Towards efficient adjoint-based mesh optimization for predictive Large Eddy Simulation

Abstract

This thesis contributes to the effective and efficient application of unsteady adjoint methods to Adaptive Mesh Refinement (AMR) for Large Eddy Simulation (LES). Three aspects, i.e., subgrid-scale model error, storage cost of high-dimensional data, and stability of the adjoint problem for turbulent flows, were studied to make adjoint-based mesh adaptation feasible for time-dependent high-fidelity simulations.
The effectiveness of adjoint-based error estimation is initially demonstrated using linear advection-diffusion problems. An adjoint-based AMR strategy is further developed and analysed for unsteady 1D Burgers problems with a multi-frequency forcing term. Then we introduce a Reduced-Order Representation (ROR), which uses the Proper Orthogonal Decomposition (POD) to replace full-order primal solutions when solving the adjoint problem backward in time. Numerical results demonstrate the effectiveness of using RORs for adjoint-based AMR.
An enhanced online algorithm for POD analysis is proposed to deal with high-dimensional LES data, resulting from the nonlinearity and unsteadiness that require us to store a time history of primal states for solving the adjoint problem. The enhanced algorithm is based on the incremental Singular Value Decomposition and exploits the decomposition of full-order solutions into reconstructed and truncated solutions. Two lower-bound estimators are proposed to equip the enhanced algorithm with a posteriori error analysis. Numerical experiments demonstrate that the algorithm can significantly improve the computational efficiency of online POD analysis while accuracy is maintained with an appropriate number for the truncation of POD modes. Furthermore, the enhanced algorithm scales well in parallel and the improvement of computing efficiency is independent of the number of processors.
The unsteady adjoint problem is investigated for 2D and 3D cylinder flows. Using RORs significantly reduces the memory requirement for storing primal flow solutions for both 2D and 3D cylinder flow. Dynamic features of the adjoint field are well presented with using RORs, although there are differences in regions around and upstream of the cylinder using a small number of POD modes. Error distributions can be well predicted with POD-based RORs, especially in regions with large errors. The exponential growth of adjoint solutions in the 3D turbulent flow is found to be attenuated when using RORs for solving the adjoint problem.