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

在本研究项目中，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，在严格的变化框架中追求新的想法，因为目前没有明显的方法可用。遵循三个主要策略。第一种策略的基础构成了经典Hellinger-Reissner公式在非线性应用中的新扩展。本文利用迭代法确定了插值应力和应变的本构关系。这将在多凸应变能函数的基础上完成。在进一步的步骤中，应力的离散化是由特殊的插值函数来完成的，这保证了牵引矢量在单元边缘上的连续性。另一种方案，包括牵引矢量的连续性，是第二种策略的一部分。在每个子域上要求动量平衡的遵从性。这导致了一个原始公式，它不允许牵引矢量的跳跃。虚拟有限元法（VEM）的扩展是第三种策略的一部分。特别是，在复杂的非线性本构行为框架下，需要对稳定方法进行进一步的研究。进一步将向量机的插值函数由线性函数扩展为二次函数，以获得更好的收敛速度。此外，为了在晶体塑性的框架内使用该方法，将对VEM进行扩展、制定和实施，以用于3D应用。特别是在这种应用中，VEM在网格生成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，为生成高效的有限元代码提供了灵活的工具。

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.