Upscaling and multilevel methods for three dimensional elasticity via element agglomeration

该项目侧重于整合数值模型中最新的理论和算法进展，这些模型描述了弹性材料在多个尺度上的行为，表现出随机行为。该项目将包括算法设计、收敛和复杂性分析，以及在现实模拟中升级和多层算法性能中出现的问题。提出的研究：(1)有助于开发新的和强大的方法，为结构力学提供可靠的计算和预测；(2)支持将这些方法移植到现实生活中的科学和工程模拟中；(3)通过研究和教育活动吸引更广泛的科学界，强调弹性材料数值建模的综合方法，从自适应离散化到鲁棒求解，然后再回来。提出的研究旨在提高对微分几何和拓扑技术之间相互作用的理解，从而导致与继承自物理/数学模型的几何和拓扑结构兼容的离散化。基于此，pi计划开发集聚方法，提供可证明的最优算法性能。该方法在材料科学和地球科学中具有潜在的应用前景。此外，精确的粗离散化可以得到线性系统的高效多级解，这些解来自于相应的线性弹性离散化。这样的求解器使模拟具有更精细的空间分辨率和/或减少必要的计算资源为这样的模拟。最后，升级技术的设计与一般离散化的设计有许多相似之处。该项目的成功将有助于精确的离散化方案和基于单元团聚的线性弹性方程的鲁棒求解。

This project focuses on the integration of recent theoretical and algorithmic advances in numerical models, which describe the behavior of elastic materials on multiple scales, exhibiting stochastic behavior. The project will include algorithmic design, convergence and complexity analysis, as well as issues that arise in the performance of the upscaling and multilevel algorithms in realistic simulations. The proposed research: (1) aids the development of new and robust methods for upscaling that provide reliable calculations and predictions in structural mechanics; (2) supports the migration of such methods into real-life scientific and engineering simulations; and (3) engages the broader scientific community through research and educational activities, highlighting the integrated approach in numerical modeling of elastic materials from adaptive discretizations to robust solvers and back.The proposed research aims to improve understanding of the interplay between the techniques from differential geometry and topology, which lead to discretizations compatible with the geometric and topological structures inherited from the physical/mathematical model. Based on this, the PIs plan to develop agglomeration methods that offer provable optimal algorithm performance. The novel efficient and accurate upscaling techniques for elasticity problems have potential applications in material sciences and geosciences. In addition, accurate coarse discretizations yield efficient multilevel solvers for the linear systems coming from corresponding discretizations of linear elasticity. Such solvers enable simulations with finer spatial resolution and/or reduce the necessary computational resources for such simulations. Finally, the design of upscaling techniques has many similarities with the design of discretizations in general. The success of the project will facilitate accurate discretization schemes and robust solvers for linear elasticity equations based on element agglomeration.