Regularized integral formulation of mixed Dirichlet-Neumann problems

Abstract

This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions-for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on use of Green functions and integral equations, and it relies on the Fourier Continuation method for regularization of all smooth-domain Zaremba singularities as well as newly derived quadrature rules which give rise to high-order convergence even around Zaremba singular points. As demonstrated in this paper, the resulting algorithms enjoy high-order convergence, and they can be used to efficiently solve challenging Helmholtz boundary value problems and Laplace eigenvalue problems with high-order accuracy.

On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

“…. , d Λ in (1) are commonly called the primitive (or periodicity) vectors of the lattice. Without loss of generality, throughout this work it is assumed that v 1 is parallel to the x 1 -axis, and that the lattice is contained in the subspace generated by the vectors x 1 , .…”

“…Through the introduction of the concept of hybrid "spatial/spectral" representations, this work shows that if a representation of G q κ is used which displays explicitly all terms that cause the divergence of G q κ as RW anomalies are approached, then high-accuracies can be obtained in the evaluation of G q κ in very close proximity (to machine precision) of the singular configuration-thus addressing the evaluation difficulty mentioned in point (1) above. Use of such representations, in turn, provide an insight into the ill-conditioning of the resulting linear systems around RW-anomalies mentioned in point (2) above, and they allow us to introduce a regularization technique, which we refer to as the "Woodbury-Sherman-Morrison (WSM) methodology", that resolves the difficulty and can be used to produce solutions at RW-anomalies using quasi-periodic Green function methods.…”

Wave Enhancement Through Optimization of Boundary Conditions

“…To locate the Zaremba eigenvalues, we have the following statement: In [2, Section 3] and [1], it is shown that every square root of a Zaremba eigenvalue is a real positive characteristic value of k → A(k) and every real positive characteristic value of k → A(k) is the square root of a Zaremba eigenvalue. We see that A(k) is invertible for k ∈ (0, ∞) not a square root of a Zaremba eigenvalue.…”

“…The eigenmodes and the associated eigenfrequencies of a cavity are sensitively dependant on the geometric properties of the domains, as well as the location of Dirichlet and Neumann boundary conditions. Many recent works have been devoted to the understanding of the effect of changing the boundary condition on the eigenmodes and the eigenfrequencies [1,2,4,5,11,13,14,16].…”

“…Then the size of the Neumann part is changed in such away that an eigenvalue of the mixed boundary value problem gets close to the operating frequency by using the monotonicity property of the eigenvalues of the mixed eigenvalue problem. The optimization needs the high-accuracy evaluation of certain boundary integral operators, and this is done using techniques from [1,2].…”