Structural optimization of biomorphic cellular silicon carbide ceramics with microstructures by homogenization modeling

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The project aims to develop efficient tools for the structural optimization of biomorphic microcellular ceramics based on homogenization modeling. The mathematical work will be supported by experimental investigations that provide both realistic model parameters as well as data for the purpose of model validation. The research is intended to be done on the basis of an interdisciplinary co-operation of applied mathematicians and materials scientists. The biomorphic ceramics were prepared by the biotemplating technique, where natural grown materials like wood, cellulose fibres, paper and cardboard structures are used as bulk templates for fast high-temperature conversion into high-strength or light-weight, microcellular ceramics and ceramic composites. Hereby, the engineering design can be significantly facilitated by the application of advanced optimization strategies and numerical solution techniques based on a physically consistent and transparent mathematical modeling taking into account the fine properties of the microstructures but on a much larger scale than that considered in the microscopic regime. Such a model can be established by the mathematical theory of homogenization. We attempt to optimize mechanical performances of the ceramic composites such as the compliance or the bending strength taking into account technological and problem specific constraints on the state variables and design parameters. This leads to a constrained nonconvex, nonlinear optimization problem which will be solved by primal-dual damped Newton-type interior-point methods involving the use of appropriately chosen merit functions for convergence monitoring. In the course of the optimization process, the inelastic deformation of the biomorphic microcellular ceramics has to be computed. This will be taken care of by fully adaptive temporal and spatial discretizations applied to a macroscale model obtained by homogenization from the dynamic and constitutive equations describing the evolving microstructure that features an interface between the organic and inorganic phase.