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A triangular cutcell adaptive method for highorder discretizations of the compressible Navier–Stokes equations
, 2007
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An a posteriori error control framework for adaptive precision optimization using discontinuous Galerkin finite element method
, 2005
"... Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG103035). 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 higherorder DG. In particular, Krzysztof Fidkowsk ..."
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Cited by 25 (0 self)
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Professor Darmofal and the generous funding provided by NASA Langley (grant number NAG103035). 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 higherorder 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...............................
Symmetric interior penalty DG methods for the compressible Navier{Stokes equations II: Goal{oriented a posteriori error estimation
 In preparation
"... Abstract. In this article we consider the development of discontinuous Galerkin nite element methods for the numerical approximation of the compressible Navier{Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the ..."
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Abstract. In this article we consider the development of discontinuous Galerkin nite element methods for the numerical approximation of the compressible Navier{Stokes equations. For the discretization of the leading order terms, we propose employing the generalization of the symmetric version of the interior penalty method, originally developed for the numerical approximation of linear selfadjoint second{order elliptic partial dierential equations. In order to solve the resulting system of nonlinear equations, we exploit a (damped) Newton{GMRES algorithm. Numerical experiments demonstrating the practical performance of the proposed discontinuous Galerkin method with higher{order polynomials are presented. Key words. Finite element methods, discontinuous Galerkin methods, compressible Navier{ Stokes equations 1.
Anisotropic mesh adaptation in computational fluid dynamics: Application to the advection–diffusion–reaction and the Stokes problems
, 2004
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An Exact Dual Adjoint Solution Method for Turbulent Flows on Unstructured Grids
 Computers & Fluids
, 2003
"... this report is for accurate reporting and does not constitute an official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics and Space Administration ..."
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Cited by 13 (6 self)
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this report is for accurate reporting and does not constitute an official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics and Space Administration
Analysis of dual consistency for discontinuous Galerkin discretizations of source terms
 M.I.T. Aerospace Computational Design Laboratory
, 2006
"... 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 t ..."
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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 onedimensional 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. 1
Adjoint Formulation for an EmbeddedBoundary Cartesian Method
, 2005
"... A discreteadjoint formulation is presented for the threedimensional Euler equations discretized on a Cartesian mesh with embedded boundaries. The solution algorithm for the adjoint and flowsensitivity equations leverages the Runge–Kutta timemarching scheme in conjunction with the parallel multig ..."
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Cited by 11 (6 self)
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A discreteadjoint formulation is presented for the threedimensional Euler equations discretized on a Cartesian mesh with embedded boundaries. The solution algorithm for the adjoint and flowsensitivity equations leverages the Runge–Kutta timemarching scheme in conjunction with the parallel multigrid method of the flow solver. The matrixvector products associated with the linearization of the flow equations are computed onthefly, thereby minimizing the memory requirements of the algorithm at a computational cost roughly equivalent to a flow solution. Threedimensional test cases, including a wingbody 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 complexgeometry problems.
Review of OutputBased Error Estimation and Mesh Adaptation in Computational Fluid Dynamics
"... Error estimation and control are critical ingredients for improving the reliability of computational simulations. Adjointbased techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement. This article reviews recent work on th ..."
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Error estimation and control are critical ingredients for improving the reliability of computational simulations. Adjointbased techniques can be used to both estimate the error in chosen solution outputs and to provide local indicators for adaptive refinement. This article reviews recent work on these techniques for computational fluid dynamics applications in aerospace engineering. The definition of the adjoint as the sensitivity of an output to residual source perturbations is used to derive both the adjoint equation, in fully discrete and variational formulations, and the adjointweighted residual method for error estimation. Assumptions and approximations made in the calculations are discussed. Presentation of the discrete and variational formulations enables a sidebyside comparison of recent work in outputerror estimation using the finite volume method and the finite element method. Techniques for adapting meshes using outputerror indicators are also reviewed. Recent adaptive results from a variety of laminar and Reynoldsaveraged Navier–Stokes applications show the power of outputbased adaptive methods for improving the robustness of computational fluid dynamics computations. However, challenges and areas of additional future research remain, including computable error bounds and robust mesh adaptation mechanics. I.
Outputbased adaptive meshing using triangular cutcells
 M.I.T. Aerospace Computational Design Laboratory
, 2006
"... This report presents a mesh adaptation method for higherorder (p> 1) discontinuous Galerkin (DG) discretizations of the twodimensional, compressible NavierStokes equations. The method uses a mesh of triangular elements that are not required to conform to the boundary. This triangular, cutcell ..."
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This report presents a mesh adaptation method for higherorder (p> 1) discontinuous Galerkin (DG) discretizations of the twodimensional, compressible NavierStokes equations. The method uses a mesh of triangular elements that are not required to conform to the boundary. This triangular, cutcell 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 outputbased error estimator and adaptive method are presented, with emphasis on appropriately accounting for highorder solution spaces in optimizing local mesh anisotropy. Accuracy on cutcell meshes is demonstrated by comparing solutions to those on standard boundaryconforming 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 initialmesh dependence study demonstrates that, for sufficiently low error tolerances, the