Robust Structured Multigrid Algorithms for Mechanics of Heterogeneous Media

随着现代计算资源的不断增加，许多科学和工程领域对高保真计算仿真工具的需求不断增加。这种需求是由理解物理过程的需求驱动的，在这种情况下，原型工程或实验室规模的实验是不切实际或难以处理的，但在这种情况下，潜在的物理过程被充分理解，可以产生精确的数学模型，以此为基础，建立计算方法。事实上，这样的模拟在其历史上一直是计算的驱动力，现代模拟工具包包括非常详细的模型，因此，与它们相关的计算成本很高。本提案的主要重点是开发、分析和实现计算机算法来加速这些计算。一类广泛的物理问题是由类似的数学方程组来建模的，重点是不可压缩流体和不可压缩固体的动力学。这些方程自然地模拟了与质量守恒约束相结合的物理能量最小化原理。虽然等效的无约束能量最小化问题是一类已知的有效模拟技术，但这里感兴趣的约束版本却不是。因此，本研究的重点是将这些方法扩展到与这些问题相关的约束最小化模型。将研究两个关键方面，重点是这些模拟的所谓最佳算法的发展，以及开发这些算法以充分利用现代计算硬件。本研究计划中的算法集中在多网格方法上，该方法以问题的多个尺度命名，用于实现有效的解决算法。对于具有可变材料特性的问题，应用多网格方法的关键挑战是开发适当的平均模型来解释远距离行为，以及开发有效的互补技术来解决短距离行为。在本提案中，主要关注的是以一种极大地利用现代高性能计算架构来产生在理论和实践中都有效的算法的方式来实现后者。所提出的研究结果有望为这些问题的学术研究提供强有力的一步，并且可以直接应用于其他科学和工程学科以及工业实践的软件。因此，预期的经济效益是显著的，特别是对高技术和生物技术部门。拟议的工作将直接导致对本科生和研究生的培训，使他们掌握新兴的计算科学和工程学科的关键技能，使他们能够在这些行业中担任高需求的角色，这对加拿大的信息经济至关重要。

With the steady increase in modern computing resources has come an increased demand for high-fidelity computational simulation tools in many areas of science and engineering. This demand is driven by the need for understanding physical processes at scales where prototype engineering or lab-scale experimentation are impractical or intractable, but where the underlying physical processes are well-enough understood to yield accurate mathematical models on which to base a computational approach. Such simulation has, indeed, been a driving force in computation throughout its history, and modern simulation toolkits include highly detailed models that, consequently, have high computational costs associated with them. The principle focus of this proposal is the development, analysis, and implementation of computer algorithms to accelerate these computations.A wide class of physical problems are modeled by a similar family of mathematical equations, focusing on the dynamics of incompressible fluids and incompressible solid bodies. These equations naturally model physical energy-minimization principles coupled with the constraint of conservation of mass. While the equivalent unconstrained energy minimization problems are in a class for which efficient simulation techniques are known, the constrained versions of interest here are not. Thus, the focus of this research is in the extension of these approaches to the constrained minimization models associated with these problems. Two key aspects will be studied, focusing on the development of so-called optimal algorithms for these simulations, and on developing these algorithms to take full advantage of modern computational hardware.The algorithms in this research program are focused on the multigrid methodology, named after the multiple scales of a problem that are used in order to achieve an efficient solution algorithm. For problems with variable material properties, the key challenges in applying the multigrid approach are in developing appropriately averaged models to account for long-distance behaviour, and in developing efficient complementary techniques for resolving short-distance behaviour. In this proposal, a major focus is given to doing the latter in a way that greatly leverages modern high-performance computing architectures to produce algorithms that are efficient both in theory and in practice.The results of the proposed research promise to provide both a strong step forward in the academic study of these problems and software that can be directly applied in other science and engineering disciplines, and in industrial practice. Thus, the expected economic benefit is significant, particularly to the high-technology and bio-technology sectors. The proposed work will directly result in the training of several students, at the undergraduate and graduate levels, in key skills in the emerging discipline of computational science and engineering, enabling them to fill high-demand roles in these industries, essential for the Canadian information economy.