203 research outputs found
Isogeometric Boundary Elements in Electromagnetism: Rigorous Analysis, Fast Methods, and Examples
We present a new approach to three-dimensional electromagnetic scattering
problems via fast isogeometric boundary element methods. Starting with an
investigation of the theoretical setting around the electric field integral
equation within the isogeometric framework, we show existence, uniqueness, and
quasi-optimality of the isogeometric approach. For a fast and efficient
computation, we then introduce and analyze an interpolation-based fast
multipole method tailored to the isogeometric setting, which admits competitive
algorithmic and complexity properties. This is followed by a series of
numerical examples of industrial scope, together with a detailed presentation
and interpretation of the results
A Nash-Hormander iteration and boundary elements for the Molodensky problem
We investigate the numerical approximation of the nonlinear Molodensky
problem, which reconstructs the surface of the earth from the gravitational
potential and the gravity vector. The method, based on a smoothed
Nash-Hormander iteration, solves a sequence of exterior oblique Robin problems
and uses a regularization based on a higher-order heat equation to overcome the
loss of derivatives in the surface update. In particular, we obtain a
quantitative a priori estimate for the error after m steps, justify the use of
smoothing operators based on the heat equation, and comment on the accurate
evaluation of the Hessian of the gravitational potential on the surface, using
a representation in terms of a hypersingular integral. A boundary element
method is used to solve the exterior problem. Numerical results compare the
error between the approximation and the exact solution in a model problem.Comment: 32 pages, 14 figures, to appear in Numerische Mathemati
Boundary Element and Finite Element Coupling for Aeroacoustics Simulations
We consider the scattering of acoustic perturbations in a presence of a flow.
We suppose that the space can be split into a zone where the flow is uniform
and a zone where the flow is potential. In the first zone, we apply a
Prandtl-Glauert transformation to recover the Helmholtz equation. The
well-known setting of boundary element method for the Helmholtz equation is
available. In the second zone, the flow quantities are space dependent, we have
to consider a local resolution, namely the finite element method. Herein, we
carry out the coupling of these two methods and present various applications
and validation test cases. The source term is given through the decomposition
of an incident acoustic field on a section of the computational domain's
boundary.Comment: 25 page
Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics
International audienceThe fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles
Fast Numerical Methods for Non-local Operators
[no abstract available
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