ArXiv (open access)
Summary
- We consider the Arnold-Falk-Winther finite element triplet to discretize the stress, displacement, and rotation variables.
- Using a spanning tree, a particular solution $\sigma_f = Sf$ is rapidly constructed that balances body and boundary forces and is weakly symmetric.
- Since $S$ is a right-inverse of $B$, the operator $(I - SB)$ is a projection onto the kernel of $B$.
- Two strategies are proposed to update the particular solution with a homogeneous solution
- A Split approach in which a neural network is trained to update the particular solution with $\tilde{\sigma}$, which is then projected onto the kernel of $B$:
\(\sigma = \sigma_f + (I - SB) \tilde{\sigma}\)
- A Corrected strategy that trains a neural network to compute the stress directly as $\hat{\sigma}$ and we apply a correction:
\(\sigma = \sigma_f + (I - SB) (\hat{\sigma} - \sigma_f)\)
In both cases, we have $B\sigma = B\sigma_f$ and thus it conserves linear and angular momentum.
- The Corrected approach is found to be more reliable overall.
