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的稳定计算的合适基础。为了实现这一目标，由于目前没有明显的方法可用，因此在严格的变分框架中追求新的想法。遵循三个主要策略。第一个策略的基础构成了经典 Hellinger-Reissner 公式到非线性应用的新颖扩展。在此，借助迭代过程确定插值应力和应变的本构关系。这将基于多凸应变能函数来完成。在进一步的步骤中，通过特殊的插值函数完成应力的离散化，这保证了单元边缘上牵引矢量的连续性。另一种方案是第二种策略的一部分，它结合了牵引向量的连续性。每个子域都要求遵守动量平衡。这导致了一个原始公式，它不允许牵引矢量的跳跃。有前途的虚拟有限元方法（VEM）的扩展是第三个策略的一部分。特别是，将在复杂的非线性本构行为框架中对稳定方法进行进一步的研究。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度。此外，VEM 还将针对 3D 应用进行扩展、制定和实施，以便在晶体可塑性框架中使用该方法。特别是在这种应用中，VEM 在网格生成方面的灵活性将带来巨大的好处。作为通用软件开发平台，AceGen 环境的应用为生成高效的有限元代码提供了灵活的工具。