56 research outputs found
Reviving the Method of Particular Solutions
Fox, Henrici and Moler made famous a "Method of Particular Solutions" for computing eigenvalues and eigenmodes of the Laplacian in planar regions such as polygons. We explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties. The crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant
Adaptive BEM with optimal convergence rates for the Helmholtz equation
We analyze an adaptive boundary element method for the weakly-singular and
hypersingular integral equations for the 2D and 3D Helmholtz problem. The
proposed adaptive algorithm is steered by a residual error estimator and does
not rely on any a priori information that the underlying meshes are
sufficiently fine. We prove convergence of the error estimator with optimal
algebraic rates, independently of the (coarse) initial mesh. As a technical
contribution, we prove certain local inverse-type estimates for the boundary
integral operators associated with the Helmholtz equation
The Factorization method for three dimensional Electrical Impedance Tomography
The use of the Factorization method for Electrical Impedance Tomography has
been proved to be very promising for applications in the case where one wants
to find inhomogeneous inclusions in a known background. In many situations, the
inspected domain is three dimensional and is made of various materials. In this
case, the main challenge in applying the Factorization method consists in
computing the Neumann Green's function of the background medium. We explain how
we solve this difficulty and demonstrate the capability of the Factorization
method to locate inclusions in realistic inhomogeneous three dimensional
background media from simulated data obtained by solving the so-called complete
electrode model. We also perform a numerical study of the stability of the
Factorization method with respect to various modelling errors.Comment: 16 page
Bempp-cl: A fast Python based just-in-time compiling boundary element library
Summary
The boundary element method (BEM) is a numerical method for approximating the solution
of certain types of partial differential equations (PDEs) in homogeneous bounded or
unbounded domains. The method finds an approximation by discretising a boundary integral
equation that can be derived from the PDE. The mathematical background of BEM is covered
in, for example, Steinbach (2008) or McLean (2000). Typical applications of BEM include
electrostatic problems, and acoustic and electromagnetic scattering.
Bempp-cl is an open-source boundary element method library that can be used to assemble all
the standard integral kernels for Laplace, Helmholtz, modified Helmholtz, and Maxwell problems.
The library has a user-friendly Python interface that allows the user to use BEM to solve
a variety of problems, including problems in electrostatics, acoustics and electromagnetics.
Bempp-cl began life as BEM++, and was a Python library with a C++ computational core.
The ++ slowly changed into pp as functionality gradually moved from C++ to Python with
only a few core routines remaining in C++. Bempp-cl is the culmination of efforts to fully
move to Python, and is an almost complete rewrite of Bempp.
For each of the applications mentioned above, the boundary element method involves approximating
the solution of a partial differential equation (Laplace’s equation, the Helmholtz
equation, and Maxwell’s equations respectively) by writing the problem in boundary integral
form, then discretising. For example, we could calculate the scattered field due to an
electromagnetic wave colliding with a series of screens by solving
∇ x ∇ x E - k2E = 0;
v x E = 0 on the screens;
where E is the sum of a scattered field Es and an incident field Einc, and � is the direction
normal to the screen. (Additionally, we must impose the Silver–Müller radiation condition to
ensure that the problem has a unique solution.) This problem is solved, and the full method
is derived, in one of the tutorials available on the Bempp website (Betcke & Scroggs, 2020a).
The solution to this problem is shown below
Computed eigenmodes of planar regions
Recently developed numerical methods make possible the high-accuracy computation of eigenmodes of the Laplacian for a variety of "drums" in two dimensions. A number of computed examples are presented together with a discussion of their implications concerning bound and continuum states, isospectrality, symmetry and degeneracy, eigenvalue avoidance, resonance, localization, eigenvalue optimization, perturbation of eigenvalues and eigenvectors, and other matters.\ud
\ud
Timo Betcke was supported by a Scatcherd European Scholarship
An OSRC Preconditioner for the EFIE
The Electric Field Integral Equation (EFIE) is a well-established tool to solve electromagnetic scattering problems. However, the development of efficient and easy to implement preconditioners remains an active research area. In recent years, operator preconditioning approaches have become popular for the EFIE, where the electric field boundary integral operator is regularised by multiplication with another convenient operator. A particularly intriguing choice is the exact Magnetic-to-Electric (MtE) operator as regulariser. But, evaluating this operator is as expensive as solving the original EFIE. In work by El Bouajaji, Antoine and Geuzaine, approximate local Magnetic-to-Electric surface operators for the time-harmonic Maxwell equation were proposed. These can be efficiently evaluated through the solution of sparse problems. This paper demonstrates the preconditioning properties of these approximate MtE operators for the EFIE. The implementation is described and a number of numerical comparisons against other preconditioning techniques for the EFIE are presented to demonstrate the effectiveness of this new technique
Boundary element methods for Helmholtz problems with weakly imposed boundary conditions
We consider boundary element methods where the Calder\'on projector is used
for the system matrix and boundary conditions are weakly imposed using a
particular variational boundary operator designed using techniques from
augmented Lagrangian methods. Regardless of the boundary conditions, both the
primal trace variable and the flux are approximated. We focus on the imposition
of Dirichlet and mixed Dirichlet--Neumann conditions on the Helmholtz equation,
and extend the analysis of the Laplace problem from the paper \emph{Boundary
element methods with weakly imposed boundary conditions} to this case. The
theory is illustrated by a series of numerical examples.Comment: 27 page
Numerical aspects of Casimir energy computation in acoustic scattering
Computing the Casimir force and energy between objects is a classical problem
of quantum theory going back to the 1940s. Several different approaches have
been developed in the literature often based on different physical principles.
Most notably a representation of the Casimir energy in terms of determinants of
boundary layer operators makes it accessible to a numerical approach. In this
paper, we first give an overview of the various methods and discuss the
connection to the Krein-spectral shift function and computational aspects. We
propose variants of Krylov subspace methods for the computation of the Casimir
energy for large-scale problems and demonstrate Casimir computations for
several complex configurations. This allows for Casimir energy calculation for
large-scale practical problems and significantly speeds up the computations in
that case
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