Author : Inga Girshfeld
Publisher :
ISBN 13 :
Total Pages : 53 pages
Book Rating : 4.:/5 (132 download)
Book Synopsis High-order Wave Tracking Strategy for Solving High-frequency Scattering Problems by : Inga Girshfeld
Download or read book High-order Wave Tracking Strategy for Solving High-frequency Scattering Problems written by Inga Girshfeld and published by . This book was released on 2021 with total page 53 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wave equations effectively model physical phenomena, applying but not limited to earthquake engineering, geophysical exploration, medical imaging, nondestructive testing, underwater acoustics, electromagnetics, etc. Extensively studied for over a century, the mathematics of wave propagation problems are relatively well-understood, but their computation poses substantial issues, especially for high-frequency regime [3]. Traditional FEM techniques require fine discretization or high order elements, resulting in the pollution effect [1] and numerical instabilities. Over the last few decades, significant efforts have been dedicated toward developing alternative techniques, including a least-squares method, plane wave discontinuous Galerkin methods, etc. Helmholtz problems, which describe time harmonic wave propagation, are well understood mathematically [3], but difficult to solve numerically in the high-frequency regime [1]. Moreover, practical applications of the Helmholtz equation demand solving systems with more than ten million complex unknowns in the mid-frequency range. Thus, reducing the computational cost and the complexity of implementation while preserving the level of accuracy and expanding the frequency regime would have far-reaching effects in the area of real-world application as well as in the computationally important infrastructure. We propose a numerical method to efficiently solve the Helmholtz problem in the high-frequency wave regime by implementing oscillating basis functions, along with a wave tracking strategy to align the basis functions with the direction of the propagating field. Thus, we are able to reduce the number of basis functions which grants access to the high-frequency regime. We use an adaptive local wave tracking strategy that implements a least-squares method. On each element of the mesh, shape functions are rotated until one aligns with the direction of the propagated wave, determined by solving a nonlinear minimization problem using Newton's method. This method is an extended effort from [2], where the distinguishing difference is the choice of basis functions. Moreover, the computation of Jacobians and Hessians that arise in the iterations of Newton's method is based on the exact characterization of the Fréchet derivatives of the field with respect to the propagation directions. Such a characterization is crucial for the stability, fast convergence, and computational efficiency of the Newton algorithm.
