Finite Element Quadrature of Regularized Discontinuous and Singular Level Set Functions in 3D Problems

Abstract

Regularized Heaviside and Dirac delta function are used in several fields of computational physics and mechanics. Hence the issue of the quadrature of integrals of discontinuous and singular functions arises. In order to avoid ad-hoc quadrature procedures, regularization of the discontinuous and the singular fields is often carried out. In particular, weight functions of the signed distance with respect to the discontinuity interface are exploited. Tornberg and Engquist (Journal of Scientific Computing, 2003, 19: 527-552) proved that the use of compact support weight function is not suitable because it leads to errors that do not vanish for decreasing mesh size. They proposed the adoption of non-compact support weight functions. In the present contribution, the relationship between the Fourier transform of the weight functions and the accuracy of the regularization procedure is exploited. The proposed regularized approach was implemented in the eXtended Finite Element Method. As a three-dimensional example, we study a slender solid characterized by an inclined interface across which the displacement is discontinuous. The accuracy is evaluated for varying position of the discontinuity interfaces with respect to the underlying mesh. A procedure for the choice of the regularization parameters is proposed.

Cracking in autoclaved aerated concrete: Experimental investigation and XFEM modeling

“…Other developments have regarded the simulation of crack propagation in composite materials [28] and the combination of XFEM with other techniques so as to increase the rate of convergence (e.g., cut off functions and geometric enrichment, [29,30]). Moreover, several researches have been devoted to the solution of numerical and technical problems, mainly related to enrichment implementation, as well as to the assembly of the stiffness matrix (which requires integra-tion of singular/discontinuous functions) and to the quadrature of the weak form (among others, e.g., [31][32][33][34]). Traditional quadrature techniques, which are successfully adopted for standard finite elements, should be indeed modified when the approximation space is enriched by singular/discontinuous functions, since inaccurate quadrature can lead to poor convergence and inaccuracy in the solution.…”

Accuracy of three-dimensional analysis of regularized singularities

“…Denote the truncated regularized delta function as where w ρ represents half the width of the truncated support of the regularized delta function. According to , the analytical error associated with truncation can be neglected, provided that the mass condition and the first moment conditions hold The first order moment condition is automatically satisfied because δ ρ is an even function . The error in the mass condition is the truncation error that depends on ρ and w ρ .…”

“…The first order moment condition (36b) is automatically satisfied because ı is an even function [39]. The error in the mass condition (36a) is the truncation error that depends on and w .…”

Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis

“…However, these functions result in discontinuous terms in the finite element stiffness matrix, leading to a non-negligible computational error when integrating using a classical quadrature rule due to the non-polynomial nature of the integrand [13]. Quadrature of terms including discontinuous and singular functions is usually achieved in this context by subdividing the elements crossed by discontinuities into quadrature subdomains [7,12,14], although alternative quadrature methods via adaptive quadrature [15], nonconvex polygons [16] and a regularised Heaviside step function [17,18] have been proposed. Nonetheless, splitting the integration domain into subdomains spoils the elegance of XFEM and somehow voids its main purpose of not requiring remeshing as quadrature subcells are to be introduced.…”