Modeling, Simulation, and Optimization of Microstructured Biomorphic Materials

We consider the modeling, simulation, and optimization of microstructured cellular biomorphic ceramics obtained by biotemplating. This is a process in biomimetics, a recently emerged discipline in materials science where engineers try to mimick or use biological materials for the design of innovative technological devices and systems. In particular, we focus on the shape optimization of microcellular silicon carbide ceramics to be used as temperature and wear resistant, highly porous filters in chemical processing or as implants in medical applications. These ceramics are derived from naturally grown wood by a two-step process involving pyrolysis of the wood, which results in a graphite-like carbon preform, and a subsequent infiltration by liquid or gaseous silicon. The mechanical behavior of the final ceramics is largely determined by the geometry of its microstructure which can be very precisely tuned during the biotemplating process. The ultimate goal is to determine these microstructural details in such a way that an optimal mechanical performance is achieved with respect to merit criteria depending on the specific application. From a mathematical point of view, this amounts to a shape optimization problem where the state variables are the displacements subject to the underlying elasticity equations, and the design variables are the geometrical quantities determining the microstructure. Since a resolution of the microstructure is numerically cost-prohibitive, we perform homogenization techniques, assuming periodically distributed microcells, and apply state-of-the-art optimization methods to the homogenized model.