Stable discretization methods and scalable solvers for embedded fiber/solid coupling

用于嵌入式光纤/固体耦合的稳定离散化方法和可扩展求解器

• 批准号: 528397555
• 负责人: Dr.-Ing. Matthias Mayr
• 金额: --
• 依托单位: Institut für Mathematik und Computergestützte Simulation
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/528397555?language=en
• 关键词: Stable; discretization; methods; scalable; solvers

Structures and systems reinforced by thin fibers play an increasingly important role in many applications. Embedded fibers can positively influence the mechanical, but also the functional properties of a component or system. For example, short steel fibers embedded in concrete components increase the tensile strength (even in the post-cracking regime) as well as the impact resistance. Further applications are found in light-weight structures in aerospace engineering or the design of medical devices. In order to quantitatively characterize the influence of such fibers on the system behavior and, above all, to predict and optimize the system’s behavior, a detailed understanding of the interaction of the fibers with the surrounding solid material is imperative. So far, the behavior of a single fiber can be determined experimentally as well as by fully resolved simulation models. However, if systems with several or even many fibers are to be analyzed, existing simulation methods reach their limits. As remedy, mixed-dimensional modeling approaches drastically reduce the size of a simulation model while maintaining comparable modeling quality, which also allows the analysis of systems with thousands of embedded fibers. The embedding solid material is still considered a volume problem, whereas the embedded fibers are modeled by slender one-dimensional structural models (beams). In this way, the meshing of the volume model and the fiber model is decoupled from each other during mesh generation. Both are coupled with each other in the numerical model by embedded mesh approaches. So far, the coupling constraints are enforced using so-called penalty methods, which penalize deviations from the exact constraint fulfillment. Penalty methods are conceptually easy to implement but come with mathematical disadvantages such as inexact constraint enforcement and pose immense challenges to the solution algorithms. This is exactly where the proposed research project comes in: By using Lagrange multipliers for constraint enforcement, the coupling conditions can be satisfied exactly and mathematical deficiencies and ill-conditioning from penalty methods are avoided. Yet, the discretization of the Lagrange multiplier field requires special consideration and demands suitable solution techniques. This project will deliver stable and robust discretization schemes for Lagrange multipliers for mixed-dimensional fiber/solid coupling. By enrichment with mechanical and discretization insight, novel and custom-built multilevel solvers and block preconditioning methods will be developed for the problem class of mixed-dimensional fiber/solid coupling. This clever combination of physical knowledge with numerics and simulation technology will allow to analyze realistic systems efficiently and scalably on modern parallel high-performance computers. This will open the door to further analyses (e.g. for the quantification of uncertainties or optimization problems).

由细纤维增强的结构和系统在许多应用中发挥着越来越重要的作用。嵌入纤维不仅可以对部件或系统的机械性能产生积极影响，还可以对其功能特性产生积极影响。例如，在混凝土构件中嵌入短钢纤维可以提高抗拉强度(即使在开裂后)以及抗冲击性。在航空航天工程或医疗器械设计的轻质结构中也有进一步的应用。为了定量表征这种纤维对体系行为的影响，尤其是为了预测和优化体系的行为，必须详细了解纤维与周围固体材料的相互作用。到目前为止，单根光纤的行为既可以通过实验确定，也可以通过完全解析的模拟模型来确定。然而，如果要分析具有几个甚至多个光纤的系统，现有的模拟方法就达到了它们的极限。作为补救措施，混合维度建模方法大大减小了仿真模型的大小，同时保持了相当的建模质量，这也允许分析具有数千个嵌入光纤的系统。嵌入的固体材料仍然被认为是一个体积问题，而嵌入的纤维是由细长的一维结构模型(梁)来模拟的。这样，在网格生成过程中，体积模型和纤维模型的网格化彼此分离。两者通过嵌入网格方法在数值模型中相互耦合。到目前为止，耦合约束是使用所谓的惩罚方法来实施的，这种方法惩罚与精确约束履行的偏差。惩罚方法在概念上易于实现，但存在数学上的缺陷，如约束执行不精确，并给求解算法带来巨大的挑战。这正是所提出的研究项目的用武之地：通过使用拉格朗日乘子来实施约束，可以精确地满足耦合条件，并避免了惩罚方法的数学缺陷和病态。然而，拉格朗日乘子场的离散化需要特殊的考虑和适当的求解技术。该项目将为混合维光纤/固体耦合的拉格朗日乘子提供稳定和健壮的离散化方案。通过丰富力学和离散化的洞察力，将为混合维纤维/固体耦合问题类开发新的和定制的多层解算器和块预处理方法。这种物理知识与数值和模拟技术的巧妙结合将允许在现代并行高性能计算机上高效和可扩展地分析现实系统。这将为进一步的分析(例如，对不确定性或优化问题的量化)打开大门。