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
increased
mechanical,
thermal,
and
electromagnetic
performance. The present paper describes a modeling and
optimization procedure based on the finite element method
(FEM) and the evolutionary genetic algorithm (GA). 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 prohibitive
and inconclusive. In this paper, a stepwise mesh refinement
technique is presented. Coupled to the multi-objective GA, 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 GA 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
and solid zones) pattern. Annicchiarico and Cerrolaza [1]
1
Copyright © #### by ASME
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, Huang and Fadel [8]
presented a one-dimensional parametric representation and
optimization process of a heterogeneous flywheel and a threedimensional 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 multimaterial optimization has been used in 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 FEM elements. The research done by Jackson et al.
[9] outlines a gradient approach created to use Solid Freeform
Fabrication (SFF), a manufacturing processes that uses layered
addition to build objects. A discrete approach was proposed by
Koenig [number] who worked toward developing optimization
tools that determine the best material distributions in twodimensional multi-material structures with multiple objectives
and set constraints. This research addresses 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 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 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 structured mesh subdivision technique can now be
explained in the following sections
2. METHODOLOGY
2.1. Genetic Algorithm
Quick description of GA (population evolve to best
individuals by natural elimination). Used SteadyState GA.
Whatever that means.
2.2. Finite Element Analysis
Commercial software used as a preprocessor to generate
the finite element models, compute and output the stiffness
matrices. Our code uses those matrices, assembles and inverts
by Gauss elimination.
2.3. Fitness Function
All individual of the populations are evaluated using the
fitness function, which is a quantity to maximize during the
optimization process. In the present work, the two conflicting
objectives are to minimize the maximum deformation of the
structure and minimize the total weight. The two objectives are
aggregated into a weighted sum written as
U
W
Fitness   al   st
(1)
U
W
where Ual and U are the maximum displacements of the
homogeneous aluminum structure (i.e. all elements are made of
aluminum) and the current individual, respectively. W st and W
are the total weight of the homogeneous steel structure and the
current individual, respectively.  and  are preselected weights
that reflect the relative importance of one objective over the
other. Notice that Ual and Wst are known quantities defined as
constants during the optimization and are used for
normalization. U and W are objective functions to minimize,
the inverse of which are to maximize.
A connectivity analysis may be introduced in the
optimization in order to reduce the checkerboard effect and
remove disconnected elements. The randomness nature of GA
may allow isolated elements that do not have much physical
meaning. Being able to identify them are remove them tends to
2
Copyright © #### by ASME
improve convergence speed. Introducing the connectivity
analysis as a third objective to be maximized, the fitness
function becomes
U
W
C
Fitness   al   st   st
(2)
U
W
C
where Cst and C are the connectivity values representing the
amount of connection between elements for the homogeneous
steel structure and the current individual, respectively. The
connectivity value is the summation over all elements of the
number of elements of same material connected by a face
(explain better!)… In other words, if a given element is made of
steel and is connected to 3 steel elements and 1 aluminum
element, its contribution to the connectivity value is equal to 3.
The effect of the connectivity analysis on the results and
the design interpretation will be illustrated and discussed in the
following in subsequent sections.
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 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
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
Did you talk about the stopping criterion in GA section?
Primary results:
Show group number 1 and 2 and 3. Gain in CPU = 87% and 37
%.
Conclusion (1): Better to do 3 steps to avoid large a step ratio in
number of elements. Comparison between group number 1 and
2 shows that the normalized weights and normalized
displacements are comparable. However, the connectivity value,
which is not included in the fitness function, is 20% higher in
group number 1. This means that the solution found by the step
method is more compact than the direct method, as shown in
figure (?). It is anticipated that the coarseness of the mesh of the
initial step has an effect on the final solution and tends to
encourage more compact solution. It must be clear that the two
3
Copyright © #### by ASME
best solutions found in group number 1 and 2 are Pareto
solutions of the bicriteria optimization problem with
comparable fitness functions (4.369 versus 4.363).
Conclusion (2): Comparing group number 2 and 3 shows that a
higher fitness function can be found with two steps and in 37%
less computational time. The gain in time is not as beneficial as
with three steps because of the large jump in number of
elements, i.e. number of design variables. In terms of similitude
between final solutions, as expected, the solution of group
number 3 is more compact than the AAO.
The representation of
4.1. Interpretation of Results
Distinction between isolated elements due to GA
randomness and other groups of elements
4.2. Smoothing
Effect of constraint:
In the previous applications, a constraint on the total weight of
the structure was applied. The constraint tends to reduce the
Pareto set. No need to talk about it since no difference with
groups 1, 2, 3.
Effect of ratio between steps:
In the next series of applications, the initial mesh is finer that
the one in series 1 (270 elements as opposed to 140), and the
final mesh is coarser (1089 as opposed to 1479). The same
conclusions can be drawn from this series of results as in the
previous series. That is
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)
(3)
Nelem
(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 (Mickey)
(5) Gain in CPU (%) = 100*(CPU(1 step)-CPU(n steps))/CPU(1 step)
3.3. Application with Connectivity Analysis
4. MANUFACTURABILITY OF THE RESULTS
Rapid prototyping techniques allow the fabrication of
functionally gradient structures. Materials are mixed at a
specific volume fractions and deposited layer by layer on the
build…
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 [reference I have the
paper] or for each element. In both cases, the coarseness of the
mesh introduces an inevitable error.
Figure 2 Final material distribution after smoothing
4.3. B-Spline Modeling for Rapid Prototyping
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
[1] Annicchiarico, W. and M. Cerrolaza, “Structural Shape
Optimization 3D Finite-Element Models Based on Genetic
Algorithms and Geometric Modeling,” Finite Elements in Analysis
and Design, Vol. 37, pp. 403-15, 2001.
[2] Beckers, M., “Topology Optimization Using a Dual Method With
Discrete Variables,” Structural Optimization, Vol. 17, pp. 14-24,
1999.
[3] Chapman, C., M. Jakiela, et al., “Genetic Algorithms as an
Approach to Configuration and Topology Design,” Journal of
Mechanical Design, Vol. 116, pp. 1005-12, Dec. 1994.
4
Copyright © #### by ASME
[4] Eby, D., R. Averill, et al., “An Injection Island GA for Flywheel
Design Optimization,” Invited Paper, Proc. EUFIT 1997, 5 th
European Congress on Intelligent Techniques and Soft Computing.
[5] Goldberg, D., Genetic Algorithms in Search, Optimization and
Machine Learning. Addison-Wesley Publishing Company, Inc.,
New York, 1989.
[6] Goodman, E., R. Averill, et al., “Parallel Genetic Algorithms in the
Optimization of Composite Structures,” Second World Conference
on Soft Computing, June 1997.
[7] Grosset, L. et al., “Genetic Optimization of Two-Material
Composite Laminates,” Proceedings of the 16th ASC Conference,
September 2001.
[8] Huang, J. and Fadel, G., “Heterogeneous Flywheel Modeling and
Optimization,” Materials and Design, Vol. 21, pp. 111-25, 2000.
[9] Jackson, T., N. Patrikalakis, et al., “Modeling and Designing
Components with Locally Controlled Compensation,” Solid
Freeform Fabrication Symposium Proceedings, pp. 259-66, 1998.
[10] Kim, H., M. Garcia, et al., “Fixed Grid Finite Element Analysis in
Evolutionary Structural Optimisation,” Proceedings of the Third
World Congress of Structural and Multidisciplinary Optimization,
Buffalo, NY, USA, 06-EVM1-3, May 17-21, 1999.
[12] Kumar, V. and Dutta, D., “Solid Model Creation of Materially
Graded Objects,” Solid Freeform Fabrication Symposium
Proceedings, pp. 613-20, 1997.
[12 bis] Kumar, V. and Wood, A., “Design of Optimally Stiff
Structures using Shape Density Function,” Proceedings DETC
1999 DAC, September 1999, Las Vegas, NE, USA.
[13] Malott, B., R. Averill, et al., “Use of Genetic Algorithms for
Optimal Design of Laminated Composite Sandwich Panels with
Bending-Twisting Coupling,” AIAA, Structures, Dynamics and
Materials, 1996.
[14] Morvan, S., MMa-Rep, a Representation for Multimaterial
Solids. Doctor of Philosophy Dissertation Presented to The
Graduate School of Clemson University, Clemson, SC, USA,
May 2001.
[15] Punch, W., A. Ronald, et al., “Design Using Genetic Algorithms –
Some Results for Laminated Composite Structures,” IEEE
Expert, Jan. 1995.
[16] Qian, X. and Dutta, D., “Physics Based B-Spline Heterogeneous
Object Modeling,” Proceedings DETC2001, Pittsburgh, PA,
2001.
[17] Qiu, D., N. Langrana, et al., “Virtual Simulation for MultiMaterial LM Process,” Solid Freeform Fabrication Symposium
Proceedings, pp. 681-88, 1997.
[18] Sigmond, O., “Systematic Design and Optimization of MultiMaterial Multi-Degree-of-Freedom Micro Actuators,” Technical
Proceedings of the MSM, International Conference on Modeling
and Simulation of Microsystems, 2000.
5
Copyright © #### by ASME