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arxiv: 2603.20120 · v2 · submitted 2026-03-20 · ⚛️ physics.comp-ph

Recognition: no theorem link

Deep learning-based phase-field modelling of brittle fracture in anisotropic media

N. Plung\.e , P. Brommer , R. S. Edwards , E. G. Kakouris
Authors on Pith no claims yet

Pith reviewed 2026-05-15 07:07 UTC · model grok-4.3

classification ⚛️ physics.comp-ph
keywords phase-field modellingbrittle fractureanisotropic mediaDeep Ritz MethodB-spline basis functionsvariational deep learningcrack propagationphysics-informed neural networks
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The pith

A variational deep learning framework solves higher-order anisotropic phase-field fracture models by minimising total energy with B-spline enriched trial spaces.

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

The paper introduces a physics-informed deep learning approach to phase-field modelling of brittle fracture that extends prior Deep Ritz Method work to anisotropic media. It defines a generalised crack density functional that incorporates higher-order anisotropic fracture surface energies and minimises the resulting total energy functional. The trial space is built from higher-order B-spline basis functions so that higher-order gradients are represented accurately and stably without automatic differentiation. Numerical tests on isotropic, cubic and orthotropic cases show that the method reproduces the expected direction-dependent crack paths.

Core claim

This work presents a variational physics-informed deep learning framework for phase-field modelling of brittle crack propagation in anisotropic media. For the first time in a variational deep learning setting it introduces a family of higher-order anisotropic phase-field models through a generalised crack density functional, solves the fracture problem by minimising the total energy with the Deep Ritz Method, and enriches the trial space with higher-order B-spline basis functions to represent higher-order gradients accurately and stably without conventional automatic differentiation. The methodology is assessed for isotropic, cubic and orthotropic fracture surface energy densities, and the 1

What carries the argument

Generalised crack density functional solved via Deep Ritz Method on a trial space enriched with higher-order B-spline basis functions

If this is right

  • Direction-dependent crack growth is reproduced in cubic and orthotropic media without mesh-dependent artifacts.
  • The same energy-minimisation procedure works for any fracture surface energy density that can be written as a generalised crack density functional.
  • Higher-order gradients are handled stably, allowing phase-field models of arbitrary order to be treated within the Deep Ritz framework.
  • The approach removes the requirement for automatic differentiation when computing variational derivatives of the energy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The B-spline enrichment could be transferred to other physics-informed neural network problems that involve fourth- or higher-order differential operators.
  • Because the method is mesh-free in spirit, it may scale more readily to three-dimensional anisotropic fracture problems than traditional finite-element phase-field codes.
  • The generalised crack density functional provides a natural route to include additional physics such as thermal or electrical coupling inside the same variational deep learning setting.

Load-bearing premise

Higher-order B-spline basis functions in the trial space represent higher-order gradients accurately and stably, eliminating the need for automatic differentiation.

What would settle it

Run the method on a standard orthotropic plate with a central crack under uniaxial tension and compare the predicted crack path angle against the analytically expected direction; mismatch in the angle would falsify the claim that the enriched trial space captures anisotropic behaviour correctly.

Figures

Figures reproduced from arXiv: 2603.20120 by E. G. Kakouris, N. Plung\.e, P. Brommer, R. S. Edwards.

Figure 1
Figure 1. Figure 1: (a) Solid body Ω with crack path Γ and (b) phase field approximation of the crack path Γ. [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Polar representation of the fracture surface energy density [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic illustration of the DRM employed in this work. Spatial coordinates are input [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Effect of the scaling coefficient r on the tanh activation function. Smaller values extend the quasi-linear regime. (b) Non-smooth piecewise constraint used to enforce admissible phase-field values. In classical finite element formulations, these effects may be alleviated through mesh refinement. In NN-based approximations, however, refinement of the discretisation is not directly applicable. Instead, … view at source ↗
Figure 5
Figure 5. Figure 5: Displacement field enforcement via distance functions. The shaded regions represent non [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Square plate under pure tension: geometry and boundary conditions: (a) homogeneous plate [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) Polar representation of the fracture surface energy density [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Results for the isotropic symmetry case. (a) Evolution of elastic and fracture energies [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Cubic material symmetry for two material orientations. Material orientation [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Orthotropic material symmetry for two material orientations. Material orientation [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Orthotropic material symmetry of −30◦ tested with network architecture introduced by Manav et al. for second-order fracture problems: (a) variation of elastic and fracture energies with applied displacement, (b) phase-field distribution obtained using the finite element method (FEM), and (c) phase-field distribution obtained using the proposed DRM at a displacement of 0.015 m. The fracture trajectory is g… view at source ↗
Figure 12
Figure 12. Figure 12: Results for the layered orthotropic material case. (a) Evolution of elastic and fracture [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Displacement step convergence study. (a) Mean absolute error and standard deviation [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Mesh convergence study. (a) Mean absolute error and standard deviation of elastic energy, [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
read the original abstract

This work presents a variational physics-informed deep learning framework for phase-field modelling of brittle crack propagation in anisotropic media. Previous Deep Ritz Method (DRM) approaches have focused on second-order, isotropic phase-field fracture formulations. In contrast, the present work introduces, for the first time within a variational deep learning setting, a family of higher-order anisotropic phase-field models through a generalised crack density functional. The resulting fracture problem is solved by minimising the total energy using the DRM. The trial space is enriched with higher-order B-spline basis functions to represent higher-order gradients accurately and stably, thereby eliminating the need for conventional automatic differentiation. The methodology is assessed for isotropic, cubic, and orthotropic fracture surface energy densities. Numerical examples demonstrate direction-dependent crack growth in anisotropic cases, highlighting the capability of the method to accurately capture this behaviour.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper presents a variational physics-informed deep learning framework based on the Deep Ritz Method for phase-field modeling of brittle fracture in anisotropic media. It introduces a generalized higher-order anisotropic crack density functional, solves the energy minimization problem via DRM, and enriches the trial space with higher-order B-spline basis functions to represent higher-order gradients without conventional automatic differentiation. The approach is assessed on isotropic, cubic, and orthotropic fracture surface energy densities, with numerical examples claimed to demonstrate direction-dependent crack growth.

Significance. If the stability and accuracy of the B-spline enrichment are rigorously established, the work would usefully extend DRM-based variational methods from isotropic second-order phase-field models to anisotropic higher-order cases. The avoidance of automatic differentiation via spline enrichment is a technically interesting choice that could improve robustness in energy minimization for fracture problems.

major comments (2)
  1. [Numerical examples] Numerical examples section: the central claim that the method 'accurately captures' direction-dependent crack growth in anisotropic cases is not supported by any reported error norms, convergence rates, or quantitative comparisons against reference solutions (e.g., established FEM implementations of the same anisotropic phase-field models). Only qualitative demonstration is described.
  2. [Method (B-spline enrichment)] Section describing the B-spline-enriched trial space: the assertion that higher-order B-splines 'represent higher-order gradients accurately and stably' and thereby eliminate the need for automatic differentiation lacks a priori stability estimates, dispersion analysis, or numerical convergence studies for the cubic/orthotropic crack-density functionals. This is load-bearing for the claim that the enriched space is suitable for the generalized anisotropic model.
minor comments (1)
  1. [Abstract] Abstract: the phrase 'for the first time within a variational deep learning setting' should be accompanied by a brief citation or clarification of the exact prior DRM works that are being extended.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the work's potential. We address each major comment below and will revise the manuscript to strengthen the quantitative support for our claims.

read point-by-point responses

  1. Referee: [Numerical examples] Numerical examples section: the central claim that the method 'accurately captures' direction-dependent crack growth in anisotropic cases is not supported by any reported error norms, convergence rates, or quantitative comparisons against reference solutions (e.g., established FEM implementations of the same anisotropic phase-field models). Only qualitative demonstration is described.

    Authors: We agree that quantitative validation is needed to support the accuracy claims. In the revised manuscript, we will add L2 error norms for the phase-field and displacement fields, crack path deviation metrics, and convergence rates with respect to network parameters and spline order, using established FEM solutions as references for the isotropic, cubic, and orthotropic cases. revision: yes

  2. Referee: [Method (B-spline enrichment)] Section describing the B-spline-enriched trial space: the assertion that higher-order B-splines 'represent higher-order gradients accurately and stably' and thereby eliminate the need for automatic differentiation lacks a priori stability estimates, dispersion analysis, or numerical convergence studies for the cubic/orthotropic crack-density functionals. This is load-bearing for the claim that the enriched space is suitable for the generalized anisotropic model.

    Authors: We acknowledge the value of additional analysis. While a full a priori theoretical stability proof is beyond the scope of this computational paper, the revised version will include numerical convergence studies, error tables, and dispersion analysis specifically for the B-spline enrichment on the cubic and orthotropic crack-density functionals to empirically demonstrate stability and accuracy. revision: partial

Circularity Check

0 steps flagged

No significant circularity; extends DRM with B-spline enrichment on independent numerical validation

full rationale

The derivation introduces a generalised anisotropic crack density functional and solves the energy minimisation via DRM with higher-order B-spline enrichment of the trial space. This enrichment is presented as a technical choice to avoid automatic differentiation, building on established DRM literature without reducing the central claim (accurate capture of direction-dependent crack growth in cubic/orthotropic cases) to a fitted parameter, self-definition, or self-citation chain. Numerical examples are used to demonstrate the behaviour rather than deriving it tautologically from the inputs. Any prior DRM citations are not load-bearing for the anisotropic extension or the stability claim, satisfying the criteria for a low circularity score with self-contained content against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review; specific functional forms, hyperparameters, and training details are not provided, so ledger entries are limited to explicitly invoked assumptions.

free parameters (1)
  • B-spline order
    Chosen to represent higher-order gradients accurately and stably
axioms (1)
  • domain assumption Minimization of total energy via DRM solves the anisotropic fracture problem
    Central variational principle assumed to hold for the generalised crack density functional

pith-pipeline@v0.9.0 · 5450 in / 1068 out tokens · 35038 ms · 2026-05-15T07:07:57.203432+00:00 · methodology

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