A mesh-free neural solver reproduced several crack patterns in numerical tests, including a symmetric Y-shaped branch, a linked staircase crack and two cracks growing from a circular hole. In tension and shear tests, its second-order version also tracked reference load responses closely. The result came with a clear trade-off: the reported cost per load increment remained roughly an order of magnitude above finite-element references.
A solver built to let cracks change shape
The work, released as an arXiv preprint dated 25 August 2026, uses one neural network for two parts of the calculation: displacement, which describes material movement, and a phase field for damage. It does not rely on a fixed mesh. Instead, it minimizes the system’s energy increment by increment, uses a multiresolution C 1 feature grid to set the finest resolvable scale, and redraws its integration points at every optimizer iteration. The framework supports both second- and fourth-order densities, along with exact boundary lifts, an exact NURBS geometry map and a domain mask.
The evaluation combined six core numerical examples with a separate benchmark of ten random configurations, producing twenty zero-shot runs. For the proposed benchmark calculations, zero-shot meant that no solved samples were used. The researchers also tested what happened when either of two ingredients was weakened: integration points were held fixed, or the encoding was made coarser than the damage-band width. In those ablations, crack propagation disappeared under either condition, while a redraw interval ten times longer was tolerated.
Close on standard load tests
In the tension case, the second-order result peaked at 97.6 newtons, compared with 98.7 N for the reference, putting the peak-load difference within 1.1%. The peak displacement was within 3.7%. The fourth-order calculation failed at 94.3 N, about 3% below the second-order result. The supplied analysis reports no inferential interval around these comparisons. For the notched-square and coalescence comparisons, the finite-element reference used a staggered second-order calculation on 512 × 512 bilinear elements with 789,507 degrees of freedom, while key problem settings were matched.
Shear produced an even closer peak match: 66.42 N against 66.35 N for the reference, within 0.1%. The model also followed the post-peak softening valley, at 43.1 N versus 42.6 N, and the second rise, at 51.1 N versus 49.5 N, within 3.2%. The fourth-order peak was 63.0 N, about 5% below second order. A separate four-seed check found that all runs completed, with mean peak load of 66.3 ± 0.3 N, valley of 42.6 ± 1.0 N and second maximum of 50.3 ± 1.3 N; these spreads are sample standard deviations.
Crack paths without explicit tracking
The model was also tested on topology changes: branching, coalescence and nucleation around a hole, without explicit crack tracking. Under isotropic driving, it produced a symmetric Y-shaped branch. Its second-order peak was 59.3 N, about 8% below matched finite elements; the fourth-order run followed nearly the same path and peaked at 57.8 N. In the coalescence case, three en-echelon cracks formed one staircase crack. The second-order peak was 195.4 N versus 184.8 N for finite elements, 5.8% higher, while fourth order reached 182.8 N, about 1% below the reference.
In the circular-hole fracture stage, damage nucleated at the two maximum-hoop-stress locations around the hole, at plus or minus 90 degrees, and formed two symmetric cracks. The load peaked at 232.7 N at a displacement of 8.0 × 10−4 mm, followed by a drop to 23.8 N in the next increment. In the curved ring case, the crack grew horizontally from a notch toward the centre and severed the wall. Reaction reached 3.91 kilonewtons at a prescribed displacement of 2.67 × 10−2 mm.
A promising benchmark result, with a heavy price tag
On the random multi-crack benchmark, the solver classified 252 of 280 seeded cracks correctly, for 90% accuracy. Ten of the 20 runs classified every crack correctly. Mean Dice scores—a measure of overlap between the computed and reference damage fields—were 0.739 in tension and 0.824 in shear. Those scores exceeded surrogate values of 0.680 and 0.733 from models trained on 800 solved samples, even though the proposed runs used no solved samples.
The performance comparison does not erase the computational cost. The authors report that the solver remained roughly an order of magnitude more expensive per load increment than the finite-element references and do not present it as faster for a single quasi-static forward solution. Fourth-order runs used the same discretization but cost 1.4 to 1.6 times as much as second-order runs; on a per-iteration basis, the ratio was 1.3. The complete-run ratio includes different early-stopping behavior.
The reported case for the method is flexibility rather than speed: it spans load-response tests, crack networks and a curved ring, while the benchmark runs used no solved training samples. It is an arXiv preprint dated 25 August 2026, according to the supplied record. Production runs used the Gadi supercomputer with computational resources from NCI Australia, an NCRIS-enabled capability supported by the Australian Government.
Paper data and sources
Original title: A mesh-free multiresolution deep energy method with phase-field modeling of brittle fracture
Authors: Han Zhang, Mehrisadat Makki Alamdari, Babak Shahbodagh et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text