A Fictitious Domain Method with Mixed Finite Elements for Elastodynamics

Abstract

Abstract. We consider in this paper the wave scattering problem by an object with Neumann boundary conditions in an anisotropic elastic body. To obtain an efficient numerical method (permitting the use of regular grids) we follow a fictitious domain approach coupled with a first order velocity stress formulation for elastodynamics. We first observe that the method does not always converge when the Q div 1 − Q 0 finite element is used. In particular, the method converges for some scattering object geometries but not for others. Note that the convergence of the Q div 1 − Q 0 finite element method was shown in [E. Bécache, P. Joly, and C. Tsogka, SIAM J. Numer. Anal., 39 (2002Anal., 39 ( ), pp. 2109Anal., 39 ( -2132 for the elastodynamic problem in the absence of a scattering object (i.e., without the coupling of the mixed finite elements with the fictitious domain method). Therefore we propose here a modification of the Q div 1 − Q 0 element following the approach in [E. Bécache, J. Rodríguez, and C. Tsogka, On the convergence of the fictitious domain method for wave equation problems, Technical report 5802, INRIA, 2006], where the simpler acoustic case was considered. To study the numerical properties of the new element we carry out a dispersion analysis. Several numerical simulations as well as a numerical convergence analysis show that the proposed method provides a good approximate solution.Key words. mixed finite elements, fictitious domain method, elastic waves, convergence AMS subject classifications. 65M60, 65M12, 65M15, 65C20, 74S05 DOI. 10.1137/060655821 1. Introduction. We consider here the scattering of elastic waves by objects or cracks with a homogeneous Neumann boundary condition. This is of interest in several applications such as ultrasonic nondestructive testing, seismic wave propagation, etc. To develop an efficient numerical method we intend to use a fictitious domain approach. Following this approach, we extend artificially the solution to a fictitious domain with a simple shape, typically a rectangle in 2D. Then the boundary condition on the complex geometry is enforced by introducing an auxiliary unknown that leaves only at the boundary of the object. The key point of the method is that the mesh for the unknowns leaving on the enlarged domain can be chosen independently of the geometry of the object. In particular, one can use regular grids or structured meshes which allows for simple and efficient computations.The fictitious domain method was initially introduced for stationary problems [3,30,24,18,20,27], and it was then successfully generalized to time-dependent problems. In particular, it was used for modeling unsteady incompressible viscous flow with fixed or moving boundaries [25,21,22,23] and in several applications of wave propagation, such as acoustic scattering problems [15,10,12], electromagnetic scattering problems [17], and musical acoustics [33,29,4]. For both stationary and time-dependent problems the fictitious domain improves the performance of the nu-

Convergence results of the fictitious domain method for a mixed formulation of the wave equation with a Neumann boundary condition

“…the bilinear forms as, 8) and the bracket w · n, μ Γ is the duality product between G and G . Note that, due to (1.3), under assumptions of Theorem 1.1, the data also belong to…”

“…Due to this enrichment of the approximation space for the pressure field, spurious modes that contaminate the discrete solution appear when using the modified element. These spurious modes can be characterized further using a dispersion analysis and it is easy to prove [7,8] that their main components belong to M r h . In order to damp this main part of the spurious modes, we introduce the following approximate problem (instead of (1.13)):…”

“…The numerical illustrations, done with a straight horizontal crack, seemed to show the convergence of the method. However, in [8], it was shown considering other configurations of cracks (e.g. diagonal cracks) that this approach did not always lead to a convergent method.…”

Mathematical and Numerical Study of Transient Wave Scattering by Obstacles with a New Class of Arlequin Coupling

“…We are now in position to present a theorem for existence/uniqueness of the problem (2,4) and guarantee its equivalence with respect to problem (2, 3):…”

“…The fictitious domain method presented in [2] is designed to take into account scattering by impenetrable obstacle. It has some of the properties mentioned above: it preserves a discrete energy and allows the use of a time step adapted to the discretization properties of the background medium.…”