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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 29 (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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Cited by 19 (11 self)
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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 15 (7 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
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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Cited by 14 (3 self)
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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.
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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Cited by 13 (3 self)
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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 12 (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.
An Adaptive Simplex CutCell Method for Discontinuous Galerkin
 Discretizations of the NavierStokes Equations,” AIAA Paper
, 2007
"... A cutcell adaptive method is presented for highorder 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 w ..."
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Cited by 11 (6 self)
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A cutcell adaptive method is presented for highorder 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. Highorder integration rules are derived for the arbitrarilyshaped areas and volumes that result from the cutting. These rules take the form of quadraturelike points and weights that are calculated in a preprocessing step. Accuracy of the cutcell method is verified in both two and three dimensions by comparison to boundaryconforming cases. The cutcell method is also tested in the context of outputbased adaptation, in which an adjoint problem is solved to estimate the error in an engineering output. Twodimensional adaptive results for the compressible NavierStokes equations illustrate automated anisotropic adaptation made possible by triangular cutcell 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 threedimensional results indicate that, for the cases tested, p = 2 and p = 3 solution approximation achieves the userprescribed error tolerance more efficiently compared to p = 1 and p = 0. I.