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31
Tetrahedral Embedded Boundary Methods for Accurate and Flexible Adaptive Fluids
"... When simulating fluids, tetrahedral methods provide flexibility and ease of adaptivity that Cartesian grids find difficult to match. However, this approach has so far been limited by two conflicting requirements. First, accurate simulation requires quality Delaunay meshes and the use of circumcentri ..."
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Cited by 17 (2 self)
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When simulating fluids, tetrahedral methods provide flexibility and ease of adaptivity that Cartesian grids find difficult to match. However, this approach has so far been limited by two conflicting requirements. First, accurate simulation requires quality Delaunay meshes and the use of circumcentric pressures. Second, meshes must align with potentially complex moving surfaces and boundaries, necessitating continuous remeshing. Unfortunately, sacrificing mesh quality in favour of speed yields inaccurate velocities and simulation artifacts. We describe how to eliminate the boundarymatching constraint by adapting recent embedded boundary techniques to tetrahedra, so that neither air nor solid boundaries need to align with mesh geometry. This enables the use of high quality, arbitrarily graded, nonconforming Delaunay meshes, which are simpler and faster to generate. Temporal coherence can also be exploited by reusing meshes over adjacent timesteps to further reduce meshing costs. Lastly, our free surface boundary condition eliminates the spurious currents that previous methods exhibited for slow or static scenarios. We provide several examples demonstrating that our efficient tetrahedral embedded boundary method can substantially increase the flexibility and accuracy of adaptive Eulerian fluid simulation.
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 15 (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.
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.
An Optimization Framework for Anisotropic Simplex Mesh Adaptation: Application to Aerodynamic Flows
"... We apply an optimizationbased framework for anisotropic simplex mesh adaptation to highorder discontinuous Galerkin discretizations of twodimensional, steadystate aerodynamic flows. The framework iterates toward a mesh that minimizes the output error for a given number of degrees of freedom by c ..."
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Cited by 10 (7 self)
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We apply an optimizationbased framework for anisotropic simplex mesh adaptation to highorder discontinuous Galerkin discretizations of twodimensional, steadystate aerodynamic flows. The framework iterates toward a mesh that minimizes the output error for a given number of degrees of freedom by considering a continuous optimization problem of the Riemannian metric field. The adaptation procedure consists of three key steps: sampling of the anisotropic error behavior using elementwise local solves; synthesis of the local errors to construct a surrogate error model in the metric space; and optimization of the surrogate model to drive the mesh toward optimality. The anisotropic adaptation decisions are entirely driven by the behavior of the a posteriori error estimate without making a priori assumptions about the solution behavior. As a result, the method handles any discretization order, naturally incorporates both the primal and adjoint solution behaviors, and robustly treats irregular features. The numerical results demonstrate that the proposed method is at least as competitive as the previous method that relies on a priori assumption of the solution behavior, and, in many cases, outperforms the previous method by over an order of magnitude in terms of the output accuracy for a given number of degrees of freedom. I.
The importance of mesh adaptation for higherorder discretizations of aerodynamic flows
, 2011
"... This work presents an adaptive framework that realizes the true potential of a higherorder discretization of the Reynoldsaveraged NavierStokes (RANS) equations. The framework is based on an outputbased error estimate and explicit degree of freedom control. Adaptation works toward the generation o ..."
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Cited by 10 (6 self)
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This work presents an adaptive framework that realizes the true potential of a higherorder discretization of the Reynoldsaveraged NavierStokes (RANS) equations. The framework is based on an outputbased error estimate and explicit degree of freedom control. Adaptation works toward the generation of meshes that equidistribute local errors and provide anisotropic resolution aligned with solution features in arbitrary orientations. Numerical experiments reveal that uniform refinement limits the performance of higherorder methods when applied to aerodynamic flows with low regularity. However, when combined with aggressive anisotropic refinement of singular features, higherorder methods can significantly improve computational affordability of RANS simulations in the engineering environment. The benefit of the higher spatial accuracy is exhibited for a wide range of applications, including subsonic, transonic, and supersonic flows. The higherorder meshes are generated using the elasticity and the cutcell techniques, and the competitiveness of the cutcell method in terms of accuracy per degree of freedom is demonstrated.
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 7 (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
An Unsteady Adaptation Algorithm for Discontinuous Galerkin Discretizations of the RANS Equations
"... An adaptive method for highorder discretizations of the Reynoldsaveraged NavierStokes (RANS) equations is examined. The RANS equations and SpalartAllmaras (SA) turbulence model are discretized with a dual consistent, discontinuous Galerkin discretization. To avoid oscillations in the solution in ..."
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Cited by 6 (4 self)
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An adaptive method for highorder discretizations of the Reynoldsaveraged NavierStokes (RANS) equations is examined. The RANS equations and SpalartAllmaras (SA) turbulence model are discretized with a dual consistent, discontinuous Galerkin discretization. To avoid oscillations in the solution in underresolved regions, particularly the edge of the boundary layer, artificial dissipation is added to the SA model equation. Two adaptive procedures are examined: a standard outputbased adaptation algorithm that requires the steady state solution to estimate the error and a new, unsteady approach that allows the mesh to be adapted without requiring a steady state solution. Results show that the combination of a dual consistent discretization with artificial dissipation and adaptation has significant promise as a practical method for obtaining highorder RANS solutions. I.
A parallel NewtonKrylov flow solver for the three dimensional Reynoldsaveraged NavierStokes equations
 In 20th Annual Conference CFD Society of
, 2012
"... This work presents a parallel NewtonKrylov flow solver employing third and fourthorder spatial discretizations to solve the threedimensional Euler equations on structured multiblock meshes. The fluxes are discretized using summationbyparts operators; boundary and interface conditions are impl ..."
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Cited by 6 (3 self)
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This work presents a parallel NewtonKrylov flow solver employing third and fourthorder spatial discretizations to solve the threedimensional Euler equations on structured multiblock meshes. The fluxes are discretized using summationbyparts operators; boundary and interface conditions are implemented using simultaneous approximation terms. Functionals, drag and lift, are calculated using Simpson’s rule. The solver is verified using the method of manufactured solutions and Ringleb flow and validated using the ONERA M6 wing. The results demonstrate that the combination of highorder finitedifference operators with a parallel NewtonKrylov solution technique is an excellent option for efficient computation of aerodynamic flows. I.
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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Cited by 5 (0 self)
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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
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