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Global cracking elements: a novel tool for Galerkin-based approaches simulating quasi-brittle fracture

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Pith's one-line read The paper claims a Galerkin finite-element method that reuses the center-node degrees of freedom of a nine-node quadrilateral element as global crack openings, eliminating remeshing, enrichment, and crack tracking in quasi-brittle fracture.

desk verdict Neat pseudo-Q9 DOF-borrowing reformulation of CEM with credible benchmarks, but the tangent symmetrization in Eq. 16 is unjustified—possibly inconsistent—and the efficiency claim is under-supported. read the letter →

arxiv 1908.06268 v1 pith:T7TMNR53 submitted 2019-08-17 cs.CE

Yiming Zhang , Herbert A. Mang This is my paper · ORCID
classification cs.CE MSC 74S0574R1065N30
keywords quasi-brittlefracturecrackingelementsmethodstandardGalerkinformulationstrongdiscontinuityembeddedapproachpseudo-Q9elementcrackpropagationtraction-separationlawBraziliandisktest
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper reorganizes the earlier Cracking Elements Method (CEM) so that it lives inside a standard Galerkin finite-element framework: each cracked element is treated as a nine-node quadrilateral whose center-node displacement degrees of freedom are reassigned to represent that element's two crack-opening components. Because the crack openings become global unknowns, the method no longer needs the CEM's inner element-level iteration. The authors argue that the resulting scheme captures crack initiation and propagation in quasi-brittle materials on irregular, relatively coarse meshes without remeshing, enrichment, or crack tracking, and with better numerical stability and fewer Newton-Raphson iterations than the CEM. They support this with an L-shaped panel test and Brazilian disk tests with single and double slots, comparing force-displacement curves, normalized peak loads, and crack patterns with experiments and with other numerical methods. The interest for a general reader is that if this holds, fracture simulation becomes as routine as ordinary nonlinear finite-element analysis, at least for problems that admit quadrilateral meshes.

What carries the argument

The load-bearing object is the pseudo-Q9 cracking element, a strong-discontinuity embedded formulation in which each crack is represented by a displacement jump inside an element rather than by additional nodes. In it, the center-node degrees of freedom of a standard nine-node quadrilateral are borrowed, not deleted: they switch from describing the displacement of the center point to describing the normal and tangential crack openings ($\zeta_n$, $\zeta_t$) of that element. This promotion of crack openings to global unknowns is what removes the inner element-level iteration of the original CEM and yields a symmetric, sparse global stiffness matrix. The element's strain is written as $\bar{\varepsilon} = B U - B_\zeta \zeta$ (with $B_\zeta$ built from the crack normal $\mathbf{n}$ and the characteristic length $l_c$), and the practical stiffness is the symmetrized form $K_{\mathrm{sym}}$ assembled from $[B, B_\zeta]$, with the traction-separation tangent added in the crack-opening block. That symmetrized matrix carries the Newton-Raphson iteration in the numerical examples.

What would settle it

A direct check would be to run the method on a few-element specimen under non-proportional loading (load, unload, then rotate the principal stress direction) and compare the Newton iteration counts and final crack openings with those obtained from the non-symmetric tangent version of the same element; if the symmetric version needs significantly more iterations or diverges, the central stability claim is not general.

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Extended reading notes

Core claim

The central claim is that a strong-discontinuity embedded approach to quasi-brittle fracture can be written so that the only extra unknowns are global, element-wise crack openings, and that this form is more robust and cheaper than its predecessor. The device is a 'pseudo-Q9' element: the perimeter nodes of a nine-node quadrilateral keep their displacement unknowns, and the center node's two displacement degrees of freedom are replaced by the normal and tangential crack openings $\zeta_n$ and $\zeta_t$ of that element. From then on the cracked element is an ordinary Galerkin finite element with a symmetric, sparse stiffness matrix, so no remeshing, enriched basis, or crack-tracking strategy is required. Crack orientation is chosen locally from the eigenvector of the strain, and propagation is found by expanding a propagation region rather than by following a crack tip. The numerical tests show the method reproducing the benchmark force-displacement curves and peak loads of the L-shaped panel and slotted Brazilian disks, with small mesh dependence and fewer Newton iterations than the CEM.

Load-bearing premise

The method's stability and speed depend on an unproved simplification in which a non-symmetric matrix is replaced by a symmetric one in the element formulation; if that simplification is inaccurate for some loading paths, the claimed Newton-Raphson advantages may disappear.

Editorial extensions

If this is right

  • Crack initiation and propagation in quasi-brittle materials can be simulated with no remeshing, no enrichment, and no crack-tracking strategy, using only standard Galerkin finite-element machinery and irregular meshes.
  • Because crack openings are global unknowns, the Newton-Raphson iteration needs fewer steps than in the original CEM; the paper reports this consistently across the benchmark meshes.
  • The pseudo-Q9 form keeps a symmetric sparse stiffness matrix, so existing sparse linear solvers and standard nonlinear finite-element codes can adopt the method without new global data structures.
  • The formulation can be changed to an enriched form of the crack-opening field when desired, since the pseudo-Q9 structure leaves a clear place for enrichment.
  • Presently only quadrilateral elements with nonlinear displacement interpolation are usable, so the method is not yet available for triangular or linear-element meshes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the center-node borrowing suggests a general recipe: any element with re-assignable degrees of freedom could carry kinematic enrichment inside a standard Galerkin code, so the idea may extend to shear bands or delamination without changing the global solver.
  • Beyond the paper, the loading paths shown are monotonically increasing, so an untested consequence is how the symmetric-tangent replacement behaves under unloading-reloading cycles or load-path rotations; those are exactly the situations where a non-symmetric tangent may matter.
  • Beyond the paper, a quantitative mesh-convergence study across several element sizes, rather than the two meshes shown, would pin down the claimed negligible mesh dependency.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes the Global Cracking Elements Method (GCEM), a reformulation of the Cracking Elements Method (CEM) for quasi-brittle fracture. The key idea is to treat element-wise crack openings as global unknowns by borrowing the degrees of freedom of the center node of a Q9 element, thereby producing a pseudo-Q9 formulation that fits the standard Galerkin/FEM assembly framework. The kinematics, mixed-mode traction-separation law, and crack-initiation/propagation criteria are adopted from the authors' previous CEM publications. The new element is presented in matrix form, and numerical tests include an L-shaped panel and Brazilian disk tests with single and double slots, compared with experiments, XFEM, CEM, phase-field, and peridynamics results. The reported advantages are no remeshing, no enrichment, no crack tracking, and fewer Newton iterations than the original CEM.

Significance. If the formulation is fully justified, GCEM is a potentially attractive method for quasi-brittle fracture: the center-node borrowing idea is elegant, the implementation appears straightforward, the benchmark results match independent solutions and experiments, and the mesh-sensitivity evidence is favorable. The paper is also honest about current limitations, namely quadrilateral elements only and no direct crack-tip tracking. However, the central numerical claim, improved stability and efficiency over CEM, currently rests on an unproved replacement of the consistent tangent by a symmetric matrix, and the internal consistency of the tangent derivation is questionable. Thus the significance is conditional on resolving the tangent issue and on providing a more quantitative cost comparison.

major comments (3)
  1. [Section 2.3, Eq. (16)] The paper replaces K_s,new from Eq. (15) with the symmetric matrix K_sym in Eq. (16) with only the statement 'Fortunately, K_s,new can be replaced by the following symmetric matrix.' No derivation from the residual, no proof of solution invariance, and no convergence study are provided. This replacement is load-bearing because the Newton-Raphson iteration and the claimed efficiency and stability advantage in Section 3 (Figs. 7-8) depend on it. Please add a proof that K_sym corresponds to a consistent tangent or an equivalent potential, and a numerical study comparing convergence behavior and solutions obtained with K_sym against those obtained with the unsymmetric tangent.
  2. [Section 2.3, Eqs. (14)-(15)] The statement that K_s,new is not symmetric is inconsistent with the quadrature implied by the formulation. With B^(e),1 evaluated at the center point and B_zeta constant, the off-diagonal blocks of K_new in Eq. (15) are transposes of each other, and the added diagonal block is symmetric, so K_s,new as printed is symmetric. If the intended matrix is instead the one in Eq. (14) with the lower-left block scaled by -l_c, then K_sym in Eq. (16) is not the linearization of the residual. Please clarify which matrix is actually assembled and resolve this sign and symmetry inconsistency; the current text leaves the reader unable to reproduce the tangent.
  3. [Section 3, Figs. 7-8] The efficiency claim is supported only by Newton-iteration counts for the L-shaped panel, not by wall-clock time and not by comparison with a solver that handles the unsymmetric tangent. Because GCEM moves the crack-opening unknowns into the global system, its per-iteration cost may be higher than CEM's element-level condensation. Please provide a cost comparison such as CPU time or total linear solves, or restrict the claim to iteration counts.
minor comments (5)
  1. [Section 2.1, Eq. (2)] The threshold fracture energy is fixed as Gf,0 = 0.001 Gf without comment or sensitivity study; since this parameter enters the initial stiffness and the tangent matrix D, please state the basis for this value and its influence on the reported results.
  2. [Section 3.1, Eq. (22)] There is a typo 'if if' before the convergence criterion; please also specify how the criterion is applied when the total elastic energy becomes small or changes sign.
  3. [Section 2.4, Eq. (18)] The crack-orientation criterion is adopted from [1]; it would be helpful to state explicitly whether n^(e) is updated during Newton iterations for an already-cracked element and to quantify the claim that the rotation of n^(e) is very small.
  4. [Section 4] The sentence 'Unlike methods based on equivalent-type theories. The GCEM does not introduce bonds/lattices/links...' is grammatically incomplete and should be rewritten.
  5. [General] The paper would benefit from a short derivation showing how Eq. (12) and the residual in Eq. (14) follow from the weak form, in particular the appearance of the factor V/l_c in Eq. (15).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the GCEM reuses prior constitutive/kinematic ingredients from self-citations, but the new global-DOF formulation is fully specified here and its benchmarks are checked against external experiments and independent methods; the Eq. 16 tangent substitution is an unproved approximation, not a circular reduction.

full rationale

This paper is not circular. The new ingredient—turning element-wise crack openings into global unknowns by borrowing the Q9 center-node DOFs—is fully specified in the present work: the unknown vector U_pseudo-Q9 and B_pseudo-Q9 in Eq. 17, the residual and tangent matrices in Eqs. 14–16, and the crack initiation/propagation algorithm in Section 2.4. The traction-separation law (Eq. 2) and the strong-discontinuity kinematics (Eqs. 4–6) are inherited from the authors' earlier papers [1,15,30,31], but they are inputs with prior validation rather than quantities re-derived from the GCEM and re-labeled as predictions; the current paper's benchmarks are independent checks against external experimental data ([35], [36]) and against other published methods (XFEM [30], phase-field [37], peridynamics [38]). No material or algorithmic constant is fitted to the benchmark targets (Gf,0 = 0.001 Gf is a stated constitutive assumption, not a calibration), so the force-displacement and crack-path results are not forced by construction. The one step that deserves a correctness flag, not a circularity flag, is the replacement of the nonsymmetric consistent tangent K_s,new by the symmetric K_sym in Eq. 16: the paper simply states 'Fortunately, K_s,new can be replaced by the following symmetric matrix' without proof or a convergence study, and the noted sign/block-transpose inconsistency could invalidate the claimed Newton efficiency under some load paths. That is a missing-support or robustness issue, not circularity. Because the central GCEM formulation is not equivalent to its inputs and the validation is external, the circularity score is 0.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The central claim rests on the prior CEM formulation (self-citations [1,15]) plus an unproved symmetric tangent replacement. No parameters are fitted to the benchmark data; the material parameters are taken from the cited experimental and prior studies.

free parameters (1)
  • Gf,0 threshold fracture energy = 0.001 Gf
    Assumed in Section 2.1 after Eq. 2; sets the threshold opening zeta0 = 2 Gf,0/ft in the traction-separation law. Not fitted to the benchmark data in this paper, but chosen by hand.
assumptions (6)
  • domain assumption The kinematic decomposition of the strong discontinuity (Eqs. 4-6) is valid.
    Taken from [1,15]; the paper does not re-derive the enhanced strain field but builds the pseudo-Q9 matrix on it.
  • domain assumption The center-point strain representation holds for the whole element (Eq. 7).
    Inherited from [1,15]; the stress-strain state at the center Gauss point is assumed to represent the element, which is central to the formulation.
  • ad hoc to paper The symmetric tangent K_sym can replace the actual non-symmetric tangent without changing the solution.
    Eq. 16 replaces K_s,new with K_sym without proof; only a reference to Appendix A of [15] for stress locking is given.
  • ad hoc to paper Gf,0 = 0.001 Gf is a fixed threshold fracture energy.
    Set in Section 2.1 after Eq. 2; this is a handmade constant, not fitted or measured.
  • domain assumption Crack orientation follows the principal-strain eigenvector via Eq. 18.
    Local criterion from [1]; standard in embedded element methods and used to set n(e) for each cracking element.
  • domain assumption Small-strain, quasi-static, plane-stress conditions apply.
    Section 3 states plane-stress assumptions; all examples are 2D quasi-static.

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Pith. "Pith review of Global cracking elements: a novel tool for Galerkin-based approaches simulating quasi-brittle fracture." pith.science (2026). https://pith.science/paper/T7TMNR53

@misc{pith2026190806268,
  author       = {Pith},
  title        = {Pith review of: Global cracking elements: a novel tool for Galerkin-based approaches simulating quasi-brittle fracture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T7TMNR53}},
  note         = {Machine review of arXiv:1908.06268}
}
read the original abstract

Following the so-called Cracking Elements Method (CEM), recently presented in \cite{Yiming:14,Yiming:16}, we propose a novel Galerkin-based numerical approach for simulating quasi-brittle fracture, named Global Cracking Elements Method (GCEM). For this purpose the formulation of the original CEM is reorganized. The new approach is embedded in the standard framework of the Galerkin-based Finite Element Method (FEM), which uses disconnected element-wise crack openings for capturing crack initiation and propagation. The similarity between the proposed Global Cracking Elements (GCE) and the standard 9-node quadrilateral element (Q9) suggests a special procedure: the degrees of freedom of the center node of the Q9, originally defining the displacements, are "borrowed" to describe the crack openings of the GCE. The proposed approach does not need remeshing, enrichment, or a crack-tracking strategy, and it avoids a precise description of the crack tip. Several benchmark tests provide evidence that the new approach inherits from the CEM most of the advantages. The numerical stability and robustness of the GCEM are better than the ones of the CEM. However, presently only quadrilateral elements with nonlinear interpolations of the displacement field can be used.

Figures

Figures reproduced from arXiv: 1908.06268 by the authors.

Figure 1
Figure 1. Traction-separation curve Teq (ζeq ) From equations 1 and 2, the following relations can be obtained D =   ∂Tn/∂ζn ∂Tn/∂ζt ∂Tt/∂ζn ∂Tt/∂ζt   =    ft ζ0   1 0 0 1   , loading, ζeq ≤ ζ0 − Teq ζ 2 eq       ζ 2 n ζeq + ft ζ 2 n Gf − Gf,0 − ζeq ζn ζt ζeq + ft ζn ζt Gf − Gf,0 ζn ζt ζeq + ft ζn ζt Gf − Gf,0 ζ 2 t ζeq + ft ζ 2 t Gf − Gf,0 − ζeq       , loadi… view at source ↗
Figure 2
Figure 2. Determination of the value of lc of element e, with the volume V (e) and the effective cracking area A(e) (based on a parallel crack passing through the center point) used for representation of the stress/strain state of the whole element. Considering the equivalence of forces of discrete and embedded models [15], the following relations exist not just at the center point, but in the whole cracking element: σ (e) = … view at source ↗
Figure 3
Figure 3. Separation of the uncracked elements into the prop [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Calculation procedure within one N-R iteration st [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Set-up and material parameters of L-shaped panel t [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Meshes for the L-shaped panel test (unit: cm) [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: Force-displacement curves of the L-shaped panel fo [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Comparison of the number of iterations for the GCEM [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Deformation and crack-opening plots of the L-shap [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Deformation and crack-opening plots of the L-sha [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Model of Brazilian disk tests with single or doubl [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Meshes for Brazilian disk tests with single or dou [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: Force-displacement curves of disk tests with a si [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: Comparison of normalized peak loads of disk tests [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
Figure 15
Figure 15. Figure 15: Deformation and crack-opening plots of disk test [PITH_FULL_IMAGE:figures/full_fig_p017_15.png]
Figure 16
Figure 16. Figure 16: Deformation and crack-opening plots of disk test [PITH_FULL_IMAGE:figures/full_fig_p018_16.png]
Figure 17
Figure 17. Figure 17: Deformation and crack-opening plots of disk test [PITH_FULL_IMAGE:figures/full_fig_p018_17.png]
Figure 18
Figure 18. Figure 18: Comparison of results of disk tests with double sl [PITH_FULL_IMAGE:figures/full_fig_p018_18.png]
Figure 19
Figure 19. Figure 19: Deformation and crack-opening plots of disk test [PITH_FULL_IMAGE:figures/full_fig_p019_19.png]
Figure 20
Figure 20. Figure 20: Deformation and crack-opening plots of disk test [PITH_FULL_IMAGE:figures/full_fig_p019_20.png]
Figure 21
Figure 21. Figure 21: Deformation and crack-opening plots of disk test [PITH_FULL_IMAGE:figures/full_fig_p019_21.png]
Figure 22
Figure 22. Figure 22: Deformation and crack-opening plots of disk test [PITH_FULL_IMAGE:figures/full_fig_p020_22.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.