Numerical algorithms for the simulation of finite plasticity with microstructures

微结构有限塑性模拟的数值算法

• 批准号: 35736987
• 负责人: Professor Dr. Carsten Carstensen
• 金额: --
• 依托单位: Institut für Mathematik
• 依托单位国家: 德国
• 项目类别: Research Units
• 财政年份: 2007
• 资助国家: 德国
• 起止时间: 2006-12-31 至 2014-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/35736987?language=en
• 关键词: Numerical; algorithms; simulation; finite; plasticity

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).

固体力学中微结构的出现，特别是有限塑性中微结构的出现，可以归因于基础能量势的凸性的丧失。当材料在宏观上变形时，微观上会出现剪切带、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拟凸化。研究组内的项目要么涉及建模，要么涉及模拟。相反，这个项目的目标是通过分析离散化和设计收敛的自适应网格细化算法来证明计算机模拟的合理性。数学论证涉及到微观尺度(A)、宏观尺度(B)、时间演化微结构(C)的数值模拟。在第一个资助期，(A)建立了单晶有限塑性数值松弛的有效算法。(B)中(准)凸化变分模型的退化性质要求在适应的有限元网格上采用新的稳定化技术。在(A)和(B)的组合中考察了计算的宏观能量密度W的微扰的影响。在摄动最小化问题的分析中，主要结果保证了用于渐近精确计算能量的自适应网格加密算法的收敛。该项目继续研究全时空离散化的速率无关发展问题的数值模拟的收敛(C)。第二个资助期将调查(A)、(B)和(C)中非标准有限元方法的改进情况。