Robust Structured Multigrid Algorithms for Mechanics of Heterogeneous Media

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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.