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Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework

Sundararajan Natarajan, D. Roy Mahapatra, Stephane PA Bordas

arXiv:1107.4732v1math.NAcs.NA

TL;DR

Discontinuous XFEM stiffness integrands are commonly integrated by subdividing cut elements into aligned quadrature subcells. The paper applies a two-dimensional Schwarz-Christoffel conformal-mapping integration technique to avoid that subdivision; on the reported examples, it slightly outperforms conventional subcell integration at similar integration-point counts, while both methods converge to analytical SIFs with mesh refinement.

  • Problem

    Integrating XFEM stiffness matrices with discontinuous or singular integrands commonly requires aligned subcells, which impose a mesh-alignment requirement and are costly and less accurate for curved discontinuities.

    Source: §1 · p. 3, §2 · p. 8

  • Method

    The proposed 2D technique conformally maps discontinuity-cut element parts to a unit disk and uses midpoint quadrature to integrate the stiffness matrix without quadrature subcells.

    Source: §1 · p. 4, §3 · p. 9

  • Results

    With similar numbers of integration points, the proposed method slightly outperforms conventional integration, and both techniques converge to analytical SIFs with mesh refinement.

    Source: §4.1 · p. 14, §5 · p. 30

  • Takeaways & Limitations

    The technique provides an alternative to subcell integration for 2D XFEM stiffness calculations involving strong or weak discontinuities.

    Source: §1 · p. 4, §5 · p. 30

    SIF convergence rates are suboptimal in the reported setup, which enriches only the tip element and applies no blending correction.

    Source: §4.1 · p. 15

Abstract

from arXiv · show

Partition of unity methods, such as the extended finite element method (XFEM) allow discontinuities to be simulated independently of the mesh [1]. This eliminates the need for the mesh to be aligned with the discontinuity or cumbersome re-meshing, as the discontinuity evolves. However, to compute the stiffness matrix of the elements intersected by the discontinuity, a subdivision of the elements into quadrature subcells aligned with the discontinuity is commonly adopted. In this paper, we use a simple integration technique, proposed for polygonal domains [2] to suppress the need for element subdivision. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics and a multi-material problem, show that the proposed method yields accurate results. Owing to its simplicity, the proposed integration technique can be easily integrated in any existing code.

1. INTRODUCTION

The introduction motivates partition-of-unity enrichment as a way to represent problem-specific behavior, including discontinuities, without changing the underlying mesh. It presents a two-dimensional conformal-mapping integration technique intended to avoid quadrature subcells in XFEM stiffness calculations.

1. INTRODUCTION

  • XFEM extends the approximation basis with enrichment functions carrying solution features, supporting fracture problems with minimal remeshing.Partition-of-unity enrichment also allows moving-boundary problems to be modeled without changing the underlying mesh as the interface evolves.Source: §1 · p. 2
  • Discontinuous or non-polynomial element approximations require special numerical integration, and low solution continuity can reduce derivative accuracy near high-gradient regions, especially in 3D.The cited derivative-accuracy concern has motivated adaptivity work in GFEM and XFEM, as well as meshfree methods.Source: §1 · p. 3
  • Ventura’s equivalent-polynomial approach is exact on triangles and tetrahedra, but adds approximation for quadrilaterals whose opposite sides are not parallel.Source: §1 · p. 3
  • Strain smoothing offers another integration approach, but its convergence, stability, and accuracy remain to be understood.It combines SFEM with XFEM and transforms surface integration into equivalent boundary integration.Source: §1 · pp. 3–4
  • The proposed 2D technique maps cut element parts conformally to a unit disk with Schwarz-Christoffel mapping and uses midpoint quadrature, avoiding integration subcells.The method is proposed for computing the stiffness matrix of elements intersected by discontinuities.Source: §1 · p. 4
  • The paper evaluates efficiency and convergence on linear-elastic-fracture benchmark problems and a multi-material problem.Source: §1 · p. 4

2. BASICS OF PARTITION OF UNITY METHODS FOR DISCONTINUITIES AND SINGULARITIES

Partition-of-unity methods let XFEM represent cracks independently of the finite-element mesh by enriching the approximation locally. The resulting discontinuous stiffness integrands motivate aligned integration subcells, while the section introduces a polygonal-domain integration alternative.

2. BASICS OF PARTITION OF UNITY METHODS FOR DISCONTINUITIES AND SINGULARITIES

  • XFEM uses a regular mesh independent of crack geometry, avoiding the mesh conformity and remeshing required by standard FEM as a discontinuity evolves.Source: §2 · p. 4
  • For fracture mechanics, Heaviside enrichment captures the crack-face jump and near-tip branch functions represent asymptotic crack-tip fields.Nodes whose supports are split by the crack receive Heaviside enrichment; nodes in crack-tip elements receive near-tip enrichment.Source: §2 · p. 6
  • Adding enrichment enlarges the problem to solve, with element stiffness contributions formed from standard and enriched strain-displacement matrices and the material matrix.Source: §2 · pp. 6–7
  • Standard Gauss quadrature is unsuitable for discontinuous enriched integrands, so elements are split into aligned subcells where the integrands are continuous and differentiable.The usual triangular subcells require alignment to the discontinuity and can be costly and less accurate when it is curved.Source: §2 · pp. 7–8, Figure 2 · p. 7
  • The section proposes polygonal-domain numerical integration for elements intersected by discontinuities as an alternative to subcell integration.Source: §2 · p. 8

3. SCHWARZ-CHRISTOFFEL CONFORMAL MAPPING

A Schwarz-Christoffel map conformally maps a canonical domain to a polygon. The paper applies this mapping to integrate over XFEM elements intersected by discontinuities.

  • A Schwarz-Christoffel map conformally maps a canonical domain, such as a half-plane or unit disk, to a closed polygon.Conformal transformations preserve local angles.Source: §3 · p. 8
  • A bijective holomorphic map sends the upper half-plane to a polygon’s interior and maps the real axis to its edges.Source: §3 · pp. 8–9
  • In the mapping formula, K is a constant and a, b, c, … are real-axis values corresponding to polygon vertices.Source: §3 · p. 9
  • The authors previously used Schwarz-Christoffel mapping with disk cubature rules to integrate arbitrary polygons.The cited approach uses midpoint or Gauss-Chebyshev rules to obtain integration points on the unit disk.Source: §3 · p. 9
  • For 2D XFEM elements with discontinuous approximations or derivatives, each part is mapped onto a unit disk for numerical integration using the technique in.The figure illustrates the procedure for split and tip elements.Source: §3 · p. 9, Figure 5 · p. 12
  • Elements not intersected by a discontinuity use standard isoparametric mapping with four integration points per element.Source: §3 · p. 9

4. NUMERICAL EXAMPLES

The numerical examples test the integration method on fracture-mechanics and bimaterial benchmarks, including crack growth. Across the reported cases, results are compared with analytical solutions or published crack paths, and the bimaterial strain energy converges with mesh refinement.

4. NUMERICAL EXAMPLES

  • The examples cover linear elastic fracture mechanics and a multi-material problem, using quadrilateral elements and topological enrichment unless otherwise noted.For standard XFEM and the proposed method, the study uses 13 integration points per subcell and a similar total for each tip element; only tip elements receive near-tip enrichment.Source: §4 · p. 13

4.1. Infinite plate under tension

  • At equal integration-point counts, the proposed method slightly outperforms standard XFEM; both approach the analytical SIF, while convergence rates are 0.4 with the stated enrichment choices.On irregular meshes, the methods give comparable results, and the proposed method avoids subdividing split and tip elements. The rate reflects tip-only enrichment and the absence of blending correction.Source: §4.1 · pp. 14, 15, 17

4.2. Edge crack under tension

  • For the edge-crack plate, both methods approach the analytical SIF with mesh refinement, the proposed method performs slightly better, and both have a convergence rate of 0.4.The passage identifies geometric enrichment or modified enrichment functions in blending elements as possible ways to improve the rate.Source: §4.2 · p. 18

4.3. Inclined crack in tension

  • For inclined cracks, the computed stress intensity factors are comparable with the analytical solution; polygonal split regions are mapped to a unit disk rather than triangulated.The crack angle is varied, and for 0 < β < 90 the split elements contain a polygonal region.Source: §4.3 · p. 21, Figure 13 · p. 21

4.4. Multiple cracks in tension

  • As H/L increases, crack interaction decreases and normalized mode I SIF curves approach the single-center-crack analytical value, but do not equal it.The reference analytical value is for a plate with one center crack, unlike the two-crack setup.Source: §4.4 · pp. 22, 24, Figure 16 · p. 24
  • As both crack angles increase, mode I SIF decreases toward zero at 90°, while mode II SIF peaks at 45° and then declines.These angle trends are reported for equal crack lengths of 0.2 and spacing H = 0.1, L = 0.2.Source: §4.4 · p. 24, Figure 17 · p. 25

4.5. Bimaterial bar

  • For the bimaterial bar, displacement error decreases with mesh refinement and the L2 convergence rate is 1.34; strain energy converges for three interface configurations.The reported displacement convergence rate is suboptimal, which the authors attribute to untreated blending elements.Source: §4.5 · p. 26, Figure 19 · p. 28

4.6. Double cantilever beam

  • In the double-cantilever-beam crack-growth example, the simulated crack path qualitatively agrees with published results.The quasi-static growth uses the maximum hoop stress criterion and is simulated for eight steps.Source: §4.6 · pp. 29–30

5. CONCLUSION

The proposed integration method avoids subdividing elements cut by strong or weak discontinuities or containing a crack tip. Across the reported examples, it slightly outperforms conventional subdivision-based integration at similar integration-point counts, while both methods converge toward analytical stress intensity factors.

5. CONCLUSION

  • The polygon-based integration technique eliminates subdivision of elements cut by strong or weak discontinuities or containing a crack tip.Source: §5 · p. 30
  • With similar numbers of integration points, the proposed technique slightly outperforms conventional subdivision-based integration.Source: §5 · p. 30
  • With mesh refinement, both integration techniques converge toward analytical stress intensity factors.Source: §5 · p. 30
  • The authors suggest that extending the technique to 3D discontinuous approximations may be possible and identify it as future work.Source: §5 · p. 30
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