Combining isogeometric analysis (IGA), finite element methods (FEM) and embedded mesh (EM) coupling for contact problems

接触过程和通常与之相关的物理现象（例如摩擦、磨损）仍然是基于有限元方法（FEM）的土木工程建模和数值模拟中的最大挑战之一。在所涉及的体的体积中和在接触表面上关于可用的离散化技术、单元尺寸和单元形状的FEM网格生成之间的看似不可分割的联系是一个特别的障碍。虽然在离散接触边界处，特别是在曲面几何形状的情况下，形状函数的高度连续性和精细边界层网格被证明是有利的，但是强烈期望结构化的、理想地均匀的笛卡尔六面体网格和良好建立的有限元技术的灵活可重用性，例如以避免在体积内锁定。本研究项目致力于通过开发一种新的离散化方法，为一般的非线性三维接触问题的有限元，等几何分析（伊加）和所谓的嵌入式网格耦合方法的组合的基础上，完全解决这一困境。新方法由以下核心构建块组成：所涉及的机构的接触表面是完全从CAD几何模型中使用等几何方法与非均匀有理B样条（NURBS），并满足固有的高连续性（至少C1-连续性）。在域内生成规则六面体网格，与接触边界处的过程无关，这保证了在选择单元技术（例如经典的C 0连续FEM或NURBS）和最佳单元形状（3D体素）方面的完全灵活性，至少在未变形的初始配置中。为了连接这两个部分，面向表面的边界层网格首先生成从离散的NURBS接触表面通过挤出。该项目的重点是两个新的技术，一致耦合伊加的表面网格和体积网格。由于这两个网格重叠，大多数经典的方法不能使用，没有进一步的麻烦，由于其违反离散稳定性条件。相反，将开发基于martor/拉格朗日乘子法以及Nitsche法的适当嵌入网格耦合方法。与目前可用的方法，如FEM平滑方法，完整的3D NURBS网格或NURBS丰富的方法，所得到的接触制定将首次联合收割机的所有优点，独立的，并在每种情况下，根据当地的要求，最佳生成的边界层网格和体积网格的接触问题。

Contact processes and the physical phenomena typically associated with them (e.g. friction, wear) are still among the greatest challenges in modelling and numerical simulation in civil engineering based on finite element methods (FEM). The seemingly indissoluble link between FEM mesh generation in the volume of the bodies involved and on the contact surfaces with regard to usable discretization techniques, element sizes and element shapes is a particular obstacle. While at the discrete contact boundaries, especially in the case of curved surface geometries, a high continuity of the shape functions and fine boundary layer meshes prove to be advantageous, there is a strong desire for structured, ideally even Cartesian hexahedral grids and a flexible reusability of well-established finite element technology, for example to avoid locking, inside the volume. This research project is dedicated to the complete resolution of this dilemma by developing a novel discretization method for general nonlinear 3D contact problems based on a combination of FEM, isogeometric analysis (IGA) and so-called embedded mesh coupling methods. The new approach consists of the following core building blocks: the contact surfaces of the bodies involved are taken exactly from the CAD geometry model using isogeometric approaches with non-uniform rational B-splines (NURBS) and fulfill inherently high continuity properties (at least C1-continuity). A regular hexahedral grid is generated inside the domain, independent of the processes at the contact boundaries, which guarantees complete flexibility in the choice of element technology (e.g. classical C0-continuous FEM or again NURBS) and an optimal element shape (3D voxel) at least in the undeformed initial configuration. In order to connect these two parts, a surface-oriented boundary layer mesh is first generated from the discrete NURBS contact surface by extrusion. The project focuses on two new techniques to consistently couple IGA surface mesh and volume mesh. Since these two meshes overlap, most classical methods cannot be used without further ado due to their violation of discrete stability conditions. Instead, suitable embedded mesh coupling methods based on mortar / Lagrange multiplier methods as well as Nitsche methods will be developed. In contrast to currently available approaches, such as FEM smoothing methods, complete 3D NURBS meshing or NURBS enrichment methods, the resulting contact formulation will for the first time combine all the advantages of independent and in each case according to local requirements optimally generated boundary layer meshes and volume meshes for contact problems.