Higher order isogeometric BEM for high accuracy simulations in acoustics

Abstract

In the present work we developed five different not approximated BEM higher{ order approaches for the solution of the 2D and 3D acoustic problem, validating them against analytical solution, where available, and performing a convergence analysis. The different approaches explored in this work are: -2D Hermite (classic approach) -2D Hermite Dual -3D Hermite Coons -3D Hermite Coons Dual (preliminary implementation) -2D Nurbs We explored algorithms based mostly on Hermite splines. The presence of the nodal derivatives in the representation of the variables, has been taken into account both with a \classic approach", where those derivatives are expressed with a finite differences scheme in terms of the nodal values and with a so called \dual approach" where the same derivatives are considered as unknown of the problem and the linear system is completed by the tangential gradient of the integral equation. The Hermite based higher order representations, also coupled with Coons patches to avoid the introduction of the mixed derivatives, although achieving a higher rate of convergence than O(h), are limited to a single order. In order to overcome the single{order limitation of the Hermite based BEM methods we also explored an approach based on Non Uniform Rational Basis{ Splines (NURBS). By using a global representation of the variables based on NURBS, the order of the basis functions can be improved at runtime when needed by the specific application at hand. This is a consequence of the iterative definition of the NURBS, which makes possible the increase of the NURBS degree simply by changing an input parameter. The use of the NURBS for the representation of curves and surfaces is a common technique in the CAD community for the model- ing of complex geometries with strict requirements of smoothness and continuity between patches. Their use in the numerical solution of BIE is not new, although relatively recent. NURBS have been used to develop boundary element solutions of integral equations in elastostatics, in radiation and diffraction problems, and in potential aerodynamics. The peculiarity of this approach is in the use of the NURBS for the development of a global isogeometric approach aimed at the mesh- less numerical solution of the BVP. The control points used for the representation of the dependent variables are obtained through the h-refinement of the optimal NURBS representation of the geometry. In the work we outline the underlying theory for the acoustic problem and for the BEM methodology, we illustrate the above said approaches in detail and we show how the high order methods described in this work are an effective tool to the simulation of high accuracy phenomena such as the ones involving the acoustic analysis of phononic crystals.