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

在这个研究项目中，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，在严格的变分框架中追求新的想法，因为目前没有明显的方法。三个主要战略如下。第一个策略的基础构成了一个新的扩展的经典Hellinger-Reissner制定非线性应用。在这里，本构关系的插值应力和应变的确定与迭代过程的帮助。这将在多凸应变能函数的基础上进行。在进一步的步骤中，通过特殊的插值函数来完成应力的离散化，这保证了单元边缘上的牵引矢量的连续性。第二种策略的一部分是另一种公式，其中包含了牵引力矢量的连续性。在每个子域上要求符合动量平衡。这导致了一个原始的公式，它不允许跳跃的牵引矢量。扩展的有前途的虚拟有限元法（VEM）的第三个战略的一部分。特别是在复杂非线性本构行为的框架下，需要对稳定化方法进行进一步的研究。此外，VEM的插值函数将从线性扩展到二次函数，以获得更好的收敛速度。此外，VEM将被扩展，制定和实施的3D应用程序，以使用晶体塑性的框架中的方法。特别是在这种应用中，VEM在网格划分方面的灵活性将构成巨大的优势。AceGen环境作为通用的软件开发平台，为高效的有限元代码的生成提供了灵活的工具。

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.