Proceedings of ASME

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: / Vincent Y. Blouin
Mechanical Engineering
Clemson University
Clemson, SC 29634-0921, USA
Email:
Georges M. Fadel
Mechanical Engineering
Clemson University
Clemson, SC 29634-0921, USA
Email:

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 one-dimensional 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] 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 two-dimensional 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. The geometry is generated and meshed using the IDEAS-SDRC FEM software package. The FEM mesh is composed of four-node 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.

Table 1. Fitness terms

Term / Definition
 / Preselected weight for displacement objective
 / Preselected weight for weight objective
U / Displacement of current structure
Ual / Displacement of homogeneous aluminum structure
W / Weight of current structure
Wst / 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 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

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.