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

该项目旨在开发基于均匀化建模的生物形态微孔陶瓷结构优化的有效工具。数学工作将得到实验研究的支持，这些实验研究既提供了现实的模型参数，也提供了用于模型验证的数据。这项研究是在应用数学家和材料科学家的跨学科合作的基础上进行的。生物形态陶瓷是通过生物模板技术制备的，其中使用木材、纤维素纤维、纸和纸板结构等自然生长的材料作为主体模板，快速高温转化为高强度或轻质微孔陶瓷和陶瓷复合材料。因此，通过应用先进的优化策略和基于物理上一致和透明的数学模型的数值解技术，可以显著地促进工程设计，该数学模型考虑了微结构的优良特性，但比在微观区域中考虑的尺度要大得多。这样的模型可以用齐次化的数学理论来建立。我们试图优化陶瓷复合材料的力学性能，如柔度或弯曲强度，考虑到工艺和问题对状态变量和设计参数的特定约束。这导致了一个受约束的非凸、非线性最优化问题，该问题将用原始-对偶阻尼顿型内点方法来求解，其中涉及到使用适当选择的价值函数来进行收敛监测。在优化过程中，需要计算生物形态微孔陶瓷的非弹性变形。这将通过将完全自适应的时间和空间离散应用于通过从描述以有机相和无机相之间的界面为特征的微结构演变的动力学和本构方程的均化而获得的宏观模型来处理。

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.