Proceedings of ASME:
28th Design Automation Conference
September 29 – October 2, 2002, Montreal, Canada
DETC2002/DAC-####
GA-BASED MULTI-MATERIAL 3D STRUCTURAL OPTIMIZATION
USING STEPWISE MESH REFINEMENT
Jacob Y. Neal
Mechanical Engineering
Clemson University
Clemson, SC 29634-0921, USA
Email: jacobn@clemson.edu
Vincent Y. Blouin
Mechanical Engineering
Clemson University
Clemson, SC 29634-0921, USA
Email: vblouin@clemson.edu
Georges M. Fadel
Mechanical Engineering
Clemson University
Clemson, SC 29634-0921, USA
Email: gfadel@clemson.edu
ABSTRACT
Optimally designed multi-material structures offer the
potential for increased functionality. The present paper
describes a modeling and optimization procedure based on the
finite element method (FEM) combined with an evolutionary
genetic algorithm (GA). The GA offers the possibility of finding
the global optimum in a multi-modal design space. This
advantage, however, is counterbalanced by the high
computational expense of utilizing many FEM evaluations, as is
often required in structural optimization. Furthermore, in the
context of multi-material optimization, the large number of
material possibilities for each finite element can render the
conventional GA-based optimization prohibitively costly in
computational time and inconclusive. In this paper, a stepwise
mesh refinement technique is presented. Coupled to the GA
using a multi-objective function, the method is shown to have a
significantly lower computational time and leads to satisfactory
designs of heterogeneous objects of arbitrary shapes. Design
issues related to the use of this method are discussed and
exemplified with the design of a three-dimensional
heterogeneous connector.
INTRODUCTION
Computer Aided Design and Manufacturing (CAD/CAM)
packages have become industry standards. These tools coupled
with advances in rapid prototyping technologies have reduced
the need for costly machining operations that were previously
required to translate the digital domain into the physical. To
fully benefit from these advances in manufacturing, design
engineers need tools for optimizing material composition as
well as topology. The presence of composite materials in
everyday products highlights this increased complexity.
Production methods such as Computer Numerically Controlled
(CNC) machine tools, three-dimensional printers and Rapid
Prototyping (RP) have drastically decreased time to market and
overhead cost related to production scale up and tooling. These
developments have made it possible to create gradient and
discrete material distributions from a variety of metals [14].
However, while much work has been done in the field of
topology optimization, design tools specifically created for
multi-material applications are still lacking. The present
research is directed toward this engineering void. Utilizing the
evolutionary genetic algorithm (GA) optimization method, a
stepwise mesh refinement technique is developed to optimize
material distribution in objects of arbitrary shapes.
During the GA process, a diversified population of
solutions is created which samples the entire design space,
increasing the probability of arriving at a global optimum [5].
This characteristic makes GAs particularly useful in the
optimization of multi-modal problems that cannot be solved
with conventional gradient-based methods. Examples of
topology optimization using homogeneous material have been
covered in the work of Beckers et al. [2], who used a dual
method and Kim et al. [10] who used fixed grid FEM in
evolutionary optimization. Chapman et al. [3] used onedimensional binary string chromosomes with one gene per FEM
element to map the design space, which is split into quadrants
using a hierarchical subdivision method. The fitness function is
based on a simple stiffness-to-weight ratio and a connectivity
analysis is used to remove any checkerboard (alternating void
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and solid zones) pattern. Annicchiarico and Cerrolaza [1]
utilized GA in conjunction with geometric modeling programs
and B-spline surfaces for three-dimensional shape optimization.
Eby et al. [4] conducted research using injection island GA
(iiGA), which employs multiple fitness functions and
populations (or islands) to find shape variations that increase
the specific energy density for elastic flywheels. Computational
expense is reduced by first evaluating low refinement meshes
and “injecting” the results into a more refined population.
Although GA is a very powerful optimization tool, many
other methods exist and should be briefly mentioned. Beckers
[2] used the Dual Method for topology optimization of
continuous structures in static linear elasticity. A predetermined
design domain with a fixed FEM mesh was utilized to
efficiently solve problems with few constraints and many design
variables. The “checkerboard” phenomenon is overcome by
using a perimeter filter method to introduce a global constraint
that acts only on the void-material interfaces. An evolutionary
structural optimization (ESO) scheme similar to GA was
developed by Kim et al. [10]. Using fixed grid FEM, the ESO
process removes the least stressed elements from the model,
thereby creating a more fully stressed design.
Other research efforts involving multi-material applications
include: Kumar and Dutta [12] who proposed a solid modeling
scheme for materially graded objects, Kumar and Wood [12
bis], who defined the material distribution at the nodes of the
finite element model and the corresponding shape functions,
and Huang and Fadel [8] who presented a one-dimensional
parametric representation and optimization process of a
heterogeneous flywheel. A three-dimensional parametric
modeling and design approach for arbitrary heterogeneous
objects using B-splines was given by Qian and Dutta [16]. Most
research using GA to conduct multi-material optimization has
been used in the optimization of laminate structures, examples
include Punch et al. [15], Grosset et al. [7], Goodman et al. [6]
and Malott et al. [13].
In general, modeling methods geared towards developing
design tools for multi-material structures can be grouped into
gradient and discrete distribution methods. Gradient structures
are modeled as having gradual boundaries between materials
while discrete compositions have sharp separations. Both
approaches seek to encompass material and shape data. The
research presented in this paper uses a discrete method to create
heterogeneous models composed of many homogeneous
isotropic finite elements. The research done by Jackson et al.
[9] outlines a gradient approach created to use Solid Freeform
Fabrication (SFF), a manufacturing process that uses layered
addition to build objects. A discrete approach was proposed by
Koenig [] who worked toward developing optimization tools
that determine the best material distributions in twodimensional multi-material structures with multiple objectives
and set constraints. The preceding research examples address
the need to model part geometry, topology and composition.
In this paper, the development of a robust stepwise mesh
refinement technique is explained and the resulting significant
decreases in computational expense are examined. The method
assigns each homogeneous element of the finite element model
a random material property. A GA is used to optimize the initial
coarse structure and the information is incorporated into the
following runs to increase the accuracy while lowering overall
runtimes.
The previous section has outlined many research areas that
pertain to the work done in multi-material structural
optimization using GA, FEM and mesh subdivision techniques.
The review has served the purpose of validating our approach
as well as providing insight into other topics. The details and
results of the stepwise mesh refinement technique is explained
in the following sections
2. METHODOLOGY
2.1. Genetic Algorithm
Developed by Holland [] and later refined by Goldberg [],
GA is an adaptive method of solving search and optimization
problems utilizing the biological principal of natural selection.
Each solution in the design space is represented by a
chromosome, which is composed of a finite number of genes.
The genes can have a range of values depending on the type of
chromosome, i.e. binary, integer, float and character. In this
research the gene values are chosen from a predefined allele set
of integer values representing sets of material properties. To
thoroughly sample the design domain an initial random
population is created. The quality of the individuals (or
chromosomes) is calculated using a fitness function, which if
coded properly will favor individuals that have suitable
phenotypes (physical appearances). Individuals deemed most fit
are given a higher probability for reproduction, causing the
population as a whole to “evolve” towards the global optimum.
The primary reproductive operators are crossover and mutation.
These operators use selected “parent” chromosomes to create
“offspring.” This “survival of the fittest” type approach makes
GA a compromise between gradient or “strong” methods, which
rapidly seek optima in an informed manor and random or
“weak” methods, which are computationally expensive but
often times better at finding global optima [Goldberg].
This research uses a C++ library of GA functions (GAlib)
written by Wall [] at MIT. The GAlib function
GASteadyStateGA is used to control the GA process, this
function uses the standard GA setup mentioned above with the
addition of an overlapping population. A temporary population
is created at each generation, then this population is added to
the original and the worst individuals are removed. The percent
overlap, crossover and mutation are set for the evolution. The
GA2DArrayAlleleGenome operator contains the genome data,
with a predefined allele set of integer values available for each
gene in the genome.
2.2. Finite Element Analysis
Optimization of material distribution within the design
domain is accomplished by first creating a homogeneous model.
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The geometry is generated and meshed using the IDEAS-SDRC
FEM software package. The FEM mesh is composed of fournode tetrahedral elements, which were selected for their ability
to mesh complex three-dimensional geometries and the high
element-to-node ratio. The lower number of nodes allows for a
more effective bandwidth reduction prior to solving. The model
data is then processed with ABAQUS to generate the element
stiffness matrices for each material used in the model. The
ABAQUS results can also be utilized for error analysis. The
percent error for the displacement magnitude at the loaded
nodes was found to be only 0.0002% as compared to the
ABAQUS results. [before that, need to talk about the inversion
and the postprocessing of the matrices] The stiffness matrices
and model data are then used to do displacement analyses of the
individuals generated by the GA.
A C++ FEM code was written to carry out the displacement
analysis. This code is optimized to decrease model runtimes and
all applicable data is preprocessed before the GA begins.
During the optimization process the material distribution for
each individual in the population is mapped into FEM. This is
accomplished by combining the preprocessed element stiffness
matrices (for the set of materials) to create a heterogeneous
global stiffness matrix that represents the genome’s material
distribution. Loads and constraints are then applied to the
global stiffness matrix prior to bandwidth reduction. The
reduced matrix is then solved using gauss elimination. Runtime
for a single 1479 element model analysis [does it include
assembly, bandwidth reduction?] is approximately 300
milliseconds on a Dual Rack Onyx2 Infinite Reality Silicon
Graphics system.
As mentioned earlier, FEA is advantageous over parametric
representation because of the relative simplicity of analyzing
structural responses such as static deformation and stresses. The
disadvantage, however, is its discrete nature, which requires the
material distribution to be discretized in space by specifying the
material properties either at the nodes [12 bis] or for each
element. The advantage of specifying the material properties for
each element is that effective properties are not needed since
each element is made of 100 percent of a given material. The
discrete nature of this representation, however, may not capture
the real behavior of the corresponding functionally gradient
material structure. This issue, which is related to the coarseness
of the mesh, introduces inaccuracies. In this research, however,
it is assumed that the comparative nature of GA allows for this
type of inaccuracy. In other words, the optimum found using
inaccurate analyses is assumed to be same as the one that would
be found using accurate analyses. With this in mind, the finite
element representation is justified.
2.3. Fitness Function
During the optimization process the GA attempts to
maximize the fitness of the overall population. Each individual
is evaluated according to the same fitness function. The
structure’s deformation at the loaded nodes and total weight are
simultaneously minimized by this bi-criteria fitness function.
The two conflicting objectives are placed into the denominators
of the two term weighted sum equation, thereby making it
desirable to maximize the fitness function. The function is
given below with the terminology defined in table 1.
Fitness  
Term


U
Ual
W
Wst
U al
W
  st
U
W
(1)
Table 1. Fitness terms
Definition
Preselected weight for displacement objective
Preselected weight for weight objective
Displacement of current structure
Displacement of homogeneous aluminum structure
Weight of current structure
Weight of homogeneous steel structure
Notice that Ual and Wst are known quantities defined as
constants before the optimization to normalize the fitness
results. The normalization of the fitness function allows the
same function to be applied to different models without the use
of penalty terms and the values for andwhich were
determined empirically, also work for a range of models.
The multi-objective nature of the optimization means that a
potentially large number of pareto solutions exist. This point
will be also discussed later.
2.4. Stepwise Mesh Refinement Technique
The initial mesh is purposely gross in order to decrease the
number of design variables and reduce the convergence time for
the model.
The technique is briefly outlined in the following steps: 1)
Create an initial coarse mesh, 2) Optimize that structure using
GA, 3) Incorporate (or inherit) the element material values into
a refined mesh, 4) Repeat optimization process and 5) Loop
until satisfying mesh refinement is achieved. This simple
algorithm reduces the complexity of the mesh refinement to that
of a straightforward element partitioning routine.
The main goal of the stepwise mesh refinement technique is
to achieve the same or better results with less computational
time. Also, one of the shortcomings of the conventional GA
procedure (i.e. in one step), is that for large genomes (i.e. large
number of finite elements) the speed of convergence becomes
prohibitive. Using the stepwise mesh refinement technique
accelerates the convergence process and ultimately allows an
optimum to be found.
The idea is to start from a coarse mesh. In this paper, a
stepwise mesh subdivision technique is proposed. The goals
are, first, to decrease the computational expense associated with
using GA in conjunction with FEA, and second, to help the GA
evolve properly using large genomes. The technique is briefly
outlined in the following steps: (1) Create an initial coarse
mesh, (2) Create an initial population randomly generated, (3)
Optimize the structure using GA, (4) Inherit the element
material distribution of the best individual into a subdivided (or
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finer) mesh to generate a new initial population, (5) Go to step
(3) and loop until satisfying mesh refinement is achieved.
Depending on the geometric complexity of the structure
and the number of materials, two, three of four steps are
generally sufficient to obtain satisfactory results. This point is
illustrated in the following section.
The initial mesh is purposely coarse in order to decrease
the number of design variables and reduce convergence time.
The coarseness of the initial mesh, which may prevent the
exploration of some solution paths, and how the information is
passed from one step to the next are critical aspects in the
success of the optimization.
How is the information passed from one step to the next?
By location of the volumetric center of each element.
How coarse is coarse? What is the effect of the coarseness
of the initial mesh? To issues must be considered: accuracy of
the finite element analysis and the capacity of mesh to capture
the spatial changes material distribution.
3. APPLICATIONS
3.1. Model Description
A simple connector is used to illustrate the method and is
described in Fig. 2. Solid tetrahedrons are used as finite
elements. Three meshes are initially created, with 270, 532, and
1089 elements. The planes of symmetry of the geometry and the
loading conditions allow the use of only a quarter of the
structure. The connector is clamped on one side and pulled in
the longitudinal direction as shown. The distribution of
constrained and loaded nodes correspond to nodes common to
the three finite element models in order to minimize the effect
of differences between meshes.
The two planes of symmetry for the geometry, loading, and
boundary conditions allow to consider a quarter of the total
structure.
Figure 1 Finite element model of the full structure and a
quarter by use of planes of symmetry
3.2. Application without Connectivity Analysis
Case Nstep(1)
1
2
3
4
5
6
1
2
3
1
2
3
Nelem(2)
1479
140 / 1479
140 / 455 / 1479
1089
270 / 1089
270 / 532 / 1089
Ratio
CPU(4)
Gain(5)
Nelem(3) (min)
74
10.6
47
36%
3.3 / 3.3
9
87%
32
4.0
16
50%
2.0 / 2.0
17
47%
(1) Nstep = Number of steps
(2) Number of finite elements in each of the steps
(3) Ratio of number of elements between finer mesh and current mesh
(4) On what machine (Dual Rack Onyx2 Infinite Reality)
(5) Gain in CPU (%) = 100*(CPU(1 step)-CPU(n steps))/CPU(1 step)
4. MANUFACTURABILITY OF THE RESULTS
Rapid prototyping techniques allow the fabrication of
discrete and continuous heterogeneous structures, also called
functionally gradient material (FGM) structures. The fabrication
process is an additive method where material of various volume
fractions is laid down point by point and layer by layer until the
object is built. This process assumes that the material
composition is known at every point, the size of which depends
on the characteristics of the rapid prototyping process and is
much smaller than finite elements. Therefore, the solutions
found by the present method must be post-processed in order to
obtain a smooth distribution of material composition. This
smoothing process is also required for strength and durability of
the structure, since in most cases the materials mixed together
have different coefficients of thermal expansion. Hence abrupt
change in material composition may not be feasible.
Notice that the representation and the optimization
technique presented in this research are independent of the
smoothing process required for manufacturing of the best
solution. This is one of the disadvantages of this method since
there is no guaranty that optimality is maintained by the
smoothing process. It is believed, however, that the error
introduced by the smoothing process is negligible in most
engineering applications.
4.1. Interpretation of Results
As mentioned earlier, several solutions show isolated steel
elements within the aluminum medium, see figure #. This
element appears clearly to be a result of the randomness of the
GA. In other words, it is likely to disappear after a few more
generations. On the contrary, a cloud of disconnected steel
elements within the aluminum medium, as seen in figure #, is
interpreted as a gradient material with volume fraction equal to
the ratio between the numbers of steel and aluminum elements
present in a given volume. The same applies to isolated
aluminum elements within a steel medium.
Distinction between isolated elements due to the GA
randomness and clouds of isolated elements is made by
examining the aforementioned ratio as well as the presence of
the same isolated element in the rest of the population. If the
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presence of a given element is assumed to be due to the
randomness of the GA, the element material is switched to
material of the dominant medium before the smoothing process.
4.2. Smoothing Process
The smoothing process comprises two steps. First, discrete
material properties are computed at each finite element node.
This is done by averaging the material composition within a
given volume centered at each node. The size of the volume,
which may be a sphere of radius between one and three times
the size of a finite element, determines the smoothness of the
final solution. Second, an interpolation is performed in order to
evaluate the material composition everywhere within the
structure. The type of interpolation—linear, quadratic or higher
order—creates a secondary smoothing. The final solution ready
for slicing and rapid prototyping is shown in Figure 2, for which
a radius of one finite element size and a linear interpolation
were used.
5. CONCLUSION
A stepwise mesh refinement technique for GA-based
optimization of multi-material objects was presented. Since the
computational effort is directly related to the number of finite
elements, the idea is to increase incrementally the level of mesh
refinement during the optimization process. The method was
shown to reduce significantly the computational time. The
geometrical variation of material composition is controlled by
the connectivity analysis, which quantifies the amount of
disconnected elements and by the coarseness of the initial mesh.
The latter was shown to have an effect on the final solution in
the case of multi-objective optimization where several pareto
solutions exist.
ACKNOWLEDGMENTS
This research was supported by the Automotive Research
Center (ARC), a U.S. Army TACOM Center of Excellence for
Modeling and Simulation of Ground Vehicles at the University
of Michigan. The views presented here do not necessarily
reflect those of our sponsor whose is gratefully acknowledged.
REFERENCES
Figure 2 Final material distribution after smoothing
A three-dimensional B-spline interpolation is expected to
be more appropriate for this type of representation. The volume
fraction at each point (u,v,w) of the parametric domain is
defined as follows
n m
l
(u, v, w )     N i,p (u )N j,q ( v) N k ,r ( w ) i, j,k
(3)
i  0 j0 k  0
where i,j,k are the volume fractions at the control points; n, m
and l are the number of control points in u, v and w direction,
respectively; Ni,p, Nj,q, Nk.r are the p-th degree, q-th degree and
r-th degree shape functions in u, v and w directions,
respectively and are defined by
1 if u i  u  u i 1
N i ,0 ( u )  
(4)
0 otherwise
N i, p (u ) 
u i  p 1  u
u  ui
N i,p 1 (u ) 
N i 1,p 1 (u ) (5)
u ip  u i
u i  p 1  u i 1
Since the parametric domain (u,v,w) varies within a regular
set of control points, i.e. a three-dimensional array of size n by
m by l, the discrete material composition known at the finite
element nodes must be mapped into the parametric domain at
the control points.
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