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MCWASP IOP Publishing
IOP Conf. Series: Materials Science and Engineering 84 (2015) 012036 doi:10.1088/1757-899X/84/1/012036

Application of Numerical Optimization to Aluminum Alloy

Wheel Casting

J Duan1, C Reilly1, D M Maijer1, S L Cockcroft1 and A B Phillion2

1
The Department of Materials Engineering, The University of British Columbia,
Vancouver, BC, Canada V6T 1Z4
2
School of Engineering, The University of British Columbia, Kelowna, BC, V1V 1V7

Abstract. A method of numerically optimizing the cooling conditions in a low-

pressure die casting process from the standpoint of maintaining good directional
solidification, high cooling rates and reduced cycle times has been developed for the
production of aluminum alloy wheels. The method focuses on the optimization of
cooling channel timing and utilizes an open source numerical optimization algorithm
coupled with an experimentally validated, ABAQUS-based, heat transfer model of the
casting process. Key features of the method include: 1) carefully designed constraint
functions to ensure directional solidification along the centerline of the wheel; and 2)
carefully formulated objective functions to maximize cooling rate. The method has
been implemented on a prototype production die and the results have been tested with
plant trial test.

1. Introduction
The aluminum wheel industry is very competitive and manufacturers are constantly under pressure to
improve quality and to reduce cost. This is achieved by increasing production rates while attempting to
also maintain or reduce scrap rates. One of the defects that is a significant challenge to manufacturers
is shrinkage porosity. Shrinkage porosity occurs when liquid or semi-solid metal is cut off from a
source of liquid metal, which is needed to compensate for the volume change associated with
solidification (the volume change is in the range of 5 to 7% depending on the aluminum alloy). In the
case of wheel casting, achieving controlled progress of the solidification front, or “directional
solidification”, is the key to avoiding shrinkage porosity. As illustrated in Figure 1, good directional
solidification is achieved when solidification starts high in the inboard-rim flange, progresses down
through the rim, and then across the spokes, ending just below the hub in the sprue. To achieve this
solidification pattern, both proper die design and careful management of the cooling conditions within
a cast cycle are required.
Historically, the design of a die and the associated operational parameters, were determined based on a
combination of experience and trial-and-error optimization. The methodology typically involves long
design lead times, prototyping, several preproduction trials and can result in high scrap rates and less
than optimum production rates. The move to adopt water-cooling in place of air cooling in order to
reduce cycle times, improve productivity and improve wheel fatigue performance represents a
significant challenge to the foundry engineer as there is a limited experiential base to draw on. This
manuscript describes a methodology in which the trial-and-error optimization of the operational
parameters (die cooling timing) is eliminated from the preproduction trial process. It combines a
casting process model with an advanced optimization algorithm that is applied to achieve the optimum
set of operating conditions that produce a combination of minimum cycle time and limited or no
shrinkage porosity in a given die design.

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MCWASP IOP Publishing
IOP Conf. Series: Materials Science and Engineering 84 (2015) 012036 doi:10.1088/1757-899X/84/1/012036

Figure 1. Illustration of desired solidification direction in wheel

2. Background
Numerical design optimization provides a systematized and versatile procedure for arriving at a design
solution. It can offer a reduction in design time, removes experiential biases from the design process
and virtually always yields some design improvement [1]. Numerical optimization first found
application in structural design [2, 3] and is now increasingly applied in other areas such as, cast
product and casting process design [4-7].
The numerical optimization process is essentially a procedure for finding the minimum or maximum
of an objective function, which mathematically describes some attribute of the process or product
being optimized. Additionally, constraints are normally added to account for additional limitations
imposed on the design. The general mathematical expressions of the objective function and the
constraint functions are presented in Equations (1) and (2).
max (or min) f(x) x = (x1, x2, … , xn) (1)  

subject to gi(x) = ci for i = 1, … , n (2)  

hj(x) ≤ dj for j = 1, … , m

where x is a vector containing all the design variables that may be modified in order to achieve the
optimization goal, f(x) is the objective function, and gi(x) and hj(x) are equality and inequality
constraints, respectively.
The objective and constraint functions are generally evaluated based on the results of a numerical
model. In this work, a comprehensive and experimentally validated model of a low pressure die
casting (LPDC) process for the production of A356 automotive wheels has been used. Details related
to the casting process model have been discussed previously in the literature [8, 9]. The current work
presents the application of the optimization algorithm using two versions of the casting process model
(see Sections 0 and 0) in order to highlight both the utility of the optimization algorithm and the
importance of the process boundary conditions in the model utilized within the optimization algorithm.

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MCWASP IOP Publishing
IOP Conf. Series: Materials Science and Engineering 84 (2015) 012036 doi:10.1088/1757-899X/84/1/012036

3. Architecture of the optimization tool

Figure 2 shows the overall structure of the optimization algorithm or “tool”, which consists of three
major modules. The first one is the casting process model, which has been developed in the
commercial finite-element package ABAQUSTM. The casting process model simulates the casting
process and calculates the thermal field within the wheel and die. Due to the large thermal mass of the
die, each change in cooling timing requires several casting cycles for the full effect of the change to
propagate through the die and for a new cyclic steady-state condition to be achieved. Once the new
steady state condition is achieved, the necessary data is obtained from the ABAQUS output file and
processed with the analysis module developed in Python. Specifically, the analysis module evaluates
the objective and the constraint functions and periodically, based on a set of criteria, evaluates the
sensitivity of these functions to the design variables. This information is then passed to the optimizer.
The optimizer is based on an open-source Python package (scipy.optimize), which has a FORTRAN-
based core wrapped in Python. The optimizer determines if the design criteria, subject to the imposed
constraints has been reached. If the criteria are not met, it then determines how to change the design
variables, which are then passed back to the casting process model. If met, the optimal design
parameters are output. The whole process iterates until an optimum solution is reached or until the
user-defined maximum number of iterations is reached.

Figure 2. Architecture of the optimization tool

The strategy used to move from one iteration to the next in the design variable space distinguishes one
algorithm from another. The package utilized in this study uses the line search method. The line search
approach first finds a descent direction along which the objective function is reduced and then
computes a step length that determines how far the update should move along that direction. In the
line search method, the design variables are updated as follows [1]:

𝒙!!! = 𝒙! + α𝑆   (3)

where 𝑥! is the design at the kth iteration and 𝑥!!! is the new design at the (k+1)th iteration, 𝑆 is the
search direction, and α is the step length.
The descent direction is determined using the quasi-Newton method in the current algorithm. The
Quasi-Newton method is reported to have better convergence behavior than the Steepest Descent
method, and it is more efficient than Newton’s method, as it does not require information on second-
order derivatives. Detailed discussion of these three methods can be found in reference [1]. The step
length can be determined either exactly, by finding the minima of the objective function in the given
direction 𝑆, or inexactly, by finding the value of α that leads to an acceptable descent amount.
4. Application to a 2D Axisymmetric Prototype Die Model
The die structure used in the application presented in this paper is proprietary and cannot be disclosed.
Hence, only the wheel and the approximate locations of the water-cooling channels are shown in

TM
ABAQUS is a trademark of Dassault Systèms

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MCWASP IOP Publishing
IOP Conf. Series: Materials Science and Engineering 84 (2015) 012036 doi:10.1088/1757-899X/84/1/012036

Figure 3. There are seven cooling channels in the die design: three in the top die (TD_CC_1 ~ 3), two
in the side die (SD_CC_1 ~2), and two in the bottom die (BC_CC_1 ~ 2).

Figure 3. Cross-section of the wheel geometry with approximate locations of the seven cooling
channels and showing the locations used for evaluation of the objective (P1-6) and the constraint
(white dots) functions.

The design optimization strategy is based on a combination of maximizing the cooling rate in the
wheel while achieving directional solidification. The former is desired to both increase productivity
(reduce the overall cycle time) and produce a finer structure with better overall fatigue performance.
The latter is needed to eliminate/reduce shrinkage porosity, which, if present, would negate the
positive benefits of a reduced cycle time on fatigue performance.
The objective and constraint functions applied in the present work are given in Equations (4-5).
!
 
𝑓 𝒙 = 𝑎! ×(−  𝑇!!"#!$#"%&' )  
!!!   (4)

ℎ!! 𝒙 = 𝑡!!!
!"!℃
−   𝑡!!"!℃   ≥ 0  ,      𝑗 = 1,2,3, … 35 (5)

ℎ! 𝒙 = (𝑡!"#$%#& −   𝑡!"#$%$&$'()$"* )/𝑡!"#$%#&   ≥ 0 (6)

where f(x) is the objective function, x is a vector containing the design variables (the on and off
timings of the 7 water cooling channels), ai are constants used to adjust the relative weights of the
objective function, 𝑇!!"#!$#"%&' is the calculated average cooling rate within the solidification range at
the ith point, ℎ!! 𝑥 is the first type of constraint imposed at the jth constraint point, ℎ! 𝑥 is a second
the type of constraint function, 𝑡!"#$%#& is the time the die opens, and 𝑡!"#$%$&$'()$"* is the time taken
for the cast to solidify to the middle of the hub (a solid shell forms at this time to maintain the wheel
structure).
Careful selection of the objective and constraint functions is crucial to obtaining good results. In
Equation (4), the cooling rate is maximized when the optimization algorithm minimizes f(x). The
calculated cooling rates are assessed at six points distributed at different regions within the wheel,
indicated by points P1 through P6 in Figure 3. The first type of equality/inequality constraints are
based on the time to reach 575°C, which is the eutectic temperature of A356 [10]. As shown in
Equation (5), the constraints are formulated to require that the difference in time between two adjacent
points, j+1 and j, to reach 575°C is greater than or equal to zero. The constraint functions are
evaluated at 36 points, which are shown as the white dots in Figure 3. j = 1 is located at the top of the
inboard rim flange, the first place to solidify, and j = 36 is located at bottom of the hub, the preferred
last place to solidify. Referring back to Equation 5, the j+1 point is located further along the path than

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MCWASP IOP Publishing
IOP Conf. Series: Materials Science and Engineering 84 (2015) 012036 doi:10.1088/1757-899X/84/1/012036

the jth point in the desired direction for progress of the solidification front – i.e. upstream in the
direction of the compensatory flow necessary to feed solidification shrinkage. This constraint is
utilized to ensure directional solidification.
A second type of equality/inequality constraint is applied that requires the wheel to be solidified
before the die opens (Equation (6)). The solidification requirement constraint is normalized to bring it
into the same magnitude range as the other constraints as numerical difficulties often arise when one
constraint function is of a different magnitude, or changes more rapidly, than the others and dominates
the optimization process [1].

4.1 First Application – Base-Case Version of the Model

The optimization tool was first applied to a 2-D axisymmetric base-case version of the casting process
model. The large computational overhead associated with repetitively running the thermal model
necessitated that the initial application be based on a 2-D axisymmetric version of the LPDC process
model in order to minimize computational time during development of the optimizer. The model
includes the different stages in the casting process – i.e. die filling, wheel cooling and solidification,
die open, wheel removal and die closing. A few additional features are listed below:

1. The wheel and die material properties are temperature dependent;

2. Die filling is not simulated however, the interfacial cooling between the die and the wheel is
activated based on the expected filling sequence, to approximate the effect of filling on heat
transfer;
3. The initial distribution of temperature in the wheel is assumed to vary from 620 °C at the top
of the rim to 700 °C at the inlet from the sprue (this is done to approximate the heat loss from
the wheel to the die as the liquid metal is filling the die cavity);
4. The die/wheel interface boundary conditions vary with temperature to account for gap
formation during solidification and are also different for the different die sections – top, side
and bottom dies – to account for differences in the evolution of the gap at the interface;
5. The heat transfer in the water cooling channels is assumed to be constant when on and is
described using a single-phase convection heat transfer coefficient; and
6. Each model is capable of being run in a cyclic mode – i.e. the initial condition for the die at
the beginning of each cycle uses the results from the previous one – and therefore can obtain
the cyclic steady state condition for a given set of process cooling conditions.

The results showing the solidification patterns obtained for two sets of cooling timings are shown in
Figure 4. Each row of images shows color contours of the temperature on a cross-sectional slice of the
wheel at different times in the cycle for a given set of cooling conditions. The 572 and 575 °C
isotherms are added on each contour to delineate the lines representing approximately 75 and 50%
solid fraction for A356, respectively. The top row shows the results using the initial cooling times,
which are based on trial-and-error optimization using the model. As can be seen at 49s there is a
region of potential liquid/semi-solid encapsulation in the top of the rim indicated by the 50% fraction
solid contour, which could result in solidification shrinkage. This is a well-known problem area in the
water-cooled version of the LPDC process (note: this is occurring in an area that will be accurately
described using the 2-D axisymmetric model). There is however, no liquid encapsulation predicted in
the rim-spoke junction (73s), which is also an area prone to solidification shrinkage [9,11]. The results
obtained with the numerical optimizer are presented in the lower row of images. As can be seen, these
results show the elimination, or reduction, in the tendency to form shrinkage porosity in the top of the
rim while maintaining similar solidification time.

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MCWASP IOP Publishing
IOP Conf. Series: Materials Science and Engineering 84 (2015) 012036 doi:10.1088/1757-899X/84/1/012036

Figure 4. The initial (top) and the numerically optimized (bottom) solidification sequence in a water-
cooled die.

Based on the optimized cooling timing, a casting trial was run at a commercial wheel plant (note: the
water-cooling timing actually used in the industrial casting trial was further refined using a 3-D
model of the LPDC casting process to address limitations in the 2-D axi-symmetric model). The
results from this trial revealed fine, distributed, shrinkage porosity persistent in the upper rim – see
Figure 5.

Figure 5. X-ray image showing distributed shrinkage porosity in the top rim at the top of the wheel
rim

4.2 Second Application – Updated Wheel/Die Interface and Water Channel Boundary Conditions
To better understand why the industrial trial failed to produce a shrinkage free wheel, the thermal
model used in the optimization tool was re-examined to assess the veracity of some of the key
boundary conditions. On going work at UBC has revealed that: 1) the behaviour of the wheel die
interface during a casting cycle is complex with gap formation in some regions, the development of
pressure (no gap) in others and both gap formation followed by gap closure and pressure development
in still others[12]; and 2) the heat transfer behaviour in the cooling channels is also complex exhibiting

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MCWASP IOP Publishing
IOP Conf. Series: Materials Science and Engineering 84 (2015) 012036 doi:10.1088/1757-899X/84/1/012036

transient periods of two phase boiling channel flow when the water first enters the channel and again
when the flow is terminated to the channel [13]. The results of re-running the model with improved
boundary conditions are presented in Figure 6. The upper set of images in Figure 6 (labeled “initial”)
shows the results obtained with the previously optimized timing applied in the model with the updated
boundary conditions. The lower image shows the results of re-running the optimizer to obtain a new
optimum (labeled “optimized”). Focusing first on the initial results, the model now predicts shrinkage
porosity in the upper rim consistent with what was observed in the industrial trial – i.e. the
encapsulation of the 75% solid fraction isotherm would lead to distributed, relatively fine, shrinkage
porosity. Note: some shrinkage was also predicted in the spoke approximately mid-way between the
rim and the hub, which was not observed in the wheel. This is an artefact of the 2D axi-symmetric
analysis, as the shrinkage is not predicted using a 3D version of the model. Thus, the initial boundary
conditions lacked the sophistication needed for the optimizer to reach a valid solution. Turning to the
optimized results, re-running the optimizer failed to produce a solution that was free of encapsulation /
shrinkage in the upper rim owing to the elongated 75% solid fraction isotherm; however, it did reduce
the severity of the encapsulation. The prediction of mid-spoke shrinkage was also eliminated. The
conclusion from this result was that the current design of the dies from the standpoint of size and
placement of the cooling channels in the vicinity of the top rim would not yield a wheel free of
porosity without the addition of a riser to the top of the rim.

Figure 6. The initial (the upper row) and the optimized solidification conditions (the lower row) for
the model run with the updated boundary conditions.

5. Conclusions
This paper discusses the development and the application of an optimization methodology to an LPDC
process used for the commercial production of A356 automotive wheels. The optimization
methodology has been successfully applied to optimize the on and off cooling timings of up to 7
water-cooling channels both from the standpoint of eliminating shrinkage based porosity associated
with encapsulation of liquid and reducing the overall solidification time (process cycle time). The
initial application of the results of the optimizer in an industrial setting revealed an inconsistency in
the prediction of porosity in the upper rim area with what was observed in the cast wheel. One of the
primary reasons for the inconsistency was the overly simplified boundary conditions used in the
thermal model that was utilized within the optimizer. However, reapplication of the optimizer on a

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MCWASP IOP Publishing
IOP Conf. Series: Materials Science and Engineering 84 (2015) 012036 doi:10.1088/1757-899X/84/1/012036

model with updated boundary conditions revealed limitations in the underlying design of the dies with
respect to the placement and size of some of the cooling channels. A key and perhaps obvious finding
of the study is that the utility of numerical optimizers applied to casting processes in an industrial
setting hinges critically on an accurate description of the process conditions. Looking forward, design
of dies for the production of wheels will ultimately require the development and application of
topological optimization algorithms in conjunction with process timing optimization, a formidable
computational challenge.

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