High-Order Asymptotic and Numerical Techniques for the Simulation of Wave Scattering Processes

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

The PI proposes a development of effective and accurate algorithms for simulation of electromagnetic and acoustic wave scattering processes. The project includes the development of efficient, error-controllable schemes based on asymptotic expansions and on direct discretizations. While the computational emphasis will be on efficiency, error-controllability will demand careful mathematical analysis; both requirements will call for a number of intellectual innovations. Among the methods based on asymptotic expansions, the PI proposes to: (i) further advance his work on high-order geometric perturbation methods. Here, effort will concentrate on the design of alternative schemes with improved stability properties; and (ii) the search for an innovative (rigorous) high-frequency (integral-equation) solver for scattering by surfaces and bounded bodies that exhibits the advantages of asymptotic high-frequency treatments (i.e. a frequency-independent number of degrees of freedom) while allowing for error control at any fixed, finite frequency. A combination of the developments in (i) and (ii) will also be sought in the design of a "multi-scale" algorithm. The program on direct methods, on the other hand, shall be concerned with the development of algorithms for the treatment of penetrable bodies. For this, it is proposed to develop (iii) accelerated high-order methods for (volumetric) integral equations; novelties in the proposed approach include the design of specialized high-order quadrature rules for volume potentials, and their incorporation into a general strategy for their fast evaluation, based on (planar arrays of) "equivalent sources". Finally, the proposal presents a project on (iv) certain specific aspects of (Runge-Kutta) discontinuous Galerkin finite element approaches, including the implementation of exact radiation conditions, the design of optimal inter-element fluxes and time-integration strategies, and the derivation of a-posteriori error estimates for adaptive error control. Electromagnetic and acoustic devices are ubiquitous in present day technology. Indeed, electromagnetism and acoustics have found and continue to find applications in a wide array of areas, encompassing both civilian and military purposes. Among the former, applications of current interest include those related to communications (e.g. transmission through optical fiber lines), to biomedical devices and health (e.g. ultrasound, tomography, power-line safety, etc), to circuit or magnetic storage design (electromagnetic compatibility, hard disc operation), to geophysical prospecting, and to non-destructive evaluation (e.g. crack detection), to name but just a few. Applications in defense, on the other hand, include the design of military hardware with decreased signatures ("virtual prototyping"); automatic target recognition (e.g. bunkers, mines and buried ordnance, etc); propagation effects on communications and radar systems (e.g. over complex terrains, through the atmosphere, etc); tactical antenna design; etc. Although the principles of acoustic and electromagnetic wave propagation are well understood, their application to practical configurations of current interest, such as those that arise in connection with the examples above, is significantly complicated and far beyond manual calculation in all but the simplest aspects. These complications typically arise from geometrical and/or compositional complexity in the underlying structures, from the intricacies of the electromagnetic fields, or from both. The significant advances in computer modeling of acoustic and electromagnetic wave interactions that have taken place over the last two decades, on the other hand, have made it possible to shift the classical "trial and error" design paradigm for devices that work on such interactions to one that heavily relies on computer simulation. Still, the sheer complexity of some applications of current interest and the ever-increasing industrial demand for faster and more reliable software, present significant challenges to state-of-the-art numerical solvers. This proposal is focused on the design and testing of improved algorithms that, based on sophisticated mathematical techniques and developments, will result in implementations that will substantially expand on the capabilities of presently available simulators.