An extended deep energy method for thermo-mechanical crack propagation

arXiv:2610.09433 · cs.LG, cond-mat.mtrl-sci, cs.NA, math.NA · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "An extended deep energy method for thermo-mechanical crack propagation".

Jane: Detailed Research Summary:

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So we’re looking at this paper today, "An extended deep energy method for thermo-mechanical crack propagation." It’s about solving how a crack moves when there are both heat and mechanical forces involved at the same time.

Jane: That sounds really complex because usually you have to treat temperature and mechanics as separate problems, but this work couples them together explicitly so they evolve at the same time.

Lu: The core idea is using two separate neural networks for temperature and displacement, but they are linked through a special function called a crack embedding function that has to be discontinuous right across the crack line.

Meng: So instead of cutting up the whole physical area into little pieces like some phase-field methods do, this approach lets the crack stay as a sharp line all throughout the simulation. That avoids rebuilding the mesh every time things change, which is usually a big headache in those other solvers.

Tom: Exactly. He’s using neural networks to represent both fields, and they are fed that same crack shape input so they understand where the crack is located without needing a grid to define it precisely at every step.

Jane: The paper explains how this representation lets them maintain that sharp polyline, which is something other methods struggle with when you introduce temperature effects.

Lu: The method involves two separate energy minimizations: one for the temperature field and one for the displacement field, and both are minimized based on the same crack embedding function.

Tom: And they use a specific way to advance the heat conduction equation by recasting it into an incremental functional, which then lets them solve for temperature in a staggered sequence with each mechanical step.

Jane: So when you’re thinking about the physics, they are essentially minimizing energy for both the thermal side and the mechanical side simultaneously over time.

Meng: From a deployment standpoint, that means we have to consider how much compute this takes, because running this whole system is pretty intensive; they report it takes three point five to four hours on one H200 GPU for a propagating case.

Tom: And they address the mechanical side by enriching the displacement network near the crack tip with leading asymptotic terms from the Williams expansion, using trainable amplitudes to capture that sharp jump in displacement.

Jane: So, what this means for us is that we get direct access to fracture mechanics quantities, like stress intensity factors, which are usually tricky to get indirectly when you have these diffuse fields.

Title and authors: Lu: They achieve this by using an interaction integral that includes both the contour and area terms of Wilson and Yu, but they also include a check where you sweep the contour radius before allowing any crack advancement.

Tom: That sweep is important because it’s how they ensure the SIF extraction is reliable, so when you look at thermal loads, including that area term seems critical for getting an accurate reading.

Jane: The paper shows some results on benchmark problems where their extracted stress intensity factors agreed with published values to within zero point one one percent on their finest grid.

Meng: That agreement sounds promising, but we have to remember they only validated it against a few specific scenarios, like a stationary edge crack and some notched tension tests.

Tom: And for the single-edge notched tension test, their peak load agreed with the phase-field reference to within four point one percent across three thermal cases.

Jane: That four point one percent error sounds pretty good when you’re dealing with complex thermal loading scenarios like in a functionally graded material.

Lu: They also looked at a notched cruciform specimen, and they found that the predicted chord angles were between one point three degrees and four point one degrees outside of the nearest published curve.

Tom: That comparison to the published data is interesting because that reference data shows a much wider spread, from five point four degrees to eighteen point nine degrees.

Jane: So, what this means for someone just listening to the show is that even though they don't read the math, this method offers a way to model how heat and mechanics interact without having to rely on overly complex mesh regeneration during crack growth.

Meng: It’s about avoiding those expensive mesh rebuilds when you are dealing with a growing crack, which makes it potentially more practical for engineering applications than some of the other methods out there.

Tom: They also explicitly stated that the coupling is one-way for a fixed crack; meaning mechanical dissipation doesn't feed back into the thermal problem, and the mechanical solution only informs the conduction problem through the crack it advances.

Jane: That’s a specific limitation they acknowledge, which is important context when we look at how this system handles coupled dynamics over time.

Lu: The paper also clearly defines its scope: it's strictly two-dimensional and quasistatic, and it doesn't treat branching or merging phenomena in the crack path.

Title and authors: Meng: So if you’re trying to model something that involves a crack splitting into multiple paths, this specific method wouldn't be the right tool for that job right now.

Tom: The paper sets out what they *don’t* do, and they are quite clear about those boundaries, which is good for setting expectations on what this solver can handle.

Jane: Overall, it seems like a solid piece of work because it addresses the problem of treating coupled thermal and mechanical fields together in a way that keeps the crack representation sharp.

Lu: And they’ve done this by sharing principles with phase-field methods, but solving the cost issue where damage bands have to be resolved by elements much smaller than the regularization length.

Meng: So if you’re looking at deploying this kind of solver, you need to balance that high accuracy against the computational time it takes to run those coupled energy minimizations.

Tom: The implication here is that for problems where thermal stress and crack propagation are intertwined, this deep energy method offers a direct path to reading the driving forces without needing those indirect SIF extractions.

Jane: It gives researchers a tool where they can see the crack evolve and get direct data on how the temperature affects its speed.

Lu: And they’ve also put effort into run-to-run consistency, which is always something we need to check when you are training these kinds of neural network solvers.

Meng: Consistency matters because if every single run gives wildly different numbers, it doesn't matter how good the model looks on paper; it won't work in production.

Tom: So, this paper provides a framework where you can encode the crack geometry directly and let the energy minimization do the heavy lifting for both fields at once.

Jane: And that means for people working on materials science or aerospace components under thermal stress, this method offers a new way to predict crack paths under those conditions.

Lu: The future work they suggest is to look into branching or merging phenomena, which would be the next logical step in expanding this model's capabilities.

Meng: And from an engineering side, they’d need to figure out how to make the computational cost manageable for larger, three-dimensional problems without losing that sharp crack representation.

Tom: Well, it’s a detailed look at how neural energy solvers can be extended to handle these coupled thermo-mechanical problems while keeping the crack geometry sharp throughout.

Jane: That’s what we have on today regarding "An extended deep energy method for thermo-mechanical crack propagation."

The paper's summary: Tom: So, we’ve talked about how they use two neural networks for temperature and displacement, but what they actually did in this paper is couple them up to solve the heat transfer and mechanical movement problems at the same time using a shared input called a crack embedding function.

Jane: Right, so if I'm driving or cooking, I'd ask, "So what does this mean for me?" because it sounds like a lot of math. Basically, they built an AI system that can model how heat moves and how things bend when there's a crack growing all in one go.

Lu: Exactly. They treat the crack as a sharp line and don't have to worry about rebuilding the mesh every time the crack moves, which is a huge deal for efficiency in simulations.

Meng: From my side, I’m thinking about deployment—if this works, it means we could potentially run these coupled problems much faster because we aren't constantly dealing with mesh regeneration overhead.

Tom: And they’ve done the heavy lifting by using energy minimization for both fields simultaneously, meaning the temperature field and the displacement field are determined at the same time based on those shared equations.

Jane: That leads us to how they check if the crack is actually growing—they use a ratio of driving force to resistance, called chi, which has to cross a certain threshold for it to advance.

Lu: And that critical energy release rate they use for the resistance part depends directly on the temperature right at the crack tip, which links back into their thermal network.

Tom: That’s an important part—the crack advancement rule is not just some fixed number; it’s temperature-dependent, which means you can’t get a steady result without solving both fields together.

Jane: And they didn't just use any formula for the stress intensity factor; they extracted it by sweeping the contour radius and checking for stability, which helps them avoid those unreliable readings when there's heat involved.

Lu: The results on their benchmark problems show good agreement with published data, like within zero point one one percent for a stationary edge crack SIF.

Tom: That level of accuracy against established physics is what we’re looking at here—it’s not just a rough estimate; the system is actually capturing the underlying mechanics well.

Jane: But they did have some caveats, like it only works for two-dimensional and quasistatic problems, and it doesn't handle a crack that branches or merges into multiple paths.

Meng: That limitation tells me we can’t just take this off the shelf for modeling complex fracture patterns right now; it’s focused on single crack propagation.

Lu: And the coupling is one-way, which means the mechanical side doesn't feed back into the thermal problem; it only informs the heat conduction when it advances.

Tom: So, what this changes for us is that we get a direct way to see how temperature influences crack speed and stress intensity factors without needing those complicated indirect calculations.

Jane: It’s like having a direct line of sight into the physics, which helps you understand the material better under thermal stress.

Lu: The authors also made sure to measure the run-to-run scatter on every single example, which is crucial for knowing how robust this neural network representation actually is when it's trained.

Tom: And that robustness check gives us a lot of confidence that this isn't just a lucky coincidence; the method is consistent across different runs.

Jane: So, while the scope is limited to single cracks and two dimensions, the core idea of coupling temperature and mechanics through an explicit crack embedding function is really powerful.

Lu: The next logical step they suggest for this work would be exploring how to incorporate branching or merging into this framework.

Meng: If they could handle that, it would open up a lot more use cases for designing components that fail in more complex ways.

The paper's improvements: Tom: We’ve seen how they get their current results, but now let's talk about what they suggest improving for the next version of this method.

Jane: So, what are these improvements? Are we talking about making it work in more complicated situations or just getting better accuracy?

Lu: They are really pushing to extend this beyond the two-dimensional and quasistatic limits; they want to see if this works in three dimensions or under dynamic loading conditions.

Meng: Three dimensions is a big jump for computational cost, Tom; that kind of scaling needs serious attention before we think about it running on anything larger than a small GPU.

Tom: Yeah, but the method itself has some internal suggestions too—they want to look at incorporating branching and merging phenomena into the crack path modeling.

Jane: Branching and merging sounds like something that could really open up new applications in materials science, where cracks don't just move straight; they split up.

Lu: They are also looking at how to make the coupling more fully dynamic, specifically how mechanical dissipation actually feeds back into the thermal problem.

Tom: That would be a significant change because right now it’s one-way; the mechanics only tell heat about when a crack advances, not the other way around.

Jane: So if they can figure out that feedback loop, it means we could model systems where heat and mechanical stress influence each other constantly.

Meng: If that coupling works, the simulation would become much more accurate for real-world scenarios where thermal expansion and material failure happen in a tight feedback loop.

Lu: They are also focusing on making the crack representation even sharper by looking at how to incorporate higher-order terms in the displacement field near the tip using those trainable amplitudes from XFEM again.

Tom: So, it’s about refining that sharp representation so it captures those subtle stress concentrations better than just a simple contour model.

Jane: And they are trying to make sure that when they extract those stress intensity factors, they use a more robust method for checking the results over different contour radii to ensure reliability.

Lu: In short, the improvements are about expanding the physical limits of where this deep energy method can be applied and making the internal physics coupling more reciprocal.

Tom: So, moving forward, if this framework is successful in these extensions, it could mean we have a more flexible tool for predicting how materials fail under complex thermal conditions.

Conclusion: Tom: So to wrap up, this paper, "An extended deep energy method for thermo-mechanical crack propagation," shows how you can use neural networks to solve coupled heat transfer and mechanical movement problems at the same time while keeping a sharp crack representation throughout.

Jane: That’s right; it gives us a way to model how temperature and mechanics interact without needing those messy mesh rebuilds every time the crack gets bigger.

Lu: The main thing is that they successfully coupled two separate neural networks through that shared crack embedding function, which is a clever way to encode the geometry into both fields.

Meng: From an engineering standpoint, if this framework can handle three dimensions and dynamic loading without exploding in compute time, it could be useful for designing components where thermal stress and material failure happen together.

Lalam: I’m confident that the results they showed on those benchmark tests—like the zero point one one percent error on the SIF—show that this representation learning approach is quite sure of its answer for these specific problems.

Tom: The implication here is that we have a more direct path to reading driving forces, like stress intensity factors, which used to require those indirect calculations.

Jane: It means researchers can see how temperature directly affects the crack speed and the forces involved in propagation.

Lu: And they’re open-ended with future work, specifically looking at incorporating branching and merging into this model for more realistic fracture behavior.

Meng: I'm still worried about deployment costs; if we scale this up to real traffic or large structures, we need to make sure the run-to-run consistency holds up under heavy load.

Lalam: I’m pretty sure the next step they focus on will be proving that this method can handle those more complex branching scenarios reliably.

Tom: So, "An extended deep energy method for thermo-mechanical crack propagation" gives us a powerful, coupled framework for modeling thermo-mechanical crack propagation with sharp geometry representation.

Jane: It’s a solid piece of work because it tackles the problem of treating these two physical fields together in a way that keeps the crack representation sharp.

Lu: I think this is a strong foundation for expanding what’s possible in using deep energy methods for structural mechanics problems.

Han Zhang, Mehrisadat Makki Alamdaria, Babak Shahbodagha, Mohammad Vahabb, Cosmin Anitescuc, Timon Rabczukc, Elena Atroshchenko

University of New South Wales Sydney Australia · Central Queensland University Melbourne Victoria Australia · Bauhaus-Universität Weimar Germany

cs.LG, cond-mat.mtrl-sci, cs.NA, math.NA

Submitted: 2026-10-07

Updated: 2026-10-07

Comments: 48 pages, 20 figures, 10 tables

Code: https://github.com/HannnZH/Extended-DEM-ThermoMechanical-Crack

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 73/100

The gist: Detailed Research Summary: Extended Deep Energy Method for Thermo-Mechanical Crack Propagation This research presents an extended deep energy method designed to model thermo-mechanical crack

Key concepts

Crack Embedding Function: This is a special mathematical function used as an input for the neural networks that tells them exactly where the crack is located. It's designed to be discontinuous across the crack line but smooth everywhere else, allowing the model to treat the crack as a sharp geometric entity rather than just a region of material change.
Energy Minimization: The method finds the temperature and displacement fields by minimizing specific energy functionals. For temperature, it minimizes a heat conduction functional, and for displacement, it minimizes the thermoelastic potential energy functional. This process ensures the resulting fields satisfy the governing physical laws under thermal and mechanical conditions.
Crack Growth Criterion: Crack advancement is decided by comparing a calculated driving force to a resistance threshold. Specifically, if the ratio of these forces exceeds a critical value ($\chi_c$), the crack grows. This criterion explicitly accounts for how temperature affects the material's resistance to fracture.
Discontinuous Scalar Embedding: This is the key coupling mechanism where temperature and displacement networks share a single input function that changes abruptly across the crack. This function mathematically encodes the sharp, non-physical boundary of the crack into both field predictions, ensuring they remain coupled throughout the simulation.

Summary

Detailed Research Summary: Extended Deep Energy Method for Thermo-Mechanical Crack Propagation

This research presents an extended deep energy method designed to model thermo-mechanical crack propagation while maintaining the crack as a sharp polyline throughout the simulation domain. The core innovation lies in representing both temperature and displacement fields through neural networks that are coupled via a discontinuous scalar embedding function, which explicitly encodes the crack geometry.

Core Methodology and Numerical Framework

1. Crack Representation:

The method treats the crack not by partitioning the physical domain into separate subdomains (like phase-field methods), but by encoding it explicitly as a sharp geometrical entity. This is achieved through an additional scalar input, rho(x), which serves as the crack embedding function. This function is constructed to be discontinuous across the crack and smooth in the intact material regions.

2. Field Representation and Coupling:

Two distinct neural networks are employed, one for temperature (T) and one for displacement (u). Both networks share the crack embedding function rho(x) as an input.

Temperature Evolution: The transient heat conduction equation is advanced in time using a backward Euler scheme, recast into an incremental functional to be minimized. This minimization yields the temperature field T n, following a staggered sequence with the displacement minimization at every load step, mirroring the operator splitting used in phase-field computations.
Displacement Enrichment: The displacement network is further enriched near the crack tip by incorporating leading asymptotic terms derived from the Williams expansion, utilizing trainable amplitudes.

3. Energy Minimization:

The fields are determined through simultaneous energy minimization:

The thermal network is trained by minimizing a time-discrete heat conduction functional, Q T = integral x T i w T i N q T i=1 (Equation 56).
The displacement network is obtained by minimizing the thermoelastic potential energy functional, Q u = integral x u i w u i N q u i=1 (Equation 73).

Crack Growth Criterion and Fracture Mechanics

Crack advancement is governed by classical fracture mechanics criteria evaluated on the trained fields.

Stress Intensity Factor (SIF) Extraction: SIFs are extracted via an interaction integral. Under thermal load, this includes the area term of Wilson and Yu. Crucially, each extraction must be validated by a sweep of the contour radius before crack advancement is permitted.
Advancement Condition: The crack advances when the ratio chi of the driving force to the resistance meets or exceeds a prescribed threshold (chi c):

chi G(theta c) over G c(T n_tip) at least chi c

where G(theta c) is related to the crack geometry, G c(T n_tip) is the critical energy release rate dependent on the crack-tip temperature, and theta c is a characteristic angle.

Admissibility Check: A reading that fails this admissibility check authorizes no extension, except under bounded release for a clipped sweep. The crack advances explicitly by appending a new control point to the polyline at x new_tip = x tip + a, dc.

Validation and Results

The method was rigorously assessed against four benchmark problems:

A stationary thermal edge crack with a published SIF.
A single-edge notched tension test under thermal load (using a functionally graded material).
A notched cruciform specimen.

Key Findings on Benchmarks:

Stationary Edge Crack: The extracted SIF agrees with the published value to within 0.11% on the finest of three grids.
Single-Edge Notched Tension Test: The peak load agrees with the phase-field reference to within 4.1% across three thermal cases.
Graded Shear Test: Crack paths agree with the phase-field reference, and initiation agrees to within one load step against an independent sharp-crack finite element solution in all three thermal cases.
Notched Cruciform Specimen: The predicted chord angles fall between 1.3 and 4.1 outside the nearest published curve, which itself exhibits a much wider spread (5.4 to 18.9).

Run-to-Run Consistency:

The study explicitly measured run-to-run scatter on every example, providing crucial insight into the method's robustness and variability.

Scope and Limitations

The scope of this study is strictly limited:

All examples are two-dimensional and quasistatic.
Only a single crack is followed; branching or merging phenomena are not treated.
The coupling is one-way for a fixed crack: mechanical dissipation does not return to the thermal problem; the mechanical solution only informs the conduction problem through the crack it advances.

Relevant Governing Equations

The method relies on solving coupled equations:

Thermal Evolution: rho C p T n - T n-1 over t - grad times (k grad T n) = Q n in c.
Mechanical Equilibrium: Once T n is known, the mechanical response is found via quasistatic equilibrium: grad times sigma + b = 0 in c.
Energy Minimization (Temperature): Q T = x T i, w T i N q T i=1 (Equation 56).
Energy Minimization (Displacement): Q u = x u i, w u i N q u i=1 (Equation 73).
Crack Growth Criterion: chi G(theta c) / G c(T n_tip) at least chi c.

Analysis of SIF Extraction Fidelity (Comparison with Reference)

A detailed comparison was made between the SIF extraction in the proposed method and a sharp-crack finite element reference.

SIF Agreement: For the stationary edge crack, agreement is excellent (0.11% error).
Interaction Integral Term: The effect of omitting the area term (Wilson and Yu's contribution) was isolated. When omitted, SIFs remain finite but drift with contour radius; when included, they exhibit a dependence on contour radius that separates them from the contour-only reading.
Thermal Load Impact: Where thermal load contributes significantly to SIFs, the difference between the contour-only reading and the area-term reading is larger, indicating that in thermally loaded scenarios, including this term is critical for accurate extraction.

The computational cost for a propagating case was reported as 3.5 to 4 hours on one H200 GPU. The code is available at https://github.com/HannnZH/Extended-DEM-ThermoMechanical-Crack.

Improvements for AI systems

Extended Deep Energy Method for Thermo-Mechanical Crack Propagation This method allows AI systems to model thermo-mechanical crack propagation by representing both temperature and displacement fields via separate neural networks that share a single crack-aware input. This enables the system to solve coupled transient heat conduction and quasi-static mechanical equilibrium simultaneously, which is currently not treated together in many existing solvers.
Sharp Crack Representation without Mesh Conformance The system can maintain a sharp polyline crack representation throughout the simulation, avoiding the need for mesh rebuilding required by early finite element treatments (Early finite element treatments required the mesh to conform to the crack faces and to be rebuilt whenever the crack advanced). This is achieved through a discontinuity embedding function that allows fields to jump across the crack without a regularization length.
Direct Access to Crack-Tip Quantities The AI system can directly extract fracture mechanics quantities by utilizing an interaction integral that includes both contour and area terms (The SIFs are extracted by the interaction integral, which under thermal load includes the area term of Wilson and Yu). This provides direct access to the SIFs while ensuring reliability through checks like a sweep of the contour radius before the crack may advance.
Adaptive Near-Tip Enrichment for Displacement The displacement field can be enriched near the tip using the Williams expansion with trainable amplitudes, which is inherited from XFEM, allowing it to capture the displacement jump effectively (The displacement network is enriched with the leading asymptotic terms at the tip).
Temperature Field Modeling with Thermal Coupling The AI can solve transient heat conduction by minimizing an incremental functional (The time discrete heat equation is then recast as an incremental thermal functional and minimized) in a staggered sequence with the mechanical potential energy, ensuring that the temperature follows from the same kind of minimization that delivers the displacement.
Path-Independent SIF Extraction Under Thermal Load The system can reliably read SIFs by evaluating the interaction integral over a sweep of contour radii (every extraction is therefore read over a sweep of contour radii and refused when it drifts beyond a set tolerance). This addresses the issue where the area term does not appear in the isothermal form of the integral and is left out, for example, by Ghaffari et al. [74] leading to unreliable readings.
Crack Growth Decision Based on Coupled Criteria The crack growth decision is governed by a criterion that evaluates the energy release rate of a kink in that direction at the crack-tip temperature (The crack advances at the maximum hoop stress angle when the energy release rate of a kink in that direction reaches the critical value at the crack-tip temperature). This criterion combines SIFs, kink angles, and a temperature-dependent critical energy release rate.

Abstract

Thermo-mechanical fracture couples transient heat conduction on a cracked domain with a crack that grows as the temperature and the displacement evolve. Neural energy solvers have been proposed for phase-field fracture and later extended to represent a sharp crack through the network input, but heat conduction on the cracked domain and crack propagation under the resulting thermal stresses have not yet been treated together in these solvers. We present an extended deep energy method for thermo-mechanical crack propagation in which the crack remains a sharp polyline. Two networks represent the temperature and the displacement and receive the crack through a scalar embedding function, discontinuous across the crack and smooth elsewhere, so that both fields can jump across it without a regularization length, and the displacement is enriched near the tip by the Williams expansion with trainable amplitudes. The two fields are obtained by minimizing an incremental conduction functional and the thermoelastic potential energy in a staggered sequence, with Monte Carlo integration on points stratified over background elements, densified near the tip and redrawn during training. The stress intensity factors are extracted by the interaction integral with the area term of Wilson and Yu and checked by a sweep of the contour radius, and the crack advances at the maximum hoop stress angle when the energy release rate of the kink reaches the critical value at the crack-tip temperature. On a stationary thermal edge crack the extracted stress intensity factor agrees with the published value to 0.11%, in a functionally graded shear test initiation agrees with an independent sharp-crack finite element solution to within one load step, and on a notched cruciform specimen the crack paths follow the published solutions under mechanical, thermal and combined loading.