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

在这个研究项目中，主要目标是发展新的有限元公式，作为现代各向同性和各向异性材料的稳定计算的合适基础，这些材料具有复杂的非线性材料行为。为了实现这一目标，在严格的变化框架内追求新的想法，因为目前没有明显的办法可用。以下是三个主要策略。第一种策略的基础构成了经典Hellinger-Reissner公式向非线性应用的新扩展。在此，通过迭代程序确定了插值点的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内插函数完成的，该函数保证了单元边缘上的牵引力矢量的连续性。第二种策略的一部分是包含牵引向量连续性的另一种表述。在每个子域上都要求遵守动量平衡。这导致了一个不允许牵引矢量跳跃的原始公式。有希望的虚拟有限元方法(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关系的框架下，对稳定化方法进行了进一步的研究。此外，VEM的插值函数将从线性函数扩展为二次函数，以获得更好的收敛速度。此外，VEM将针对3D应用进行扩展、制定和实施，以便在晶体塑性的框架下使用该方法。特别是在这种应用中，VEM在网格划分方面的灵活性将是一个巨大的好处。作为通用的软件开发平台，AceGen环境的应用为高效的有限元代码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.