This repository contains a TensorFlow 2 implementation of a Physics-Informed Neural Network (PINN) to solve a 2D solid mechanics problem. The project simulates a quasi-static tensile test of a hyperelastic dog-bone specimen, demonstrating how deep learning can be used to solve complex non-linear partial differential equations (PDEs).
The core of this work is a PINN that directly solves the strong form of the equilibrium equation, with a hybrid optimization scheme that combines ADAM and L-BFGS for efficient and accurate training.
Figure 1: The architecture of the Physics-Informed Neural Network.
The PINN is trained to find a displacement field
The simulation solves the static equilibrium equation in its strong form, where body forces are neglected:
Here,
The material is modeled as hyperelastic. The relationship between deformation and stress is derived from the strain energy density function. The key kinematic quantities and the constitutive law are:
-
Deformation Gradient:
$\mathbf{F} = \mathbf{I} + \nabla \mathbf{u}$ -
PK1 Stress: The code implements a compressible Neo-Hookean-like model, where the PK1 stress tensor
$\mathbf{P}$ is given by:$$\mathbf{P} = \mu \mathbf{F} + (\lambda \ln(J) - \mu) \mathbf{F}^{-T}$$ where$J = \det(\mathbf{F})$ , and$\mu$ and$\lambda$ are Lamé's first and second parameters.
The dog-bone specimen is subjected to the following boundary conditions:
-
Prescribed Displacement: A tensile displacement,
$u_x = u_{\text{applied}}$ , is applied incrementally to the right-hand face of the specimen. -
Fixed/Symmetry Condition: The left-hand face is fixed in the x-direction (
$u_x = 0$ ). -
Traction-Free: All other boundaries are traction-free, meaning the forces on these surfaces are zero:
$$\mathbf{P} \cdot \mathbf{N} = \mathbf{0}$$ where$\mathbf{N}$ is the outward normal vector to the surface.
The neural network is trained by minimizing a composite loss function that includes the physics residual and the boundary conditions:
where:
-
$L_{PDE}$ is the mean squared error of the equilibrium equation residuals at interior collocation points.$$L_{PDE} = \frac{1}{N_f} \sum_{i=1}^{N_f} \left| \nabla \cdot \mathbf{P}(\mathbf{X}_i) \right|^2$$ -
$L_{Traction}$ is the mean squared error of the traction forces on the free boundaries.$$L_{Traction} = \frac{1}{N_t} \sum_{i=1}^{N_t} \left| \mathbf{P}(\mathbf{X}_i) \cdot \mathbf{N}_i \right|^2$$ -
$L_{Displacement}$ is the mean squared error for the prescribed displacement boundary condition.
This project employs a two-phase training strategy for each displacement increment:
- ADAM Optimizer: The network is first trained for a number of epochs using the ADAM optimizer to quickly find a good region in the loss landscape.
- L-BFGS Optimizer: Subsequently, the
scipy.optimize.minimizefunction with the L-BFGS algorithm is used to fine-tune the network parameters. L-BFGS is a quasi-Newton method that can achieve higher precision, which is often crucial for solving physics problems. Theutils/scipy_loss.pymodule provides a wrapper to make the TensorFlow model compatible with SciPy's optimizer.
- Hyperelastic Model: Implements a non-linear, hyperelastic constitutive model suitable for large deformations.
- Hybrid Optimization: Combines the ADAM and L-BFGS optimizers for robust and precise training.
- Incremental Loading: Simulates a quasi-static analysis by applying displacement in small, discrete steps, allowing the model to solve a sequence of non-linear problems.
- Strong-Form PINN: Solves the strong form of the PDE, with Dirichlet boundary conditions enforced directly in the network's output layer.
- Modular Code: The project is organized into clear modules for solvers, plotting, and utilities.