Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide

Abstract

International audienceThis work concerns the numerical finite element computation, in the frequency domain, of the diffracted wave produced by a defect (crack, inclusion, perturbation of the boundaries, etc.) located in a 3D infinite elastic waveguide. The objective is to use modal representations to build transparent conditions on some artificial boundaries of the computational domain. This cannot be achieved in a classical way, due to non-standard properties of elastic modes. However, a biorthogonality relation allows us to build an operator, relating hybrid displacement/stress vectors. An original mixed formulation is then derived and implemented, whose unknowns are the displacement field in the bounded domain and the normal component of the normal stresses on the artificial boundaries. Numerical validations are presented in the 2D case

On the use of perfectly matched layers in the presence of long or backward propagating guided elastic waves

“…We present a few numerical results to validate the new method and assert its pertinence for the computation of solutions in a two-dimensional (infinite) plate by comparing it to a reference solution, which either has a closed form or is obtained numerically using a finite element method and the extension of the Dirichlet-to-Neumann approach proposed in [14]. In these numerical experiments, the values ρ = 2700 kg m −1 for the mass density, E = 69 GPa for the Young modulus and ν = 0.33 for the Poisson ration are used, corresponding to a plate made of aluminium.…”

“…The numerical solutions produced the PML method and the novel approach are thus compared with an approximation obtained by using a similar finite element discretization and the transparent boundary conditions developed in [14]. The configuration is identical to that of the first of the previous simulations, except for the plate, which is now locally perturbed by a single rectangular slot.…”

“…In the particular cases of a homogeneous plate (the Rayleigh-Lamb modes) or of a homogeneous circular rod (the Pochhammer-Chree modes), closed-form expressions of the modes are available and their determination reduces to a search for the zeros of a dispersion relation. It follows (see [14] for instance) from properties of the functional operators involved in the problem that the set of complex numbers β associated with the set of modes is both discrete and symmetric with respect to Re(β) = 0 and Im(β) = 0, a mode being said to be propagative if β ∈ R, and evanescent if Im(β) = 0.…”

“…Now, assuming the wavenumber β is a smooth function of the pulsation ω, this hypothesis implies that the group velocity ∂ω ∂β of a propagative mode does not vanish (see Lemma 4 and Remark 1 in [14]), which allows to define the propagation direction of the considered mode. We thus say that a propagative mode is rightgoing (resp.…”

“…However, this technique is sometimes discarded in elastic waveguides, because it fails when backward propagating modes are present (see [8] for instance). Alternative solutions, which use the modal representation, have been developed [20,14,21], but they may require specific implementation efforts. In section 4, we propose a strategy which combines the simplicity of the PML methods with the accuracy of the modal ones.…”