A Hierarchical Multiscale Method for Nonlocal Fine-scale Models via Merging Weak Galerkin and VMS Frameworks

通过合并弱伽辽金和 VMS 框架的非局部细尺度模型的分层多尺度方法

• 批准号: 1620231
• 负责人: Arif Masud
• 金额: $ 15万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1620231&HistoricalAwards=false
• 关键词: Hierarchical; Multiscale; Method; Nonlocal; Fine

Many problems in the natural sciences and engineering involve phenomena that possess a spectrum of material, spatial, and temporal scales which correspond to coarse and fine scale physics. Very often fine scale physics is also nonlocal and therefore these problems pose a great challenge to the current computational techniques. Nonlocality of fine-scales and advecting sharp gradients are two key ingredients in the modeling of several classes of fluid dynamics problems. This research effort focuses on the development of variationally based methods for problems with steep gradients and discontinuities in the underlying fields and wherein fine scales have a nonlocal feature. Of specific interest are propagating steep fronts for which interacting discontinuities challenge the stability of the numerical methods. Typical examples are chemically reacting fluids permeating through porous elastic solids where fast reaction rates produce steep concentration fronts. Such problems arise in (i) high temperature injection molding of fibrous composites in micro and nanomaterials engineering, and (ii) enhanced oil recovery and secondary shale gas recovery processes in petroleum engineering. Another class of problems from mathematical physics is advection dominated viscous flows leading to anisotropic turbulence. The PI will develop a unique and novel, simultaneous top-down and bottom-up multiscale approach for consistent representation of both the hierarchy of scales as well as the structure of inter-scale coupling operators for complex fluid mechanics problems. It blends ideas from the Variational Multiscale (VMS) method that helps decompose the governing system of equations into coarse-scale and fine-scale sub-problems and then employs Weak Galerkin (WG) ideas at the fine-scale variational level to extract models for the finer physics. Weak continuity of functions that is facilitated by the Weak Galerkin method results in fine-scale models that are nonlocal. A further generalization of WG via Discontinuous Galerkin (DG) ideas provides a framework to develop methods for problems with sharp gradients on sound mathematical basis. This notion leads to variationally derived Discontinuity Capturing (DC) methods that are independent of the user-defined or user-designed parameters. Emphasis is placed throughout on variationally consistent interscale coupling with rigorous treatment of the continuity conditions that are critical for the mathematical and algorithmic stability. The resulting computational algorithms will be ideal for massively parallel computing on distributed systems where message passing traditionally has been a bottle neck. The variational structures underlying the new methods will increase local-solves that are cost effective because of local resident memory on the new generation of processors while substantially reducing global communication between processors, thereby leading to efficient and economic computations. The mathematical frameworks and computational algorithms emanating from this work will be broadly disseminated by publication in high-quality archival journals, and by presentations at high impact conferences.

自然科学和工程中的许多问题都涉及到具有物质、空间和时间尺度光谱的现象，这些尺度对应于粗尺度和精细尺度的物理。细尺度物理通常也是非局域的，因此这些问题对当前的计算技术构成了巨大的挑战。细尺度的非局域性和平流的尖锐梯度是模拟几类流体动力学问题的两个关键因素。这项研究工作集中于发展基于变分的方法来解决下垫场中具有陡峭梯度和不连续性的问题，其中细尺度具有非局部特征。特别令人感兴趣的是传播陡峭的锋面，相互作用的不连续性对数值方法的稳定性提出了挑战。典型的例子是化学反应流体渗透到多孔弹性固体中，其中快速的反应速度会产生陡峭的浓度前沿。这些问题出现在(I)微纳米材料工程中纤维复合材料的高温注射成型，(Ii)石油工程中的强化采油和页岩气二次开采工艺。数学物理中的另一类问题是平流主导的粘性流动，导致各向异性湍流。PI将开发一种独特而新颖的、同时自上而下和自下而上的多尺度方法，以一致地表示复杂流体力学问题的尺度层次和尺度间耦合算子的结构。它融合了变分多尺度(VMS)方法的思想，该方法将控制方程系统分解为粗尺度和细尺度的子问题，然后在细尺度的变分水平上使用弱Galerkin(WG)思想来提取更精细的物理模型。由弱Galerkin方法促成的函数的弱连续性导致了非局部的精细模型。通过不连续Galerkin(DG)思想对WG的进一步推广，为在合理的数学基础上发展具有尖锐梯度问题的方法提供了一个框架。这一概念导致了与用户定义或用户设计的参数无关的变分派生的不连续捕获(DC)方法。始终强调变分一致的尺度间耦合与对数学和算法稳定性至关重要的连续性条件的严格处理。由此产生的计算算法将是分布式系统上大规模并行计算的理想选择，在分布式系统中，消息传递传统上一直是一个瓶颈。基于新方法的变分结构将增加由于新一代处理器上的本地驻留存储器而具有成本效益的局部解，同时显著减少处理器之间的全局通信，从而导致高效和经济的计算。这项工作产生的数学框架和计算算法将通过在高质量的档案期刊上发表和在影响较大的会议上发表演讲来广泛传播。