Output Error Estimation for Unsteady Flows Using Reconstructed Solutions

Abstract

Unsteady numerical simulation has been proven to be an essential tool for research. The quality of the results can be improved by using mesh adaptation. Mesh adaptation uses error indicators to refine the mesh in regions with high errors. The error indicators used are output errors with the most accurate output error estimation method being adjoint-based error estimation. However, for this method, the primal solution needs to be stored, which is storage intensive, especially for large unsteady simulations. The method proposed in this thesis uses a neural network autoencoder to compress and reconstruct the primal solution. This solution is compared to a reconstructed solution using Proper Orthogonal Decomposition (POD).

The one-dimensional unsteady Burgers equation is used as validation for the methods using a manufactured solution while the lid-driven cavity flow is investigated using the proposed method. The manufactured solution of the one-dimensional Burgers case could be exactly reconstructed using two POD modes. For the autoencoder a small latent space was used. For low resolutions, the small latent space did not prove to be a problem as the primal and residual could be captured accurately. However, for higher resolutions, the reconstruction error of the autoencoder became dominant for the residuals and resulted in erroneous adjoint-based error estimates while the primal remained qualitatively similar.

For the lid-driven cavity flow, the POD was still able to capture the solution using a low number of modes due to the smoothness of the solution. This resulted in an unfair comparison between the POD and autoencoder reconstructed solutions. The reconstructed autoencoder error estimates for lower resolutions were more accurate due to the latent space being large enough to capture the residual of the discrete primal accurately enough. When moving to higher resolutions, the autoencoder was not able to reconstruct the residual accurately enough leading to erroneous error estimates. Therefore, the latent space of autoencoders should be sufficiently large in order to gain an accurate reconstruction of the residual. If the latent space is large enough, the error estimate is accurate and the local error estimates can be used as a first iteration error indicator for mesh refinement.