Impact of stability and surface area on fracture toughness of biological structures

Nature is the master of creating material systems with a remarkable combination of properties. Biosystems are tough, strong, resilient, fast and efficient. Bioinspired system design is an emerging framework in structural engineering for the realization of engineered structures that are tough, resilient, light, and multi-functional. Over the last few decades, numerical topology optimization has developed into a very powerful design tool to achieve this goal. This work focuses primarily on trabecular bone, which naturally optimizes for external loading conditions as described by Wolff's Law. Topology optimization is applied to uncover the underlying design principles for the trabecular structure. The phase field method, with a novel preconditioner, is subsequently used to characterize the fracture resistance of bone-inspired structures. Specifically, this thesis is divided into two major themes of work: Theme 1: Multigrid methods for large-scale topology optimization with application to trabecular bone-inspired structures: We evaluate the computational cost involved in large-scale topology optimization problems encompassing multiple objectives. Some of these objectives, such as stability maximization, require multiple solutions of a generalized eigenvalue problem at each optimization iteration. We then show that the cost may be most effectively managed using multigrid methods in the linear algebra routines and identify conditions under which geometric or algebraic multigrid solvers are more effective for topology optimization. The techniques for managing the cost are then applied to use topology optimization to generate bone-like structures at unprecedented scales in 3D. Theme 2: Phase field method and network analysis for unraveling connections between topological design and failure patterns in the bone-inspired structures. The phase field method is appealing for its variational basis and its lack of mesh dependence; however, it does introduce numerous challenges numerically. The most well-known challenge is the indefinite matrices it generates, which leads to a breakdown of any nonlinear solver unless some measures are taken to stabilize it. The most common approach is to use a staggered scheme that decouples the damage and displacement fields in the solver. We instead show that better performance can be obtained by using this staggered approach only as a preconditioner for the full linear system. In addition to the phase field method, we also use a network model to convert the continuum structures to discrete lattices for analysis. The discrete structures are modeled with a linear-elastic regime and a von Mises stress failure criterion. We show that the results of these separate analyses agree well with those performed using phase field on the continuum, despite the limitations associated with the conversion from continuum to discrete models.U of I OnlyAuthor requested U of Illinois access only (OA after 2yrs) in Vireo ETD syste