1,050 research outputs found
Solution of the 3D-Helmholtz equation in exterior domains using spherical harmonic decomposition
AbstractThis work is devoted to a finite element formulation for the Helmholtz equation in exterior domains. The proposed formulation uses a separation of variables, combining a 2D FE discretization on an intermediate spherical boundary and an ‘a priori’ analytical pattern for the radial direction. Using the analytical radial pattern and the series expansion of trial and test functions in terms of spherical harmonics, an efficient semi-analytical technique is obtained for the direct calculation of the global FE matrices. The accuracy and reliability of the formulation are illustrated through numerical examples of radiation and scattering in the exterior domain
Spectral method for matching exterior and interior elliptic problems
A spectral method is described for solving coupled elliptic problems on an
interior and an exterior domain. The method is formulated and tested on the
two-dimensional interior Poisson and exterior Laplace problems, whose solutions
and their normal derivatives are required to be continuous across the
interface. A complete basis of homogeneous solutions for the interior and
exterior regions, corresponding to all possible Dirichlet boundary values at
the interface, are calculated in a preprocessing step. This basis is used to
construct the influence matrix which serves to transform the coupled boundary
conditions into conditions on the interior problem. Chebyshev approximations
are used to represent both the interior solutions and the boundary values. A
standard Chebyshev spectral method is used to calculate the interior solutions.
The exterior harmonic solutions are calculated as the convolution of the
free-space Green's function with a surface density; this surface density is
itself the solution to an integral equation which has an analytic solution when
the boundary values are given as a Chebyshev expansion. Properties of Chebyshev
approximations insure that the basis of exterior harmonic functions represents
the external near-boundary solutions uniformly. The method is tested by
calculating the electrostatic potential resulting from charge distributions in
a rectangle. The resulting influence matrix is well-conditioned and solutions
converge exponentially as the resolution is increased. The generalization of
this approach to three-dimensional problems is discussed, in particular the
magnetohydrodynamic equations in a finite cylindrical domain surrounded by a
vacuum
Domain Decomposition Method for Maxwell's Equations: Scattering off Periodic Structures
We present a domain decomposition approach for the computation of the
electromagnetic field within periodic structures. We use a Schwarz method with
transparent boundary conditions at the interfaces of the domains. Transparent
boundary conditions are approximated by the perfectly matched layer method
(PML). To cope with Wood anomalies appearing in periodic structures an adaptive
strategy to determine optimal PML parameters is developed. We focus on the
application to typical EUV lithography line masks. Light propagation within the
multi-layer stack of the EUV mask is treated analytically. This results in a
drastic reduction of the computational costs and allows for the simulation of
next generation lithography masks on a standard personal computer.Comment: 24 page
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
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