Rapidly-convergent, high-performance PDE solvers for materials-science and engineering applications: theory, implementation and applications.

California Institute Of Technology, Pasadena CA

Investigators

Abstract

Bruno DMS-1008631 The investigator develops and analyzes novel high-performance, highly accurate numerical algorithms for computing solutions of partial differential equations (PDE), with application to a wide range of problems in materials science, engineering and medicine. The PDE solvers apply to problems involving: (i) Various physical observables (elastic and electromagnetic fields, linear and nonlinear acoustic fields, thermal fields, fluid-flow) within and around (ii) Complex structures (photonic or electronic devices, singular geometries with corners, edges or cracks, anatomic geometries, air, water or land vehicles built from metals or modern composite materials), and containing (iii) Fluids or solid materials -- including composite elastic media, dielectrics, perfect and lossy conductors, compressible and incompressible fluids, as well as media leading to dispersion and frequency-dependent absorption. The computational methodology underlying the proposed work is based on a class of numerical solvers and surface-representation and meshing methodologies developed in recent years under the direction of the investigator. These are Fourier-based integral and differential solvers that can produce solutions with high-order accuracy, unconditional stability, and no numerical dispersion, for realistic engineering geometries including features such as full aircraft, complex anatomic configurations for medical applications, complex photonic or electronic devices, etc. In practice, these types of solvers have demonstrated up to one-thousand times faster numerics, for a given accuracy, than some of the most competitive solvers otherwise available: the new methods can enable solution of previously intractable problems. The project impacts upon a variety of areas of societal interest, including medicine (diagnostic and therapeutic ultrasound with application to, e.g., tumor detection, kidney stone destruction, and targeted drug delivery), electrical engineering (optics, electronics), military and civilian remote sensing and communications (radar, sonar, stealth, antennas), design of efficient air, water or land vehicles, etc. The new algorithms, which in a number of challenging case studies have demonstrated up to one-thousand times faster numerics than previous approaches, enable solution of previously intractable problems in areas such as those mentioned above. The project also trains graduate and undergraduate students and postdoctoral associates. Undergraduate students, for example, carry out summer-long research projects in an integrative environment involving graduate students and postdocs as well as the investigator and some of the industrial and lab researchers who propose specific engineering problems.