Solution of viscoelastic scattering problems in linear acoustics using hp Boundary/Finite Element Method

The interaction of acoustic waves with submerged structures remains one of the most di cult and challenging problems in underwater acoustics. Many techniques such as coupled Boundary Element (BE)=Finite Element (FE) or coupled In nite Element (IE)=Finite Element approximations have evolved. In the present work, we focus on the steady-state formulation only, and study a general coupled hp-adaptive BE=FE method. A particular emphasis is placed on an a posteriori error estimation for the viscoelastic scattering problems. The highlights of the proposed methodology are as follows: (1) The exterior Helmholtz equation and the Sommerfeld radiation condition are replaced with an equivalent Burton{Miller (BM) boundary integral equation on the surface of the scatterer. (2) The BM equation is coupled to the steady-state form of viscoelasticity equations modelling the be- haviour of the structure. (3) The viscoelasticity equations are approximated using hp-adaptive FE isoparametric discretizations with order of approximation p>5 in order to avoid the `lockingʹ phenomenon. (4) A compatible hp superparametric discretization is used to approximate the BM integral equation. (5) Both the FE and BE approximations are based on a weak form of the equations, and the Galerkin method, allowing for a full convergence analysis. (6) An a posteriori error estimate for the coupled problem of a residual type is derived, allowing for estimating the error in pressure on the wet surface of the scatterer. (7) An adaptive scheme, an extension of the Texas Three Step Adaptive Strategy is used to manipulate the mesh size h and the order of approximation p so as to approximately minimize the number of degrees of freedom required to produce a solution with a speci ed accuracy. The use of this hp-scheme may exhibit exponential convergence rates. Several numerical experiments illustrate the methodology. These include detailed convergence studies for the problem of scattering of a plane acoustic wave on a viscoelastic sphere, and adaptive solutions of viscoelastic scattering problems for a series of MOCK0 models. Copyright