A Fictitious Domain Approach for Wave Propagation Problems in Unbounded Domains

Abstract

Abstract. We consider the scattering of waves by an obstacle O ⊂ R 2 having a sufficiently smooth boundary Γ. By using a fictitious domain approach, we artificially extend the solution inside the obstacle O. Then we solve the original problem in the simpler domain that includes O and that is artificially bounded by a perfectly transparent boundary B, delimiting the region of interest. We prescribe on B a Non Reflecting Boundary Condition (NRBC) which is based on a space-time integral equation and that defines a relationship that the solution of the differential problem and its normal derivative must satisfy on B (see [2]). Further, we enforce the Dirichlet boundary condition on Γ in a weak sense, by means of Lagrange multipliers. The NRBC is discretized on B by combining a special second order (in time) convolution quadrature and a standard collocation method in space. In the enlarged domain, this discretization is coupled with an unconditionally stable ODE time integrator and a FEM in space. The constraint on Γ is imposed by a matrix B h that represents a discrete trace operator.A particularly useful application of this approach is the scattering of a wave by rotating rigid bodies. In this case the method avoids the complexity of constructing at each time step a new finite element computational mesh and requires only the construction of the discrete trace operator B h . We present some results we have obtained for rotating (even multiple) scatterers and non trivial data. 959