Iterative methods for scattering problems in isotropic or anisotropic elastic waveguides

Abstract

We consider the time-harmonic problem of the diffraction of an incident propagative mode by a localized defect, in an infinite elastic waveguide. We propose several iterative algorithms to compute an approximate solution of the problem, using a classical finite element discretization in a small area around the perturbation, and a modal expansion in the unbounded straight parts of the guide. Each algorithm can be related to a socalled domain decomposition method, with an overlap between the domains. Specific transmission conditions are used, so that at each step of the algorithm only the sparse finite element matrix has to be inverted, the modal expansion being obtained by a simple projection, using a bi-orthogonality relation. The benefit of using an overlap between the finite element domain and the modal domain is emphasized. An original choice of transmission conditions is proposed which enhances the effect of the overlap and allows us to handle arbitrary anisotropic materials. As a by-product, we derive transparent boundary conditions for an arbitrary anisotropic waveguide. The transparency of these new boundary conditions is checked for twoand three-dimensional anisotropic waveguides. Finally, in the isotropic case, numerical validation for twoand three-dimensional waveguides illustrates the efficiency of the new approach, compared to other existing methods, in terms of number of iterations and CPU time.

The Half-Space Matching method for elastodynamic scattering problems in unbounded domains

“…As mentioned in the introduction, the case of isotropic backgrounds (C satisfying (9)) is widely studied in the literature whereas general backgrounds still raise theoretical and numerical questions.…”

“…The equations (1) have to be completed by a condition at infinity to select the "outgoing" or the "radiating" solution. For isotropic backgrounds (C satisfying (9)), this can be achieved through the Kupradze-Sommerfeld radiation condition, which writes as folllows…”

“…Let us now describe how to get an analytical representation of the solution in each half-space in terms of its trace. Namely, let for all Φ determine U j (Φ) the outgoing solution of (37) where σ(U j ) is linked to the displacement field by the Hooke's law (5) in the isotropic case (9) and where by (7), the coefficients ρ, λ and µ are constant in Ω j . The meaning of outgoing is given below.…”

The Half-Space Matching Method for Elastodynamic Scattering Problems in Unbounded Domains

“…Note that this γ j is the operator appearing in the first-order absorbing boundary conditions used in [55]. Let us mention that the choice of b and of γ j may be more important when solving the problem with an iterative algorithm, using a sparse approximation as a preconditioner, in the spirit of [9]. In that case, as in domain decomposition methods [30], increasing the overlap b − a and using for γ j the approximation of the DtN map in normal incidence enable to speed up the convergence of iterative algorithms.…”

Physics-based modelling and simulation of reverberating reflections in ultrasonic guided wave inspections applied to welded rail tracks

“…Jezzine et al [29] mounted a simple transducer model accounting only for normal stresses at the end of a semi-nite 1D waveguide and computed the guided wave reections from a normal crack. Later they developed a method to compute scattering from more general structural features and defects with complex geometry [30,31]. This method has been integrated into the analysis software called CIVA [32].…”