Computational homogenization of non-linear and inelastic material laws in the tensor train format (TT-Hom)

现代3D成像技术，如微计算机断层扫描，提供了材料的极其详细的微观结构图像。为了直接在这些图像上进行计算，已经建立了专门用于规则网格的计算技术。例如，基于FFT的计算均匀化，它避免了存储刚度矩阵，并且仍然稳健地处理非线性和非弹性问题。为了在这些图像上进行计算，一方面需要存储解向量，另一方面必须评估材料定律。第一点涉及内存需求，而第二点涉及计算时间。如果关注大量的加载步骤，高分辨率或参数研究，则传统的计算均匀化方法将被拉伸到其极限。例子可以在蠕变行为、疲劳、塑性或损伤/断裂力学中的细剪切带中找到。数学上的困难源于问题的复杂性，该问题由（自由度）x（加载步数）x（计算次数）从下而上限定。该提案的目标是巧妙地将张量压缩方法（如张量trainformat）应用于数字图像的计算均匀化。利用低秩表示的计算复杂性可以降低到低于明显的barrier.From方法的观点，有效的压缩真实的微观结构和有效的表示的微分和积分算子的弹性张量列车格式的中心。此外，应用非线性弹性，粘弹性和弹塑性处理。特别是，需要使用时空预处理器，一些功能张量列车表示需要推广到非光滑options.The拟议的项目目标：-计算有效的材料法律-复杂的图像为基础的微观结构-材料法律与内部变量-多轴加载结合负载反转-高循环计数-使用张量数据压缩技术-结合现代非线性求解器拟议的项目将产生的知识，可以分为以下几类。首先，始终使用张量压缩方法可以长期存档微观结构和模拟结果。其次，我们的目标出现的处理问题，由于大量的自由度或许多负载步骤，难以处理与传统的方法，如复合材料的多尺度疲劳计算。

Modern 3D imaging techniques like micro-computed tomography provide extremely detailed microstructure images of materials. To compute directly on these images, computational techniques specialized for regular grids have been established. For example, FFT-based computational homogenization, which avoids storing the stiffness matrix and still treats non-linear and inelastic problems robustly.State-of-the-art micro-computed tomographs produce volume images consisting of several billion voxels. To compute on these images, on the one hand the solution vector needs to be stored, and, on the other hand, material laws have to be evaluated. The first point concerns memory demands, whereas the second point involves computational time.Conventional computational homogenization methods are stretched to their limits if a high number of load steps, a high resolution or parameter studies are of concern. Examples can be found in creeping behavior, fatigue, fine shear bands in plasticity or in damage/fracture mechanics. The mathematical difficulty is rooted in the complexity of the problems which is bounded from below by (degrees of freedom)x(number of load steps)x(number of computations).The proposal targets applying tensor compression methods (like the tensor trainformat ) cleverly to computational homogenization on digital images. Exploiting low rank representations the computational complexity can be reduced below the apparent barrier.From a methodical point of view the efficient compression of real microstructures and the efficient representation of the differential and integral operators of elasticity in the tensor train format are central. Furthermore, the application to non-linear elasticity, viscoelasticity and elastoplasticity is addressed. In particular, space-time preconditioners need to be used, and some functional tensor train representations need to be generalized to non-smooth operations.The proposed project targets:- computing effective material laws for - complex image-based microstructures - material laws with internal variables - multiaxial loading incorporating load reversal - high cycle count - using tensor data compression techniques - incorporating modern non-linear solversThe proposed project shall generate knowledge, which can be classified as follows. Firstly, consistently using tensor compression methods enables archiving microstructure and simulation results in a long-term manner. Secondly, we target handling problems that appear, due to a large number of degrees of freedom or many load steps, untractable with conventional methods, like multi-scale fatigue computations for composites.