Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide. (English) Zbl 1405.35121
Summary: This 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.
MSC:
| 35P25 | Scattering theory for PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 74J20 | Wave scattering in solid mechanics |
| 74R10 | Brittle fracture |
| 76Q05 | Hydro- and aero-acoustics |
| 78A50 | Antennas, waveguides in optics and electromagnetic theory |
| 74S05 | Finite element methods applied to problems in solid mechanics |
Keywords:
elastic waveguide; modal decomposition; finite elements; Dirichlet-to-Neumann map; biorthogonality; scatteringReferences:
| [1] | Bécache, E.; Bonnet-Ben Dhia, A.-S.; Legendre, G., Perfectly matched layers for the convected Helmholtz equation, SIAM J. Numer. Anal., 42, 409-433 (2004) · Zbl 1089.76045 |
| [3] | Fraser, W. B., Orthogonality relation for the Rayleigh-Lamb modes of vibration of a plate, J. Acoust. Soc. Am., 59, 215-216 (1976) · Zbl 0324.73022 |
| [4] | Gregory, R. D., A note on bi-orthogonality relations for elastic cylinders of general cross section, J. Elasticity, 13, 351-355 (1981) · Zbl 0522.73052 |
| [5] | Pagneux, V.; Maurel, A., Lamb wave propagation in inhomogeneous elastic waveguides, Proc. R. Soc. Lond. A, 458, 1913-1930 (2002) · Zbl 1056.74030 |
| [6] | Pagneux, V.; Maurel, A., Scattering matrix properties with evanescent modes for waveguides in fluids and solids, J. Acoust. Soc. Am., 116, 1913-1920 (2004) |
| [7] | Pagneux, V.; Maurel, A., Lamb wave propagation in elastic waveguides with variable thickness, Proc. R. Soc. A, 462, 1315-1339 (2006) · Zbl 1149.74348 |
| [8] | Kirrmann, P., On the completeness of Lamb modes, J. Elasticity, 37, 39-69 (1995) · Zbl 0818.73018 |
| [9] | Besserer, H.; Malischewsky, P. G., Mode series expansions at vertical boundaries in elastic waveguides, Wave Motion, 39, 41-59 (2004) · Zbl 1163.74317 |
| [12] | Lowe, M. J.S.; Diligent, 0., Low frequency reflection characteristics of the \(s_0\) Lamb wave from a rectangular notch in a plate, J. Acoust. Soc. Am., 111, 1, 64-74 (2002) |
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