Novel finite elements - Mixed, Hybrid and Virtual Element formulations at finite strains for 3D applications

新颖的有限元 - 用于 3D 应用的有限应变下的混合、混合和虚拟元素公式

• 批准号: 255432295
• 负责人: Professor Dr.-Ing. Jörg Schröder
• 金额: --
• 依托单位: Institut für Mechanik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/255432295?language=en
• 关键词: Novel; finite; elements; Mixed; Hybrid

In this research project, the main goal is to develop new finite-element formulations as a suitable basis for the stable calculation of modern isotropic and anisotropic materials with a complex nonlinear material behavior. In order to achieve this goal new ideas are pursued in a strict variational framework since there is no obvious approach available at the moment. Three main strategies are followed. The fundament of the first strategy constitutes a novel extension of the classical Hellinger-Reissner formulation to non-linear applications. Herein, the constitutive relation of the interpolated stresses and strains is determined with help of an iterative procedure. This will be done on the basis of polyconvex strain energy functions. In a further step the discretization of the stresses is done by special interpolation functions, which guarantee the continuity of the traction vector over element edges. An alternative formulation, incorporating continuity of the traction vector, is part of the second strategy. The compliance of the balance of momentum is demanded on each subdomain. This leads to a primal formulation, which does not admit jumps of the traction vector.The extension of the promising virtual finite element method (VEM) is part of the third strategy. Particularly, further investigations in the stabilization method will be done, which are needed in the framework of complex nonlinear constitutive behavior. Furthermore the interpolation functions for the VEM will be extended from linear to quadratic functions to obtain better convergence rates. In addition, the VEM will be extended, formulated and implemented for 3D applications in order to use the method in the framework of crystal plasticity. Especially in this application the flexibility of the VEM regarding the mesh generation will constitute a huge benefit.As a common software development platform the AceGen environment is applied providing a flexible tool for the generation of efficient finite element code.

在该研究项目中，主要目标是开发新的有限元公式，作为具有复杂非线性材料行为的现代各向同性和各向异性材料稳定计算的合适基础。为了实现这一目标，由于目前尚无明显的方法，因此在严格的各种框架中追求新想法。遵循三种主要策略。第一种策略的基本原理构成了经典的Hellinger-Reissner配方到非线性应用的新型扩展。在此，插值应力和菌株的组成性关系是在迭代程序的帮助下确定的。这将根据多凸应变能函数进行。在进一步的一步中，应力的离散化是通过特殊的插值函数来完成的，这可以保证牵引向量在元素边缘上的连续性。结合牵引载体的连续性的另一种配方是第二种策略的一部分。每个子域都需要保持动量平衡的依从性。这导致了一种原始公式，该公式不承认牵引向量的跳跃。有希望的虚拟有限元方法（VEM）的扩展是第三个策略的一部分。特别是，将在稳定方法中进行进一步的研究，这是在复杂的非线性本构行为框架中需要的。此外，VEM的插值函数将从线性函数扩展到二次函数，以获得更好的收敛速率。此外，将为3D应用程序扩展，配制和实施VEM，以便在晶体可塑性框架中使用该方法。尤其是在本应用程序中，VEM关于网格生成的灵活性将构成巨大的好处。作为通用软件开发平台ACEGEN环境的应用，为生成有效的有限元代码提供了灵活的工具。