First-order system least squares finite elements for finite elasto-plasticity

提出的研究项目的目标是基于一阶系统的有限弹塑性领域的混合有限元公式的开发、构建和分析。由于目前还没有仿真工具能够为这些问题提供可靠的解决方案，因此本项目的目标是为该领域的现代离散化技术提供新的途径。最小二乘有限元法（LSFEM），特别是产生应力的精确近似，这通常是固体力学问题的主要兴趣。这种方法避免了对应力和位移的有限元空间相容条件的要求，从而导致更一般的组合。此外，还得到了拟不可压缩材料的鲁棒数值特性。在众多的调查中，申请人在数学和力学之间的合作支持下，就LSFEM及其在固体力学中的应用取得了长期的联合经验基础。在这次合作取得的进展的基础上，现在有可能共同解决有限塑性的问题。项目的主要任务可以分为三个步骤。首先，针对弹性子问题开发该方法的分量，将其推广到有限乘法弹塑性模型。在此之后，所得的公式将通过优先方案提供的基准问题以及通过计算实际复杂的微非均质结构，例如双相钢，在数值上加以验证。为了提高算法的效率和鲁棒性，对算法进行了自适应、逼近空间的选择和适当的加权等方面的研究。本研究项目旨在为有限弹塑性框架下结构力学问题的非常规离散化方法制定新的质量标准。

The goal of the proposed research project is the development, construction and analysis of mixed finite element formulations in the field of finite elasto-plasticity on the basis of first-order systems. Since there are no simulation tools available at the moment that are capable to provide reliable solutions for such problems, the ambition of this project is to provide a new access with modern discretization techniques in this field. The least squares finite element method (LSFEM), in particular produces accurate approximations of stresses which are often of primary interest in the context of solid mechanical problems. This approach avoids the requirement of a compatibility (inf-sup) condition for the finite element spaces for stresses and displacements leading to more general combinations. Furthermore, a robust numerical behavior particularly for quasi-incompressible materials is obtained.In numerous investigations the applicants achieved a long-term joint experience basis with respect to the LSFEM and its application to solid mechanics supported by a cooperation between mathematics and mechanics. On the basis of the advances obtained within this cooperation it is now possible to jointly address problems of finite plasticity.The main tasks of the project can be divided into three steps. Firstly, the components of the method will be developed for the elastic subproblem in order to extend them to the model of finite multiplicative elasto-plasticity. After that, the resulting formulations will be validated numerically by means of the benchmark problems provided by the Priority Programme as well as by computations of real complex microheterogeneous structures as for instance of dual-phase steels. In order to improve the efficiency and robustness, these computations are guided by several investigations concerning e.g. adaptivity, choice of the approximation spaces and suitable weighting.This research project aims at setting new standards of quality in the field of non-conventional discretization methods for structural mechanical problems in the framework of finite elasto-plasticity.