{"id":"ba083cbe-da6b-4d5d-bfc3-2815abcde3ee","arxiv_id":"2509.01827","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A multiple crack-tip tracking algorithm added to the 2D Crack Element Model reproduces crack branching and fragmentation in benchmark dynamic fracture tests.","lead":"This paper adds a multiple crack-tip tracking algorithm to a two-dimensional crack simulation method, letting it reproduce crack branching and fragmentation in brittle materials. It matters because dynamic branching often signals structural failure, and a GPU-accelerated simulation that captures it could improve damage prediction in engineering.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3's MCT rule advances only the single max-G free-surface quadrature point, so the algorithm as written cannot create the multiple simultaneously-growing crack tips that the branching/fragmentation examples require.","rationale":"Good-faith reading: the paper proposes an engineering extension of 2D CEM and supports it with qualitative pattern matches, fragment-count comparisons, and an acknowledged limitation that dissipated energy runs high and Neumann cases are less stable. Those issues justify the reader's conditional verdict. The most load-bearing gap, however, is prior to the local-G validity question: Section 3 describes an algorithm that scans all free-surface quadrature points and advances only the single largest-G candidate. Section 2.2's crack-advance rules also always produce one new path from one current tip. No operation in the text creates a second active tip, updates a set of tips, or nucleates a branch from a crack face. Thus the branched and fragmented patterns shown in the examples are not derivable from the algorithm as written. Resolving this requires pseudocode or source code and a count of how many candidates advance per time step. I agree with the reader that the local G formula is under-validated, but the missing multi-tip update rule is the more fundamental obstruction. I do not see fraud; this is a reproducibility and description gap, so the conditional verdict should stand unchanged.","tokens_in":18384,"tokens_out":9815,"duration_ms":115701,"concrete_test":"Ask the authors to release pseudocode/source; instrument the inner crack-advance loop and count how many edge quadrature points are fractured in a single time step during the first branching event of Section 4.2. If the count is >1, Section 3's 'picks ... the current crack tip' is an inaccurate description and must be amended; if it is 1, run the algorithm to completion and verify whether the branched pattern of Fig. 14 still emerges. If no code is released, implement the literal single-max rule on the Section 4.2 Neumann plate with the 9287-element mesh; absence of branching would confirm the gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires the MCT algorithm to produce and advance multiple crack tips, but Section 3 describes a global selection of one tip: 'selects the edge quadrature points satisfying fracture criteria as current crack-tip candidates and picks the edge quadrature point with the largest fracture energy release rate among current crack-tip candidates as the current crack tip.' The element-level rules in Section 2.2 are likewise single-path: every case ends with 'a new crack path from G0 to G1 forms.' Nowhere is there an operation that splits one tip into two active tips, a rule for updating a set of tips per time step, or a rule for nucleating a branch off an existing crack face. The branching/fragmentation figures (Figs. 14-17, 20-23, 26) therefore cannot be inferred from the described algorithm. This is an internal gap, not merely a missing convergence study: if the implementation actually advances all qualifying candidates, Section 3 misdescribes it; if it advances only the global max, the demonstrated multi-front patterns are unexplained. Either way the central claim is not reproducible from the text.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes an extension of the two-dimensional Crack Element Model (CEM) from single crack-tip tracking to Multiple Crack-tips Tracking (MCT-2D-CEM). The method considers edge quadrature points on all free surfaces, evaluates a local fracture energy release rate based on edge stretch and normal stress, and selects new crack paths among candidate tips. The authors validate the approach on four dynamic fracture benchmarks: the Kalthoff-Winkler plate, crack branching under Neumann loading, crack branching under Dirichlet loading, and fragmentation of a pressurized cylinder. They report qualitative agreement with experimental and reference numerical crack patterns, present mesh-dependence studies for some benchmarks, and compare dissipated energy and fragment counts with existing methods. GPU acceleration is used in all simulations.","tokens_in":18631,"tokens_out":3574,"duration_ms":44573,"significance":"If the algorithm is correctly specified and reproducible, the work would be a useful contribution: it offers a finite-element-based, crack-tracking alternative to phase-field and peridynamic methods for dynamic branching and fragmentation, with the practical advantage of GPU acceleration. The qualitative match with experiments and with independent phase-field/cohesive results, and the absence of artificial modulus perturbations in the cylinder example, are genuine strengths. The paper also honestly reports mesh-sensitivity and excess micro-crack dissipation. However, the central algorithmic claim is not yet reproducible from the text, and several load-bearing numerical ingredients are either unspecified or unvalidated in branched configurations.","major_comments":[{"comment":"The algorithm as written cannot produce branching or fragmentation. The text states that MCT 'considers all edge quadrature points along the free surfaces, selects the edge quadrature points satisfying fracture criteria as current crack-tip candidates and picks the edge quadrature point with the largest fracture energy release rate among current crack-tip candidates as the current crack tip.' This is a global selection of a single tip. The example in Figure 4(b) also picks only one of the three candidates. Nowhere is there an operation that turns one active tip into two, a rule for updating a set of active tips, or a rule for nucleating a branch off an existing crack face. Sections 4.2–4.4, however, present branching and fragmentation patterns that require multiple simultaneously advancing tips. This internal contradiction means the figures cannot be inferred from the described algorithm","section":"Section 3, 'Multiple Crack-tips Tracking'"},{"comment":"The driving-force measure G = delta_d * sigma_perp / 2 is carried over from a single-crack-tip formulation, but no evidence is given that this local quantity remains valid at interacting or branching crack tips. In branched configurations, the stress state at a candidate tip is influenced by the other branch, and a simple edge-stretch/sigma_perp product may mis-rank competing branches. The paper provides no convergence study, no J-integral comparison, and no experimental measurement of, say, branch angles or crack speeds to validate this measure in the branched regime. Since the MCT selection rule depends entirely on this quantity, this is a load-bearing point. A quantitative verification, e.g., comparison with J-integral or stress-intensity-factor-based criteria on a static branched geometry, and a mesh-convergence study of branch patterns, would be needed.","section":"Section 2.2 and Section 3, Eq. for G"},{"comment":"The split ratio gamma is never given a numerical value and appears in inequalities that are questionable as written. For instance, the condition G1 > gamma*(G1 - G2) in Section 2.2 is automatically satisfied for many positive G1, G2 when gamma is small; with gamma in (0,1) the inequality is trivially true because the right-hand side is smaller than G1 whenever G2 > 0. The paper does not state gamma's range, its default value, or its sensitivity. Section 3 repeats the same ambiguity. Without a concrete value or an explicit calibration, the branching criterion is not a well-defined function of the computed fields, and the reported patterns cannot be reproduced or assessed for robustness.","section":"Section 3, split-ratio criteria"},{"comment":"The validation is mostly qualitative visual comparison. The dissipated-energy comparisons in Figures 18 and 24 show large differences from reference methods, which the paper attributes to extra micro-cracks, but no quantitative convergence of crack path, branch angle, number of branches, or fragment size distribution with mesh refinement is provided. The fragment-count comparison in Table 1 spans a wide range across methods, and the present results agree with one reference at one mesh and another at another mesh, so the fragment count is not a decisive validation. The paper should report quantitative metrics of the crack patterns and demonstrate that the MCT selection rule converges or is at least stable as the mesh is refined.","section":"Sections 4.2–4.4, quantitative validation"}],"minor_comments":[{"comment":"Typo: 'MCR-2D-CEM' should be 'MCT-2D-CEM'.","section":"Abstract"},{"comment":"The text refers to 'Griffth' in the introduction; correct to 'Griffith'.","section":"Section 2.2"},{"comment":"The text says 'including green points, orange points and the red point', but the colors in the figure are not clearly labeled. It would help to mark the candidate numbers directly on the figure.","section":"Section 3, Figure 4"},{"comment":"The problem dimensions and notch geometry for the Kalthoff-Winkler plate are not given explicitly in the text; only boundary conditions and material properties are provided. A dimensioned figure or table would aid reproducibility.","section":"Section 4.1"},{"comment":"The table lists fragment counts for various methods but does not define 'major fragments' or specify the criterion used to count them. This should be clarified.","section":"Section 4.4, Table 1"},{"comment":"The final paragraph correctly identifies the need for a 'more robust and effective stress regularization scheme'. This limitation should be moved to the front of the validation discussion, because it directly affects the reliability of the branching patterns reported in Sections 4.2–4.4.","section":"Conclusions"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on two self-cited arXiv preprints (Xie et al. 2025a,b) for the underlying CEM formulation. The present work would be difficult to evaluate without those preprints, and they are not yet published. This is a scope/reproducibility concern, not a novelty concern. The main issue is that the algorithm description is internally inconsistent with the demonstrated multi-tip results; this is fixable with a precise algorithmic specification and additional validation, but it is a substantial revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a genuine algorithmic extension of the authors' own 2D CEM, and the benchmark figures look plausible. But the central algorithm as written cannot produce the multi-front patterns the paper shows. Section 3's MCT rule says: take all free-surface edge quadrature points, keep those satisfying fracture criteria, then pick the single point with the largest energy release rate as 'the current crack tip.' That is a global argmax. Nowhere is there an operation that advances multiple tips in one time step, a rule for nucleating a branch from an existing crack face, or a set-update for active tips. The branching and fragmentation results (Figs. 14-17, 20-23, 26) therefore don't follow from the text. This is not a missing convergence study; it's a missing algorithm. Either the implementation advances all qualifying candidates (and Section 3 misdescribes it), or it really advances one tip at a time (and the multi-front results are unexplained). Either way, the paper is not reproducible.\n\nWhat's actually new: replacing single-tip tracking with a global scan of free-surface quadrature points is a reasonable idea and, as far as I can tell, not in the earlier CEM papers. The examples cover Kalthoff-Winkler, Dirichlet and Neumann branching, and a pressurized cylinder, with comparisons to phase-field and cohesive references. The fragment counts in Table 1 land in the expected range. The GPU acceleration is a practical plus. The authors also honestly report that their dissipated energy is higher than the references and that one mesh arrests the main crack.\n\nSoft spots beyond the algorithm gap: the split ratio gamma appears in the branching inequality but no value is ever given; there's no pseudocode or implementation detail; no code or data are provided. The approach builds directly on two self-cited preprints, which is fine in itself, but it makes the missing specification worse.\n\nWho this is for: people working on element-based dynamic fracture, especially CEM/ES-FEM, will find the benchmark comparisons and the Dirichlet-versus-Neumann discussion useful. It's not ready as-is.\n\nRecommendation: send to peer review, not desk reject. A serious referee can push for a precise algorithm description and, ideally, code and data. This deserves referee time, but it needs major revision.","headline":"A real but under-specified extension of 2D CEM; the algorithm as written can't produce the branching and fragmentation figures.","tokens_in":19103,"tokens_out":4157,"would_cite":false,"duration_ms":43908,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["74R10","74S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single energy-based tracking rule now reproduces single cracks, branching, and fragmentation in 2D dynamic fracture.","keywords":["crack branching","fragmentation","Crack Element Model","multiple crack-tip tracking","dynamic fracture","energy release rate","edge-based smoothed FEM","GPU acceleration"],"falsifier":"Run the branching benchmarks on a sequence of refined meshes and compute a path-independent J-integral around each candidate branch; if the J-integral ranking of two competing crack paths ever reverses the G_{G1-G0} ranking that the MCT rule uses, the central claim fails. Alternatively, quantitatively compare simulated PMMA branch angles and crack speeds with high-speed experimental photographs.","tokens_in":18269,"feed_emoji":"💥","tokens_out":5465,"duration_ms":58682,"temperature":0.7,"pith_summary":"This paper extends the two-dimensional Crack Element Model from following one crack tip to tracking every crack tip at once. The proposed Multiple Crack-tips Tracking (MCT) algorithm inspects all edge quadrature points on free surfaces, keeps those whose fracture energy release rate exceeds the critical value, and advances the crack at the candidate with the largest rate. With this one set of rules, the method reproduces a Kalthoff-Winkler single crack, crack branching under both traction and displacement loading, and fragmentation of a pressurized cylinder—without the elastic-modulus perturbations other methods use to break symmetry. If the local energy-based ranking is reliable, complicated dynamic fracture patterns no longer require problem-specific branching criteria or added microstructural noise; the same tracking rule drives them all.","feed_headline":"One energy rule now drives crack branching and fragmentation","feed_subtitle":"Scanning every free-surface edge point by release rate reproduces single cracks, branches, and breakup in 2D.","key_machinery":"Multiple Crack-tips Tracking (MCT) algorithm: a scanning rule over edge quadrature points in the two-dimensional Crack Element Model. It evaluates the local fracture energy release rate G_{G1-G0} = delta_d * sigma_perp / 2 at all free-surface edge quadratures, applies the critical Gc criterion, then picks the surviving candidate with the largest G, subject to the split-ratio inequality that smooths stress-wave fluctuations. It carries the argument by making crack branching an emergent consequence of ranking local energy release at every possible crack exit, rather than a separately imposed branching criterion.","core_discovery":"The central claim is that one algorithmic rule—the Multiple Crack-tips Tracking (MCT) algorithm—can drive the two-dimensional Crack Element Model through single-crack propagation, crack branching, and fragmentation. At each step the algorithm scans every edge quadrature point on all free surfaces, keeps those with fracture energy release rate above the critical value Gc, and advances the crack tip to the candidate with the largest rate, using a split ratio to avoid false choices caused by stress oscillations. The same rule reproduces the Kalthoff-Winkler crack with micro-cracks, branching in a pre-notched plate under both Neumann and Dirichlet loading, and breakup of an internally pressurize","pith_inferences":["If the local energy ranking is valid at branch points, the same free-surface scanning strategy could extend to three-dimensional crack fronts, where a surface quadrature point with maximum G would play the role of each crack-tip candidate; the paper does not test this.","The absence of modulus perturbations raises a question the paper leaves open: whether the discretization itself supplies the symmetry-breaking perturbation that real microstructures do, or whether perfectly symmetric structured meshes would spuriously branch.","A direct comparison of G_{G1-G0} with a path-independent J-integral at a branched tip would tell whether the ranking survives where interacting stress fields dominate; the paper reports no such comparison.","Because the method introduces micro-cracks even in the single-crack Kalthoff case, its dissipated energy should be treated as an upper bound; recalibrating Gc downward may be needed for quantitative energy matching."],"forward_implications":["The same MCT rule reproduces single cracks, branching, and fragmentation without a velocity threshold or user-defined branching criterion.","Micro-cracks along the main path appear automatically and consume extra energy, which the paper identifies as the reason its dissipated-energy curves lie above phase-field and cohesive references.","Under displacement (Dirichlet) loading the computed crack patterns agree more closely with experiments than under traction (Neumann) loading, attributed to steadier stress fields.","The pressurized-cylinder fragmentation benchmark yields 11–13 major fragments on two unstructured meshes with no modulus perturbation, matching phase-field and cohesive reference counts.","All simulations run with GPU acceleration, suggesting the method can move into industrial-scale dynamic fracture workflows."],"supporting_citations":[{"why":"Supplies the original two-dimensional Crack Element Model and single crack-tip tracking algorithm that MCT extends.","marker":"Xie et al. 2025b"},{"why":"The three-dimensional CEM whose spontaneous branching motivated the two-dimensional MCT algorithm.","marker":"Xie et al. 2025a"},{"why":"Provides the edge-based smoothed finite element framework and edge quadrature points on which CEM and MCT operate.","marker":"Liu, Nguyen-Thoi and Lam 2009"},{"why":"Phase-field reference used for the Neumann branching benchmark and dissipated-energy comparison.","marker":"Borden et al. 2012"},{"why":"Local damage model used as another reference in branching and fragment-count comparisons.","marker":"Bui et al. 2022"},{"why":"Cohesive fracture model used for dissipated-energy and fragment-count comparison.","marker":"Hirmand and Papoulia 2019"},{"why":"Experimental compact-tension results used for the Dirichlet branching benchmark.","marker":"Ožbolt, Bošnjak and Sola 2013"},{"why":"Phase-field reference for the pressurized-cylinder fragmentation benchmark and fragment counts.","marker":"Geelen et al. 2019"},{"why":"Cracking node model reference used for fragment counts and for the modulus-perturbation practice that MCT avoids.","marker":"Song and Belytschko 2009"}],"fun_headline_variants":["Single energy rule tracks multiple crack tips for branching","Crack branching and breakup? One algorithm does both","From one crack tip to many: a unified tracking rule","One release-rate rule now predicts crack branching and fragmentation"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The whole scheme rests on one local formula—the fracture energy release rate at an edge quadrature, developed for a single crack tip—still ranking competing branches correctly when many crack tips interact, and no convergence or J-integral check is given for branched configurations.","fun_headline_variants_meta":{"raw":{"variants":["Single energy rule tracks multiple crack tips for branching","Crack branching and breakup? One algorithm does both","From one crack tip to many: a unified tracking rule","One release-rate rule now predicts crack branching and fragmentation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000641,"raw_usage":{"total_tokens":2740,"prompt_tokens":648,"completion_tokens":2092,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":392,"completion_tokens_details":{"reasoning_tokens":2029}},"tokens_in":392,"tokens_out":2092,"duration_ms":17631,"temperature":1.0,"reasoning_tokens":2029,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T12:08:24.032178+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the branching benchmarks on a sequence of refined meshes and compute a path-independent J-integral around each candidate branch; if the J-integral ranking of two competing crack paths ever reverses the G_{G1-G0} ranking that the MCT rule uses, the central claim fails. Alternatively, quantitatively compare simulated PMMA branch angles and crack speeds with high-speed experimental photographs.","supporting_citations":[{"cited_title":", author Nguyen-Thoi, T","cited_arxiv_id":null,"evidence_quote":"Provides the edge-based smoothed finite element framework and edge quadrature points on which CEM and MCT operate."},{"cited_title":", author Verhoosel, C.V","cited_arxiv_id":null,"evidence_quote":"Phase-field reference used for the Neumann branching benchmark and dissipated-energy comparison."},{"cited_title":", author Tran, H.T","cited_arxiv_id":null,"evidence_quote":"Local damage model used as another reference in branching and fragment-count comparisons."},{"cited_title":", author Papoulia, K.D","cited_arxiv_id":null,"evidence_quote":"Cohesive fracture model used for dissipated-energy and fragment-count comparison."},{"cited_title":", author Liu, Y","cited_arxiv_id":null,"evidence_quote":"Phase-field reference for the pressurized-cylinder fragmentation benchmark and fragment counts."},{"cited_title":", author Belytschko, T","cited_arxiv_id":null,"evidence_quote":"Cracking node model reference used for fragment counts and for the modulus-perturbation practice that MCT avoids."}],"review_version":1}