ALE-DGSEM approximation of wave reflection and transmission from a moving medium

Abstract

We derive a spectrally accurate moving mesh method for mixed material interface problems modeled by Maxwell's or the classical wave equation. We use a discontinuous Galerkin spectral element approximation with an arbitrary Lagrangian–Eulerian mapping and derive the exact upwind numerical fluxes to model the physics of wave reflection and transmission at jumps in material properties. Spectral accuracy is obtained by placing moving material interfaces at element boundaries and solving the appropriate Riemann problem. We present numerical examples showing the performance of the method for plane wave reflection and transmission at dielectric and acoustic interfaces.

Introduction

Modeling the reflection and transmission of a wave from a moving material interface is important for a variety of applications in electromagnetism and acoustics. Examples include radio communication with moving spaceships [35], and ultrasound scanners, e.g., imaging the human lung [32]. Further examples include wave scattering from vibrating surfaces in electromagnetism [5], [7], [11], [16], [17], [28], acoustics [23], [36], [38], and biology [3], [4], [12], [22], [24], [31].

The reflection and transmission of waves from moving material interfaces differ from reflection and transmission at stationary interfaces. The problem of wave reflection and transmission from a moving dielectric medium has been examined theoretically by many authors [19], [34], [39], [40], [42], [43], [45], [48], whereas the classical wave equation in a moving medium has received limited attention [23], [29], [41]. Interestingly, the physics of the problem changes since the motion introduces Doppler shifts in the frequencies, shifts in the amplitudes, and the possible compression of the medium [7], [11].

Computationally, electromagnetic scattering from a uniformly moving body has been treated by Harfoush et al. [16] with the FDTD method applied directly to Maxwell's equations. Ho [17] reported a computational solution of the scattering from moving boundaries and in a moving dielectric medium [18] in one dimension. All these approaches used rigid meshes.

A common way to approximate solutions to problems with moving boundaries is to use an arbitrary Lagrangian–Eulerian (ALE) formulation [1]. In the ALE formulation, one maps a time dependent domain ${\Omega }_t$ that has moving boundaries onto a fixed reference domain Ω. In the process, conservation law equations in the original domain are transformed to conservation law equations in the reference domain [15], [30], [33]. In the numerical approximation on the reference domain, the new set of equations depends on the mesh velocity.

To design methods to solve wave propagation problems on a moving domain, we will subdivide ${\Omega }_t$ into K elements, use an ALE formulation on each element, and discretize on the reference domain with a high-order DG spectral method. For the DG method to approximate the solution for reflection and transmission of a wave from a material interface with spectral accuracy, one simply ensures that an element boundary is placed on the material discontinuity [26]. The development and application of DG spectral methods for static domains are reviewed in [6], [8], [26].

The only coupling between adjacent elements in the DG approximation arises from the flux at the boundaries. One uses a numerical flux to compute the flux at moving element boundaries, which is found from the solution of the appropriate Riemann problem [47]. Generally, the material properties on either side of a moving element boundary may differ.

The derivations and results presented in this paper extend the previous work by Acosta Minoli and Kopriva who studied high-order wave propagation problems on moving meshes [1] and wave scattering from moving boundaries, e.g., a moving mirror [2]. This work is the next logical step where we consider wave reflection and transmission from moving internal interfaces rather than from purely reflective moving boundaries considered in [2].

We provide a detailed description of the physics as well as the mathematics when we derive the exact upwind numerical fluxes for moving material interfaces. The computation of the numerical flux at a moving material interface or within a moving object must incorporate the altered physics of wave reflection and transmission from the moving material. As was noted above, this physics differs from classical physics [37], [43]. To test the derived numerical fluxes for wave scattering models, we present numerical studies using Maxwell's equations and the classical wave equation. These studies include reflection and transmission of transverse electric (TE) or sound waves from a plane material interface moving at constant velocity, as well as the pure reflection of TE or sound waves from a moving plane mirror.

This paper is organized as follows: In Section 2 we review the discontinuous Galerkin spectral element method (DGSEM) when the arbitrary Lagrangian–Eulerian (ALE) mapping is used to incorporate the motion of the mesh. In Section 3 we derive the augmented physical fluxes for the ALE formulation as well as the exact, upwind numerical fluxes to describe wave reflection and transmission in a moving material. In Section 4 we present the numerical results. Finally, we present a summary and conclusions in Section 5. Appendix A outlines the test solutions used in the time step convergence studies.