause
numerical issues and the optimization process often converges to a local
minimum. To reduce the gradients, the penalization coefficient is typically
interpolated such that the sensitivity of the flow field with respect to the
density is reduced. Borrvall and Petersson (2003) proposed the following
interpolation:
1 − ρ̃
α̂ = α̂U + (v α̂L − α̂U ) (1 + q) . (18)
1 − ρ̃ + q
The influence of the interpolation penalty parameter q is illustrated in Fig-
ure 1c. As shown in Figure 1d, the gradients of the drag are significantly
reduced by the interpolation (18) when compared to a linear interpolation
which is marked by the dashed line.
Based on the interpolation (18) two-dimensional and three-dimensional
flow topology optimization problems have been solved using either Stokes or
Navier-Stokes flow models. However, the parameters of the optimization al-
gorithms and the interpolation penalty q in (18) need to be chosen carefully.
To reduce the tendency of the optimization process converging to a local
minimum, the enforcement of the volume constraint needs to be controlled
such that constraint violations are tolerated initially in the optimization
process. The interpolation penalty should be as low as possible but high
enough to prevent elements with intermediate porosities from appearing in
the final design.
5 Numerical Examples
To illustrate the potential of topology optimization applied to flow problems,
Figure 2 displays some frequently studied problems. In all examples, the
objective is to minimize the pressure drop while constraining the volume
that can be occupied by fluid. The pressure drop is predicted by a finite
element model of the incompressible Navier-Stokes equations augmented by
the Brinkman penalization. The flow field is discretized by standard 4-node
(2D) and 8-node (3D) elements. The SUPG and PSPG stabilization schemes
of Tezduyar et al. (1992) are used to prevent node-to-node oscillations. At
the inlet the flow is prescribed and the outlet is assumed to be traction-free.
416 K. Maute
inlet
outlet
The density distribution is discretized by the same mesh as used for the flow
analysis. The optimization variables are the nodal densities; the Brinkman
penalization parameter is constant within one element and computed via the
interpolation scheme (18) using the averaged nodal densities. The gradients
of the pressure drop with respect to the optimization variables are computed
by an adjoint approach. The optimization problem is solved via the globally
convergence method of moving asymptotes (GCMMA) of Svanberg (2002).
Details on these examples can be found in Kreissl and Maute (2011) and
Kreissl (2011).
The figure at the beginning of this chapter shows yet another interesting
application of topology optimization; here the design problem is to create a
so-called leaky valve. The objective is to minimize the pressure drop for a
flow entering the design domain at the top of the left edge and exiting the
domain at the bottom of the right edge and to maximize the pressure drop
for the reverse flow direction. More details on the formulation and solution
of this problem can be found in Pingen et al. (2008) and Pingen (2008).
Topology Optimization of Flows: Stokes and Navier-Stokes Models 417
6 Concluding Remarks
While topology optimization applied to flow problems has advanced sig-
nificantly, there are numerous unresolved issues that need to be addressed
in future research. Most studies focus on low-Reynolds number applica-
tions with laminar flows at steady-state conditions. Most engineering flow
problems, however, involve high-Reynolds number, turbulent, and unsteady
flows. A few of the many issues one encounters when applying established
techniques to unsteady flows are discussed in Kreissl and Maute (2011). So
far, the objective is typically a measure of the energy dissipation and ad-
ditional constraints on the fluid volume are needed to obtain “0-1” results.
This particular combination of objective and constraints is overly restrictive
and does not allow for tackling a broad range of flow design problems. An
example of this issue is discussed in Kreissl and Maute (2012).
7 Acknowledgment
The author acknowledges the support of the National Science Foundation
under grants CMMI–1235532 and CBET–1246854. The opinions and con-
clusions presented in this paper are those of the authors and do not neces-
sarily reflect the views of the sponsoring organization.
Bibliography
M. P. Bendsøe and O. Sigmund. Topology Optimization: Theory, Methods
and Applications. Springer, 2003.
T. Borrvall and J. Petersson. Topology optimization of fluids in Stokes
flow. International Journal for Numerical Methods in Fluids, 41(1):77–
107, 2003. ISSN 0271-2091.
V. Challis and J. Guest. Level set topology optimization of fluids in stokes
flow. International Journal for Numerical Methods in Engineering, 79
(10):1284–1308, 2009.
A. L. Cunha. A Fully Eulerian Method for Shape Optimization with Appli-
cation to Navier-Stokes Flows. PhD thesis, Carnegie Mellon University,
Pittsburgh, PA 15213, September 2004.
X. Duan, Y. Ma, and R. Zhang. Optimal shape control of fluid flow us-
ing variational level set method. Physics Letters A, 372(9):1374—1379,
2008a.
X. Duan, Y. Ma, and R. Zhang. Shape-topology optimization for navier-
stokes problem using variational level set method. Journal of computa-
tional and applied mathematics, 222(2):487–499, 2008b.
418 K. Maute
*
Kurt Maute
*
Department of Aerospace Engineering Sciences,
University of Colorado Boulder, Boulder, CO, USA
1 Introduction
The performance of many engineering systems often depends on multiple
physical phenomena belonging to different engineering disciplines, such as
solid mechanics, fluid mechanics, and heat transfer. Topology optimization
methods have been developed and applied to problems that are dominated
∂ u
j + q u = 0 in Ω jiu ni = Qu on Γqu u = û on Γu , (1)
∂xi i
∂ v
j + q v = 0 in Ω jiv ni = Qv on Γqv v = v̂ on Γv , (2)
∂xi i
with ji denotes the diffusive fluxes, q are external volumetric fluxes, and
Q are external surface fluxes. The fields are coupled via the constitutive
laws and/or the external volumetric fluxes:
In both cases, the fields are coupled in each material point or a subset of
points within the volume of the body. Examples for this type of coupling
include Joule heating for electro-thermal problems, thermal expansion in
thermo-elastic problems, and piezo-electric coupling, among others.
The other basic form of interaction between multiple physical fields is
the coupling through interface conditions. In this case the physical fields
are only defined over non-overlapping subsets Ωu and Ωv with Ω = Ωu ∪ Ωv .
The interface Γuv is defined by Γuv = Ωu ∩ Ωv with nuv being the normal
vector on Γuv pointing from Ωu to Ωv . The governing equations for this
Topology Optimization of Coupled Multi-Physics Problems 423
with regards to the solution of the governing equations and the parametriza-
tion of the geometry in topology optimization. This is in particular the
case for problems involving deforming structures, which represents a class
of problems that is of great interest to the structural design optimization
community. The main issues of surface coupled problems can be summa-
rized as follows:
where σij is the elastic stress tensor and dj is the electric displacement
vector. For the sake of a compact presentation of the aspects relevant
to topology optimization, we consider only the static case, neglect body
force in the structural equilibrium equations, and omit the definition of the
boundary conditions. The level of difficulty of extending the static case to
free and forced vibration problems depends on the particular application,
in particular whether the dynamics of the electric circuitry attached to the
piezo-electric device needs to be included in the model; see, for example,
Rupp et al. (2009).
The constitutive equations accounting for piezo-electric coupling are:
where Cijkl is the elastic tensor and εkl is the strain tensor. The dielectric
tensor is denoted by Djk and the electric field by ek . The coupling of the
mechanical and electric problems is described by the piezo-electric coupling
tensor Eijk . In electrostatics, the electric field is ek = −(∂φ)/(∂xk ), with φ
being the electric potential.
Applying a density method, the material properties need to be inter-
polated as functions of the density. This can be done via homogenization
426 K. Maute
where em denotes the Young’s modulus and de the dielectric constant. Note
it may not be necessary to interpolate all material properties; in the example
above, the Poisson ratio was chosen to be independent of the density.
The interpolation rules need to be designed such that the material distri-
bution converges to a discrete material distribution; for example, to “0-1”
solution in the case of two-phase, solid-void design problems. In density
methods, a SIMP interpolation scheme is often adopted and an additional
mass constraint is introduced. Optimizing the stiffness of a structure, the
interplay between an nonlinear interpolation of the material stiffness and
the linear interpolation of the density leads to an implicit penalization of
intermediate densities and favors a “0-1” design. However, optimizing piezo-
electric devices, the SIMP approach may or not lead to an implicit penal-
ization of intermediate densities, depending on the design objective. As
the physical phenomena become more complex it is typically not trivial to
find interpolation rules for all parameters such that a convergence to a “0-
1” solution can be achieved. If no appropriate interpolation rules can be
found, formulations explicitly penalizing intermediate densities may need to
be considered and used in combination with continuation methods. How-
ever, for multi-physics problems, one needs to ensure that the interpolations
are designed such that for intermediate densities the response of the prob-
lem is still “physical”, driving the optimization process toward a physically
meaningful solution.
S
∂ σ̂ij
static equilibrium: − = 0 in ΩS , (14)
∂ x̂j
F
∂v̂i ∂ σ̂ij
fluid momentum equation: ρ̂ v̂j − − α̂vˆi = 0 in ΩF , (15)
∂ x̂j ∂ x̂j
∂v̂i
incompressibility condition: = 0 in ΩF , (16)
∂ x̂i
where ρ̂ and vˆi denote the fluid density and velocity, respectively. The
S F
stress tensor of the solid is σ̂ij and in the fluid σ̂ij . The subscripts i, j
define the spatial directions, and ˆ indicates dimensional quantities. The
last term in the fluid momentum equations (15) stems from a Brinkman
model, approximating the flow through porous material. This term can be
used to enforce a stick condition by increasing the impermeability coefficient
α̂. The reader is referred to the chapter on flow topology optimization for
details of the Brinkman penalization method and its application to topology
optimization. Note the structural equilibrium equations are defined in a
Lagrangian observer frame while the fluid equations in an Eulerian one.
The constitutive equations for the solid and fluid are:
F 1 ∂ ûk ∂ ûl
Solid: σ̂ij = Cijkl + (17)
2 ∂xl ∂xk
∂v̂ i ∂v̂ j
F
Fluid: σ̂ij = −p̂δij + μ̂ + , (18)
∂ x̂j ∂ x̂i
where ûi denote the structural displacements, p̂ the fluid pressure, and μ̂
the dynamic viscosity. For the sake of simplicity, we assume a linear elastic
behavior of the solid material and a constant dynamic viscosity.
430 K. Maute
Both coupling concepts, the ALE formulation of the fluid as well as im-
mersed boundary techniques have been integrated into topology optimiza-
tion approaches; see, for example, the work by Maute and Allen (2004),
Maute and Reich (2006), James et al. (2008),Stanford and Ifju (2009), Yoon
(2010), and Kreissl et al. (2010). These approaches can be further classified
by the fluid and structural models used and by the design changes possi-
ble in the optimization process. Most often a linear elastic model assuming
small deformations is adopted to model the structural response, and the flow
is predicted by either a potential flow, an Euler or a Navier-Stokes model.
Depending on the combination of models, computing the design sensitivities
may pose a major challenge; see for example the work by Maute et al. (2003)
and Barcelos and Maute (2008) on adjoint sensitivity analysis methods for
compressible Navier-Stokes models. The complexity of the structural and
fluid models may further complicate the extension onto transient problems.
actuator skin
pressure
F
F FS
∂ σ̂ij
σ̂ij nj dΓ = κ(ρ) dΩ, (23)
ΓF S Ω ∂ x̂j
4 Concluding Remarks
Topology optimization provides an exciting tools for systematically study-
ing multi-physics design problems as these problems are often dominated by
nonlinear phenomena and benefit from non-intuitive design solutions. Den-
sity methods are well-suited for problems that are coupled via constitutive
equations, but face largely unresolved challenges when the physical domains
interact along interfaces. Robust and efficient topology optimization meth-
ods for complex multi-physics problems that require an accurate resolution
of the individual physical fields along the interfaces, such as transient FSI
problems at higher Reynolds numbers, are currently lacking. While level-
set methods have been applied successfully to multi-physics problems, they
have not expanded yet the range of design problems that can be solved by
topology optimization.
5 Acknowledgment
The author acknowledges the support of the National Science Foundation
under grants CMMI–1235532 and CBET–1246854. The opinions and con-
clusions presented in this paper are those of the authors and do not neces-
sarily reflect the views of the sponsoring organization.
Bibliography
M. Barcelos and K. Maute. Aeroelastic design optimization for laminar and
turbulent flows. Computer Methods in Applied Mechanics and Engineer-
ing, 197(19):1813–1832, 2008.
M. Bendsøe. Optimal shape design as a material distribution problem.
Structural and Multidisciplinary Optimization, 1(4):193–202, 1989.
M. P. Bendsøe and O. Sigmund. Topology Optimization: Theory, Methods
and Applications. Springer, 2003.
R. C. Carbonari, E. C. N. Silva, and G. H. Paulino. Topology optimiza-
tion design of functionally graded bimorph-type piezoelectric actuators.
Smart Materials and Structures, 16:2605–2620, 2007.
A. Donoso and O. Sigmund. Optimization of piezoelectric bimorph actua-
tors with active damping for static and dynamic loads. Structural and
Multidisciplinary Optimization, 2008.
E. H. Dowell and K. C. Hall. Modeling of fluid-structure interaction. Annual
Review of Fluid Mechanics, 33(1):445–490, 2001.
C. A. Felippa, K. Park, and C. Farhat. Partitioned analysis of coupled
mechanical systems. Computer methods in applied mechanics and engi-
neering, 190(24):3247–3270, 2001.
Topology Optimization of Coupled Multi-Physics Problems 435
*
Kurt Maute
*
Department of Aerospace Engineering Sciences,
University of Colorado Boulder, Boulder, USA
1 Introduction
Topology optimization is concerned with finding the geometry and the ma-
terial layout of a body such that an objective is minimized (or maximized)
and a set of constraints is satisfied. As discussed in other chapters, the
geometry within a design domain Ω can be defined in two ways, which
are illustrated in Figure 1. The geometry and material layout is uniquely
defined by describing the phase boundaries and by determining whether a
point is inside or outside a domain, which is enclosed by an interface and ex-
ternal boundaries. Alternatively, one can define an integer-valued indicator
function I(xi ) at all points xi ∈ Ω, with the value of the indicator func-
tion being the phase id. While both approaches are equivalent and allow
Note the volumetric and surfaces fluxes can only be applied to the solid
domain; otherwise the analysis problem is ill-posed.
The Extended Finite Element Method 441
where nj denotes the normal on the phase boundaries Γ(12) , pointing from
phase 2 to phase 1. Note the latter formulation allows for modeling Neu-
mann boundary conditions and applying Dirichlet boundary conditions on
the phase boundaries Γ(12) .
For now we assume that no Dirichlet or non-homogeneous Neumann in-
terface conditions on Γ(12) exist, and we study the application of discretiza-
tion schemes to the two formulations, (3) and (4). Subsequently, we focus
in the particular on the finite element method. We consider an element
that is intersected by Γ(12) as shown in Figure 2. If we do not have explicit
information of the phase boundary, but the interface geometry is defined
implicitly via the indicator function, we need to approximate the response
within the element by standard shape functions and integrate numerically
the weak form of the governing equation over the element, evaluating at
each integration point the integrand (see Figure 2a). For the particular
case of “void-solid” in the absence of Dirichlet and non-homogenous Neu-
mann boundary conditions, standard finite element interpolation schemes
are sufficient to approximate the response as it is smooth within the solid
phase and along the phase boundary. However, it cannot be guaranteed
that standard integration schemes are sufficient to properly integrate the
governing equation, which may result in singular systems (see Figure 2b).
a. b. c.
2 Level-set method
To keep this chapter as self-contained as possible, we will first outline the
particular formulation of the level-set method, which is used in the subse-
quent discussions. Note other level-set methods can be used together with
the XFEM.
The level-set method, first developed by Osher and Sethian (1988), uses
the level-set function Φ to describe the interface within a design space; see
Figure 3a. The interface Γ(f s) , here separating the fluid and solid subdo-
mains, is implicitly defined through the zero level-set contour:
The value of the scalar function Φ defines whether a point xi is fluid, i.e. xi ∈
Ω(f ) , solid, i.e. xi ∈ Ω(s) , or located on the fluid-structure interface, i.e. xi ∈
Γ(f s) :
The level-set optimization scheme employed in this study follows the idea
by de Ruiter and van Keulen (2004). Contrary to classical level-set opti-
mization methods that advance the level-set field via the solution of the
Hamilton-Jacobi equation, here the parameters of the discretized level-set
field are explicitly defined as functions of the optimization variables sk :
Φ = Φ (sk ). The main advantage of this approach is that the resulting
optimization problem can be solved by standard nonlinear programming
444 K. Maute
©d y
©c
f
Á(x)
z
s
s
©a ¡i
f
± = ¡i x
± = ¡i
©b
(a) Interface Γ(f s) implicitly defined through (b) Elemental interface po-
Φ(x) = 0 sition, based on the nodal
Φi -values.
where rij is the distance between the i-th and the j-th node, rs is the relative
smoothing radius and h denotes the edge length of one element. Note, in
contrast to density methods, smoothing the level-set function is insufficient
to control locally the geometry and guarantee that the design converges as
the discretization of the level-set field is refined; Sigmund and Maute (2013)
provide further details on this issue.
For the optimization problems considered in this study, the interface
geometry is varied by manipulating the level-set field in the optimization
process. The interface geometry is only dependent on the design variables
that lie within a band of width 2rs around the interface; see (10). The design
sensitivities of variables outside of this band vanish. Therefore the design
can only grow across existing boundaries, i.e. boundaries can merge, but no
The Extended Finite Element Method 445
μ̂ 1
μ= = , (12)
L̂ref v̂ ρ̂ Re
where L̂ref , v̂, ρ̂ are the dimensional reference length, reference velocity, and
reference density, respectively. The Reynolds number is denoted by Re.
with
1 ∀ x ∈ Ω(f ) ,
ψ (x) = (14)
0 ∀ x ∈ Ω(s) ,
The Extended Finite Element Method 447
¡i ¡i
1 1 1 1 1 1
(a) Standard shape func- (b) Heaviside enrichment (c) Product of standard
tion, N function, ψ shape and enrichment
function, N ψ
where Ni (xi ) is the standard shape function and ψ (xi ) is the enrichment
function; f˜i denotes regular and f¯i enriched degrees of freedom. The en-
riched degrees of freedom are used to describe the flow field in intersected
elements; the location of the intersection is defined by the zero level-set.
Note enriching the approximation of the flow field by no or only one
additional set of shape functions, as suggested by (13), may be insufficient
and lead to an artificial coupling of geometrically disconnected flows, which
may cause numerical instabilities and artifacts in the optimized geometry.
To properly approximate the flows in domains separated by less than two
“void” elements, additional enrichment functions may need to be intro-
duced. The reader is referred to Makhija and Maute (2013) for details on
a generalized enrichment strategy and its application to topology optimiza-
tion.
For two-phase problems, such as “solid-void” or “fluid-void” configura-
tions, the XFEM may also suffer from ill-conditioning of the discretized
governing equations if the size of the physical domain in intersected ele-
ments approaches zero; see Figure 2c. This problem can be mitigated by
eliminating the degrees of freedom, which cause the ill-conditioning, from
the discrete system and/or by applying appropriate projections schemes.
The reader is referred to Lang et al. (2013) for further details on this issue.
5 Interface Conditions
To enforce the stick condition along the fluid-solid interface,
L
where the contraction of the assumed stress field σij and the normal nj
L
serves as a Lagrange multiplier. The test function corresponding to σij is
L
denoted by γij . The strain rate tensor, ij , associated with the assumed
L
stress field, σij , is defined as:
1 L
L
ij = σij + δij p . (17)
2μ
The first term in (16) enforces the constraint, the second one represents
the contribution of the surface traction associated with the assumed stress
field to the flow momentum equation, and the third term ensures the com-
patibility of the true and assumed strain fields. The Lagrange multiplier
degrees of freedom can be eliminated on an elemental level if element-wise
continuous shape functions for Lagrange multiplier are chosen. Thus, the
interface condition is enforced without introducing additional unknowns to
the residual equations.
L
In this study we chose constant shape function for both γij and σij .
In addition, we introduce the scaling factor k in the compatibility term in
(16). Setting k to zero, leads to a classical Lagrange multiplier formula-
tion, which would require finding an appropriate Lagrange multiplier space.
While increasing k improves the numerical stability, it leads to a less ac-
curate enforcement of the boundary condition. Due to the structure of the
compatibility term, boundary conditions are less strongly enforced as the
Reynolds number (see (12)), and the elemental fluid area increase. There-
fore, we define the scaling parameter k as follows:
Ωe − Ωef
k=η , (18)
Ωe
The Extended Finite Element Method 449
where Ωef defines the fluid area and Ωe the total area of one element. The
scalar η depends on the Reynolds number and the spatial discretization.
6 Example
To demonstrate the utility of the combination of level-set topology opti-
mization and the XFEM for optimizing the geometry of two-dimensional
incompressible flow problems at steady-state, we consider the following two
problems. The first example shows the ability of the proposed approach to
reproduce results found by density methods using a Brinkman penalization
approach. The second example illustrates the advantage of the proposed
XFEM approach over the material interpolation approach.
In the following examples, the weak form of the incompressible Navier-
Stokes equations (11) is discretized by four-node finite elements, i.e. the
velocity and pressure fields are approximated piecewise by bilinear, equal-
order interpolations. To avoid numerical instabilities we employ the SUPG/PSPG-
stabilization scheme introduced by Tezduyar et al. (1992). All examples are
solved by the Globally Convergent Method of Moving Asymptotes (GCMMA)
of Svanberg (1995). The design sensitivities are computed by an adjoint
method. Both examples are described in detail in Kreissl and Maute (2012).
6.1 Pipe-bend
The pipe-bend problem was originally introduced by Borrvall and Pe-
tersson (2003) and is a standard example for fluid topology optimization. It
was also studied by Duan et al. (2008) and Pingen et al. (2007), among oth-
ers. The design domain as well as the initial design are depicted in Figures
5a and 5b. The objective is to minimize the difference in total pressure,
2
ptot = 1/2ρ v + p, between the inlet and outlet, subject to a constraint
that allows only 25% of the area of the design domain to be fluid:
in − pout ),
min z = (ptot tot
sk
Af luid (19)
s.t. g = 0.25 − ≥ 0,
L2
smin ≤ sk ≤ smax .
The total inlet and outlet pressures, ptot tot
in and pout , are averaged over the
inlet and outlet ports. The boundary conditions, depicted in Figure 5a, are
a parabolic inlet velocity, vin , and a constant static outlet pressure, pout .
A stick condition is enforced at the domain boundaries. The initial design
consists of an all-fluid design domain with four solid circles, as shown in
Figure 5b.
450 K. Maute
5L
L
vin
design
domain 5L
3.5L
3.5L L
pout
©=0 ©=0
In order to guarantee proper flow through the inlet and outlet ports,
the level-set values coinciding with the nodes on the inlet and outlet are
fixed at Φi = −1 and are omitted from the smoothing operation defined
in (10). Discretizing the design domain by 40 × 40 elements, the level-set
distributions of the initial and optimized designs are depicted in Figure 6.
The optimized geometry is shown in Fig. 7a.
For comparison we optimize the problem (19) using a density approach
for two different levels of mesh refinement: 40 × 40 and 80 × 80 elements.
Here, the material distribution defines the penalty coefficient of the Brinkman
model. Following the work by Borrvall and Petersson (2003), the penalty
coefficient is interpolated via a convex function (see chapter on flow topology
optimization).
Figures 7b and 7c show the zero level-set of the optimized XFEM ge-
ometry on top of the optimized porosity distribution. Both designs agree
well with the one obtained from the XFEM based optimization. Figure 7b
illustrates that for the same mesh refinement the proposed XFEM approach
has higher boundary resolution compared to the porosity approach.
The Extended Finite Element Method 451
0 s 1 0 s 1
(a) XFEM (40 × 40 mesh) (b) Zero level-set contour (c) Zero level-set contour
on 40 × 40 porosity distri- on 80 × 80 porosity distri-
bution bution
The figure shown at the beginning of this chapter shows the initial, an
intermediate, and the final result of a topology optimization process that
minimizes the pressure drop of a diffuser via a level-set approach. As in the
pipe-bend example above, the flow is predicted by the XFEM. The diffuser
problem was originally introduced by Borrvall and Petersson (2003) and
studied by a density method. Challis and Guest (2009) and Pingen et al.
(2007) solved the problem by level-set methods.
L
vin design
L L pout
domain
L
6L
0.787
7 Concluding Remarks
The XFEM is a promising approach for analyzing designs within level-set
topology optimization approaches. In contrast to Ersatz material tech-
niques, the XFEM preserves the crispness of the geometry description pro-
vided by level-sets. Furthermore, it allows enforcing Dirichlet and non-
The Extended Finite Element Method 453
8 Acknowledgment
The author acknowledges the support of the National Science Foundation
under grants CMMI–1201207 and EFRI–1038305. The opinions and conclu-
sions presented in this paper are those of the authors and do not necessarily
reflect the views of the sponsoring organization.
Bibliography
S. Amstutz and H. Andrä. A new algorithm for topology optimization using
a level-set method. Journal of Computational Physics, 216(2):573–588,
454 K. Maute
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flow. International Journal for Numerical Methods in Fluids, 41(1):77–
107, 2003.
M. Burger, B. Hackl, and W. Ring. Incorporating topological derivatives
into level set methods. Journal of Computational Physics, 194(1):344–
362, 2004.
V. Challis and J. Guest. Level set topology optimization of fluids in stokes
flow. International Journal for Numerical Methods in Engineering, 79
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branched and intersecting cracks with the extended nite element method.
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tion Approach. PhD thesis, Technische Universiteit Delft, 2005.
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The Extended Finite Element Method 455
*
Kurt Maute
*
Department of Aerospace Engineering Sciences,
University of Colorado Boulder, Boulder, CO, USA
1 Introduction
The general goal of engineering design is maximizing the utility of a sys-
tem or a device while simultaneously minimizing its life-cycle costs. The
latter includes the costs for developing, manufacturing, and maintaining
the system. This task is significantly complicated by the inherently non-
deterministic nature of the system and the conditions under which it oper-
ates. Since in general simultaneously maximizing the utility and minimizing
the life-cycle costs lead to a conflict, a compromise needs to be found which
balances the utility and cost over the life-time of the system. The reader
is referred to the textbook by Hazelrigg (1996) for an introduction into the
fundamentals of engineering design.
A design, which meets the goals defined above, is often labeled “robust”
and/or “reliable”. These terms have different flavors in the context of en-
gineering design. On a conceptual level, the robustness or reliability of a
system can be improved through identifying, understanding, and, if pos-
sible, eliminating basic failure mechanisms. Focusing on uncertainty, the
reliability can be associated with the probability that a failure occurs, and
optimizing for robustness can be understood as minimizing the sensitivity
of the utility and costs with respect to random variations. Identifying and
understanding failure mechanisms is at the core of traditional engineering
disciplines, such as structural and fluid mechanics. Optimizing for minimum
failure probability or maximizing the robustness of non-deterministic sys-
tems is an increasingly important field in design optimization and discussed
in this chapter in the context of topology optimization.
In recent decades, numerous optimization methods have been developed
to address the above design challenge. At the beginning of this development,
the non-deterministic, uncertain nature of engineering design problems was
often ignored and a perfect system under clearly predictable operation con-
ditions assumed. However, designs found by deterministic approaches are of-
ten sensitive to variations of system and operating parameters, and therefore
of limited value in practice. To mitigate this issue, safety factors are tradi-
tionally introduced into the formulation of the design optimization problem,
often leading to unknowingly unsafe or overly conservative designs.
Today, it is widely recognized that optimization methodologies should
account for the stochastic nature of engineering systems. These methodolo-
gies can be classified into two groups: robust design optimization (RDO)
and reliability-based design optimization (RBDO). Based on mainly deter-
ministic models, RDO methods attempt to simultaneously maximize the
deterministic performance and to minimize the sensitivity of the perfor-
mance with respect to random parameters. This approach leads to a multi-
objective optimization problem, often capturing the impact of uncertainties
only in a qualitative sense. Unlike RDO approaches, RBDO methods aim
at designing for a specific risk and target reliability level, accounting for
various sources of uncertainty in a quantitative sense. Furthermore, RBDO
approaches are based on stochastic analysis methods and, therefore, from
a theoretical and an algorithmic perspective, more challenging and com-
putationally more expensive than deterministic approaches. The reader is
referred to the books by Ang and Tang (1984), Melchers (1985), Kleiber and
Hien (1992), Ghanem and Spanos (1991), and Nikolaidis et al. (2005) for a
detailed introduction into stochastic modeling and analysis techniques. For
an overview of RBDO and RDO methods the following review papers are
recommended: Frangopol and Maute (2003), Frangopol et al. (2007), Beyer
Topology Optimization Under Uncertainty 459
2 Stochastic Modeling
A stochastic model describes the uncertainty in a system, which can be
either aleatory or epistemic. The first refers to inherent, irreducible un-
certainty, such as random variations in geometry, material properties, and
operating conditions. Epistemic uncertainty is due to a lack of knowledge
about the system, which can be reduced as more information becomes avail-
able. Depending on the type of uncertainty and the amount of information,
different modeling approaches are used, such as interval methods, fuzzy
theory, Dempster-Shafer theory, and the probability theory.
In the context of topology optimization, interval and probabilistic meth-
ods are likely the most relevant approaches and therefore we will focus on
these modeling methods. Interval methods simply require knowledge about
the upper and lower bounds of the stochastic variations of a quantity of
interest. The worst and best case scenarios for given intervals are computed
to characterize the impact of uncertainty on the system. Typically interval
arithmetic or optimization techniques are used for this purpose. Proba-
bilistic methods require significantly more information about the system in
order to construct probability distributions, such as uniform and Gaussian
distributions that describe the randomness in fields and parameters. The
complexity of developing accurate probabilistic models is often underesti-
mated, in particular the need to derive such models from sufficiently rich
data sets, i.e. observations and measurements. If there is not sufficient data
to accurately construct a probabilistic model, a stochastic analysis based
on inaccurate models might lead to coarsely incorrect results. As stochas-
tic analysis methods for probabilistic models are often significantly cheaper
460 K. Maute
slower
i ≤ si ≤ supper
i , (7)
Only in special cases, this convolution integral has a closed-form solution, for
example if the random variables rj are normal (Gaussian) and the limit state
function f is linear in the random variables. In this case, the probability of
failure Pf is:
Pf = Φ(−β), (12)
large, the number of random parameters is small, and direct sampling lead
to unacceptable computations costs.
In general, a compromise between accuracy and numerical efficiency
needs to be found. In addition, if the design optimization procedure is
driven by a gradient-based optimization algorithm, probabilistic analysis
methods need to be chosen that allow for evaluating efficiently the design
sensitivities of probabilistic design criteria. For example, first and second
order reliability methods and, more recently, stochastic projections schemes
are frequently used within design optimization schemes. The reader is re-
ferred to the studies of Hohenbichler and Rackwitz (1986), Chiralaksanakul
and Mahadevan (2005), Maute et al. (2009), Zhao et al. (2011), and Tootk-
aboni et al. (2012).
H(r) = μH + σH r, (16)
where the uncertain variable r has a standard normal distribution, i.e. zero
mean and a standard deviation of one.
The strain energy can be expressed as an explicit function of the load:
1
Π(r) = F H 2, (17)
2
where F is the coefficient of the inverse of the stiffness matrix associated
with the degree of freedom at which the load is applied. The limit state
function can be written for a given design as:
1
f (r) = Π̄ − F H 2. (18)
2
Topology Optimization Under Uncertainty 465
2 3 4
(a) PF 10 (b) PF 10 (c) PF 10
5 6 7
(a) PF 10 (b) PF 10 (c) PF 10
The most probable point (MPP) of failure is the point in the standard
normal space at which the limit state function vanishes:
1
f (r) = 0 → rM P P = (2/F )Π̄. (19)
σH
The associated probability of failure can then be easily computed by:
∞ −rM P P
Pf = h(r)dr = h(r)dr + h(r)dr = 2Φ (−rM P P ) . (20)
f <0 rM P P −∞
We need to account for the fact that the failure region is both r > rM P P
and r < −rM P P . As the probability of failure needs to be less than P̄ we
can re-write the probabilistic constraint as follows:
−1 P̄
Pf ≤ P̄ → rM P P ≥ −Φ , (21)
2
where Φ−1 is the inverse cumulative distribution function. Note Φx < 0∀0 <
x < 1.
As we have solved the reliability problem analytically, we can convert
the RBDO problem into a standard deterministic topology optimization
problem as follows:
min ρdΩ (22)
ρ
1 P̄
subject to : (2/F )Π̄ + Φ−1 ≥0 (23)
σH 2
ρlower
i ≤ ρi ≤ 1 (24)
This problem can be solved in the same way as standard deterministic prob-
lems with a mass objective and a constraint on maximum strain energy.
9
466 K. Maute
Figure 2 shows solutions for the above problem for different reliability re-
quirements, i.e. maximum feasible failure probabilities. The results were
obtained with a SIMP model using nodal densities as design variables. As
the reliability requirements increase the angle between the bars and the
thickness of the bars increase. Rozvany and Maute (2011) show that these
results are in excellent agreement with analytical results obtained for a two-
bar truss problem.
In general, the reliability problem cannot be solved analytical and nu-
merical reliability analysis methods need to be used. However, as shown,
for example, in Maute and Frangopol (2003) for a First Order Reliability
Method (FORM), the RBDO method can be integrated into the topology
optimization procedure without the need for modifying the geometry de-
scription or the overall optimization procedure of the optimization method.
a. b.
2/3 L
L
c. d.
U non-filtered/projected U filtered/projected
1 1
0 0
x x
shape variation
4 Concluding Remarks
The ability to account for uncertainty and imperfections and to increase the
robustness of the design is an important feature for any design optimization
tool. Integrating topology optimization into RDO and RBDO frameworks
is straight forward as long as the random fields and parameters are de-
sign independent. In this case and if the optimization process is driven
by gradients, RDO and RBDO approaches should be preferred which allow
computing the design sensitivities efficiently by adjoint methods.
Accounting for uncertainty in the geometry of continuous structures is
more challenging, in particular for density methods. Today’s topology meth-
ods that account for geometric uncertainty are promising. However, it is not
Topology Optimization Under Uncertainty 469
clear yet whether the stochastic models these methods are based on accu-
rately capture the shape variability. Reliability-based topology optimization
with geometric uncertainty is a large unexplored field.
5 Acknowledgment
The author acknowledges the support of the National Science Foundation
under grants CMMI–1201207 and CMMI–1235532. The opinions and con-
clusions presented in this paper are those of the authors and do not neces-
sarily reflect the views of the sponsoring organization.
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