The Generation of Arbitrary Order Curved Meshes For 3D Finite Element Analysis
The Generation of Arbitrary Order Curved Meshes For 3D Finite Element Analysis
The Generation of Arbitrary Order Curved Meshes For 3D Finite Element Analysis
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The generation of arbitrary order curved meshes for 3D finite element analysis
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Abstract
A procedure for generating curved meshes, suitable for high–order finite el-
ement analysis, is described. The strategy adopted is based upon curving
a generated initial mesh with planar edges and faces by using a linear elas-
ticity analogy. The analogy employs boundary loads that ensure that nodes
representing curved boundaries lie on the true surface. Several examples, in
both two and three dimensions, illustrate the performance of the proposed
approach, with the quality of the generated meshes being analysed in terms
of a distortion measure. The examples chosen involve geometries of par-
ticular interest to the computational fluid dynamics community, including
anisotropic meshes for complex three dimensional configurations.
Keywords: mesh generation, high–order elements, curved finite elements,
element distortion, element stretching, computational fluid dynamics
1. Introduction
The last decade has seen an increase in interest in the development of
high–order discretisation methods within the finite element community [29,
15, 11, 16]. The advantages that high–order methods bring, in terms of accu-
racy and efficiency, have been object of intensive study [17, 5] and, as higher
order approximations are considered, the effect of an appropriate boundary
representation of the domain has been identified as being critical [24, 26, 27].
The use of curved elements becomes mandatory in order to obtain the ad-
vantages of using high–order approximations [8, 1, 18, 37, 25] and the lack
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Preprint of
Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
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Preprint of
Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
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Preprint of
Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
(a) (b)
(c) (d)
Figure 1: Illustration of steps involved in the proposed method for generating high–order
curved triangular elements in two dimensions: (a) initial mesh with straight–sided el-
ements; (b) high–order nodal distribution on the straight–sided elements; (c) imposed
displacement at nodes on the curved portion of the boundary; (d) final curved high–order
mesh.
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
distribution, {ξ}p+1
k=1 ∈ [0, 1], the lengths of the subintervals [ξk , ξk+1 ] are de-
fined by
lk = ξk+1 − ξk k = 1, . . . , p
The parametric coordinates, {λk }pk=2 , of internal edge nodes are found by
using a standard root finding algorithm to solve the set
k−1
1 λk 0
Z X
|C (λ)|dλ − lj = 0 k = 2, . . . , p
L λ1 j=1
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
Figure 2: Illustration of the proposed high–order mesh generation for a surface patch: (a)
the initial linear triangular surface mesh; (b) the high–order nodal distribution constructed
on each edge of the patch boundary; (c) the high–order nodal distribution constructed on
each internal edge.
is assumed for each curved boundary. For edges on the boundary of a surface
patch, the desired nodal distribution is generated in the physical space, as
shown in Figure 2 (b). Again, an equally–spaced distribution or the Fekete
point location is normally employed. The procedure described above for
generation of edge nodal distributions in two dimensions can be directly used,
as the surface parameterisation is, when restricted to a patch boundary, just
a parametric curve in three dimensions. In this example, an equally–spaced
nodal distribution for a degree of approximation p = 3 has been selected. For
an edge that does not lie on the boundary of a surface patch, the geodesic
connecting the two edge vertices is approximated and the appropriate nodal
distribution is generated in the physical space, as shown in Figure 2 (c).
The approximation of the geodesic connecting two edge vertices x1 and
x2 is performed by iteratively constructing a list of points in the parametric
space, such that the image of these points in the boundary surface param-
eterisation S approximates the geodesic. As a first step, the parametric
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
r12
λ2 λ2 λ2
r23 r45
r13
λ5 λ5
λ1 λ3 λ1 λ3 λ1 λ3
λ4 λ4
Figure 3: Illustration of the procedure for approximating the geodesic between two points
in the parametric space: (a) selection of a point λ3 on the perpendicular bisector between
λ1 and λ2 such that equation (5) is satisfied; (b) selection of points λ4 and λ5 ; (c)
correction of λ3 using the perpendicular bisector to the segment joining λ4 and λ5 .
S(λi ) = xi i = 1, 2
Then, the point λ3 in the parametric space is determined, such that it belongs
to the perpendicular bisector of the segment connecting λ1 and λ2 , say r12 ,
and it satisfies
n o
d x1 , S(λ3 ) + d S(λ3 ), x2 = min d x1 , S(λ) + d S(λ), x2 (5)
λ∈r12
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
x2 x2
α α2
α3 α4
x1 x1
(a) (b)
Figure 4: Two triangular faces on a curved boundary showing (a) a curved edge that
forms an angle, α1 , less than the specified limit; (b) correction of the edge for a cubic
interpolation
Finally, the angle between neighboring edges is checked to ensure that the
minimum angle is not less than a specified lower limit. For each edge on a
curved boundary, e.g. the edge connecting vertices x1 and x2 in Figure 4 (a),
the angles, αi for i = 1, . . . , 4, between this edge and its neighboring edges
are measured. If any angle αi is less than the specified lower limit, a new
edge connecting the vertices x1 and x2 is defined, in such a way that the
four angles αi for i = 1, . . . , 4 are acceptable. For instance, if a degree of
approximation p = 3 is considered, the new edge is defined by a cubic curve,
that contains the two vertices x1 and x2 , with the derivative at the initial
and final vertices imposed in such a way that the minimum angle between
edges is not less than the specified limit, as shown in Figure 4 (b). For a
degree of approximation p = 4, the curvature on one edge vertex is also
imposed and, for a degree of approximation p = 5, the curvature at both
edge vertices is icluded. Extra conditions can be devised when using higher
degrees of approximation, but these are not considered here.
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
η η
γ =γ k
γ =γ k ξk ξk
α = αk ξ α = αk ξ
β =βk β =βk
(a) (b)
Figure 5: Reference triangle showing the barycentric coordinates of an internal point for
(a) an equally–spaced nodal distribution for p = 3; (b) a Fekete-nodal distribution for
p=5
the same as that for the corresponding node in the reference element.
The procedure for placing high–order interior nodes for triangular faces
is considered first. A p-th degree nodal distribution is defined on a reference
element, e.g. Figures 5 (a) and (b) show an equally–spaced nodal distri-
bution of degree p = 3 and a Fekete–nodal distribution of degree p = 5
respectively in the reference triangle. The barycentric coordinates, denoted
by (α, β, γ), for an internal node with local coordinates ξ k = (ξk , ηk ), are
given by (αk , βk , γk ) = (1 − ξk − ηk , ξk , ηk ), as illustrated in Figure 5. Note
that the barycentric isolines intersect the boundary of the reference element
at the location of the nodes, if the nodal distribution is equally–spaced, as
shown in Figure 5 (a). However, this is not the case in general, as shown in
Figure 5 (b) for the case of a Fekete nodal distribution.
The barycentric coordinates of an internal node can be used to define
three geodesics, g α , g β and g γ , in the physical space. The geodesic, g α , joins
the points xα12 and xα13 , where xαIJ lies on the geodesic, gIJ , connecting the
vertices xI and xJ . In this case,
dS (xI , xαIJ ) = (1 − αk )dS (xI , xJ )
where dS denotes the distance function over the surface parametrised by S,
i.e. the distance along the geodesic between two points. Similarly, g β joins
the points xβ21 and xβ23 , where xβIJ ∈ gIJ and
dS (xI , xβIJ ) = (1 − βk )dS (xI , xJ )
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
x1
xγ31
xβ21 gβ gγ xα
13
gα
xα
12 x3
xβ23
x2 xγ32
(a) (b)
Figure 6: Surface patch showing (a) a curved face; (b) the three geodesics used to determine
the position of the internal node.
and g γ joins the points xγ31 and xγ32 , where xγIJ ∈ gIJ and
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
ξk η= ηk
ξ = ξk
ξ
Figure 7: Reference square, with a Fekete-nodal distribution, for p = 5 and the local
coordinates for an interior node
g ξ , joins the points xξ12 and xξ43 , where xξIJ ∈ gIJ , and
Similarly, the geodesic g η joins the points xη23 and xη14 , where xηIJ ∈ gIJ , and
which is the projection over the true surface of the average position of xξ
and xη .
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
x4
η
x1 x14
ξ
gη x43
gξ
ξ
x12 x3
η
x2 x23
(a) (b)
Figure 8: Surface patch showing (a) a curved quadrilateral face; (b) the two geodesics
used to determine the position of the internal node.
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
5. Implementation Examples
Several examples, in both two and three dimensions, are considered to
illustrate the potential of the proposed methodology. The examples that
have been selected are of particular interest to the aerospace community, as
they involve isotropic and anisotropic high–order curved meshes suitable for
the computation of external flows around aerodynamic shapes.
The elastic parameter values E = 10 and ν = 0.4 are used for all the
examples. It has been found that this combination allows the minimum
scaled Jacobian to be maximised and, at the same time, allows the con-
dition number of the linear system to be minimised. The scaled Jacobian
is found to be independent of the value of E and highly dependent on ν.
Since the scaled Jacobian is a measure of the volumetric deformation, it is
expected that, with lower values of ν, compressibility of the material will re-
sult in highly distorted elements near curved boundaries. The best element
quality is expected for values of ν approaching the incompressible limit, as
the imposed boundary displacement then propagates into the computational
domain. This behaviour is confirmed in Figure 9 (a), which illustrates the
influence of the elastic parameters on the scaled Jacobian. Figure 9 (b) shows
that the condition number of the system matrix deteriorates rapidly as the
value of E is increased. Values of ν between 0 and 0.4 are found to have
little effect. For higher values of ν, i.e when the material approaches the in-
compressible limit, the effect is more pronounced and the condition number
of the system matrix also deteriorates. Note that the exact nature of the
plots shown in Figure 9 will depend upon the geometry under consideration.
However, it is important to note that the same qualitative behavior has been
consistently observed in practice for a number of different geometries.
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
(a) (b)
Figure 9: Influence of the elastic material parameters on (a) the scaled Jacobian; (b) the
logarithm of the condition number of the system matrix.
vicinity of the leading edge of the aerofoil is shown in Figure 10 (c). This
detail shows some of the curved elements and the high–order Fekete-nodal
distribution on each element. The mesh has 370 vertices, 650 elements and
50 edges on the curved boundary. After introducing a high–order nodal
distribution, appropriate for an approximation of degree p = 5 over each
element, the resulting high–order mesh has 8 350 nodes. An example of
a high–order curved quadrilateral mesh generated for this configuration is
given in Figure 11.
Figure 12 (a) displays a histogram of the scaled Jacobian, I, for the
high–order triangular mesh. It shows the percentage of elements for a given
scaled Jacobian in intervals of 0.05. For this simple isotropic case, 99% of
the elements are such that I > 0.95 and the minimum value of the scaled
Jacobian is 0.83.
To illustrate the optimal properties of meshes generated in this fashion,
the interpolation error estimate for a smooth function is checked, on a se-
ries of curved triangular and quadrilateral high–order meshes with degree
ranging from p = 1 up to p = 7. In order to measure the interpolation
error, the nodal values of the solution are set by using the smooth function
f (x, y) = x cos(y)+y sin(x). Then, the error between the approximated solu-
tion, interpolated from the nodal values, and the exact solution is computed
at each quadrature point in order to compute the L2 (Ω) error. Figure 12 (b)
shows the evolution of this error, as a function of the square root of the
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
Figure 10: NACA0012 aerofoil showing (a) an initial mesh with straight–sided triangular
elements; (b) a curved high–order mesh; (c) a detail of the curved mesh near the leading
edge with a high–order Fekete-nodal distribution on each element.
(a) (b)
Figure 11: NACA0012 aerofoil: detail near the leading edge showing (a) the initial mesh
with straight–sided quadrilateral elements and (b) the curved high–order mesh
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
100 -1
Triangles
-2 Quadrilaterals
80 -3
60 -5
-6
10
40 -7
-8
20
-9
-10
0 0 50 100 150 200
0 0.2 0.4 0.6 0.8 1
I n1/2
dof
(a) (b)
Figure 12: NACA0012 aerofoil showing (a) the scaled Jacobian; (b) an illustration of the
optimality of the mesh for finite element analysis.
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
Figure 13: Isotropic curved high–order surface mesh for a generic Falcon aircraft.
(a) (b)
Figure 14: Isotropic mesh for a Falcon aircraft showing (a) a detail of a view of the surface
mesh with the nodal distribution near the engine intake; (b) a detail of a cut through the
interior volume mesh.
100
80
Percentage of elements
60
40
20
0
0 0.2 0.4 0.6 0.8 1
I
Figure 15: Scaled Jacobian for the generic Falcon isotropic mesh
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
(a) (b)
Figure 16: Anisotropic mesh for a generic Falcon aircraft showing the form of the mesh
(a) near the engine intake; (b) near the wing tip.
which I < 0.5. As might be expected, these elements are located in critical
regions of the mesh, such as the leading edge of the wings and the engine
intake. The low quality of these elements can be mainly attributed to the low
resolution provided by the cubic approximation when attempting to capture
the large deformations in regions with high curvature. It is worth recalling
that, although the linear elastic model has been selected for its efficiency,
cases such as this can violate the small deformation hypothesis inherent in
the linear model. Refining the initial linear mesh, or increasing the degree
of the approximation employed, may alleviate this problem, especially in the
vicinity of regions with high curvature.
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
100 100
80 80
Percentage of elements
Percentage of elements
60 60
40 40
20 20
0 0
0 100 200 300 400 0 0.2 0.4 0.6 0.8 1
Stretching I
Figure 17: Anisotropic mesh for a generic Falcon aircraft showing (a) the stretching; (b)
the scaled Jacobian.
elements in the mesh are shown in Figure 17 (b). For this complex configura-
tion, more than 91% of the elements have a scaled Jacobian value I > 0.95.
However, the minimum value of I is now 0.08 and 6 938 elements are such
that I < 0.5.
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
Figure 18: Anisotropic mesh generation for the F6 configuration showing a detail of the
surface mesh near the engine intake and the leading edge of the wing.
Figure 19: Anisotropic mesh generation for the F6 configuration showing views of a cut
through the volume mesh near the engine intake and near the leading edge of the wing.
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Z. Q. Xie, R. Sevilla, O. Hassan and K. Morgan
The generation of arbitrary order curved meshes for 3D finite element analysis
Computational Mechanics, 51 (3); 361-374, 2013
100 100
80 80
Percentage of elements
Percentage of elements
60 60
40 40
20 20
0 0
0 50 100 150 200 250 300 350 0 0.2 0.4 0.6 0.8 1
Stretching I
6. Conclusions
An a posteriori strategy for obtaining high–order curved meshes, suit-
able for finite element analysis in both two and three dimensions, has been
described. The method is based on deforming an initial mesh with planar
faces and edges using a linear elasticity model. The proposed methodology
is valid for any element topology and hybrid meshes, containing different
types of element, can be handled. Special attention has been paid to the
construction of high–order nodal distributions on edges and faces on curved
boundaries. The quality of the resulting meshes was analysed in terms of the
scaled Jacobian, which is a standard distortion measure for curved elements.
Several examples, involving geometries of complex shape in both two and
three dimensions, have been considered to demonstrate the potential of the
proposed methodology. Special emphasis has been placed on constructing
high–order curved meshes for geometries that are of particular interest to the
aerospace community. In this area, anisotropic meshes suitable for analysing
viscous flow over two complex aircraft configurations have been presented.
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