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A Meshfree Method with Fundamental Solutions for Inhomogeneous Elastic Wave Problems

Valtchev, Svilen S. ; Alves, Carlos J. S.; Martins, Nuno F. M.

Proceedings of the 6th International Conference on Numerical Analysis (NumAn 2014), September 2-5, Chania,Greece, (2014),

We consider the numerical solution of the inhomogeneous Cauchy-Navier equations of elastodynamics, assuming time-harmonic variation for the displacement field U(x; t) = u(x)exp(-iwt) of an isotropic material with Lamé constants \lambda and \mu and density \rho. The resulting elliptic PDE, posed in a bounded simply connected domain \Omega is coupled with Dirichlet boundary conditions and solved trough a meshfree method, based on the Method of Fundamental Solutions (MFS).

In particular, an extension, from the scalar [1, 2] to the vector case, of the MFS is applied and the displacement field u is approximated in terms of a linear combination of fundamental solutions (Kupradze tensors) of the corresponding homogeneous PDE with different source points and test frequencies. The applicability of the numerical method is justified in terms of density results [3]. The high accuracy and the convergence of the proposed method will be illustrated through 2D numerical simulations. Convex and non-convex domains and different sets of boundary data and body forces will be considered. Interior elastic wave scattering problems will also be addressed.