In this project, the complex Hashin-Shtrikman bounds are considered as approximations for computationally designing the effective bulk, shear moduli and volume fraction of second phase in an isotropic two-phase microstructure. In addition to the macroscopic properties, stress – strain concentration tensors are measured to ensure relative smooth internal fields, corresponding to the matrix material alone. Genetic algorithm have been used to solve this project and also find the error norms between the analytical and numerical solutions using Newton’s methods.

Genetic algorithm is a search heuristic that is used for finding optimized solutions for problems. As seen from the graph, after every generation, the randomized values are evaluated for cost and the top 10 (minimum cost) are sorted and mated to form two children for every pair of parents. These 20 minimum cost associated variable values along with 80 random variable values are run through the system again to get the next top 10 minimum costs. After every generation, the minimum cost keeps reducing, until it reaches the tolerance set.

 

The difference in the two graphs is due to the fact that in the first one, 20 variable values are predetermined minimum giving samples from previous generation while in the second experiment its only 10 from the previous generation. Thus by selecting appropriate samples after every generation, the desired cost function can be forced to reach the tolerance required faster.

 

The best result is obtained by including the parents along with the children, thus giving a monotonically reducing cost function rather than a staggered result at in graph 2.

Newton’s method is used for minimizing a cost function.  With increase in the delta value, the error norm for both gradient and hessian increases.