114 research outputs found
Computing the demagnetizing tensor for finite difference micromagnetic simulations via numerical integration
In the finite difference method which is commonly used in computational
micromagnetics, the demagnetizing field is usually computed as a convolution of
the magnetization vector field with the demagnetizing tensor that describes the
magnetostatic field of a cuboidal cell with constant magnetization. An
analytical expression for the demagnetizing tensor is available, however at
distances far from the cuboidal cell, the numerical evaluation of the
analytical expression can be very inaccurate.
Due to this large-distance inaccuracy numerical packages such as OOMMF
compute the demagnetizing tensor using the explicit formula at distances close
to the originating cell, but at distances far from the originating cell a
formula based on an asymptotic expansion has to be used. In this work, we
describe a method to calculate the demagnetizing field by numerical evaluation
of the multidimensional integral in the demagnetization tensor terms using a
sparse grid integration scheme. This method improves the accuracy of
computation at intermediate distances from the origin.
We compute and report the accuracy of (i) the numerical evaluation of the
exact tensor expression which is best for short distances, (ii) the asymptotic
expansion best suited for large distances, and (iii) the new method based on
numerical integration, which is superior to methods (i) and (ii) for
intermediate distances. For all three methods, we show the measurements of
accuracy and execution time as a function of distance, for calculations using
single precision (4-byte) and double precision (8-byte) floating point
arithmetic. We make recommendations for the choice of scheme order and
integrating coefficients for the numerical integration method (iii)
Electric potential and field calculation of charged BEM triangles and rectangles by Gaussian cubature
It is a widely held view that analytical integration is more accurate than
the numerical one. In some special cases, however, numerical integration can be
more advantageous than analytical integration. In our paper we show this
benefit for the case of electric potential and field computation of charged
triangles and rectangles applied in the boundary element method (BEM).
Analytical potential and field formulas are rather complicated (even in the
simplest case of constant charge densities), they have usually large
computation times, and at field points far from the elements they suffer from
large rounding errors. On the other hand, Gaussian cubature, which is an
efficient numerical integration method, yields simple and fast potential and
field formulas that are very accurate far from the elements. The simplicity of
the method is demonstrated by the physical picture: the triangles and
rectangles with their continuous charge distributions are replaced by discrete
point charges, whose simple potential and field formulas explain the higher
accuracy and speed of this method. We implemented the Gaussian cubature method
for the purpose of BEM computations both with CPU and GPU, and we compare its
performance with two different analytical integration methods. The ten
different Gaussian cubature formulas presented in our paper can be used for
arbitrary high-precision and fast integrations over triangles and rectangles.Comment: 28 pages, 13 figure
Computation of the magnetostatic interaction between linearly magnetized polyhedrons
In this paper we present a method to accurately compute the energy of the
magnetostatic interaction between linearly (or uniformly, as a special case)
magnetized polyhedrons. The method has applications in finite element
micromagnetics, or more generally in computing the magnetostatic interaction
when the magnetization is represented using the finite element method (FEM).
The magnetostatic energy is described by a six-fold integral that is singular
when the interaction regions overlap, making direct numerical evaluation
problematic. To resolve the singularity, we evaluate four of the six iterated
integrals analytically resulting in a 2d integral over the surface of a
polyhedron, which is nonsingular and can be integrated numerically. This
provides a more accurate and efficient way of computing the magnetostatic
energy integral compared to existing approaches.
The method was developed to facilitate the evaluation of the demagnetizing
interaction between neighouring elements in finite-element micromagnetics and
provides a possibility to compute the demagnetizing field using efficient fast
multipole or tree code algorithms
Numerical cubature using error-correcting codes
We present a construction for improving numerical cubature formulas with
equal weights and a convolution structure, in particular equal-weight product
formulas, using linear error-correcting codes. The construction is most
effective in low degree with extended BCH codes. Using it, we obtain several
sequences of explicit, positive, interior cubature formulas with good
asymptotics for each fixed degree as the dimension . Using a
special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain
an equal-weight -cubature formula on the -cube with O(n^{\floor{t/2}})
points, which is within a constant of the Stroud lower bound. We also obtain
-cubature formulas on the -sphere, -ball, and Gaussian with
points when is odd. When is spherically symmetric and
, we obtain points. For each , we also obtain explicit,
positive, interior formulas for the -simplex with points; for
, we obtain O(n) points. These constructions asymptotically improve the
non-constructive Tchakaloff bound.
Some related results were recently found independently by Victoir, who also
noted that the basic construction more directly uses orthogonal arrays.Comment: Dedicated to Wlodzimierz and Krystyna Kuperberg on the occasion of
their 40th anniversary. This version has a major improvement for the n-cub
Constructing Cubature Formulas of Degree 5 with Few Points
This paper will devote to construct a family of fifth degree cubature
formulae for -cube with symmetric measure and -dimensional spherically
symmetrical region. The formula for -cube contains at most points
and for -dimensional spherically symmetrical region contains only
points. Moreover, the numbers can be reduced to and if
respectively, the later of which is minimal.Comment: 13 page
Cubature formulas of multivariate polynomials arising from symmetric orbit functions
The paper develops applications of symmetric orbit functions, known from
irreducible representations of simple Lie groups, in numerical analysis. It is
shown that these functions have remarkable properties which yield to cubature
formulas, approximating a weighted integral of any function by a weighted
finite sum of function values, in connection with any simple Lie group. The
cubature formulas are specialized for simple Lie groups of rank two. An optimal
approximation of any function by multivariate polynomials arising from
symmetric orbit functions is discussed.Comment: 19 pages, 4 figure
Analytical computation of moderate-degree fully-symmetric cubature rules on the triangle
A method is developed to compute analytically fully symmetric cubature rules
on the triangle by using symmetric polynomials to express the two kinds of
invariance inherent in these rules. Rules of degree up to 15, some of them new
and of good quality, are computed and presented.Comment: 13 pages, submitted to Journal of Computational and Applied
Mathematic
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