Towards Automated, Metric-Conforming, Mesh Optimization For High-Order, Finite-Element Methods

High-resolution flow simulations are becoming important for increasingly many engineering applications. While high-resolution simulations give us the accuracy needed for engineering purposes, they generally demand high computational power and suitable meshes. Generating high-quality meshes itself is by no means a trivial problem. This dissertation presents a meshing algorithm that aims to eliminate the reliance on user’s experience, which has constantly compromised accuracy, created robustness issues, and caused difficulties in producing high-quality meshes. The novelty of our meshing algorithm lies in the optimal placement of the high-order geometry nodes within an element for more accurate and robust computational fluid dynamics (CFD) simulations. We first introduce a new mesh refinement method, called warped-element refinement (q adaptation), in which the high-order geometry nodes are automatically placed at their optimal locations given fixed locations of an element’s primary vertices. The benefits of this refinement method are demonstrated in an adaptive framework with a high-order, discontinuous Galerkin discretization and an a posteriori output error estimate. For fluid flows governed by the Euler, Navier-Stokes, and Reynolds-Averaged Navier-Stokes equations, for example, the drag output error can be reduced by at least a factor of two compared to the initial h-adapted mesh without adding much computational cost and software complexity. The optimal placement of the high-order geometry nodes also opens up the possibility of using coarser meshes with a high-order discretization. In order to better couple the optimal placement of an element’s vertices and its high- order geometry nodes, we turn to a Riemannian metric field to encode the mesh resolution. The three key steps in this process are: 1) obtain the desired higher-order, Riemannian metric field, 2) generate a linear, metric-conforming mesh, and 3) place the high-order geometry nodes such that the mesh-implied metric conforms to the desired metric field. We propose two methods to obtain a higher-order metric field and explore several metric-based mesh generation procedures for one-dimensional problems. Preliminary two-dimensional results further emphasize the importance of the high-order geometry nodes. These findings serve as fundamental knowledge for building a high-order, metric-conforming mesh generator in the future.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/151619/1/dsanjaya_1.pd