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Dynamic Mesh Deformation for Adaptive Grid Refinement with Staggered Spectral Difference and Finite Volume Mesh

Better computational efficiency and reduced computational c for high order CFD method are areas of active research. In this study, we investigate the potential of using dynamic deforming mesh to locally refine a region of interest in the flow domain. With given number of nodes or degree of freedoms, this has the advantage of achieving an optimal distribution of mesh points, concentrating the computational resource on areas of high gradients, and leaving areas of low gradient with less points. This can be very useful for high order finite element methods that usually have difficulties capturing clean shocks due to the large size of the mesh element, despite the large number of solution and flux poitns within the element. With deforming mesh, the mesh elements can be deformed and gathered near solution with steep gradient, hence leading to high resolution solution of discontinuity. In low gradient regions, despite the diminishing cells, the high order methods are very efficient at resolving the flow with small number of mesh cells. Traditional computational fluid dynamics (CFD) methods are very efficient, robust and reliable, proven by years of industrial applications, but in general limited to second order accuracy. High order finite element methods such as Discontinuous Galerkin and its other variants have somewhat the opposite characteristics at the current stage of development, i.e. the order of accuracy can be arbitrarily high leading to mininum dissipassive methods, while on the other hand quests to improve their robustness, reliablility, stability, timestep restriction, and computational efficiency are still areas of ongoing research. In this paper, we propose to improve the computational efficiency of the high order spectral difference (SD) method by formulating the dynamic mesh deformation problem as an optimization problem that optimally redistributes the mesh points according to flow solution. We also employ a staggered CFD solver structure that combines the traditional finite volume method and the high order method, allowing us to take advantages of the efficiency and the flexibility of the finite volume mesh, and the high order of accuracy of the high order method, leading to a flexible and accurate hybrid method. In short, the finite volume method will be augmented by higher accuracy; the high order method will be augmented by greater flexibility and efficiency. This sets the theme for the paper.