A collocation IGA-BEM for 3D potential problems on unbounded domains

Abstract

In this paper the numerical solution of potential problems defined on 3D unbounded domains is addressed with Boundary Element Methods (BEMs), since in this way the problem is studied only on the boundary, and thus any finite approximation of the infinite domain can be avoided. The isogeometric analysis (IGA) setting is considered and in particular B-splines and NURBS functions are taken into account. In order to exploit all the possible benefits from using spline spaces, an important point is the development of specific cubature formulas for weakly and nearly singular integrals. Our proposal for this aim is based on spline quasi-interpolation and on the use of a spline product formula. Besides that, a robust singularity extraction procedure is introduced as a preliminary step and an efficient function-byfunction assembly phase is adopted. A selection of numerical examples confirms that the numerical solutions reach the expected convergence orders.

IgA-BEM for 3D Helmholtz problems on multi-patch domains using B-spline tailored numerical integration

“…Note that in our case the right hand side f in both ( 9) and ( 11) is just the right hand side respectively of ( 8) and (10) and also that clearly it is in any case φ = φ(x, κ).…”

“…Thus, a discrete version of the given variational Dirichlet (8) or Neumann (10) problem is obtained by approximating φ in the finite dimensional composite space S h,d . The applied collocation method leads to a linear system…”

“…is an elegant and effective tool to deal with singular integrals for 2D potential problems [5,11]. Despite being considered in some recent publications for 3D potential problems [12,10], we do not consider it in this study due to increased difficulty to properly approximate the singular kernel of the double layer potential. Owing to the division step, the regularized kernel becomes singular if U m s has roots inside the integration domain.…”

“…The last example is a standard acoustic problem interior to a torus; see for example [25,23]. The toroidal surface Γ has inner and outer radii respectively equal to 1 and 3 and is exactly represented in a 16-patch quadratic NURBS form, see Figure 3(b) and refer to [10] for details.…”