A multiple level set approach to prevent numerical artefacts in complex microstructures with nearby inclusions within XFEM

Abstract

SUMMARYIn this paper, we highlight that when the extended finite element method (XFEM) is employed to model a microstructure in which inclusions are involved and the distance between two inclusions is small enough to be comparable with the mesh size, three numerical artefacts are induced, significantly affecting the convergence and accuracy of the numerical solution to the problem with such a microstructure. These artefacts are: (a) an artificial percolation of nearby inclusions; (b) an artificial distortion of phase domains; and (c) an enrichment deficiency. We propose to improve the XFEM/Level set method so as to avoid these artefacts. The new technique leading to this improvement uses one level set function for each inclusion and adds additional enrichment in an element whose support is cut by several interfaces. A local description of the multiple level sets is provided to avoid the storage of all level set functions. A simple integration rule is employed for numerical quadrature in elements cut by several interfaces. We show that the artefacts mentioned hereinbefore are circumvented in this framework. The performances of the method are demonstrated through benchmarks and examples applied to the homogenization of concrete materials in 2D and 3D cases.

Equivalent-inclusion approach for the conductivity of isotropic matrix composites with anisotropic inclusions

“…For numerical computations, the extended finite elements method [20][21][22][23][24] will be used. In this framework, the level-set function is involved to describe the interface of inclusions.…”

“…The bounds containing the properties and volume fractions of the component materials are not very useful in the case of high contrast of matrix-inclusion properties. The numerical methods [4][5][6][7][8][9], such as finite element one, fast Fourier transformation may give better results. However, they require much computer resources and computational time when the microstructure is complex and the inclusions are close to each other.…”

XFEM/Level-set for modelling of coated inclusion composites

“…We can opt to choose the signed distance function for φ k (x), which is given in [21]. For a spherical coated inclusion, φ k (x) is expressed by…”

High order X-FEM and levelsets for complex microstructures: Uncoupling geometry and approximation

“…Indeed, if only one enrichment is considered for node I, the approximation is not rich enough to make displacement jump vanish along both interfaces. It was shown in Tran et al [51] that an entity should be enriched with respect to all the interfaces crossing its support. In this case, the authors proposed to define separate enrichment functions for each of the interfaces.…”

“…Considering the local form of the eigenvalue problem would decrease the cost of the procedure. Note that, as advocated by Tran et al [51], multiple enrichments are considered in the approximation (see section 3.4). Without this approach, the convergence is strongly degraded.…”