An hp-Adaptive Petrov-Galerkin Method for Steady-State and Unsteady Flow Problems
A Dissertation Presented for the Doctor of Philosophy in Computational Engineering, The University of Tennessee at Chattanooga
Behzad Reza Ahrabi, August 2015
After several decades of development, higher-order finite-element methods are now being considered for realistic and large scale Computational Fluid Dynamics (CFD) simulations. This necessitates further studies on utilization of mesh adaptation techniques in order to reach reliable solutions at minimal computation cost. In this study, adaptation capabilities have been developed within a Petrov-Galerkin (PG) finite-element method. The mesh modification mechanisms include h-, p-, and combined hp-adaptations which are performed in a non-conforming manner. The constrained approximation method has been utilized in order to retain the continuity of the solution space in presence of hanging nodes. Hierarchical basis functions have been employed to facilitate the implementation of the constrained approximation method. The adaptive methodology has been demonstrated on numerous cases using the Euler and Reynolds Average Navier-Stokes (RANS) equations, equipped with a modified Spalart-Allmaras (SA) turbulence model. Also, a PDE-based artificial viscosity has been added to the governing equations, to stabilize the solution in the vicinity of shock waves. For accurate representation of the geometric surfaces, high-order curved boundary meshes have been generated and the interior meshes have been deformed through the solution of a modified linear elasticity equation. A fully implicit linearization has been utilized within a Newton-type algorithm to advance each iteration or time-step, for steady-state or unsteady simulations, respectively. In order to navigate the adaptation process, adjoint-based and feature-based techniques have been employed in the steady-state and unsteady problems, respectively. It was shown that weak implementation of the boundary conditions and the use of a modified functional are required to obtain a smooth adjoint solution where Dirichlet Boundary conditions are imposed. Failure to utilize both results in a non-smooth adjoint solution. To accelerate the error reduction, an enhanced h-refinement has been used in the vicinity of singularity points. Several numerical results illustrate consistent accuracy improvement of the functional outputs and capability enhancements in resolving complex viscous flow features such as shock boundary layer interaction, flow separation, and vortex shedding.
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