Accuracy of three-dimensional analysis of regularized singularities

Abstract

SUMMARYIn computational mechanics, the quadrature of discontinuous and singular functions is often required. To avoid specialized quadrature procedures, discontinuous and singular fields can be regularized. However, regularization changes the algebraic structure of the solving equations, and this can lead to high errors. We show how to acquire accurate and consistent results when regularization is carried out. A three-dimensional analysis of a tensile butt joint is performed through a regularized extended finite element method. The accuracy obtained via Gaussian quadrature is compared with that obtained by means of CUBPACK adaptive quadrature FORTRAN tool. The use of regularized functions with non-compact and compact support is investigated through an error evaluation procedure based on the use of their Fourier transform. The proposed procedure leads to the remarkable conclusion that regularized delta functions with non-compact support exhibit superior performance.

Crack-Tracking In The Regularized XFEM: A Viable Alternative To Nonlocal And Cohesive Zone Models

“…The support of δ ρ noncompact, and it is necessary to introduce a truncation length beyond which the value of δ ρ is not to be evaluated. We found [19] that a truncated support length of 40ρ provides a satisfying compromise between computational burden and accuracy, at least for the present H ρ . In the following developments, the lengths ℓ ρ,M and ℓ ρ,m will denote the width of the truncated supports of δ ρ for ρ = ρ M and ρ = ρ m , respectively.…”

A mesh-independent framework for crack tracking in elastodamaging materials through the regularized extended finite element method

“…(54), δ ρ,M is computed in terms of ρ M , so that the interaction radius equals ρ,M /2, corresponding to averaging over a volume of width 40ρ M . We have heuristically verified that ρ M provides a satisfying compromise between computational burden and accuracy [20]. Remarkably, both the local stress ( 53) and the nonlocal stress ( 54) are computable in any element whether enriched or not, and this is useful when the element under consideration has not yet been enriched.…”

“…where Being the support of δ ρ noncompact, it is necessary to introduce a truncation length beyond which δ ρ is not evaluated. The approximation issues related to the choice of the truncation length were investigated in [20]. In particular, it was found that a truncated support length of 40ρ provides a satisfying compromise between computational burden and accuracy, at least for the present H ρ .…”

“…The final effect of using the regularized functions H ρ and δ ρ is the same as that of ramp functions to handle blending elements in standard extended finite element method. The regularized delta function is such that it is almost zero in the standard elements, except for the truncation error [20], and satisfies the partition of unit finite element property in fully enriched elements, while decaying continuously to zero in the blending elements. Analogously, the regularized Heaviside tends to a constant profile in the blending elements.…”

On regularizations of the Dirac delta distribution

“…Their approach to constructing polynomial regularizations using the Chebyshev basis has a similar flavor to our approach, as will be seen in Section 3. In a different approach, Benvenuti et al [5] study the case of regularizations that are not compactly supported but have rapidly decaying Fourier transforms in the context of extended finite element methods (XFEM) [4]. The authors demonstrate that such regularizations lead to lower numerical errors since they can be integrated using common quadrature methods such as Gauss quadrature.…”