Adjoint-Based Error Estimation for Unsteady Problems

Abstract

Among error estimation methods, adjoint-based approaches are considered the most accurate but have the highest computational cost. For unsteady non-linear problems such as the Navier-Stokes equations, substantial storage requirements arise, as the full primal solution must be stored to solve the adjoint problem. As a result, careful management of storage resources is essential. The purpose of this research was twofold: to develop a surrogate model for the primal solution and to compute an accurate adjoint-based error estimate with the developed surrogate primal.
This study compared three methodologies to create a surrogate model of the primal solution while reducing the storage requirements of unsteady adjoint-based error estimation: a Convolutional AutoEncoder (CAE), an Echo State Network (ESN) and a combination of the first two, referred to as CAE-ESN. A benchmark Proper Orthogonal Decomposition (POD) served as a baseline for comparison with the deep learning techniques. Three numerical test cases were analyzed, where the finite element method was used for spatial discretization and implemented with the \texttt{FEniCS} computational framework. The first test case involved a manufactured solution to verify the implemented solver and methodologies. The remaining test cases used a turbulent channel flow dataset to force the unsteady viscous Burgers' equations in 1D (wall-normal component) and 2D (spanwise and wall-normal components).
For the manufactured solution, the ESN and POD outperformed the remaining approaches for the lowest and highest spatial resolutions, respectively. The success of the ESN was linked to its training being a linear regression problem. As established in previous studies, the smooth nature of the solution rendered the POD optimal. For the 1D case, the CAE was optimal, particularly for lower spatial resolutions. This method offered equivalent compression ratios to the POD while being more efficient in terms of computational cost and accuracy. In contrast, the ESN-based methods failed to accurately capture the error estimate, as they were not able to accurately compute the primal residual. However, these methods offered a higher compression than other approaches, along with a decrease in accuracy. Moreover, the error indicators produced by the ESN-based methods continued to effectively pinpoint the elements required for mesh adaptation. In the 2D case, only the POD and CAE were investigated. The ESN was excluded due to the high-dimensional nature of the test case. The CAE-ESN was not applied because of the limited time interval, which provided insufficient data for training. The CAE again proved optimal due to its efficiency and higher compression capabilities than the POD. While both methods provided accurate error estimates and indicator fields, the CAE outperformed the POD due to its superior compression. The CAE was also able to compute the adjoint solution and primal residual more accurately than the POD for most spatial resolutions. This research highlighted the potential for the CAE to outperform more conventional methods, such as POD.