PINN-elasticity

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PINN-elasticity

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# Physics-Informed-Neural-Networks for linear elasticity
### Jing Hu (UT Austin)

In this project, the linear elasticity problem is solved with Physics Informed Neural Networks (PINN). The governing equations for isotropic linear elasticity are

where <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;\sigma_{ij}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;\sigma_{ij}" title="\tiny \sigma_{ij}" /></a> is the stress tensor, <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;\epsilon_{ij}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;\epsilon_{ij}" title="\tiny \epsilon_{ij}" /></a> is the strain tensor, u is the displacement vector, f is the body force vector, <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;\lambda" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;\lambda" title="\tiny \lambda" /></a> and <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;\mu" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;\mu" title="\tiny \mu" /></a> are the Lamé parameters and Einstein summation applies. Now we consider the linear elasticity problem on a square domain <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;[0,1]\times[0,1]" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;[0,1]\times[0,1]" title="\tiny [0,1]\times[0,1]" /></a>. We consider the following boundary conditions:

Top wall

Right wall

Left wall

Bottom wall

where <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;t_\bullet" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;t_\bullet" title="\tiny t_\bullet" /></a> is the traction prescribed on the boundaries and the Q is the load magnitude. We consider the following body force:

The analytical solution to this problem is [ <a href="https://www.codecogs.com/eqnedit.php?latex=\large&space;\cos(\2\pi&space;x)\sin(\pi&space;y)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\large&space;\cos(\2\pi&space;x)\sin(\pi&space;y)" title="\large \cos(\2\pi x)\sin(\pi y)" /></a>, <a href="https://www.codecogs.com/eqnedit.php?latex=\large&space;\sin(\pi&space;x)Q&space;y^4/4" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\large&space;\sin(\pi&space;x)Q&space;y^4/4" title="\large \sin(\pi x)Q y^4/4" /></a> ].

In the numerical implementation, we output both displacements and stresses and adopt the following loss function to prescribe boundary conditions and enforce the governing equation.

For model parameters: <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;\lambda=1" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;\lambda=1" title="\tiny \lambda=1" /></a>, <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;\mu=0.5" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;\mu=0.5" title="\tiny \mu=0.5" /></a> and <a href="https://www.codecogs.com/eqnedit.php?latex=\dpi{300}&space;\tiny&space;Q=4" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\dpi{300}&space;\tiny&space;Q=4" title="\tiny Q=4" /></a>, we have the following results

The decrease of the loss function is shown in the following figure.

> **Note:** The implementations were developed and tested on colab with TensorFlow 1.9 and hardware accelerator chosen as GPU.

PINN-elasticity

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