New Finite Element Techniques for Simulating Flows and Waves

Portland State University, Portland OR

Investigators

Abstract

Computer simulations of various natural and technological processes require accurate and efficient numerical approximation of flows and waves. This project advances the state of the art by developing unconventional approaches that improve numerical techniques at the heart of such simulations. In addition to the anticipated mathematical inventions, the project also plans to develop a public-domain open-source software product implementing the new techniques and use the product to solve simulation problems in varied application domains including geological hazard mitigation and material science. Through inclusion of under-represented minorities, the project activities contribute to the foundation's goals to widen participation of all in science. The technical work on the project is divided into two lines of inquiry. One leads to new methods for capturing wave solutions of hyperbolic systems, methods that are expected to excel on the emerging many-core architectures. Another line of inquiry leverages a new mathematical ingredient to obtain structure-preserving numerical approximations of viscous flows. The first line of inquiry is motivated by the observation that when hyperbolic solutions can be advanced in time by varying amounts at varying spatial locations, the allocation of computational resources is optimized. When this can also be done concurrently, much faster simulations are possible. New fast methods on unstructured spacetime meshes of causal spacetime tents are the projected outcome. The advances from this project benefit simulation of various wave propagation systems as well as inviscid compressible flows. Viscous incompressible flows are targeted by the second line of inquiry. Using a Sobolev space of matrix-valued functions, novel formulations are constructed that yield optimal fluid stress approximations, exact mass conservation, and pressure robustness. Work in both lines of inquiry involve answering several technical questions, shown to be fundamental, and of potential disciplinary impact beyond the confines of this project. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.