Browsing by Author "Martins, Nuno F. M."
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- Extending the method of fundamental solutions to non-homogeneous elastic wave problemsPublication . Alves, Carlos J. S.; Martins, Nuno F. M.; Valtchev, SvilenTwo meshfree methods are developed for the numerical solution of the non-homogeneous Cauchy–Navier equations of elastodynamics in an isotropic material. The two approaches differ upon the choice of the basis functions used for the approximation of the unknown wave amplitude. In the first case, the solution is approximated in terms of a linear combination of fundamental solutions of the Navier differential operator with different source points and test frequencies. In the second method the solution is approximated by superposition of acoustic waves, i.e. fundamental solutions of the Helmholtz operator, with different source points and test frequencies. The applicability of the two methods is justified in terms of density results and a convergence result is proven. The accuracy of the methods is illustrated through 2D numerical examples. Applications to interior elastic wave scattering problems are also presented.
- Solving boundary value problems on manifolds with a plane waves methodPublication . Alves, Carlos J. S.; Antunes, Pedro R. S.; Martins, Nuno F. M.; Valtchev, Svilen S.In this paper we consider a plane waves method as a numerical technique for solving boundary value problems for linear partial differential equations on manifolds. In particular, the method is applied to the Helmholtz–Beltrami equations. We prove density results that justify the completeness of the plane waves space and justify the approximation of domain and boundary data. A-posteriori error estimates and numerical experiments show that this simple technique may be used to accurately solve boundary value problems on manifolds.