Quadrature rules for triangular and tetrahedral elements with generalized functions

Abstract

SUMMARYQuadrature rules are developed for exactly integrating products of polynomials and generalized functions over triangular and tetrahedral domains. These quadrature rules greatly simplify the implementation of finite element methods that involve integrals over volumes and interfaces that are not coincident with the element boundaries. Specifically, the integrands considered here consist of a quadratic polynomial multiplied by a Heaviside or Dirac delta function operating on a linear polynomial. This form allows for exact integration of expressions obtained from linear finite elements over domains and interfaces defined by a linear level set function. Exact quadrature rules are derived that involve fixed quadrature point locations with weights that depend continuously on the nodal level set values. Compared with methods involving explicit integration over subdomains, the quadrature rules developed here accommodate degenerate interface geometries without any need for special consideration and provide analytical Jacobian information describing the dependence of the integrals on the nodal level set values. The accuracy of the method is demonstrated for a simple conduction problem with the Neumann and Robin-type boundary conditions.

The extended/generalized finite element method: An overview of the method and its applications

“…These polynomials are defined so to give the exact result, using coefficients that are function of the location of the discontinuity in the element. A related approach has been proposed by Holdych et al [55]. Here, starting with a given integration rule, quadrature weights are found, depending on the position of the discontinuity inside the element.…”

“…Here, starting with a given integration rule, quadrature weights are found, depending on the position of the discontinuity inside the element. The methods proposed in [106,55] gives the exact element stiffness, so it follows that the weights determined in [55] will coincide with the values of the equivalent polynomials [106] at the quadrature points.…”

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

“…For some of the existing methods for integrating discontinuous functions, see Refs. [32,[37][38][39].…”

High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles

“…Standard quadrature schemes are then applied to the surface and volume elements (e.g., triangles or tetrahedra) and result in a second-order accurate approximation of the surface and volume integrals; see, e.g., [15,11,31]. For increased accuracy, subdivision techniques are often used to locally refine the mesh geometry.…”