Eigenmode Analysis of Boundary Conditions for the One-Dimensional Preconditioned Euler Equations

The effect of local preconditioning on boundary conditions is analyzed for the subsonic, one-dimensional Euler equations. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions and different preconditioners whose intent is to accelerate low Mach number computations. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective when used with preconditioning, and asymptotically, at low Mach numbers, initial disturbances do not decay. Other boundary conditions are shown to be perfectly non-reflective in conjunction with preconditioning. Two-dimensional numerical results confirm the trends predicted by the one-dimensional analysis

Eigenmode Analysis of Boundary Conditions for One-Dimensional Preconditioned Euler Equations

An analysis of the effect of local preconditioning on boundary conditions for the subsonic, one-dimensional Euler equations is presented. Decay rates for the eigenmodes of the initial boundary value problem are determined for different boundary conditions. Riemann invariant boundary conditions based on the unpreconditioned Euler equations are shown to be reflective with preconditioning, and, at low Mach numbers, disturbances do not decay. Other boundary conditions are investigated which are non-reflective with preconditioning and numerical results are presented confirming the analysis

Anisotropic Grid Adaptation for Multiple Aerodynamic Outputs

Anisotropic grid–adaptive strategies are presented for viscous flow simulations in which the accurate prediction of multiple aerodynamic outputs (such as the lift, drag, and moment coefficients) is required from a single adaptive solution. The underlying adaptive procedure is based on a merging of adjoint error estimation and Hessian-based anisotropic grid adaptation. Airfoil test cases are presented to demonstrate the various adaptive strategies including a single element airfoil at cruise conditions and a multi-element airfoil in high-lift configuration with flow separation. Numerical results indicate that the lift, drag and moment coefficients are accurately predicted by all of the output–based strategies considered, although slightly better accuracy is obtained in the output(s) for which a particular strategy is specifically designed. Furthermore, the output-based strategies are all shown to be significantly more efficient than pure Hessian-based adaptation in terms of output accuracy for a given grid size

On Dual-Weighted Residual Error Estimates for p-Dependent Discretizations

This report analyzes the behavior of three variants of the dual-weighted residual (DWR) error estimates applied to the p-dependent discretization that results from the BR2 discretization of a second-order PDE. Three error estimates are assessed using two metrics: local effectivities and global effectivity. A priori error analysis is carried out to study the convergence behavior of the local and global effectivities of the three estimates. Numerical results verify the a priori error analysis

Output-based Adaptive Meshing Using Triangular Cut Cells

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 final adapted mesh is relatively insensitive to the starting mesh

Analysis of Dual Consistency for Discontinuous Galerkin Discretizations of Source Terms

The effects of dual consistency on discontinuous Galerkin (DG) discretizations of solution and solution gradient dependent source terms are examined. Two common discretizations are analyzed: the standard weighting technique for source terms and the mixed formulation. It is shown that if the source term depends on the first derivative of the solution, the standard weighting technique leads to a dual inconsistent scheme. A straightforward procedure for correcting this dual inconsistency and arriving at a dual consistent discretization is demonstrated. The mixed formulation, where the solution gradient in the source term is replaced by an additional variable that is solved for simultaneously with the state, leads to an asymptotically dual consistent discretization. A priori error estimates are derived to reveal the effect of dual inconsistent discretization on computed functional outputs. Combined with bounds on the dual consistency error, these estimates show that for a dual consistent discretization or the asymptotically dual consistent discretization resulting from the mixed formulation, O(h2p) convergence can be shown for linear problems and linear outputs. For similar but dual inconsistent schemes, only O(hp) can be shown. Numerical results for a one-dimensional test problem confirm that the dual consistent and asymptotically dual consistent schemes achieve higher asymptotic convergence rates with grid refinement than a similar dual inconsistent scheme for both the primal and adjoint solutions as well as a simple functional output.This work was supported by the U. S. Air Force Research Laboratory (USAF-3306-03-SC-0001) and The Boeing Company

Review of Output-Based Error Estimation and Mesh Adaptation in Computational Fluid Dynamics

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/90641/1/AIAA-53965-537.pd

Output-Adaptive Tetrahedral Cut-Cell Validation for Sonic Boom Prediction

A cut-cell approach to Computational Fluid Dynamics (CFD) that utilizes the median dual of a tetrahedral background grid is described. The discrete adjoint is also calculated, which permits adaptation based on improving the calculation of a specified output (off-body pressure signature) in supersonic inviscid flow. These predicted signatures are compared to wind tunnel measurements on and off the configuration centerline 10 body lengths below the model to validate the method for sonic boom prediction. Accurate mid-field sonic boom pressure signatures are calculated with the Euler equations without the use of hybrid grid or signature propagation methods. Highly-refined, shock-aligned anisotropic grids were produced by this method from coarse isotropic grids created without prior knowledge of shock locations. A heuristic reconstruction limiter provided stable flow and adjoint solution schemes while producing similar signatures to Barth-Jespersen and Venkatakrishnan limiters. The use of cut-cells with an output-based adaptive scheme completely automated this accurate prediction capability after a triangular mesh is generated for the cut surface. This automation drastically reduces the manual intervention required by existing methods

Hierarchal visualization of three-dimensional vortical flow calculations

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1991.Includes bibliographical references (leaves 113-115).by David L. Darmofal.M.S