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

该研究项目涉及一种非标准离散化方法。它基于相对于中心缩放边界曲面的基本范例。这导致了有限元网格与星形凸面单元，允许悬挂节点，凹角和弯曲的边缘。对于得到的多边形或类多面体结构，推导了相应的单元公式。为了实现一般应用，这些都是为几何和物理非线性的2D和3D问题而设计的。我们的目标是一种结合了有限元方法的优点和对边界几何的精确描述的单元公式。元素允许任意数量的边。直边或平面可以是直边或平面，也可以是曲线边或曲面，例如非均匀有理B样条线。结合快速四叉树或八叉树算法和局部求精策略，得到一种递归离散化方法。最初，它从实体的边界表示开始，然后自动继续结构的块划分。首先，通过四叉树或八叉树将区域分解为星形凸子区域或单元，提供粗略的初始离散化。为了减小应力和位移的逼近误差，需要进一步的网格细化。有两个优化选项。一种是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。另一种考虑了单元的参数化，其中细化要么沿边界的圆周方向进行，要么沿径向进行。请注意，此步骤可能完全限于子域或正在考虑的元素。由于所谓的悬挂节点，没有必要精炼相邻元素。因此，它是一种局部细化，可以以不同的方式应用。可以在单元的边界边缘和/或内部引入额外的节点，或者可以在单元级别增加多项式次数。不同的细化策略需要根据有意义的准则进行评估。嵌入单元格式的离散化概念导致了一种通用的数值有限元方法，该方法适用于分析含有夹杂物和孔洞的非均匀材料。该项目的目的是为可靠和稳健的数值分析做出贡献，以满足现代混杂复合材料设计的要求。

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.