Multi-Model Multi-Domain Computational Methods: Project Overview

A numerical laboratory that employs domain decomposition in two fundamentally distinct and equally important ways will be developed and applied to Grand Challenge problems in fluid dynamics. The two domain decomposition paradigms are medium-grained data decomposition and coarse-grained task decomposition. Data decomposition, inherited by partitioning a domain that is homogeneous with respect to the relevant local governing equations, is a well established route to efficient parallelism on a single task. Task decomposition, in which different governing phenomena in different regions are addressed with different algorithms while interfacial compatibility is maintained, is a natural approach to multi-physics applications. A rich theory undergirds the former paradigm. The latter is less fully understood from a theoretical viewpoint, though the algorithmic tools applied are broadly the same: iterative solution on overlapping subdomains or reduction to an interfacial problem. Besides the implementation of these algorithmic tools in publicly available software, we are endeavoring to undergird theoretically the task decomposition form of domain decomposition for two specific applications with multiple physics, aerodynamics and acoustics, and to combine both data and task paradigms into a common theory.

The past ten years have seen extensive development in the theory and practice of domain decomposition methods. However, the most fruitful and creative years for their advancement are just ahead, as the methods are passed from developers to computational engineers and scientists in need of the memory capacities and processing rates of distributed computing for applications. Most such applications can be formulated in a hierarchical series of models, ranging from simple (scalar and sometimes even linear) to complex (multiple fields and nonlinear). At any level of the mathematical model hierarchy, these problems are CPU- and memory-intensive in industrial contexts, and are becoming more so in multidisciplinary combinations.

Problems in aerodynamics and acoustics, as well as fluid-structure problems with two-way interactions, are usually characterized by a complex nearfield and a simple farfield, with a transition zone between them whose location is a priori unknown. Because of difficulties both of mathematical formulation and of code implementation in managing multiple models, either of two suboptimal approaches is often adopted: a simple model is used everywhere and patched up by source terms or ad hoc post-processing interpretations, or a complex model is used everywhere and limited in resolution by finite memory or slow convergence rate. In expert hands, simple models, such as the potential equation of aerodynamics or the convective wave equation of aeroacoustics, can be coaxed with inhomogeneous sources to yield valuable practical results; such tools are employed daily in the industrial setting of one of the co-PIs. By the same token, with enough computing power, complex models can be impressively employed in demonstration calculations. One of our aims is to systematically combine such models to obtain the strengths of each -- routine overall computability and guaranteed physical reliability that comes from using a high fidelity model where required. Our other chief aim is to deliver respectable (near scalable) and portable parallelism for such problems through data-parallel domain decomposition. Our software, will take primarily the form of application modules built upon the the Argonne National Laboratory PETSc library, which will be useful for workstations, workstation networks, and massively parallel computers.

Effective multi-domain parallel solutions to multi-model coupled field problems will have widespread usefulness, particularly in light of the growing emphasis on multidisciplinary analysis and optimization. The current pioneers of multidisciplinary optimization are finding it necessary to revisit the lower levels of the constituent modeling hierarchies. It is not practical to simultaneously upgrade all of the low-level models employed inside of optimization loops to the complex models used in the state-of-the-art of the individual disciplines, for reasons of complexity and severe ill-conditioning. This creates an obvious niche for the deliverables of this proposal -- highly performant solvers with built-in support for multi-model analysis.