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

该项目旨在开发基于均质化建模的生物形态微细胞陶瓷结构优化的有效工具。数学工作将得到实验调查的支持，实验调查提供了现实的模型参数以及模型验证的数据。这项研究是在应用数学家和材料科学家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，其中天然生长的材料如木材、纤维素纤维、纸和纸板结构被用作大块模板，用于快速高温转化为高强度或轻质的微细胞陶瓷和陶瓷复合材料。因此，先进的优化策略和数值求解技术的应用可以显著促进工程设计，这些优化策略和数值求解技术基于物理一致和透明的数学模型，考虑到微观结构的精细特性，但在更大的范围内考虑微观制度。这种模型可以用均匀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，同时考虑到技术和问题对状态变量和设计参数的具体限制。这将导致一个约束的非凸非线性优化问题，该问题将通过原始对偶阻尼牛顿型内点方法来解决，该方法涉及使用适当选择的优点函数进行收敛监测。在优化过程中，需要计算仿生微细胞陶瓷的非弹性变形。这将通过应用于宏观尺度模型的完全自适应时间和空间离散化来解决，该模型是通过均质化获得的，该模型来自描述有机相和无机相之间界面的演变微观结构的动态和本构方程。

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.