Transfer learning enhanced physics informed neural network for phase-field modeling of fracture (1907.02531v1)

Abstract: 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.

• The paper introduces a variational energy framework in a PINN that simplifies boundary condition enforcement and reduces computational complexity.
• It employs Gauss-Legendre quadrature and transfer learning to enhance numerical integration and improve simulation efficiency for fracture mechanics.
• Empirical results validate the approach’s accuracy and robustness across brittle fracture cases, highlighting its potential for broader engineering applications.

Overview of Transfer Learning Enhanced PINN for Fracture Modeling

The paper introduces a physics-informed neural network (PINN) algorithm aimed at solving brittle fracture problems using phase-field modeling. Unlike traditional approaches, this method minimizes the variational energy of the system rather than the residuals of the governing partial differential equations (PDEs), offering two significant advantages: a more straightforward imposition of boundary conditions and reduced computational complexity due to lower-order derivatives.

Key Contributions

1. Variational Energy Approach: Diverging from traditional PINNs, the proposed method employs a variational energy framework, reducing the complexity involved in calculating higher-order derivatives inherent in residual-based methods.
2. Enhanced Boundary Condition Handling: By modifying the neural network output to inherently satisfy boundary conditions, the method eliminates the need for penalty terms, streamlining the optimization process and improving robustness.
3. Efficient Numerical Integration: The research employs a Gauss-Legendre quadrature scheme over geometries defined using NURBS to enhance accuracy and computational efficiency. This approach overcomes the inefficiencies in traditional integration methods, particularly in capturing non-smooth features like fractures.
4. Transfer Learning Implementation: To address the computational challenges associated with training the PINN at every load step, the paper leverages transfer learning. By partially retraining the network while retaining most weights, computational efficiency is significantly enhanced.

Empirical Results

The proposed approach is validated against four fracture mechanics problems, with results showcasing close agreement with existing literature:

• Accuracy: The variational approach demonstrated improved accuracy over conventional residual-based PINNs, especially in handling complex fracture paths.
• Computational Efficiency: Utilizing Gauss-Legendre quadrature, the method required fewer discretized elements compared to traditional finite element methods, indicating a potential application as an efficient surrogate model.
• Robustness: The transfer learning technique proved effective in reducing computational load, highlighting its potential for use in complex, real-world fracture simulations.

Implications and Future Directions

The authors have set a foundation for utilizing PINNs in fracture mechanics, presenting a method that balances theoretical rigor and computational efficiency. However, the approach remains in its nascent stages, with several avenues for future exploration:

• Adaptive Refinement: Development of adaptive mesh refinement strategies could further optimize computational resources by dynamically concentrating efforts on regions of interest, such as crack paths.
• Broader Applicability: Extending the framework to heterogeneous materials and more complex loading scenarios would enhance its utility in practical engineering applications.
• Integration with Traditional Methods: Combining this PINN approach with established computational mechanics techniques may yield hybrid models that capitalize on the strengths of each method.

In conclusion, the paper provides a significant contribution to the field of fracture mechanics through a novel PINN framework that caters to the efficiency and accuracy needs of modern computational simulations. The utilization of transfer learning within this context not only reduces computational burdens but also opens up new potential for AI-driven solutions in structural mechanics.