Adaptive isogeometric modeling of discontinuities in complex-shaped heterogeneous solids

创新产品的开发需要多材料轻量化设计和复杂的异质局部材料结构。他们的计算机辅助工程依赖于本构模型，特别是裂纹扩展的数值模拟。基础的数值技术必须考虑界面和块状材料的破坏以及它们以裂纹分支和合并形式的相互作用。为了通过模拟提供真实的预测，必须捕获问题的真正3D本质。为此，本项目开发了新的数值模型和方法，将来自等几何分析（IGA）的基于自适应样条的近似与裂纹扩展的相场模型相结合。相场方法大大放松了必要的界面和裂纹的分辨率，从而避免了重网格划分的关键问题。IGA框架弥补了与复杂结构的计算机辅助设计（CAD）之间的差距，当与自适应相结合时，可以在相场建模的背景下准确地表示扩散界面。这一建议涉及到在复杂形状的非均质三维固体中破坏的计算建模的这种有前途的方法的增强。关键思想是将一个精确的、连续的表面描述与一个不一致的结构体网格连接起来，以一种隐含的、扩散的方式代表局部的材料结构。该方法的优点将通过接触损伤和断裂的数值分析来证明。同样重要的是基础数学，以预测和评估其在实践中的效率和可靠性。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有关，例如，非结构化（水密）样条曲面的表示和自适应细化，样条曲面与结构体网格的可行耦合，非均质材料的正则化建模，以及前渐近状态下的严格误差分析和控制。这些目标的实现将建立在工程师和数学家在第一个资助期建立的合作基础上，这将导致令人兴奋的发展，例如IGA中适合分析的网格特征和最优自适应网格细化，其通过bsamizier提取的高效算法实现，以及其在脆性和韧性断裂相场模型中的应用。进一步的基本结果考虑到所设想的方法的竞争力，特别是与标准有限元相比，IGA推测的频谱优势的数学证明。

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.