Computer Program Implementation

of three different modules: homogenization, structural analysis and optimization.

In the homogenization module, first we need to establish a model of microstructure. Then our aim is to develop the relationship between effective material properties and the geometrical shape parameters of the microstructure. One way to achieve this goal is to carry out a series of finite element analysis for the different geometrical shape parameters of the microstructure. Subsequently, the polynomials for the elasticity matrix of the homogeneous solid are obtained. Most of microstructures can be calculated by this way. With high speed computer technology development, more and more calculations of microstructure properties will use numerical method. Another way is using analytical formulae method, such as in the case of ranked layered one-material and bi-material microstructures, the effective material properties generally can be derived analytically. In such an analytical approach, explicit expressions for the effective elastic tensor can be obtained by establishing the optimal upper and lower bounds for the complementary elastic energy density of the porous material. These microstructures are known as “extremal” microstructures in the sense that they achieve optimality in the Hashin-Shtrikman bounds on the effective properties of composite materials. This method can be applied to both two-dimensional and three-dimensional layered material cell of finite rank. The third way is using a simple formula and combining it with penalty method, for example, Power-law material microstructure (Bendsøe and Sigmund, 2002) and

artificial model (Hassani and Hinton, 1998).

In the structural analysis module, first a finite element model of microstructures needs to be established. It is common practice to use a regular mesh of elements, each element is a microstructure. Then for a given boundary and loading conditions, a finite element analysis is carried out based on the effective material properties that have been obtained by homogenization method. Stress, strains and displacements can be calculated.

In the optimization module, considering the shape parameters of the microstructure model in finite elements as design variables, the total potential energy as objective function, the volume of material as global constraint, by using optimality criteria method and filter technique, the topology optimization program can be implemented.

The HDM computer program developed in this research was built on a computer Pentium III, 128MB memory, 32MB DDR Nvidia GeForce2 GTS Graphics cards, using Windows 2000 Operating System. The visual graphic was developed by Delphi 5.0 software. The mesh development uses Strand 7 standard finite element software by G+D Computing Pty. Ltd. Australia, which permits user to build models, add loads and constraints very easily and quickly. The following facilities are available in strand7.

tools, automatic meshing and unlimited undo.

• Organise a complicated model into a simple set of parts using the Group Tree.

• Define your own coordinate systems and beam cross-sections.

• Check mesh quality with aspect ratio and warping contours and free edge detection.

However, the solver of the finite element was developed and included in the optimization program by the author. Four and eight nodes elements are available. The optimization code is a combination of C++ and FORTRAN 95. All the programs are finally controlled by a HDM.bat batch file. Typically, for a 496 nodes, 450 meshes, single load problem, the analysis time is 12 minutes for 200 numbers of iterations.

The topology optimization algorithm is as follows:

Step 1 Program start and greeting

Step 2 Draw a suitable reference domain and discretize the reference domain by generating a finite element mesh for analysis, define surface tractions, fixed boundaries, loads, and material properties, etc. by using Strand7 software.

Step 3 Choose a microstructure model out of fifteen models available and optimization parameters.

orientation value θ is set to zero.

Step 5 Compute the effective material properties of the composite, using homogenization theory. This gives a functional relationship between the density of material in the composite (i.e. sizes of holes) and the effective material properties.

Step 6 Carry out structure analysis to obtain stress, strain and displacement.

Step 7 Evaluate the objective function.

Step 8 Use filter technique to modify ϕ ψ ω φ, , , to ϕ ψ ω φ', ', ', '

Step 9 Resize the design variables and orientation value

Step 10 Check the volume constraint, if it is active, continue, otherwise update it and go back to Step 9

Step 11 Form a new design based on the new set of design variables for each element

Step 12 Check if solution has converged; if it is, go to next step, otherwise update design variables and go back to step 5

Step 13 Output the image layout of new design

The algorithm is illustrated in Figure 5.3 and a typical run of HDM software is presented in Appendix B.

5.6 Summary

microstructure models, a computer program named HDM (Homogenization with Different Microstructures) has been developed. This program includes five new one-material microstructure models and four new bi-material microstructure models. The new one-material models are: cross shape microstructure model, circular microstructure model, triangular multi-voids microstructure model, rectangular multi-voids microstructure model, and square multi-voids microstructure model. The new bi-material models are:

cross shape bi-material model, square bi-material model, rectangular bi-material model and triangular bi-material model.

The program also includes some existing microstructure models for which the microstructures optimization program codes are not available in the literature. These models are: ranked layered model, triangular microstructure model, hexagon microstructure model, power-law one-material model, power-law bi-material model and ranked layered bi-material model.

A filtering program for checkerboard pattern control was established. The filter method is very useful not only for checkerboard pattern control but also for mesh-dependence problem, which easily appeared in SIMP method.

The program uses optimality criteria method for updating design variable and principal stress method for updating orientation variable. A simple convergence criterion was used.

In the next chapter, we use the program developed in this chapter to investigate some benchmark problems of topology optimization.

Chapter 6

COMPARISONS BETWEEN ALGORITHMS USING