Abstract
The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain. The use of asymptotic models is very efficient and the purpose of this work is to provide a rigorous justification of a new asymptotic model for low-cost numerical simulations. This model is based on asymptotic near-field and far-field developments that are then matched by a key procedure that we describe and demonstrate. We show that it is enough to focus on the regular part of the wave field to rigorously establish the complete asymptotic expansion. For that purpose, we provide an error estimate which is set in the whole space, including the transition region separating the near-field from the far-field area. The proof of convergence is established through Kondratiev’s seminal work on the Laplace equation and involves the Mellin transform. Numerical experiments including multiple scattering illustrate the efficiency of the resulting numerical method by delivering some comparisons with solutions computed with a finite element software.
Highlights
The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain
We have proposed a new solution methodology for solving 3D multiple scattering problems when the size of the obstacles is small with respect to the characteristic wavelength
This work contains two key results. It validates an asymptotic representation of the field diffracted by a small obstacle illuminated by an incident acoustic wave of very large characteristic length in front of the radius of the obstacle
Summary
Mechanical wave simulations are of great interest in many applications due to their capability of transporting information in the medium they propagate into. They are stable only by respecting the Courant–Friedrichs–Levy condition which shows a linear dependency of the time step with the space step Applying this condition to a direct numerical simulation based on a locally refined grid leads to increased computational costs since the time step will be adapted to the smallest cell. This is disabling when the domain has only a few small obstacles. We begin with describing the matching procedure for the construction of the asymptotic model This requires defining the near-field and far-field approximations which are matched to give a representation of the diffracted field throughout the space, disregarding the obstacle which should no longer be taken into account in the computational method. The analysis consists in establishing a convergence result which uses the Mellin transform (see [7]) in the formalism of Kondratiev spaces [24]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: ESAIM: Mathematical Modelling and Numerical Analysis
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
