scholarly journals Generalized impedance boundary conditions and shape derivatives for 3D Helmholtz problems

2016 ◽  
Vol 26 (10) ◽  
pp. 1995-2033 ◽  
Author(s):  
Djalil Kateb ◽  
Frédérique Le Louër
Keyword(s):  
Thin Layer ◽  
Three Dimensional ◽  
Transmission Problem ◽  
Impedance Boundary Condition ◽  
Constant Thickness ◽  
Laplacian Operator ◽  
Shape Sensitivity ◽  
Impedance Boundary ◽  
First Order ◽  
Exterior Boundary

This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three-dimensional sound-soft or sound-hard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called generalized impedance boundary condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first-order GIBCs characterizing the presence of an interior thin layer with a constant thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non-vanishing jumps across the exterior and the interior boundary of the thin layer. We show that we can interchange the first-order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlight the various theoretical results.

scholarly journals RELAXATION APPROXIMATION OF THE KERR MODEL FOR THE THREE-DIMENSIONAL INITIAL-BOUNDARY VALUE PROBLEM

2009 ◽  
Vol 06 (03) ◽  
pp. 577-614 ◽  
Author(s):  
GILLES CARBOU ◽  
BERNARD HANOUZET
Keyword(s):  
Boundary Value Problem ◽  
Response Time ◽  
Boundary Value ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Debye Model ◽  
Impedance Boundary Condition ◽  
Impedance Boundary ◽  
Initial Boundary Value

The electromagnetic wave propagation in a nonlinear medium is described by the Kerr model in the case of an instantaneous response of the material, or by the Kerr–Debye model if the material exhibits a finite response time. Both models are quasilinear hyperbolic and are endowed with a dissipative entropy. The initial-boundary value problem with a maximal-dissipative impedance boundary condition is considered here. When the response time is fixed, in both the one-dimensional and two-dimensional transverse electric cases, the global existence of smooth solutions for the Kerr–Debye system is established. When the response time tends to zero, the convergence of the Kerr–Debye model to the Kerr model is established in the general case, i.e. the Kerr model is the zero relaxation limit of the Kerr–Debye model.

A Study of Three-Dimensional Acoustic Scattering by Arbitrary Distribution Multibodies Using Extended Immersed Boundary Method

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Yongsong Jiang ◽  
Xiaoyu Wang ◽  
Xiaodong Jing ◽  
Xiaofeng Sun
Keyword(s):  
Boundary Condition ◽  
Computational Model ◽  
Acoustic Field ◽  
Three Dimensional ◽  
Acoustic Scattering ◽  
Impedance Boundary Condition ◽  
Immersed Boundary ◽  
Complex Geometries ◽  
Impedance Boundary ◽  
Wall Boundary Conditions

A three-dimensional computational model for acoustic scattering with complex geometries is presented, which employs the immersed boundary technique to deal with the effect of both hard and soft wall boundary conditions on the acoustic fields. In this numerical model, the acoustic field is solved on uniform Cartesian grids, together with simple triangle meshes to partition the immersed body surface. A direct force at the Lagrangian points is calculated from an influence matrix system, and then spreads to the neighboring Cartesian grid points to make the acoustic field satisfy the required boundary condition. This method applies a uniform stencil on the whole domain except at the computational boundary, which has the benefit of low dispersion and dissipation errors of the used scheme. The method has been used to simulate two benchmark problems to validate its effectiveness and good agreements with the analytical solutions are achieved. No matter how complex the geometries are, single body or multibodies, complex geometries do not pose any difficulty in this model. Furthermore, a simple implementation of time-domain impedance boundary condition is reported and demonstrates the versatility of the computational model.

An Unpreconditioned Boundary-Integral for Iterative Solution of Scattering Problems with Non-Constant Leontovitch Impedance Boundary Conditions

2014 ◽  
Vol 15 (5) ◽  
pp. 1431-1460 ◽  
Author(s):  
D. Levadoux ◽  
F. Millot ◽  
S. Pernet
Keyword(s):  
Electromagnetic Scattering ◽  
Three Dimensional ◽  
Iterative Solution ◽  
Accurate Solution ◽  
Boundary Integral ◽  
Impedance Boundary Condition ◽  
Scattering Problems ◽  
Impedance Boundary ◽  
New Approach ◽  
Impedance Condition

AbstractThis paper concerns the electromagnetic scattering by arbitrary shaped three dimensional imperfectly conducting objects modeled with non-constant Leontovitch impedance boundary condition. It has two objectives. Firstly, the intrinsically well-conditioned integral equation (noted GCSIE) proposed in [30] is described focusing on its discretization. Secondly, we highlight the potential of this method by comparison with two other methods, the first being a two currents formulation in which the impedance condition is implicitly imposed and whose the convergence is quasi-optimal for Lipschitz polyhedron, the second being a CFIE-like formulation [14]. In particular, we prove that the new approach is less costly in term of CPU time and gives a more accurate solution than that obtained from the CFIE formulation. Finally, as expected, It is demonstrated that no preconditioner is needed for this formulation.

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