Robust Structured Multigrid Algorithms for Mechanics of Heterogeneous Media

随着现代计算资源的稳步增长，在科学和工程的许多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要驱动的，在这种情况下，原型工程或实验室规模的实验是不切实际的或棘手的，但在这种情况下，底层的物理过程被充分理解，以产生精确的数学模型，并以此为基础的计算方法。事实上，这种模拟在其整个历史中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。该建议的主要重点是计算机算法的开发、分析和实现，以加速这些计算。广泛的一类物理问题由类似的数学方程组建模，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物理能量最小化原理与质量守恒的约束。虽然等价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不是。因此，本研究的重点是在扩展这些方法的约束最小化模型与这些问题。两个关键方面将进行研究，集中在所谓的最佳算法，这些模拟的发展，并在发展这些算法，以充分利用现代计算硬件。在这个研究计划中的算法集中在多重网格方法，命名后的多个尺度的问题，用于实现一个有效的解决方案的算法。对于具有可变材料特性的问题，应用多重网格方法的关键挑战是开发适当的平均模型来考虑长距离行为，以及开发有效的补充技术来解决短距离行为。在这个建议中，一个主要的重点是做后者的方式，大大利用现代高性能计算架构，产生算法，是有效的，在理论和实践中，所提出的研究的结果承诺提供一个强有力的一步，在学术研究这些问题和软件，可以直接应用于其他科学和工程学科，并在工业实践。因此，预期的经济利益是巨大的，特别是对高技术和生物技术部门。拟议的工作将直接导致几个学生的培训，在本科和研究生阶段，在计算科学和工程的新兴学科的关键技能，使他们能够填补这些行业的高需求的角色，加拿大信息经济必不可少。

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.