Numerical algorithms for the simulation of finite plasticity with microstructures

在固体力学中，特别是在有限塑性中，微观结构的出现可以归因于潜在能量势的凹凸性的损失。当材料在宏观上变形时，在微观尺度上出现剪切带、裂纹或层状结构。这些例子的共同之处在于，它们的宏观模拟必须基于能量泛函的准自象化。研究小组中的项目要么涉及建模，要么涉及模拟。相比之下，本项目的目标是通过分析离散化和设计收敛自适应网格细化算法来证明计算机模拟的合理性。数学论证涉及微观尺度(a)、宏观尺度(b)和随时间变化的微观结构(c)的数值模拟。在第一期资助中，建立了单晶有限塑性数值松弛的有效算法(a)。(b)中（拟）凸变分模型的退化性质要求在自适应有限元网格上采用新的稳定技术。在(a)和(b)的组合中考察了计算得到的宏观能量密度W的扰动的影响。分析扰动最小化问题的主要结果保证了用于能量渐近精确计算的自适应网格细化算法的收敛性。该项目继续研究速率无关演化问题的全时空离散化数值模拟的收敛性(c)。第二个资助期将研究(a)、(b)和(c)项中非标准有限元方法的改进。

The occurrence of microstructures in solid mechanics and, in particular, in finite plasticity can be attributed to a loss of the convexity of the underlying energy potentials. While the material deforms macroscopically, structures in the form of shear bands, cracks or lami- nates arise on microscopic scales. Common to these examples is that their macroscopic simulations have to be based on the quasiconvexification of the energy functional.The projects within the research group either concern the modelling or the simulation. In contrast, the object of this project is the justification of computer simulations with an analysis of discretisation and design of converging adaptive mesh-refining algorithms. The mathematical justification concerns numerical simulations on the microscopic scale (a), on the macroscopic scale (b), for time-evolving microstructure (c).In the first funding period, an efficient algorithm on the numerical relaxation in single- crystal finite plasticity has been established in (a). The degenerate nature of the (quasi-) convexified variational model in (b) required novel stabilisation techniques on adapted finite element grids. The influence of the perturbation of the computed macroscopic energy density W is examined in the combination of (a) and (b). The main result in the analysis of perturbed minimisation problems guarantees convergence of an adaptive mesh-refining algorithm for asymptotically exact computation of energies.The project continues to investigate the convergence of numerical simulations of rate- independent evolution problems for the full time-space discretisation (c). The second funding period shall investigate the improvements of nonstandard finite element methods in (a), (b), and (c).