Coupling Finite and Boundary Element Methods Using a Localized Adaptive Radiation Condition for the Helmholtz’s Equation

Abstract

In this paper, we are interested in impenetrable surfaces with relatively large size on which a heterogeneous object of relatively small size is posed. In this case, a straightforward FEM-BEM (finite and boundary element methods) coupling leads to a linear system of very large scale difficult to solve [7]. In this work, we propose an alternative method derived from a modification of the adaptive radiation condition approach [1, 11, 12]. This technique consists of enclosing the computational domain by an artificial truncating surface on which the adaptive radiation condition is posed. This condition is expressed using integral operators acting as a correction term of the absorbing boundary condition. However, enclosing completely the computational domain by an artificial surface in this range leads to problems with very large size, and results in very slow convergence of the iterative procedure. We propose to localize this surface only around the heterogenous region, which will generates a relatively small bounded domain dealt with by a FEM, and suitably coupled with a BEM expressing the solution on the impenetrable surface. The resulting formulation, based on a particular overlapping domain decomposition method, is solved iteratively where FEM and BEM linear systems are solved separately. The wave problem considered in this paper is stated as follows

$$\displaystyle{ \left \{\begin{array}{l} \nabla \cdot (\chi \nabla u) +\chi \kappa ^{2}n^{2}u = 0\quad \text{in}\ \varOmega, \\ \chi \partial _{\mathbf{n}}u = -f\text{ on }\varGamma, \\ \lim _{\vert x\vert \rightarrow \infty }\vert x\vert ^{1/2}(\partial _{\vert x\vert }u - i\kappa u) = 0,\end{array} \right. }$$

(1)

where Ω is the complement of the impenetrable obstacle. We indicate by Ω ₁ a bounded domain filled by a possibly heterogeneous material and posed on a slot Γ _slot on which are applied the sources producing the radiated wave u. The interface $\varSigma$ separates Ω ₁ from the free propagation domain Ω ₀, n denotes the normal to Γ or to $\varSigma$ directed outwards respectively the impenetrable obstacle enclosed by Γ or the domain Ω ₁ (see Fig. 1), χ and n indicate, respectively, the relative dielectric permittivity and the relative magnetic permeability, and κ is the wave number. Let us note finally that $\chi = n = 1$ in Ω ₀. For the sake of presentation, we express problem (1) in the form of the following system

$$\displaystyle{ \left \{\begin{array}{l} \varDelta u_{0} +\kappa ^{2}u_{0} = 0\;\text{in}\ \varOmega _{0}, \\ \partial _{\mathbf{n}}u_{0} = 0\;\text{on }\varGamma \cap \partial \varOmega _{0}, \\ \lim _{\vert x\vert \rightarrow \infty }\vert x\vert ^{1/2}(\partial _{\vert x\vert }u_{0} - i\kappa u_{0}) = 0, \end{array} \right. }$$

(2)

$$\displaystyle{ \left \{\begin{array}{l} \nabla \cdot (\chi \nabla u_{1}) +\chi \kappa ^{2}n^{2}u_{1} = 0\;\text{in}\ \varOmega _{1}, \\ \chi \partial _{\mathbf{n}}u_{1} = -f\text{ on }\varGamma \cap \partial \varOmega _{1}.\end{array} \right. }$$

(3)

These boundary-value problems are coupled on $\varSigma$ through the transmission conditions

$$\displaystyle{ u_{0} = u_{1},\quad \partial _{\mathbf{n}}u_{0} =\chi \partial _{\mathbf{n}}u_{1}. }$$

(4)