Upscaling and reliable two-scale Fourier/finite element-based simulations

在这个项目中，我们专注于确定性和随机弹性和非弹性问题的小应变区，如各向同性硬化的弹塑性的双尺度建模。在微观尺度上，我们考虑基于快速傅里叶变换（FFT）的求解器，使非常有效的三维图像为基础的研究。基于我们以前的结果变分FFT为基础的技术在傅里叶-伽辽金设置，我们认为一个计划相当于Moulinec-Suquet算法和改进的计划的基础上精确积分。对于宏观问题，我们考虑了具有FE 2类耦合过程的标准有限元方法和允许在不进行尺度分离假设的情况下处理问题的单元网格（MIEL）方法，该项目将进一步提高基于FFT的求解器的计算效率（例如预处理，通过低秩技术加速）的微尺度问题，这将与随机材料场建模。在随机设置，离散化和解决方案的程序内的变分设置将开发，提供宏观材料性能的概率描述。替代边界条件将被认为是基于FFT的求解器，使其耦合不仅在FE 2框架，而且在MIEL方法;随机性从微观到宏观尺度的转移将特别感兴趣。对于这些耦合的随机问题，我们将发展双尺度拟牛顿法，着重于线搜索和信赖域算法，从而发展双尺度非线性问题的离散化和求解程序，在微观尺度上使用基于FFT的求解器，在宏观尺度上使用基于FEM的求解器。我们预计，与捷克共和国研究小组的合作将导致显着提高效率的两个规模的模拟与现实的微观结构表示，使他们在传统的计算机平台上访问。

In this project, we focus on the two-scale modelling of deterministic and also stochastic elastic and inelastic problems in the small strain regime, such as elasto-plasticity with isotropic hardening. On the micro-scale we consider Fast Fourier Transform (FFT)-based solvers, enabling very efficient three-dimensional image-based studies. Building on our previous results on variational FFT-based techniques in the Fourier-Galerkin setting, we consider a scheme equivalent to the Moulinec-Suquet algorithm and an improved scheme based on exact integration. For a macro-scale problem, we consider the standard finite element method with FE2-like coupling procedures and also the mesh-in-element (MIEL) method, which allows to treat problems without the assumption of scale separation.The project will further increase the computational efficiency of FFT-based solvers (e.g. preconditioning, acceleration by low-rank techniques) for micro-scale problems, which will be modelled with random material fields. In the stochastic setting, discretisation and solution procedures within a variational setting will be developed, providing a probabilistic description of macro material properties. Alternative boundary conditions will be considered for FFT-based solvers to enable its coupling not only in the FE2-framework, but also in the MIEL method; the transfer of randomness from micro- to macro-scale will be of particular interest. For those coupled stochastic problems, two-scale quasi-Newton methods with emphasis on line-search and trust-region algorithms will be developed.As a result, the discretisation and solution procedures will be developed for two-scale nonlinear problems with FFT-based solvers on the micro-scale and FEM solvers on the macro-scale. We expect that the collaboration with the research group in the Czech Republic will lead to a significant increase in the efficiency of two-scale simulations with realistic microstructural representations, making them accessible on conventional computer platforms.