A finite element model for the analysis of the nonlinear mechanical behavior of hybrid composite materials

研究项目涉及非标准离散化方法。它是基于缩放边界表面相对于一个中心的基本范式。它导致了一个具有星形凸元素的有限元网格，允许悬挂节点、可重入角和弯曲边缘。对于所得到的多边形或类多面体结构，开发了相应的单元公式。为了实现一般应用，这些设计用于几何和物理非线性二维和三维问题。我们的目标是一个单元公式，它结合了有限元法的优点和边界几何的精确描述。元素允许任意数量的边。直线边或平面也可以，曲面边或曲面可以用非均匀有理B样条来描述。将快速四叉树或八叉树算法与局部细化策略相结合，形成递归离散化方法。最初，它从实体的边界表示开始，然后自动继续进行结构的块划分。通过这种方式，解决了来自不同材料或空隙的内部界面，并将曲面边界嵌入有限元网格中。首先，将区域四叉树或八叉树分解为星凸子区域或元素，提供了粗略的初始离散化。为了减小应力和位移的近似误差，需要进一步细化网格。有两个改进选项。一种递归地应用四叉树或八叉树分解来创建下一个更精细的离散化级别。另一种考虑元素的参数化，其中细化应用于沿边界的周向或径向方向。请注意，此步骤可能完全限制于子域或考虑中的元素。由于存在所谓的挂节点，不需要对相邻元素进行细化。因此，它是一种局部细化，可以以不同的方式应用。可以在边界边缘和/或元素内部引入额外的节点，也可以在元素级别增加多项式度。需要根据有意义的标准评估不同的细化策略。采用嵌入单元的离散化概念，得到了一种通用的数值有限元方法，适用于含夹杂和孔洞的非均质材料的分析。该项目的目的是促进可靠和稳健的数值分析，以满足现代混合复合材料的设计要求。

The research project deals with a non-standard discretization method. It is based on the fundamental paradigm of scaling the boundary surfaces with respect to a center. It leads to a finite element mesh with star-convex elements that allow hanging nodes, reentrant corners and curved edges. For the resulting polygon or polyhedron-like structures, the corresponding element formulations are developed. In order to enable a general application, these are designed for geometrical and physically non-linear 2D and 3D problems. We are aiming for an element formulation, which combines the advantages of the finite element method and the geometrically exact description of the boundary geometry. The element allows for an arbitrary number of sides. Straight edges or plane surfaces are possible as well as curved edges or surfaces, which are described by e.g. non-uniform rational B splines. In combination with the fast Quadtree or Octree algorithm and a local refinement strategy, it leads to a recursive discretization method. Initially, it starts with the boundary representation of a solid and then automatically continues with a block partitioning of the structure. In this manner, internal interfaces originating from different materials or voids are resolved, and the curved boundary is embedded in the finite element mesh.At first, a Quadtree or Octree decomposition of the domain into star-convex subdomains or elements provides a rough initial discretization. Further mesh refinements are needed to reduce the approximation error of the stresses and displacements. There are two refinements options. One recursively applies Quadtree or Octree decomposition to create the next finer level of discretization. The other considers the parameterization on the element, where the refinement applies either in circumferential direction along the boundary or in radial direction. Note that this step may be completely restricted to a subdomain or the element under consideration. It is not necessary to refine neighboring elements due to so-called hanging nodes. Therefore, it is a local refinement, which can be applied in different ways. It is possible to introduce additional nodes at the boundary edges and/or in the interior of an element, or it is possible to increase the polynomial degree at element level. The different refinement strategies need to be evaluated based on meaningful criteria.The discretization concept with the embedded element formulation results in a general numerical finite element method, which is suitable for the analysis of heterogeneous materials with inclusions and voids. The aim of the project is to contribute to reliable and robust numerical analysis to meet the requirements for the design of modern hybrid composites.