752 research outputs found
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Numerical methods for computing Casimir interactions
We review several different approaches for computing Casimir forces and
related fluctuation-induced interactions between bodies of arbitrary shapes and
materials. The relationships between this problem and well known computational
techniques from classical electromagnetism are emphasized. We also review the
basic principles of standard computational methods, categorizing them according
to three criteria---choice of problem, basis, and solution technique---that can
be used to classify proposals for the Casimir problem as well. In this way,
mature classical methods can be exploited to model Casimir physics, with a few
important modifications.Comment: 46 pages, 142 references, 5 figures. To appear in upcoming Lecture
Notes in Physics book on Casimir Physic
Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation
We consider the Galerkin approach for the numerical solution of retarded boundary integral formulations of the three dimensional wave equation in unbounded domains. Recently smooth and compactly supported basis functions in time were introduced which allow the use of standard quadrature rules in order to compute the entries of the boundary element matrix. In this paper, we use TT and QTT tensor approximations to increase the efficiency of these quadrature rules. Various numerical experiments show the substantial reduction of the computational cost that is needed to obtain accurate approximations for the arising integral
Fast Isogeometric Boundary Element Method based on Independent Field Approximation
An isogeometric boundary element method for problems in elasticity is
presented, which is based on an independent approximation for the geometry,
traction and displacement field. This enables a flexible choice of refinement
strategies, permits an efficient evaluation of geometry related information, a
mixed collocation scheme which deals with discontinuous tractions along
non-smooth boundaries and a significant reduction of the right hand side of the
system of equations for common boundary conditions. All these benefits are
achieved without any loss of accuracy compared to conventional isogeometric
formulations. The system matrices are approximated by means of hierarchical
matrices to reduce the computational complexity for large scale analysis. For
the required geometrical bisection of the domain, a strategy for the evaluation
of bounding boxes containing the supports of NURBS basis functions is
presented. The versatility and accuracy of the proposed methodology is
demonstrated by convergence studies showing optimal rates and real world
examples in two and three dimensions.Comment: 32 pages, 27 figure
Fast quadrature techniques for retarded potentials based on TT/QTT tensor approximation
We consider the Galerkin approach for the numerical solution of retarded boundary integral formulations of the three dimensional wave equation in unbounded domains.
Recently smooth and compactly supported basis functions in time were introduced which allow the use of standard quadrature rules in order to compute the entries of the boundary element matrix. In this paper we use TT and QTT tensor approximations to increase the effciency of these quadrature rules. Various numerical experiments show the substantial reduction of the computational cost that is needed to obtain accurate approximations for the arising integrals
Planewave density interpolation methods for 3D Helmholtz boundary integral equations
This paper introduces planewave density interpolation methods for the
regularization of weakly singular, strongly singular, hypersingular and nearly
singular integral kernels present in 3D Helmholtz surface layer potentials and
associated integral operators. Relying on Green's third identity and pointwise
interpolation of density functions in the form of planewaves, these methods
allow layer potentials and integral operators to be expressed in terms of
integrand functions that remain smooth (at least bounded) regardless the
location of the target point relative to the surface sources. Common
challenging integrals that arise in both Nystr\"om and boundary element
discretization of boundary integral equation, can then be numerically evaluated
by standard quadrature rules that are irrespective of the kernel singularity.
Closed-form and purely numerical planewave density interpolation procedures are
presented in this paper, which are used in conjunction with Chebyshev-based
Nystr\"om and Galerkin boundary element methods. A variety of numerical
examples---including problems of acoustic scattering involving multiple
touching and even intersecting obstacles, demonstrate the capabilities of the
proposed technique
On the modelling of ultrasonic testing using boundary integral equation methods
Ultrasonic nondestructive testing has important applications in, for example, the nuclear
power and aerospace industries, where it is used to inspect safety-critical parts for flaws.
For safe and reliable testing, mathematical models of the ultrasonic measurement systems
are invaluable tools. In this thesis such measurement models are developed for the ultrasonic
testing for defects located near non-planar surfaces. The applications in mind are
the testing of nuclear power plant components such as thick-walled pipes with diameter
transitions, pipe connections, etc. The models use solution methods based on frequency
domain boundary integral equation methods, with a focus on analytical approaches for the
defects and regularized boundary element methods for the non-planar surfaces. A major
benefit of the solution methods is the ability to provide accurate results both for low, intermediate
and high frequencies. The solution methods are incorporated into a framework
of transmitting probe models based on prescribing the traction underneath the probe and
receiving probe models based on electromechanical reciprocity. Time traces are obtained
by applying inverse temporal Fourier transforms, and it is also shown how calibration and
effects of material damping can be included in the models
Harmonic density interpolation methods for high-order evaluation of Laplace layer potentials in 2D and 3D
We present an effective harmonic density interpolation method for the
numerical evaluation of singular and nearly singular Laplace boundary integral
operators and layer potentials in two and three spatial dimensions. The method
relies on the use of Green's third identity and local Taylor-like
interpolations of density functions in terms of harmonic polynomials. The
proposed technique effectively regularizes the singularities present in
boundary integral operators and layer potentials, and recasts the latter in
terms of integrands that are bounded or even more regular, depending on the
order of the density interpolation. The resulting boundary integrals can then
be easily, accurately, and inexpensively evaluated by means of standard
quadrature rules. A variety of numerical examples demonstrate the effectiveness
of the technique when used in conjunction with the classical trapezoidal rule
(to integrate over smooth curves) in two-dimensions, and with a Chebyshev-type
quadrature rule (to integrate over surfaces given as unions of non-overlapping
quadrilateral patches) in three-dimensions
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