A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

Xiao, Han; Gimbutas, Zydrunas

doi:10.1016/j.camwa.2009.10.027

Cited by 175 publications

(173 citation statements)

References 6 publications

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“…For example, in the node elimination algorithm [29,30,32], the basis functions can be selected as monomials that are homogeneous functions (see Sect. 3).…”

Section: Example 2: Homogeneous Quadrature Over a Regular Hexagonmentioning

confidence: 99%

“…The resulting quadrature is exact (within the tolerance of the solution method) for the integration of the basis functions. The minimum number of integration points that can satisfy (14) is a classical problem of numerical analysis which has no solution yet [30]. Each integration point in n-dimensions, contributes n + 1 unknowns (degrees of freedom): n unknowns for its coordinate components and one for the weight.…”

Section: Moment Fitting Equationsmentioning

confidence: 99%

“…2 can be applied to calculate lhs. Moreover, can take on the form of any other set of basis functions that are linearly independent and cover the space of polynomials up to order d. For example, one may start with the monomials P d and orthogonalize them for the given weight function over the integration region via a Gram-Schmidt procedure [29,30]. It bears mention that the integration of the orthogonal polynomials can be done using integration of homogeneous functions, of course with some extra steps.…”

Section: Moment Fitting Equationsmentioning

confidence: 99%

“…In Ref. [29], moment fitting was used together with the node elimination algorithm [30], to construct and optimize quadratures for the integration of polynomials over arbitrary polygons. The moment equations contain the integration of the basis functions (polynomials of total degree up to d) over the domain.…”

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confidence: 99%

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Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

2010

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We construct efficient quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous quadratures is based on Lasserre's method for the integration of homogeneous functions over convex polytopes. We also construct quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.Keywords Numerical integration · Lasserre's method · Euler's homogeneous function theorem · Irregular polygons and polyhedrons · Homogeneous and nonhomogeneous functions · Strong and weak discontinuities · Polygonal finite elements · Extended finite element method

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“…For example, in the node elimination algorithm [29,30,32], the basis functions can be selected as monomials that are homogeneous functions (see Sect. 3).…”

Section: Example 2: Homogeneous Quadrature Over a Regular Hexagonmentioning

confidence: 99%

Section: Moment Fitting Equationsmentioning

confidence: 99%

Section: Moment Fitting Equationsmentioning

confidence: 99%

mentioning

confidence: 99%

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Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

2010

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“…Non-negative weights lend stability to the quadrature rule by preventing round-off errors [24]. Xiao and Gimbutas [19] introduce a significance index for each quadrature node (see (8)) that can be considered as a measure of contribution of the node in the evaluation of the integral over the domain. If there are no nodes outside the domain, then the one with the least significance is eliminated.…”

Section: Algorithm For Construction Of Efficient Quadraturesmentioning

confidence: 99%

Generalized Gaussian quadrature rules on arbitrary polygons

Mousavi

Xiao

Sukumar

2009

Numerical Meth Engineering

159

112

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SUMMARYIn this paper, we present a numerical algorithm based on group theory and numerical optimization to compute efficient quadrature rules for integration of bivariate polynomials over arbitrary polygons.These quadratures have desirable properties such as positivity of weights and interiority of nodes and can readily be used as software libraries where numerical integration within planar polygons is required. We have used this algorithm for the construction of symmetric and non-symmetric quadrature rules over convex and concave polygons. While in the case of symmetric quadratures our results are comparable to available rules, the proposed algorithm has the advantage of being flexible enough so that it can be applied to arbitrary planar regions for the integration of generalized classes of functions. To demonstrate the efficiency of the new quadrature rules, we have tested them for the integration of rational polygonal shape functions over a regular hexagon. For a relative error of 10 −8 in the computation of stiffness matrix entries, one needs at least 198 evaluation points when the region is partitioned, whereas 85 points suffice with our quadrature rule.

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Measure transformation and efficient quadrature in reduced‐dimensional stochastic modeling of coupled problems

Arnst

Ghanem

Phipps

et al. 2012

Numerical Meth Engineering

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Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower-dimensional space than the sources themselves. In this work, we thus propose to use a dimension reduction technique for obtaining the representation of the exchanged information, and we propose a measure transformation technique that allows subproblem implementations to exploit this dimension reduction to achieve computational gains. The effectiveness of the proposed dimension reduction and measure transformation methodology is demonstrated through a multiphysics problem relevant to nuclear engineering. 1045 which include Monte Carlo sampling techniques [3] and stochastic expansion methods. The latter typically involve the computation of a representation of the predictions as a polynomial chaos (PC) expansion. Several approaches, such as embedded projection [4,5], nonintrusive projection [5], and collocation [6][7][8][9][10], are available to calculate the coefficients in this expansion.A key challenge in the formulation and implementation of a coupled model is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. This information can comprise physical properties, energetic quantities, or solution patches, among other quantities. Although the number of sources of uncertainty can be expected to be large in most coupled problems, we believe that the exchanged information often resides in a considerably lower dimensional space than the sources themselves. Exchanged information can be expected to have a low effective stochastic dimension in multiphysics problems when this information consists of a solution field that has been smoothed by a forward operator and in multiscale problems when this information is obtained by summarizing fine-scale quantities into a coarse-scale representation.In a previous paper [11], we had proposed the use of a dimension reduction technique to represent the exchanged information: we proposed to represent the exchanged information by an adaptation of the Karhunen-Loève (KL) decomposition as this information passes from subproblem to subproblem and from iteration to iteration. The main objective of this paper is to complement this dimension reduction technique by a measure transformation technique that allow...

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A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

Cited by 175 publications

References 6 publications

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

Generalized Gaussian quadrature rules on arbitrary polygons

Measure transformation and efficient quadrature in reduced‐dimensional stochastic modeling of coupled problems

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