Upscaling and multilevel methods for three dimensional elasticity via element agglomeration

该项目的重点是在数值模型，它描述了弹性材料的行为在多个尺度上，表现出随机行为的最新理论和算法的进步的集成。该项目将包括算法设计，收敛性和复杂性分析，以及在现实模拟中的升级和多级算法的性能出现的问题。拟议的研究：（1）帮助开发新的和强大的方法来升级，提供可靠的计算和预测结构力学;（2）支持这些方法迁移到现实生活中的科学和工程模拟;以及（3）通过研究和教育活动吸引更广泛的科学界，强调了弹性材料数值模拟中从自适应离散到鲁棒求解器的综合方法。拟议的研究旨在提高对微分几何和拓扑学技术之间的相互作用，导致离散化与物理/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佳算法性能的聚集方法。 弹性问题的新型高效准确的尺度放大技术在材料科学和地球科学中具有潜在的应用。此外，精确的粗离散产生有效的多级求解器的线性系统来自相应的离散的线性弹性。这样的求解器使得模拟具有更精细的空间分辨率和/或减少用于这样的模拟的必要的计算资源。 最后，升尺度技术的设计与一般的离散化设计有许多相似之处。该项目的成功将促进精确的离散计划和强大的基于单元聚集的线性弹性方程求解器。

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.