FULL-PLANE IMPEDANCE LAPLACE PROBLEM

FULL-PLANE IMPEDANCE LAPLACE PROBLEM

In this appendix we study the perturbed full-plane or free-plane impedance Laplace problem, also known as the exterior impedance Laplace problem in 2D, using integral equation techniques and the boundary element method. We consider the problem of the Laplace equation in two dimensions on the exterior of a bounded obstacle. The Laplace equation for an exterior domain, using typically either Dirichlet or Neumann boundary conditions, is a good example to illustrate the complexity of the integral equation techniques. For a more general treatment and in order to allow a better comparison with the development performed before for half-spaces, we consider in particular an impedance boundary condition. The perturbed full-plane impedance Laplace problem is not strictly speaking a wave scattering problem, but it can be regarded as a limit case of such a problem when the frequency tends towards zero (vid. Appendix C). It can be also regarded as a surface wave problem around a bounded two-dimensional obstacle. The three-dimensional case is treated thoroughly in Appendix D. For the problem treated herein we follow mainly Ned´ elec (1977, 1979, 2001) and ´ Raviart (1991). Further related books and doctorate theses are Chen & Zhou (1992), Evans (1998), Giroire (1987), Hsiao & Wendland (2008), Kellogg (1929), Kress (1989), Muskhelishvili (1953), Rjasanow & Steinbach (2007), and Steinbach (2008). Some articles that consider the Laplace equation with an impedance boundary condition are Ahner & Wiener (1991), Lanzani & Shen (2004), and Medkova (1998). Wendland, Stephan & ´ Hsiao (1979) treat the mixed boundary-value problem. Interesting theoretical details on transmission problems can be found in Costabel & Stephan (1985). The boundary element calculations are performed in Bendali & Devys (1986). The coupling of boundary integral equations and finite element methods is done in Johnson & Ned´ elec (1980). The use of ´ cracked domains is studied by Medkova & Krutitskii (2005), and the inverse problem by ´ Fasino & Inglese (1999) and Lin & Fang (2005). Applications of the Laplace problem can be found, among others, for electrostatics (Jackson 1999), for conductivity in biomedical imaging (Ammari 2008), and for incompressible plane potential flows (Spurk 1997). The Laplace equation does not allow the propagation of volume waves inside the considered domain, but the addition of an impedance boundary condition permits the propagation of surface waves along the boundary of the obstacle. The main difficulty in the numerical treatment and resolution of these problems is the fact that the exterior domain is unbounded. We treat this issue by using integral equation techniques and the boundary element method. The idea behind these techniques is to use Green’s integral theorems to transform the problem and express it on the boundary of the obstacle, which is bounded. These methods require thus only the calculation of boundary values, rather than values throughout the unbounded exterior domain. They are in a significant manner more efficient in terms of computational resources for problems where the surface versus volume ratio is small. The drawback of these techniques is a more complex mathematical treatment and the requirement of knowing the Green’s function of the system. It is the Green’s function 371 which stores the information of the system’s physics throughout the exterior domain and which allows to collapse the problem to hold only on the boundary. The dimension of a problem expressed in a volume is therefore reduced towards a surface, i.e., one dimension less, which is what makes these methods so interesting to consider. This appendix is structured in 13 sections, including this introduction. The direct perturbation problem of the Laplace equation in a two-dimensional exterior domain with an impedance boundary condition is presented in Section B.2. The Green’s function and its far-field expression are computed respectively in Sections B.3 and B.4. Extending the direct perturbation problem towards a transmission problem, as done in Section B.5, allows its resolution by using integral equation techniques, which is discussed in Section B.6. These techniques allow also to represent the far field of the solution, as shown in Section B.7. A particular problem that takes as domain the exterior of a circle is solved analytically in Section B.8. The appropriate function spaces and some existence and uniqueness results for the solution of the problem are presented in Section B.9. By means of the variational formulation developed in Section B.10, the obtained integral equation is discretized using the boundary element method, which is described in Section B.11. The boundary element calculations required to build the matrix of the linear system resulting from the numerical discretization are explained in Section B.12. Finally, in Section B.13 a benchmark problem based on the exterior circle problem is solved numerically

Direct perturbation problem 

We consider an exterior open and connected domain Ωe ⊂ R 2 that lies outside a bounded obstacle Ωi and whose boundary Γ = ∂Ωe = ∂Ωi is regular (e.g., of class C 2 ), as shown in Figure B.1. As a perturbation problem, we decompose the total field uT as uT = uW + u, where uW represents the known field without obstacle, and where u denotes the perturbed field due its presence, which has bounded energy. The direct perturbation problem of interest is to find the perturbed field u that satisfies the Laplace equation in Ωe, an impedance boundary condition on Γ, and a decaying condition at infinity. We consider that the origin is located in Ωi and that the unit normal n is taken always outwardly oriented of Ωe, i.e., pointing inwards of Ωi .The total field uT satisfies the Laplace equation ∆uT = 0 in Ωe, (B.1) which is also satisfied by the fields uW and u, due linearity. For the perturbed field u we take also the inhomogeneous impedance boundary condition − ∂u ∂n + Zu = fz on Γ, (B.2) where Z is the impedance on the boundary, and where the impedance data function fz is assumed to be known. If Z = 0 or Z = ∞, then we retrieve respectively the classical Neumann or Dirichlet boundary conditions. In general, we consider a complex-valued impedance Z(x) depending on the position x. The function fz(x) may depend on Z and uW , but is independent of u. If a homogeneous impedance boundary condition is desired for the total field uT , then due linearity we can express the function fz as

Green’s function

 The Green’s function represents the response of the unperturbed system (without an obstacle) to a Dirac mass. It corresponds to a function G, which depends on a fixed source point x ∈ R 2 and an observation point y ∈ R 2 . The Green’s function is computed in the sense of distributions for the variable y in the full-plane R 2 by placing at the right-hand side of the Laplace equation a Dirac mass δx, centered at the point x. It is therefore a solution G(x, ·) : R 2 → C for the radiation problem of a point source, namely ∆yG(x, y) = δx(y) in D ′ (R 2 ). (B.12) Due to the radial symmetry of the problem (B.12), it is natural t

Far field of the Green’s function

The far field of the Green’s function describes its asymptotic behavior at infinity, i.e., when |x| → ∞ and assuming that y is fixed. For this purpose, we search the terms of highest order at infinity by expanding the logarithm according to ln |x − y| = 1 2 ln .

Transmission problem

We are interested in expressing the solution u of the direct perturbation problem (B.11) by means of an integral representation formula over the boundary Γ. To study this kind of representations, the differential problem defined on Ωe is extended as a transmission problem defined now on the whole plane R 2 by combining (B.11) with a corresponding interior problem defined on Ωi . For the transmission problem, which specifies jump conditions over the boundary Γ, a general integral representation can be developed, and the particular integral representations of interest are then established by the specific choice of the corresponding interior problem. A transmission problem is then a differential problem for which the jump conditions of the solution field, rather than boundary conditions, are specified on the boundary Γ. As shown in Figure B.1, we consider the exterior domain Ωe and the interior domain Ωi , taking the unit normal n pointing towards Ωi . We search now a solution u defined in Ωe ∪ Ωi , and use the notation ue = u|Ωe and ui = u|Ωi .