Upscaling and multilevel methods for three dimensional elasticity via element agglomeration

该项目致力于整合最近在数值模型方面的理论和算法进展，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，以及在现实模拟中提升和多级算法性能方面出现的问题。拟议的研究：(1)帮助发展新的和稳健的方法，以提供结构力学中可靠的计算和预测；(2)支持将这种方法移植到真实的科学和工程模拟中；以及(3)通过研究和教育活动，促进更广泛的科学界的参与，突出弹性材料的数值模拟中的集成方法，从自适应离散到稳健求解器和反向。拟议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，这些技术导致离散与从物理/数学模型继承的几何和拓扑结构兼容。在此基础上，PI计划开发提供可证明的最优算法性能的聚集方法。这种新颖、高效、精确的弹性力学升阶技术在材料科学和地球科学中具有潜在的应用前景。此外，精确的粗离散化产生了由相应的线弹性离散得到的线性系统的有效的多层解算器。这样的解算器使模拟具有更好的空间分辨率和/或减少了这种模拟所需的计算资源。最后，升级技术的设计与离散化技术的设计有许多相似之处。该项目的成功将促进基于单元聚集的线弹性方程的精确离散格式和稳健的求解。

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.