Adaptive Methods for High-Order Aerodynamic Shape Optimization

Abstract

Advancements in numerical simulations and computational power have drastically changed the engineering design process. For engineering systems that involve fluid flows, computational fluid dynamics (CFD) has become a powerful tool that allows engineers to rapidly analyze different designs. Coupling CFD with numerical design optimization provides further benefits by automating the design process. The optimizer relies heavily on a robust, accurate, and efficient computational fluid dynamics (CFD) solver. To meet the accuracy and efficiency requirements, this work focuses on high-order CFD methods; these methods have the potential to provide high accuracy at a lower cost than low-order methods. However, the benefits of high-order methods can only be obtained if the computational mesh is optimal and curved.

Without optimal curved meshes, these high-order methods face robustness issues and become less efficient. Today, curved-mesh adaptation is one of the main barriers to the widespread adoption of high-order CFD methods in engineering design optimization. Further- more, coupling high-order methods and mesh adaptation with design optimization poses many additional challenges such as deciding when and how much to adapt the mesh. The main goal of this thesis is to enable the use of high-order methods, in particular the discontinuous Galerkin methods (DG), in design optimization.

I start by first developing a strategy for coupling p-adaptation with design optimization where the convergence of the error is set to match the convergence of the optimizer. I perform a comprehensive study comparing using adaptive DG and the traditional second-order finite volume method on a fixed mesh for design optimization. The results show that adaptive DG takes more time but achieves a better optimum and with significantly fewer degrees of freedom than fixed-mesh finite volume.

To address the challenges of anisotropic curved-mesh adaptation, I develop High-Order Edge Primitive (HOEP). HOEP performs metric-based anisotropic mesh adaptation natively on curved meshes. That is, the mesh stays curved during the adaptation process and HOEP guarantees a valid curved mesh at the end. This is contrary to traditional methods for curved mesh adaptation that adapt a linear representation of the mesh and curve the final linear mesh. HOEP provides the much needed robustness at a cost similar to the current mesh curving practices. Results for a high Reynolds number test cases that require highly anisotropic boundary layer meshes show that HOEP has comparable computational efficiency and accuracy of drag prediction to existing linear mesh adaptation with recurving. These test cases highlight the improved robustness of curved-mesh adaptation as a result of eliminating the need to recurve the mesh. From a design optimization perspective, the improved robustness eliminates the previously needed user intervention due to mesh failures.

Finally, I use HOEP with design optimization and develop a new adaptation strategy that balances the additional cost of iterative mesh adaptation. The strategy I developed ensures that close to the optimum when the optimizer takes small steps the error is within a target limit and triggers adaptation the error is not met. However, small steps from the optimum are allowed and do not trigger adaptation, preventing over-refining non-optimal designs. Results for transonic airfoil optimization show that with HOEP, DG design optimizations are now more robust and computationally efficient than second-order finite volume.