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

$\tiny \begin{aligned} \sigma_{ij,j}+f_j=0\\ \sigma_{ij}=\lambda\sigma_{ij}\epsilon_{kk}+2\mu\epsilon_{ij}\\ \epsilon_{ij}=\frac{1}{2}(u_{i,j}+u_{i,j}) \end{aligned}$

where $\tiny \sigma_{ij}$ is the stress tensor, $\tiny \epsilon_{ij}$ is the strain tensor, u is the displacement vector, f is the body force vector, $\tiny \lambda$ and $\tiny \mu$ are the Lamé parameters and Einstein summation applies. Now we consider the linear elasticity problem on a square domain $\tiny [0,1]\times[0,1]$ . We consider the following boundary conditions:

Top wall

$\tiny \begin{aligned} t_x&=\mu\pi(\frac{Q}{4}\cos(\pi x) -\cos(2\pi x))\\ t_y&=\lambda Q \sin(\pi x) +2\mu Q \sin(\pi x) \end{aligned}$

Right wall

$\tiny \begin{aligned} t_x&=0\\ t_y&=\mu\pi(-\frac{Q}{4}y^4+\cos(\pi y)) \end{aligned}$

Left wall

$\tiny \begin{aligned} t_x&=0\\ t_y&=-\mu\pi(\frac{Q}{4}y^4+\cos(\pi y)) \end{aligned}$

Bottom wall

$\tiny \begin{aligned} u_x = 0\\u_y = 0 \end{aligned}$

where $\tiny t_\bullet$ is the traction prescribed on the boundaries and the Q is the load magnitude. We consider the following body force:

$\tiny \begin{aligned} f_{x} &=\lambda\left[4 \pi^{2} \cos (2 \pi x) \sin (\pi y)-\pi \cos (\pi x) Q y^{3}\right] \\ &+\mu\left[9 \pi^{2} \cos (2 \pi x) \sin (\pi y)-\pi \cos (\pi x) Q y^{3}\right]\\ f_{y} &=\lambda\left[-3 \sin (\pi x) Q y^{2}+2 \pi^{2} \sin (2 \pi x) \cos (\pi y)\right] \\ &+\mu\left[-6 \sin (\pi x) Q y^{2}+2 \pi^{2} \sin (2 \pi x) \cos (\pi y)+\pi^{2} \sin (\pi x) Q y^{4} / 4\right] \end{aligned}$

The analytical solution to this problem is [ $\large \cos(\2\pi x)\sin(\pi y)$ , $\large \sin(\pi x)Q y^4/4$ ].

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.

$\tiny \begin{aligned} \mathcal{L} &=\left|u-u^{*}\right|_{Dirichlet BC}+\left|t-t^{*}\right|_{Neumann BC} \\ &+\left|\sigma_{ij,j}+f_j^*\right|_{collocation points} \\ &+\left|\lambda\sigma_{ij}\epsilon_{kk}+2\mu\epsilon_{ij}-\sigma_{ij}\right|_{collocation points} \end{aligned}$