Aspects of adaptive mesh refinement in the spectral element method

Sammanfattning: This thesis deals with the improvement of the efficiency of numerical simulations in computational fluid dynamics. Some of the limitations of current algorithms on future exascale supercomputers are addressed with the main goal of using adaptive mesh refinement for the simulation of turbulent and three-dimensional flows. The comparison between two different error estimators is also investigated.The framework considered all throughout this thesis is Nek5000, a highly parallel code based on the spectral element method.First, the strong parallel scaling of Nek5000 is studied on three petascale machines in order to assess the scalability of the code and identify the current bottlenecks. It is found that the strong scaling limit ranges between 5,000 and 220,000 gridpoints per core depending on the machine and the case. The need for synchronized and low latency communication for fast computations is confirmed. It is also shown that, on a single core, Nek5000 is memory limited rather than compute limited.Then, a new method for the coarse grid part of the preconditioner for the pressure equation is implemented, which represents a significant improvement compared to existing solvers. We use an algebraic multigrid method from the Hypre library to perform the setup and solver parts on the fly and fully in parallel. The setup phase only amounts to a few percents of the wall clocktime for a single timestep of the solver and is therefore negligible; this is a requirement for adaptive simulations, where the setup must be performed after each adaptation. In addition, the new solver is shown to be suitable for large and complex simulations in three dimensions.Finally, the main objective of this work is to perform simulations with adaptive mesh refinement to achieve error control and efficient use of computational resources. We develop adjoint error estimators based on the dual-weighted residuals method and also consider more traditional a posteriori error indicators for comparison. Adaptive simulations are performed on test cases of increasing complexity: the steady flow in a lid-driven cavity in 2D and 3D, the steady flow past a 2D cylinder and the turbulent flow inside a constricted periodic channel in 3D. It is concluded that adjoint error estimators are preferred for flows with a strongly convective nature, while the more straightforward spectral error indicators are sufficient otherwise. Moreover, we perform an adaptive simulation of the turbulent flow past a NACA4412 profile at Reynolds number Re = 850,000, using the spectral error indicators. In comparison to cases with a fixed mesh, the aspect ratio of grid elements in the far field remains low, the convergence of the pressure solver is significantly sped up and the much larger computational domain allows true free-stream boundary conditions and thus makes the results less dependent on the boundary conditions.