215 research outputs found
Increasing the Reliability of Adaptive Quadrature Using Explicit Interpolants
We present two new adaptive quadrature routines. Both routines differ from
previously published algorithms in many aspects, most significantly in how they
represent the integrand, how they treat non-numerical values of the integrand,
how they deal with improper divergent integrals and how they estimate the
integration error. The main focus of these improvements is to increase the
reliability of the algorithms without significantly impacting their efficiency.
Both algorithms are implemented in Matlab and tested using both the "families"
suggested by Lyness and Kaganove and the battery test used by Gander and
Gautschi and Kahaner. They are shown to be more reliable, albeit in some cases
less efficient, than other commonly-used adaptive integrators.Comment: 32 pages, submitted to ACM Transactions on Mathematical Softwar
Chebfun and numerical quadrature
Chebfun is a Matlab-based software system that overloads Matlab’s discrete operations for vectors and matrices to analogous continuous operations for functions and operators. We begin by describing Chebfun’s fast capabilities for Clenshaw–Curtis and also Gauss–Legendre, –Jacobi, –Hermite, and –Laguerre quadrature, based on algorithms of Waldvogel and Glaser, Liu, and Rokhlin. Then we consider how such methods can be applied to quadrature problems including 2D integrals over rectangles, fractional derivatives and integrals, functions defined on unbounded intervals, and the fast computation of weights for barycentric interpolation
On the equivalence between the cell-based smoothed finite element method and the virtual element method
We revisit the cell-based smoothed finite element method (SFEM) for
quadrilateral elements and extend it to arbitrary polygons and polyhedrons in
2D and 3D, respectively. We highlight the similarity between the SFEM and the
virtual element method (VEM). Based on the VEM, we propose a new stabilization
approach to the SFEM when applied to arbitrary polygons and polyhedrons. The
accuracy and the convergence properties of the SFEM are studied with a few
benchmark problems in 2D and 3D linear elasticity. Later, the SFEM is combined
with the scaled boundary finite element method to problems involving
singularity within the framework of the linear elastic fracture mechanics in
2D
Efficient Construction, Update and Downdate Of The Coefficients Of Interpolants Based On Polynomials Satisfying A Three-Term Recurrence Relation
In this paper, we consider methods to compute the coefficients of
interpolants relative to a basis of polynomials satisfying a three-term
recurrence relation. Two new algorithms are presented: the first constructs the
coefficients of the interpolation incrementally and can be used to update the
coefficients whenever a nodes is added to or removed from the interpolation.
The second algorithm, which constructs the interpolation coefficients by
decomposing the Vandermonde-like matrix iteratively, can not be used to update
or downdate an interpolation, yet is more numerically stable than the first
algorithm and is more efficient when the coefficients of multiple
interpolations are to be computed over the same set of nodes.Comment: 18 pages, submitted to the Journal of Scientific Computing
Microstructure-based modeling of elastic functionally graded materials: One dimensional case
Functionally graded materials (FGMs) are two-phase composites with
continuously changing microstructure adapted to performance requirements.
Traditionally, the overall behavior of FGMs has been determined using local
averaging techniques or a given smooth variation of material properties.
Although these models are computationally efficient, their validity and
accuracy remain questionable, since a link with the underlying microstructure
(including its randomness) is not clear. In this paper, we propose a modeling
strategy for the linear elastic analysis of FGMs systematically based on a
realistic microstructural model. The overall response of FGMs is addressed in
the framework of stochastic Hashin-Shtrikman variational principles. To allow
for the analysis of finite bodies, recently introduced discretization schemes
based on the Finite Element Method and the Boundary Element Method are employed
to obtain statistics of local fields. Representative numerical examples are
presented to compare the performance and accuracy of both schemes. To gain
insight into similarities and differences between these methods and to minimize
technicalities, the analysis is performed in the one-dimensional setting.Comment: 33 pages, 14 figure
- …