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A physics-informed variational DeepONet for predicting the crack path in brittle materials

Somdatta Goswami, Minglang Yin, Yue Yu, George Karniadakis

arXiv:2108.06905v2cs.LGmath.NA

TL;DR

Brittle-fracture simulation needs surrogates that reduce repeated high-fidelity computation despite discontinuous, fluctuation-sensitive responses. The paper develops a variational, physics-informed DeepONet using governing-equation losses and labelled data, and demonstrates it on Mode-I and Mode-II benchmarks. The reported framework supports interpolation and extrapolation, including 3.17% extrapolation accuracy for Mode-II failure.

  • Problem

    Fracture analysis requires computationally demanding high-fidelity simulations, while discontinuous responses and limited labelled data challenge conventional surrogate models.

  • Method

    The paper trains V-DeepONet with a hybrid loss combining variational energy from the governing PDE and labelled high-fidelity data.

  • Results

    The surrogate is demonstrated on Mode-I and Mode-II brittle-fracture benchmarks and reports 3.17% extrapolation accuracy for Mode-II failure.

  • Takeaways & Limitations

    After training, V-DeepONet predicts failure paths, displacements, failure zones, and damage for varying crack configurations and applied displacements.

  • Takeaways & Limitations

    The method requires judicious input scaling and manual modulation of the data-loss and energy-loss weights.

Abstract

from arXiv · show

Failure trajectories, identifying the probable failure zones, and damage statistics are some of the key quantities of relevance in brittle fracture applications. High-fidelity numerical solvers that reliably estimate these relevant quantities exist but they are computationally demanding requiring a high resolution of the crack. Moreover, independent intensive simulations need to be carried out even for a small change in domain parameters and/or material properties. Therefore, fast and generalizable surrogate models are needed to alleviate the computational burden but the discontinuous nature of fracture mechanics presents a major challenge to developing such models. We propose a physics-informed variational formulation of DeepONet (V-DeepONet) for brittle fracture analysis. V-DeepONet is trained to map the initial configuration of the defect to the relevant fields of interests (e.g., damage and displacement fields). Once the network is trained, the entire global solution can be rapidly obtained for any initial crack configuration and loading steps on that domain. While the original DeepONet is solely data-driven, we take a different path to train the V-DeepONet by imposing the governing equations in variational form and we also use some labelled data. We demonstrate the effectiveness of V-DeepOnet through two benchmarks of brittle fracture, and we verify its accuracy using results from high-fidelity solvers. Encoding the physical laws and also some data to train the network renders the surrogate model capable of accurately performing both interpolation and extrapolation tasks, considering that fracture modeling is very sensitive to fluctuations. The proposed hybrid training of V-DeepONet is superior to state-of-the-art methods and can be applied to a wide array of dynamical systems with complex responses.

1. Introduction

Brittle-fracture surrogates must reduce the cost of repeated high-fidelity simulations while handling discontinuous, fluctuation-sensitive responses. The paper proposes a variational, physics-informed DeepONet trained with governing equations and labelled data.

  • High-fidelity fracture models are computationally expensive and require repeated simulations when domain, material, or operating conditions vary.
  • Fracture surrogate modeling is difficult because discontinuous responses violate the smoothness assumptions common in conventional surrogates.
  • Reduced-order approaches can provide fast emulators but may sacrifice accuracy, stability, and generalization.
  • Sparse labelled data in engineering motivates encoding domain knowledge into neural networks for learning fracture solution operators.
  • The proposed framework uses a variational energy-based DeepONet to learn generalized solutions of fracture-governing PDEs.
  • The method is demonstrated on Mode-I tensile and Mode-II shear crack-growth benchmarks using phase-field data generated through isogeometric analysis.

2. Phase field modeling of fracture

Phase-field fracture represents cracks through coupled elastic and scalar phase fields, with crack evolution obtained by minimizing a total variational energy. The formulation uses finite-width damage zones, irreversibility, and prescribed boundary conditions.

  • The phase-field formulation couples a vector-valued elastic field with a scalar phase field to represent fracture.The cracked region has φ = 1, while the undamaged region has φ = 0.
  • The total energy combines stored elastic strain energy, fracture energy, and external potential energy.
  • A finite process-zone width l0 smooths the crack representation, with sharp crack topology recovered as l0 → 0.
  • Crack paths are resolved by minimizing the energy functional, using a stress-degradation function and a tension-compression split of elastic energy.
  • Irreversibility conditions and a strain-history functional ensure crack extension over time and allow pre-existing cracks to be modeled.
  • The monolithic scheme simultaneously minimizes elastic and fracture energies to obtain displacement and phase fields.

3. DeepONet

DeepONet learns solution operators by combining a branch network encoding input functions at fixed sensors with a trunk network encoding output coordinates. This operator-learning structure is adapted to fracture’s sparse-data setting by incorporating physical laws.

  • DeepONet targets operator regression, mapping functions to functions rather than finite-dimensional vectors to vectors.
  • The branch net encodes the input function at fixed sensor points, while the trunk net represents the output location.
  • Branch inputs can represent domain shape, initial or boundary conditions, coefficients, and source terms, sampled at common sensor locations.
  • The branch and trunk networks produce q-dimensional feature embeddings that are merged through a dot product to form the solution operator.
  • A DeepONet dataset consists of branch inputs, trunk coordinates, and target solution values, with each input function repeated across evaluation points.
  • Because fracture data are expensive and crack growth is sensitive to varied parameters, the paper encodes physical laws to train DeepONet in a small-data regime.

4. Variational energy based DeepONet

V-DeepONet embeds a variational formulation into operator learning, enforces Dirichlet conditions through output modification, and trains with combined data and variational losses. Its inputs are designed to predict damage paths across initial defects and applied displacements.

  • V-DeepONet learns a solution operator for a differential equation by minimizing an energy functional derived from its variational form.
  • The framework represents the governing PDE through a differential operator K and a variational functional F over the physical domain.
  • For fracture prediction, the branch net receives prior tensile strain-energy histories, while the trunk net receives coordinates, initial crack configuration, and applied displacement.
  • The modified network output exactly satisfies Dirichlet boundary conditions, eliminating the need for a boundary-loss term.
  • The hybrid objective is L(θ) = λ1 × Ldata(θ) + λ2 × Lvar(θ), combining labelled measurements with variational energy.
  • The surrogate is designed to predict damage paths for arbitrary initial defects and applied displacements while balancing efficiency and accuracy through input and sensor selection.

5. Surrogate modeling for fracture analysis

The proposed V-DeepONet surrogate predicts displacement and phase fields to infer crack locations and paths across applied displacements and initial defect configurations. A unified model addresses sequence dependence needed when initial crack locations vary.

  • Model formulation: The two-dimensional V-DeepONet outputs displacement components and a scalar phase field at each material point.These outputs represent the coupled elastic and fracture fields used to infer failure pathways.
  • Model formulation: The surrogate targets crack location and domain deformation for fixed initial conditions across applied displacements.This is the first of two subproblems addressed by the surrogate-model framework.
  • Model formulation: A second surrogate predicts the final crack path for varying initial defect locations under fixed applied displacement.The final model integrates the displacement-step and initial-condition surrogate tasks.
  • Training and inputs: The training procedure uses sensor-point tensile strain energy, sampled output coordinates, and a hybrid loss combining variational energy and data-driven terms.The branch net receives discretized input functions, while the trunk net encodes output locations; optimized parameters minimize the hybrid loss.
  • Training and inputs: For sequential displacement prediction, the branch input at each step is the tensile energy computed from the preceding V-DeepONet displacement and phase-field outputs.Displacement gradients and strain eigenvalues are used to obtain tensile and compressive energy components.
  • Unified model challenges: Multiple initial crack locations require retaining initial-location or multi-step information because crack paths can overlap and sequence information can be lost.Using only previous-step strain energy or energy near the original defect can therefore produce erroneous predictions.

6. Data Generation

The study trains the V-DeepONet with variational and labelled high-fidelity data, then verifies predicted failure paths against a high-fidelity fracture model.

  • Data generation: The hybrid loss combines variational energy from the governing PDE with data-driven loss from labelled high-fidelity datasets.The datasets are generated using phase-field codes within an isogeometric-analysis framework.
  • Data generation: The labelled datasets are simulated with phase-field fracture codes developed within isogeometric analysis.The authors identify the IGA phase-field codes used for dataset construction.
  • Data generation: The approach can also use more general training datasets, including datasets generated by V-PINNs or experimental measurements when available.The reported study itself does not use experimental data.
  • Verification: Predicted failure paths are compared with those computed by a high-fidelity fracture-mechanics model to verify surrogate performance.This comparison is the stated verification procedure for the surrogate model.

7. Simulation Results

The V-DeepONet is evaluated on tensile and shear fracture benchmarks, including interpolation and out-of-distribution predictions of phase, displacement, and crack-path fields. Hybrid physics-and-data training supports accurate predictions with limited samples and improves performance over the data-driven DeepONet.

  • Benchmark problems: The study evaluates V-DeepONet on Mode-I tensile and Mode-II shear crack-growth benchmarks, including out-of-distribution predictions.The benchmark setup uses phase-field fracture data generated with isogeometric analysis.
  • Tensile failure: For tensile loading, six initial crack lengths and seven displacement steps train the model to predict displacement and phase fields.The tested in-distribution crack length is 0.5 mm, while the training lengths range from 0.25 to 0.55 mm.
  • Tensile failure: 0.63% averaged error in predicted φ is obtained for the tensile test with lc = 0.5 mm.The model maps initial history information and loading inputs to the displacement and phase fields.
  • Tensile failure: 1.85% averaged error on φ is obtained for the tensile out-of-distribution test at lc = 0.65 mm, beyond the training range.Predicted y-displacements are compared with high-fidelity ground truth at two x-locations and three displacement steps.
  • Shear failure: 0.67% relative error in predicted φ is reported for two shear test cases after training with 85 initial crack locations.The crack lengths span [0.3, 0.65] and crack heights span [0.2, 0.675].
  • Shear failure: 26.44%, 13.49%, 7.49% and 3.17% relative mean errors are reported as V-DeepONet training samples increase from 20 to 40, 60 and 85.Each sample count is evaluated on a corresponding out-of-distribution prediction.

8. Summary and discussion

The V-DeepONet provides a fast surrogate for brittle-fracture quantities across crack configurations and loading, using hybrid physics-and-data training. It reports Mode-II extrapolation accuracy of 3.17% but requires careful scaling and manual loss-weight tuning.

  • V-DeepONet predicts brittle-fracture failure paths, failure zones, damage, and displacements after training.The trained model can evaluate different crack-tip locations and applied displacements in a fraction of a second.
  • The hybrid loss combines physics-driven and data losses, supporting extrapolation and variational treatment of discontinuous solutions.The physics loss comes from the variational governing-PDE form.
  • The methodology is presented as independent of dimensionality and applicable to reliability analysis, uncertainty quantification, and design optimization.
  • 3.17% extrapolation accuracy is reported for Mode-II failure.The paper identifies Mode-II failure as challenging even for classical numerical methods.
  • The method requires judicious input and output scaling and manual modulation of the physics- and data-loss weights.The energy input spans [0, 10^4], while elastic and phase-field outputs are of orders O(10^-3) and O(1), respectively.

Appendix A Algorithm for constructing the proposed unified surrogate model.

The algorithm constructs and trains the V-DeepONet from sensor locations, high-fidelity responses, sampled domain points, and initial-history functions. It then predicts displacement, elastic, and phase fields while recursively updating the energy input for subsequent steps.

  • Algorithm inputs: The workflow samples p domain points and gathers u(y), v(y), and φ(y) responses for selected displacement steps.
  • Purpose: The algorithm is explicitly presented as a V-DeepONet-based surrogate model for predicting brittle-fracture crack paths.
  • Algorithm inputs: It initializes sensor locations, computes initial history functions analytically and from high-fidelity data, and prepares branch-net inputs.The branch input is formed as [H0(Xs), 0].
  • Training: The V-DeepONet is initialized, evaluated through its solution operators, and trained by minimizing the hybrid loss to obtain optimized parameters.
  • Prediction: For prediction, the trunk net receives sampled points and associated inputs, producing u*(yj), v*(yj), and φ*(yj).
  • Time stepping: The predicted solutions are used to compute the energy Hi, which becomes the input required at the next displacement step.

Appendix B Flow in heterogeneous porous media

The porous-media example learns hydraulic-head solution fields from spatially varying conductivity using a variational V-DeepONet, without labelled input-output datasets. Predictions for two permeability samples are compared with finite-element PDE solutions and their errors.

  • Problem setup: Flow is governed by a heterogeneous-conductivity PDE with zero hydraulic head on the domain boundary.K(x) is spatially varying hydraulic conductivity, while h(x) is hydraulic head.
  • Problem setup: Conductivity fields are generated as K(x) = exp(F(x)), where F(x) is a truncated Karhunen–Loève expansion of a Gaussian process.The samples retain the leading 100 KL terms.
  • Variational training: The V-DeepONet is trained by minimizing the variational energy subject to the Dirichlet boundary condition, without labelled input-output data.The objective is E = Ψh with h(x) = 0 on ∂ΩD.
  • Network and evaluation: The operator approximation uses separate 6-layer fully connected branch and trunk networks with 32 neurons per hidden layer.Training uses n = 200 permeability samples and evaluates solutions at p = 10000 randomly sampled points.
  • Results: Figure 11 compares predicted hydraulic heads with PDE-toolbox ground truth for two permeability samples and plots their difference as the error.Permeability K(x) is shown on a log scale.

Appendix C Additional results

Additional tensile-failure results evaluate an out-of-distribution crack configuration sequentially across displacement steps, comparing predicted fields with IGA ground truth and prediction errors.

  • Additional results: The model predicts the tensile-failure solution for lc = 0.65 mm, outside the training range lc ∈ [0.3, 0.55].Training uses six crack-length values and seven displacement steps.
  • Additional results: Predictions are generated sequentially at four displacement steps, with phase-field and y-displacement predictions compared against IGA simulations.The final row presents the error between predicted and ground-truth values.
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