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

提出的研究项目的目标是在一阶体系的基础上发展、构造和分析有限弹塑性领域的混合有限元格式。由于目前还没有可用的模拟工具能够为此类问题提供可靠的解决方案，因此本项目的目标是在该领域提供一种新的具有现代离散化技术的途径。特别是最小二乘有限元方法(LSFEM)，它能产生精确的应力近似值，这在固体力学问题中通常是最重要的。这种方法避免了对有限元空间的应力和位移的相容(inf-sup)条件的要求，从而导致更一般的组合。在大量的调查中，申请人获得了关于LSFEM及其在固体力学中的应用的长期联合经验基础，这是由数学和力学之间的合作支持的。在这一合作取得进展的基础上，现在可以共同解决有限塑性问题。该项目的主要任务可分为三个步骤。首先，对弹性子问题发展该方法的分量，以便将它们推广到有限乘性弹塑性模型。在此之后，将通过优先方案提供的基准问题以及对实际复杂的微观非均质结构的计算，例如双相钢，对所得到的公式进行数值验证。为了提高计算的效率和稳健性，这些计算在适应性、近似空间的选择和合适的权重等方面进行了研究。本研究项目旨在为有限弹塑性框架下的结构力学问题的非常规离散方法设定新的质量标准。

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.