The XFEM for high‐gradient solutions in convection‐dominated problems

Abstract

SUMMARYConvection-dominated problems typically involve solutions with high gradients near the domain boundaries (boundary layers) or inside the domain (shocks). The approximation of such solutions by means of the standard finite element method requires stabilization in order to avoid spurious oscillations. However, accurate results may still require a mesh refinement near the high gradients. Herein, we investigate the extended finite element method (XFEM) with a new enrichment scheme that enables highly accurate results without stabilization or mesh refinement. A set of regularized Heaviside functions is used for the enrichment in the vicinity of the high gradients. Different linear and non-linear problems in one and two dimensions are considered and show the ability of the proposed enrichment to capture arbitrary high gradients in the solutions.

A new method for fracture toughness determination of graded (Pt,Ni)Al bond coats by microbeam bend tests

“…Enriched elements are used around the notch with enhanced degrees of freedom to account for the discontinuity across the crack line and asymptotic stress fields ahead of the crack tip. The technique also sets aside the need for an extremely fine mesh at the crack tip without which convergence cannot be achieved with conventional FEM techniques [35]. The stresses, strains and displacements recovered from the XFEM user elements is used to calculate the integration point's contribution to the element's interaction integral value, which then is subsequently used to compute the SIF for the model.…”

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

“…we choose weight functions with compact support equal to the finite element size multiplied by a suitable integer number m that depends on h, namely ℓ ρ ∝ m h [1,3]. This means that whenever the mesh changes, the weight function has to be changed accordingly, otherwise non-vanishing errors for decreasing mesh size are obtained.…”

“…Regularization of singular and highly localized fields is a widely diffuse practice in several computational mechanics fields, such as thermal analysis, advection-diffusion problems [1], in Boundary Element Method applications [2], electronic structure calculations [3], for applications to modeling delamination in multilayered composites [4], and for capturing the behaviour of plastic hinges in Timoshenko beam elements [5]. Among other methods [6,7], the eXtended Finite Element Method (XFEM) [6,[8][9][10]] is a powerful technique for modelling engineering problems characterized by discontinuous and singular functions.…”

“…The use of the level set method [20] leads to a straightforward extension of regularized Heaviside and delta functions to the multidimensional case. Albeit regularization can be carried out by means of compact support weight functions, in [1,3,19,21] it has been proved that three-dimensional extensions based on compact support weight functions can lead to inconsistent results, as errors do not reduce with discretization refinement. The problem is that the moment conditions no longer hold for any shift (dilation) of the weight function with respect to the quadrature grid.…”

Accuracy of three-dimensional analysis of regularized singularities

“…One of the advantages of adopting this regularized XFEM is that blending is not required. Whereas kinematics is regularized analogously to other approaches based on the regularization of the Heaviside and delta functions , the strain fields, the constitutive modeling, and the adopted variational formulation are different.…”