In this thesis, based on a newly-defined form proposed for topological derivative, a novel topology optimization methodology was proposed to identify the optimal phase distribution in a two-phase material, an important step towards the design of advanced bioinspired materials that can mimic the mechanical behavior of tissue. The method is then applied to the optimization of the microstructure of patterned hydrogel to obtain a material system with behavior similar to that of arterial wall layers. First, a 2D two-phase finite element model for the patterned hydrogel is used, where the pattern and the base are described by materials with constitutive laws that take into account their significant differences both in terms of their stiffness and in terms of the nonlinearity exhibited in large deformations. Then, this methodology was extended to 3D problems, to match 3 orthogonal strong nonlinear target stress-stretch relations, and obtain the microscopic phase distribution for architectured (or hierarchical) composite soft material.

Identifying the microstructure is equivalent in this case to optimizing the phase distribution and is an inverse problem. The topology optimization requires an ability to morph a geometry to systematically guide the mechanical behavior of the resulting system towards the desired target, an ability of a proposed topological operation, Geometric Topology to Function Space Topology Morphing. In order to accomplish this, a formulation of the optimization problem and a methodology capable to solve it are proposed in this thesis. The problem is formulated in terms of a distance function and the solution methodology is based on a newly defined form proposed for the topological derivative and on a general result derived for a general elastostatic Boundary Value Problem. The influence of elemental phase-switch on the objective function is evaluated here in the context of strong non-linearity. A mathematical understanding of Xie's sensitivity number based on the proposed topological integration is also presented.

The proposed topology optimization methodology was implemented in Matlab and simulations to determine the mechanical response are performed with the finite element package FEAP. Numerical experiments were performed and demonstrate the ability to reach a topology of phase distribution for a composite material system that exhibits the desired mechanical response. Ultimately, these contributions facilitate a combined in silico-in vitro design process that greatly accelerates the design of engineering material systems that mimic biological tissue.