Letizia Scuderi

We consider the decomposition into scalar potentials for the simulation of transient 2D soft scattering elastic wave propagation problems in unbounded isotropic We consider the decomposition into scalar potentials for the simulation of transient 2D soft scattering elastic wave propagation problems in unbounded isotropic

In this paper, we propose and analyse a numerical method to solve 2D Dirichlet time-harmonic elastic wave equations. The procedure is based on the decoupling of the elastic vector field into scalar Pressure ($P$-) and Shear ($S$-) waves via a suitable Helmholtz-Hodge decomposition. For the approximation of the two scalar potentials we apply a virtual element method associated with different mesh sizes and degrees of accuracy. We provide for the stability of the method and a convergence error estimate in the $L^2$-norm for the displacement field, in which the contributions to the error associated with the $P$- and $S$- waves are separated. In contrast to standard approaches that are directly applied to the vector formulation, this procedure allows for keeping track of the two different wave numbers, that depend on the $P$- and $S$- speeds of propagation and, therefore, for using a high-order method for the approximation of the wave associated with the higher wave number. Some numerical tests, validating the theoretical results and showing the good performance of the proposed approach, are presented.

In this paper, we present a numerical method based on the coupling between a Curved Virtual Element Method (CVEM) and a Boundary Element Method (BEM) for the simulation of wave fields scattered by obstacles immersed in homogeneous infinite media. In particular, we consider the 2D time-domain damped wave equation, endowed with a Dirichlet condition on the boundary (sound-soft scattering). To reduce the infinite domain to a finite computational one, we introduce an artificial boundary on which we impose a Boundary Integral Non-Reflecting Boundary Condition (BI-NRBC). We apply a CVEM combined with the Crank–Nicolson time integrator in the interior domain, and we discretize the BI-NRBC by a convolution quadrature formula in time and a collocation method in space. We present some numerical results to test the performance of the proposed approach and to highlight its effectiveness, especially when obstacles with complex geometries are considered.

In this paper, initially we consider the (two- and three-dimensional) interior and exterior problems for the scalar wave equation, with a Neumann boundary condition and, in general, with nontrivial data. For these problems, we derive a space-time hypersingular boundary integral equation formulation. Then, in the two-dimensional case, we propose a numerical approach for the computation of the extra ‘volume’ integrals generated by the problem data. To solve the above integral equation in the two-dimensional case, we propose to use second-order Lubich convolution quadrature for the discretization of the time integral, coupled with a (space) ϵ-collocation boundary element method. We also analyse the numerical evaluation of all the integrals required by this method. Finally, to show the efficiency of the proposed numerical approach and to detect its convergence rate, we solve some two-dimensional test problems by applying the new method, as well as the corresponding Galerkin one. Several numerical examples are presented.