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Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG1-03035). Secondly, the effort put into Project X by faculty and students (past and present) have made it possible to carry out the computational demonstrations in higher-order DG. In particular, Krzysztof Fidkowski and Todd Oliver are to be acknowledged for their contributions towards the development of the flow solvers and also for providing some of the grids for the test cases demonstrated. Finally, thanks must go to thesis committee members Professors Peraire and Willcox as well as thesis readers Dr. Natalia Alexandrov and Dr. Steven Allmaras for the time they put into reading the thesis and providing the valuable feedbacks. 3 46 Adjoint approach to shape sensitivity 117 6.1 Introduction...............................

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Engineering problems commonly require functional outputs of computational fluid dynamics (CFD) simulations with specified accuracy. These simulations are performed with limited computational resources. Computable error estimates offer the possibility of quantifying accuracy on a given mesh and predicting a fine grid functional on a coarser mesh. Such an estimate can be computed by solving the flow equations and the associated adjoint problem for the functional of interest. An adjoint-based error correction procedure is demonstrated for transonic inviscid and subsonic laminar and turbulent flow. A mesh adaptation procedure is formulated to target uncertainty in the corrected functional and terminate when error remaining in the calculation is less than a user-specified error tolerance. This adaptation scheme is shown to yield anisotropic meshes with corrected functionals that are more accurate for a given number of grid points then isotropic adapted and uniformly refined grids.

This report presents a mesh adaptation method for higher-order (p> 1) discontinuous Galerkin (DG) discretizations of the two-dimensional, compressible Navier-Stokes equations. The method uses a mesh of triangular elements that are not required to conform to the boundary. This triangular, cut-cell approach permits anisotropic adaptation without the difficulty of constructing meshes that conform to potentially complex geometries. A quadrature technique is presented for accurately integrating on general cut cells. In addition, an output-based error estimator and adaptive method are presented, with emphasis on appropriately accounting for high-order solution spaces in optimizing local mesh anisotropy. Accuracy on cut-cell meshes is demonstrated by comparing solutions to those on standard boundary-conforming meshes. Adaptation results show that, for all test cases considered, p = 2 and p = 3 discretizations meet desired error tolerances using fewer degrees of freedom than p = 1. Furthermore, an initial-mesh dependence study demonstrates that, for sufficiently low error tolerances, the

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A discrete-adjoint formulation is presented for the three-dimensional Euler equations discretized on a Cartesian mesh with embedded boundaries. The solution algorithm for the adjoint and flow-sensitivity equations leverages the Runge–Kutta time-marching scheme in conjunction with the parallel multigrid method of the flow solver. The matrix-vector products associated with the linearization of the flow equations are computed on-the-fly, thereby minimizing the memory requirements of the algorithm at a computational cost roughly equivalent to a flow solution. Three-dimensional test cases, including a wing-body geometry at transonic flow conditions and an entry vehicle at supersonic flow conditions, are presented. These cases verify the accuracy of the linearization and demonstrate the efficiency and robustness of the adjoint algorithm for complex-geometry problems.

A cut-cell adaptive method is presented for high-order discontinuous Galerkin discretizations in two and three dimensions. The computational mesh is constructed by cutting a curved geometry out of a simplex background mesh that does not conform to the geometry boundary. The geometry is represented with cubic splines in two dimensions and with a tesselation of quadratic patches in three dimensions. High-order integration rules are derived for the arbitrarily-shaped areas and volumes that result from the cutting. These rules take the form of quadrature-like points and weights that are calculated in a pre-processing step. Accuracy of the cut-cell method is verified in both two and three dimensions by comparison to boundary-conforming cases. The cut-cell method is also tested in the context of output-based adaptation, in which an adjoint problem is solved to estimate the error in an engineering output. Two-dimensional adaptive results for the compressible Navier-Stokes equations illustrate automated anisotropic adaptation made possible by triangular cut-cell meshing. In three dimensions, adaptive results for the compressible Euler equations using isotropic refinement demonstrate the feasibility of automated meshing with tetrahedral cut cells and a curved geometry representation. In addition, both the two and three-dimensional results indicate that, for the cases tested, p = 2 and p = 3 solution approximation achieves the user-prescribed error tolerance more efficiently compared to p = 1 and p = 0. I.

When approximating the solutions of partial differential equations, it is a few key output integrals which are often of most concern. This paper shows how the accuracy of these values can be improved through a correction term which is an inner product of the residual error in the original p.d.e. and the solution of an appropriately defined adjoint p.d.e. A number of applications are presented and the challenges of smooth reconstruction on unstructured grids and error correction for shocks are discussed.