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

自然科学和工程中的许多问题都涉及到具有物质、空间和时间尺度的现象，这些尺度对应于粗尺度和细尺度物理学。很多时候，精细尺度物理也是非局域的，因此这些问题对当前的计算技术提出了很大的挑战。精细尺度的非局部性和平流梯度是几类流体动力学问题建模中的两个关键因素。这项研究工作的重点是发展变分为基础的方法的问题与陡峭的梯度和不连续性的基础领域，其中精细尺度具有非局部功能。特别感兴趣的是传播陡峭的前相互作用的不连续性挑战的数值方法的稳定性。典型的例子是渗透通过多孔弹性固体的化学反应流体，其中快速反应速率产生陡峭的浓度前沿。这样的问题出现在（i）微米和纳米材料工程中纤维复合材料的高温注射成型，和（ii）石油工程中的强化采油和二次页岩气开采过程中。另一类来自数学物理的问题是平流主导的粘性流导致各向异性湍流。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，用于一致地表示复杂流体力学问题的尺度层次结构以及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，该方法有助于将控制方程系统分解为粗尺度和细尺度子问题，然后在细尺度变分层次上采用弱伽辽金（WG）思想来提取更精细的物理模型。弱伽辽金方法所促进的函数的弱连续性导致非局部的细尺度模型。通过不连续Galerkin（DG）思想的WG的进一步推广提供了一个框架，以发展的方法与健全的数学基础上的尖锐的梯度问题。这个概念导致变分衍生的不连续性捕获（DC）方法是独立的用户定义或用户设计的参数。重点放在整个变分一致的尺度间耦合与严格的治疗的连续性条件是至关重要的数学和算法的稳定性。由此产生的计算算法将是理想的分布式系统上的大规模并行计算的消息传递传统上一直是一个瓶颈。新方法的变分结构将增加本地解决方案，这是成本效益，因为本地驻留内存的新一代处理器，同时大大减少了全球处理器之间的通信，从而导致高效和经济的计算。从这项工作中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高影响力的会议上发表而广泛传播。

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.