Adaptive isogeometric modeling of discontinuities in complex-shaped heterogeneous solids

创新产品的开发需要具有复杂异质局部材料结构的多材料轻量化设计。它们的计算机辅助工程依赖于本构模型，特别是扩展裂纹的数值模拟。基本的数值技术必须考虑界面和块体材料的失效，以及它们之间以裂纹分叉和合并形式的相互作用。为了通过模拟提供逼真的预测，必须捕捉到问题的真实3D本质。为此，该项目开发了新的数值模型和方法，该模型和方法将基于自适应样条基的等几何分析(IGA)近似与用于裂纹扩展的相场模型相结合。相场方法显著地放宽了对界面和裂纹的必要解析，从而避免了重新划分网格的关键问题。IGA框架在复杂结构的计算机辅助设计(CAD)之间架起了一座桥梁，当与自适应相结合时，它允许在相场建模的背景下准确地表示漫射界面。其关键思想是将精确、连续的曲面描述与以隐式漫反射方式表示局部材质结构的非协调结构化体积网格相连接。这种方法的好处将通过接触引起的损伤和断裂的数值分析来证明。同样重要的是，在实践中预测和评估其效率和可靠性的基本数学方法。该项目的主要目标与计算力学、数值分析和材料科学领域的基本挑战相联系，例如，非结构化(水密)样条曲面的表示和自适应细化、样条曲面与结构化整体网格的可行耦合、非均匀材料的正则化建模以及在预渐近区域中的严格误差分析和控制。这些目标的实施将建立在第一个资助期内建立的工程师和数学家的合作基础上，这些合作导致了令人振奋的发展，如适合分析的网格的表征和IGA中的最佳自适应网格细化、其通过Bézier提取的高效算法实现以及其在脆性和延性断裂的相场模型中的应用。进一步的基本结果涉及到设想的方法的竞争力，特别是，与标准有限元相比，IGA猜想的谱优势的数学证明。

The development of innovative products demands multi-material lightweight designs with complex heterogeneous local material structures. Their computer-aided engineering relies on the constitutive modelling and, in particular, the numerical simulation of propagating cracks. The underlying numerical techniques have to account for the failure of interfaces and bulk material as well as their interaction in the form of crack branching and coalescence. In order to provide realistic predictions by simulation, the true 3D nature of the problem has to be captured.For this purpose, this project develops new numerical models and methods that combine adaptive spline-based approximations from Isogeometric Analysis (IGA) with phase-field models for crack propagation. The phase-field approach significantly relaxes the necessary resolution of interfaces and cracks and, hence, avoids the critical problem of re-meshing. The IGA framework bridges the gap to the computer aided design (CAD) of complex structures and, when combined with adaptivity, allows the accurate representation of diffuse interfaces in the context of phase-field modeling.This proposal concerns the enhancement of this promising approach to the computational modeling of failure in complex-shaped heterogeneous 3D solids. The key idea is to connect an exact, continuous surface description with a non-conforming structured volume mesh that represents the local material structure in an implicit, diffuse manner. The benefits of the approach, will be demonstrated by the numerical analysis of contact induced damage and fracture. Equally important are the underlying mathematics to foresee and assess its efficiency and reliability in practice. The main goals of this project are linked to fundamental challenges in the fields of Computational Mechanics, Numerical Analysis and Material Sciences, e.g., the representation and adaptive refinement of unstructured (water-tight) spline surfaces, the feasible coupling of spline surfaces with structured bulk meshes, the regularized modeling of heterogeneous materials, and the rigorous error analysis and control in pre-asymptotic regimes. The implementation of these goals will build upon the collaboration of engineers and mathematicians established during the first funding period that lead to exciting developments such as the characterization of analysis-suitable meshes and optimal adaptive mesh refinement in IGA, its efficient algorithmic realization via Bézier extraction as well as its application to phase-field models for brittle and ductile fracture. Further fundamental results regard the competitiveness of the envisioned approach, in particular, the mathematical justification of the conjectured spectral superiority of IGA when compared with standard finite elements.