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Grid adaptation for functional outputs: application to twodimensional inviscid flows
 J. Comput. Phys
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A triangular cutcell adaptive method for highorder discretizations of the compressible Navier–Stokes equations
, 2007
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Progress in adjoint error correction for integral functionals
 COMPUTING AND VISUALIZATION IN SCIENCE
, 2004
"... 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. an ..."
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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.
OutputBased Error Estimation and Mesh Adaptation in Computational Fluid Dynamics: Overview and Recent Results
"... Error estimation an 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 the ..."
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Cited by 5 (1 self)
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Error estimation an 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 (CFD) 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 fullydiscrete 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 output error estimation using the finite volume method and the finite element method. Recent adaptive results from a variety of applications show the power of outputbased adaptive methods for improving the robustness of CFD computations. However, challenges and areas of additional future research remain, including computable error bounds and robust mesh adaptation mechanics. I
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 3 (1 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.
An Automated Reliable Method for TwoDimensional Reynoldsaveraged NavierStokes Simulations
, 2011
"... development of computational fluid dynamics algorithms and increased computational resources have led to the ability to perform complex aerodynamic simulations. Obstacles remain which prevent autonomous and reliable simulations at accuracy levels required for engineering. To consider the solution st ..."
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Cited by 2 (0 self)
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development of computational fluid dynamics algorithms and increased computational resources have led to the ability to perform complex aerodynamic simulations. Obstacles remain which prevent autonomous and reliable simulations at accuracy levels required for engineering. To consider the solution strategy autonomous and reliable, high quality solutions must be provided without user interaction or detailed previous knowledge about the flow to facilitate either adaptation or solver robustness. One such solution strategy is presented for
A HighOrder, Adaptive, Discontinuous Galerkin Finite . . .
, 2008
"... This thesis presents highorder, discontinuous Galerkin (DG) discretizations of the ReynoldsAveraged NavierStokes (RANS) equations and an outputbased error estimation and mesh adaptation algorithm for these discretizations. In particular, DG discretizations of the RANS equations with the Spalart ..."
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This thesis presents highorder, discontinuous Galerkin (DG) discretizations of the ReynoldsAveraged NavierStokes (RANS) equations and an outputbased error estimation and mesh adaptation algorithm for these discretizations. In particular, DG discretizations of the RANS equations with the SpalartAllmaras (SA) turbulence model are examined. The dual consistency of multiple DG discretizations of the RANSSA system is analyzed. The approach of simply weighting gradient dependent source terms by a test function and integrating is shown to be dual inconsistent. A dual consistency correction for this discretization is derived. The analysis also demonstrates that discretizations based on the popular mixed formulation, where dependence on the state gradient is handled by introducing additional
Moving mesh adaptation scheme for aerodynamic shape optimization
"... A method of mesh adaptation is proposed for gradientbased aerodynamic shape optimization. The method consists in coupling an equation for the mesh node coordinates with the discretized Euler equations of gas dynamics in steady state. The variational mesh equation is inspired by Winslow’s variable d ..."
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A method of mesh adaptation is proposed for gradientbased aerodynamic shape optimization. The method consists in coupling an equation for the mesh node coordinates with the discretized Euler equations of gas dynamics in steady state. The variational mesh equation is inspired by Winslow’s variable diffusion mapping. The system of mesh and flow equations is solved, instead of the flow equations alone, when performing shape optimization. The solution algorithm of the coupled equations is an approximate Newton method supplemented with an interpolation of the variable diffusivity by radial basis functions. Tests are carried out for supersonic flow over a wedge, a problem that is used here as a benchmark for the mesh adaptation and for a simple problem of inverse design. At a given design, the method of adaptation improves the accuracy of the calculated drag, a functional that is used in the construction of the inverse problem. The accuracy of the shape, obtained by inverse design, experiences similar improvements due to the mesh adaptation scheme. Key words: gradient optimization, inviscid compressible flow, variable diffusion mapping, mediandual finitevolume, radial basis functions, adjoint equations, interpolation, mesh adaptation, shape optimization 1991 MSC: 90C06, 90C90, 93C20, 76M12 1
and Direct Mesh Adaptation for Output Error
, 2009
"... including angle sizes and aspect ratios, on accuracy and stiffness is investigated for simulations of highly anisotropic problems. The results indicate that for highorder discretizations, large angles do not have an adverse impact on solution accuracy. However, a correct aspect ratio is critical fo ..."
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including angle sizes and aspect ratios, on accuracy and stiffness is investigated for simulations of highly anisotropic problems. The results indicate that for highorder discretizations, large angles do not have an adverse impact on solution accuracy. However, a correct aspect ratio is critical for accuracy for both linear and highorder discretizations. In addition, large angles are not problematic for the conditioning of the linear systems arising from discretization. They can be overcome through small increases in preconditioning costs. A direct adaptation scheme that controls the output error via mesh operations and mesh smoothing is also developed. The decision of mesh operations is solely based on output error distribution without any a priori assumption on error convergence rate. Anisotropy is introduced by evaluating the error changes due to potential edge split, and thus the anisotropies of both primal and dual solutions are taken into account. This scheme is demonstrated to produce grids with fewer degrees of freedom for a