Upscaling and multilevel methods for three dimensional elasticity via element agglomeration

通过元素聚集实现三维弹性的升级和多级方法

• 批准号: 1418843
• 负责人: Ludmil Zikatanov
• 金额: $ 18万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-15 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1418843&HistoricalAwards=false
• 关键词: Upscaling; multilevel; methods; three; dimensional

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.

该项目侧重于数值模型中最新理论和算法进展的整合，该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，以及现实模拟中升级和多级算法性能中出现的问题。拟议的研究：（1）帮助开发新的、稳健的升级方法，为结构力学提供可靠的计算和预测； (2) 支持将此类方法迁移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中从自适应离散化到鲁棒求解器和反向的综合方法。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，从而导致与从物理/数学模型继承的几何和拓扑结构兼容的离散化。在此基础上，PI 计划开发可提供可证明的最佳算法性能的聚合方法。 针对弹性问题的新型高效、准确的放大技术在材料科学和地球科学中具有潜在的应用。此外，精确的粗离散化可以为来自相应线性弹性离散化的线性系统产生有效的多级求解器。此类解算器能够以更精细的空间分辨率进行模拟和/或减少此类模拟所需的计算资源。 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。该项目的成功将促进基于单元聚集的线性弹性方程的精确离散化方案和稳健的求解器。