Generalized Gaussian quadrature rules on arbitrary polygons

Abstract

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.

The virtual element method in 50 lines of MATLAB

“…It is the use of this quadrature which produces the requirement in Section 3 that the element must contain its own centroid. Clearly, more general integration methods are possible (for example by triangulating the element or using more advanced techniques such as [19,20,24,26]), although for the sake of simplicity this is not something we pursue here. Since each basis function of V E h is defined to be 1 at a single vertex and 0 at the others, we can express…”

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

“…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).…”

“…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.…”

“…These integrations were performed algebraically in Ref. [29]. Although algebraic integration of polynomials is straightforward in two dimensions, it is not generally a well-posed problem in three dimensions, and therefore extending the technique of Ref.…”

“…Although algebraic integration of polynomials is straightforward in two dimensions, it is not generally a well-posed problem in three dimensions, and therefore extending the technique of Ref. [29] to three dimensions is unwieldy. It bears mention that for monomials one can use divergence theorem to transform the surface integrals to line integrals [8], which can be handled efficiently.…”