Source-linked AI summary
Transfer learning enhanced physics informed neural network for phase-field modeling of fracture
Somdatta Goswami, Cosmin Anitescu, Souvik Chakraborty, Timon Rabczuk
TL;DR
Brittle-fracture PINNs must represent fracture physics despite limited engineering data and the difficulty of enforcing physical laws. This paper minimizes variational energy, exactly embeds boundary conditions, and adds efficient quadrature and transfer learning. Across four examples, results match literature closely, while the first two show greater accuracy than residual-based PINNs.
Problem
Engineering neural-network applications may have limited training data and cannot guarantee satisfaction of governing physical laws, motivating physics-informed methods for brittle-fracture modeling.
Method
The paper develops an energy-based PINN that minimizes variational energy, exactly satisfies boundary conditions through modified outputs, uses spline-based geometry with Gauss quadrature, and applies transfer learning across load steps.
Results
Across four fracture-mechanics examples, results match the literature closely, and the proposed approach is more accurate than conventional residual-based PINNs for the first two problems.
Takeaways & Limitations
The approach provides an efficient PINN framework for predicting brittle-fracture crack paths using phase-field modeling.
Takeaways & Limitations
The study pre-refines the expected crack-growth path rather than refining it adaptively.
Abstract
from arXiv · showhide
We present a new physics informed neural network (PINN) algorithm for solving brittle fracture problems. While most of the PINN algorithms available in the literature minimize the residual of the governing partial differential equation, the proposed approach takes a different path by minimizing the variational energy of the system. Additionally, we modify the neural network output such that the boundary conditions associated with the problem are exactly satisfied. Compared to conventional residual based PINN, the proposed approach has two major advantages. First, the imposition of boundary conditions is relatively simpler and more robust. Second, the order of derivatives present in the functional form of the variational energy is of lower order than in the residual form. Hence, training the network is faster. To compute the total variational energy of the system, an efficient scheme that takes as input a geometry described by spline based CAD model and employs Gauss quadrature rules for numerical integration has been proposed. Moreover, we utilize the concept of transfer learning to obtain the crack path in an efficient manner. The proposed approach is used to solve four fracture mechanics problems. For all the examples, results obtained using the proposed approach match closely with the results available in the literature. For the first two examples, we compare the results obtained using the proposed approach with the conventional residual based neural network results. For both the problems, the proposed approach is found to yield better accuracy compared to conventional residual based PINN algorithms.
1 Introduction
The paper develops a PINN for brittle-fracture growth that minimizes variational energy instead of governing-equation residuals, exactly satisfies boundary conditions, and uses efficient integration and transfer learning. It positions these choices as improvements for accuracy and computational efficiency.
- Motivation: Engineering applications often have limited training data, while unconstrained neural networks may not satisfy essential physical laws.Data acquisition for complex engineering systems can be computationally expensive.
- Proposed approach: The proposed PINN minimizes the system’s variational energy rather than the residual of the governing differential equations.The formulation is motivated by energy criteria used for fracture propagation in phase-field modeling.
- Proposed approach: The variational-energy formulation requires derivatives one order lower than conventional residual formulations, which supports faster network training.The paper identifies lower derivative order as a major advantage over residual-based PINNs.
- Implementation: Gauss-Legendre quadrature is used to reduce the integration cost associated with the many points required by trapezoidal or Monte Carlo schemes.NURBS patches model complex geometries, while quad/oct-tree subdivision refines regions near the crack path.
- Implementation: Transfer learning retrains only the final-layer weights and biases after the first load or displacement step, accelerating training and supporting efficient crack-path computation.Weights and biases in the other layers remain fixed at their previously trained values.
- Contribution: The work’s novelty combines an enhanced PINN with its application to fracture growth and propagation, which the authors identify as a first use of PINNs for this problem.The paper presents numerical examples to illustrate the proposed approach’s performance.
2 Phase-field modeling for fracture
The phase-field formulation represents fracture with a finite-width process zone and a scalar damage field, then solves displacement and phase fields through energy minimization under prescribed boundary conditions. The study focuses on a monolithic solution scheme with history-dependent crack evolution.
- Phase-field representation: Phase-field fracture assumes a finite-width process zone controlled by the length scale l0, with a sharp crack recovered as l0 → 0.The scalar phase-field uses φ = 1 for cracked regions and φ = 0 for undamaged regions.
- Phase-field representation: The formulation solves for the vector-valued elastic field u and scalar-valued phase-field φ.The elastic and phase-field governing equations are enforced with their respective boundary conditions.
- Constitutive model: The elastic degradation function g(φ) decreases monotonically, with the common isotropic form g(φ) = (1 − φ)^2.The degradation function modifies the elastic response as the phase field evolves.
- Constitutive model: Only the tensile component of principal stress degrades with evolving damage, while the compressive component remains unaffected.The strain-energy functional is decomposed into tensile and compressive contributions.
- Crack evolution: The strain-history functional uses the maximum positive tensile energy and ensures monotonically increasing φ, preventing crack healing.An initial history functional can define initial cracks using the distance to a discrete crack line.
- Energy formulation: The total variational energy E = Ψe + Ψc is minimized subject to prescribed displacement on the Dirichlet boundary, with traction-free Neumann conditions satisfied automatically.Ψe denotes stored elastic strain energy and Ψc denotes fracture energy.
- Solution scheme: The paper uses a monolithic scheme that simultaneously minimizes Ψe and Ψc to obtain the displacement and phase fields.The alternative staggered scheme cycles between separate minimizations, but is not considered here.
3 Physics informed neural network
The proposed PINN learns neural-network parameters by minimizing variational energy while enforcing Dirichlet boundary conditions through modified outputs. It combines automatic differentiation with geometry-aware numerical integration for fracture-related problems.
- Physics-informed formulation: The approach constructs a neural network whose weights and biases are trained from the governing physics rather than only data.The physics is represented through a nonlinear partial differential equation and its variational formulation.
- Boundary conditions: The network output is modified so Dirichlet boundary conditions are exactly satisfied before optimization.This converts the parameter-learning problem into an unconstrained optimization problem.
- Energy minimization: The method minimizes total variational energy, computed from the network output and its automatically differentiated derivatives.The energy variation is itself represented through neural networks sharing the same parameters θ.
- Numerical integration: Gauss points are generated within geometry elements to integrate accurately near cracks and other nonsmooth local features.The domain is divided into elements because whole-domain Gauss-point generation can be inaccurate for nonsmooth fracture solutions.
- Geometrical modeling: NURBS patches model curved geometries, while local quad/oct-tree refinement concentrates elements near the crack path.NURBS can represent curved boundaries such as circles and cylinders exactly, unlike piecewise polynomials.
4 Phase-field modeling of fracture using PINN
The proposed PINN is applied to phase-field brittle-fracture modeling through displacement-increment training, variational-energy minimization, and history-field updates. Transfer learning reduces repeated training costs by retraining only the final network layer after the first displacement step.
- Incremental fracture evolution: The strain-history function is updated at every displacement step to support incremental fracture evolution and crack-path prediction.The analysis uses displacement-controlled failure with a constant displacement step Δu.
- Energy evaluation: Automatic differentiation computes displacement gradients, strain eigenvalues, phase-field gradients, and the quantities entering the fracture-energy terms.The resulting elastic and crack contributions are integrated with Gauss-Legendre quadrature and combined as Ve = Ψe + Ψc.
- Optimization: Optimization uses the Adam optimizer followed by the quasi-Newton L-BFGS method at each displacement step.The overall procedure repeats energy minimization, history prediction, and field prediction through the loading sequence.
- Phase-field PINN workflow: The phase-field PINN represents the displacement and phase-field variables with neural networks and computes total variational energy at each displacement increment.The workflow initializes the problem, generates geometry and prediction grids, represents primal variables, and minimizes the resulting loss.
- Transfer learning: Transfer learning accelerates subsequent displacement-step training by retraining only the weights and biases of the last layer.Weights and biases in the other layers remain fixed at their previously trained values.
5 Numerical examples
The numerical examples evaluate the proposed variational-energy PINN across fracture problems, using Gauss-Legendre integration, exact boundary-condition enforcement, and transfer learning. Its predictions closely match analytical or literature results and outperform residual-based PINNs in the first benchmark.
- One-dimensional elastic bar with crack: The first benchmark uses a cracked one-dimensional elastic bar with an available analytical solution to assess accuracy and compare PINN formulations.The bar is fixed at both ends and subjected to sinusoidal loading.
- One-dimensional elastic bar with crack: 4.46% and 3.61% relative L2 errors are obtained for displacement u and phase-field φ using the proposed approach.The calculation uses 336 Gauss points and 2100 iterations.
- One-dimensional elastic bar with crack: 85.87% for u and 91.55% for φ are obtained with the residual-based PINN, which fails to capture the crack-induced sharp gradient.The comparison keeps the neural-network architecture and integration-point count fixed.
- Integration strategy: Uniform integration points require architectural changes and domain subdivision to achieve accuracy comparable to Gauss-Legendre integration.The comparison reports two additional hidden layers of 50 neurons each for the alternative setup.
- Single-edge notched tension example: The single-edge notched tension problem produces a 670N failure load, close to the 687N value reported in.The crack width increases with displacement, and no analytical solution is available for this problem.
- Transfer learning: Transfer learning accelerates repeated training across displacement steps by reusing previously trained network parameters.From the second displacement step onward, only the last-layer weights and biases are retrained while earlier layers remain fixed.
- Three-dimensional example: The proposed PINN reproduces literature-like crack patterns for the three-dimensional example while using 512 elements instead of 134,567 reported in.The observed propagation pattern is similar to the corresponding two-dimensional single-edge-notch problem.
6 Conclusions and future work
The paper proposes a variational-energy PINN for phase-field fracture that enforces boundary conditions exactly, uses NURBS and Gauss-Legendre integration, and accelerates repeated training through transfer learning. Across four examples, it matches literature results and is more accurate than residual-based PINNs in the first two cases, while remaining subject to early-stage scope limitations.
- Contributions: The proposed PINN predicts brittle-fracture crack paths by minimizing variational energy instead of a residual-based loss.Its neural-network outputs are modified so boundary conditions are exactly satisfied without a boundary-loss component.
- Contributions: NURBS represents the problem geometry, while Gauss-Legendre quadrature computes total variational energy through element-wise integration.The domain is discretized to generate integration points efficiently near the fracture zone.
- Results: Four fracture-mechanics examples produce results that match the literature, while the proposed method is more accurate than conventional residual-based PINNs in the first two examples.The first two examples also demonstrate efficiency gains from Gauss-Legendre-based integration.
- Limitations: The current approach pre-refines the expected crack-growth path rather than refining it adaptively.The authors identify adaptive refinement as a future improvement and describe the framework as being at an early stage.
- Potential applications: The method could serve as a low-fidelity surrogate for high-fidelity solvers in reliability analysis, uncertainty quantification, and design optimization.Because it is trained directly from problem physics, it does not require a high-fidelity solver for training.