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Christopher J. Roy - One of the best experts on this subject based on the ideXlab platform.
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Truncation Error Based Mesh Optimization
Journal of Verification Validation and Uncertainty Quantification, 2020Co-Authors: Charles W Jackson, Christopher J. Roy, Christopher R SchrockAbstract:Abstract Truncation Error is used to drive mesh adaptation in order to reduce the discretization Error in solutions to a variety of 1D and 2D flow problems. The adaptation is performed using r-adaptation to move the mesh nodes in the domain in an attempt to reduce the Truncation Error since it is the local source of discretization Error. This work introduces a new optimization process to perform the r-adaptation along with several sets of design variables. This work compares these new optimization techniques to an equidistribution process. Some observations on the performance and behavior of the different adaptation methods and best practices for their use are presented. All of the optimization methods are shown to reduce the Truncation Error one to two orders of magnitude and the discretization Error by roughly one order of magnitude for all of the 1D problems. In two dimensions the optimization-based adaptation is able to reduce the DE by up to a factor of seven. Equidistribution achieved similar levels of improvement for much less cost.
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adjoint and Truncation Error based adaptation for finite volume schemes with Error estimates
53rd AIAA Aerospace Sciences Meeting, 2015Co-Authors: Joseph M. Derlaga, Christopher J. Roy, Tyrone S. Phillips, Jeff BorggaardAbstract:In addition to design, control, and optimization applications, adjoint methods can be used to provide discretization Error estimation and solution adaptation for solution functionals. In this paper, adaptation based on estimates of Truncation Error is coupled with adjoint-based Error estimation to provide improved estimates of discretization Error in solution functionals. Comparisons between different types of adaptation indicators and Error estimation techniques are made for the two-dimensional Euler equations.
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Structured Mesh r-Refinement using Truncation Error Equidistribution for 1D and 2D Euler Problems
21st AIAA Computational Fluid Dynamics Conference, 2013Co-Authors: Aniruddha Choudhary, Christopher J. RoyAbstract:An Error equidistribution-based r-refinement method for structured mesh adaption is implemented for 1D and 2D Euler equations. A subsonic diffuser case and a subsonic converging-diverging nozzle case are analyzed for 1D Euler equations while a 2D expansion fan case is analyzed for 2D Euler equations. Truncation Error is used as the criterion for performing mesh refinement. As the exact solutions are known for these problems, Truncation Error can be evaluated exactly and does not need to be estimated. Details and some nuances of the equidistribution process are provided for SAM (a newly developed structured adaption module). It is found that with a simple equidistribution approach it is possible to obtain about an order of magnitude reduction in discretization Error for these problems as compared to unadapted meshes.
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Finite Volume Solution Reconstruction Methods For Truncation Error Estimation
21st AIAA Computational Fluid Dynamics Conference, 2013Co-Authors: Tyrone S. Phillips, Christopher J. Roy, Joseph M. Derlaga, Jeff BorggaardAbstract:The numerical solution to differential equations results in a discrete solution space for the finite volume and finite difference discretization methods. For various reasons, it can be necessary to prolong the solution from a discrete space to a continuous space. The prolongation to a continuous space can be done using various curve-fitting methods which adds an additional level of approximation to the solution. The allowable Error of a prolongation operation depends on the specific task required by the user. In this paper we investigate various prolongation methods and identify the minimal requirements specifically for the purpose of Truncation Error estimation (the difference between the discrete and integral governing equations) for finite volume methods. The reconstruction methods investigated will include k-exact and ENO methods. Truncation Error estimation for 1D Burgers’ equation and the k-exact method suggest that the minimum polynomial order is dependent on the highest derivatives in the Truncation Error expression and, therefore, the discretization scheme. The effect of different reconstruction methods on Truncation Error estimation is investigated and the minimum polynomial order for accurate Truncation Error estimation is identified for the Euler equations and is found to be second-order for the weak formulation and third-order for the strong formulation.
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Adjoint and Truncation Error Based Adaptation for 1D Finite Volume Schemes
21st AIAA Computational Fluid Dynamics Conference, 2013Co-Authors: Joseph M. Derlaga, Christopher J. RoyAbstract:In addition to design, control, and optimization applications, adjoint methods can be used to provide discretization Error estimation and solution adaptation for solution functionals. This paper seeks to demonstrate the link between adjoint based Error estimation, Truncation Errors, residuals, and discretization Errors. The generalized Truncation Error expression is extended from finite difference schemes to finite volume schemes and is used to provide Truncation Error estimates for both 1D Burgers’ equation and the quasi-1D form of the Euler equations. Comparisons between different types of adaptation indicators and Error estimation techniques are made.
Eusebio Valero - One of the best experts on this subject based on the ideXlab platform.
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Truncation Error Estimation in the p-Anisotropic Discontinuous Galerkin Spectral Element Method
Journal of Scientific Computing, 2019Co-Authors: Andrés M. Rueda-ramírez, Gonzalo Rubio, Esteban Ferrer, Eusebio ValeroAbstract:In the context of discontinuous Galerkin spectral element methods (DGSEM), $$\tau $$ τ -estimation has been successfully used for p-adaptation algorithms. This method estimates the Truncation Error of representations with different polynomial orders using the solution on a reference mesh of relatively high order. In this paper, we present a novel anisotropic Truncation Error estimator derived from the $$\tau $$ τ -estimation procedure for the traditional DGSEM. We exploit the tensor product basis properties of the numerical solution to design a method where the total Truncation Error is calculated as a sum of its directional components. We show that the new Error estimator is cheaper to evaluate than previous implementations of the $$\tau $$ τ -estimation procedure and that it obtains more accurate extrapolations of the Truncation Error for representations of a higher order than the reference mesh. The robustness of the method allows performing the p-adaptation strategy with coarser reference solutions, thus further reducing the computational cost. The proposed estimator is validated using the method of manufactured solutions in a test case for the compressible Navier–Stokes equations.
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Quasi-A Priori Truncation Error Estimation in the DGSEM
Journal of Scientific Computing, 2014Co-Authors: Gonzalo Rubio, François Fraysse, David A. Kopriva, Eusebio ValeroAbstract:In this paper we show how to accurately perform a quasi-a priori estimation of the Truncation Error of steady-state solutions computed by a discontinuous Galerkin spectral element method. We estimate the spatial Truncation Error using the $$\tau $$?-estimation procedure. While most works in the literature rely on fully time-converged solutions on grids with different spacing to perform the estimation, we use non time-converged solutions on one grid with different polynomial orders. The quasi-a priori approach estimates the Error while the residual of the time-iterative method is not negligible. Furthermore, the method permits one to decouple the surface and the volume contributions of the Truncation Error, and provides information about the anisotropy of the solution as well as its rate of convergence in polynomial order. First, we focus on the analysis of one dimensional scalar conservation laws to examine the accuracy of the estimate. Then, we extend the analysis to two dimensional problems. We demonstrate that this quasi-a priori approach yields a spectrally accurate estimate of the Truncation Error.
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the estimation of Truncation Error by tau estimation for chebyshev spectral collocation method
Journal of Scientific Computing, 2013Co-Authors: Gonzalo Rubio, François Fraysse, J. De Vicente, Eusebio ValeroAbstract:In this paper we show how to accurately estimate the local Truncation Error of the Chebyshev spectral collocation method using $$\tau $$ -estimation. This method compares the residuals on a sequence of approximations with different polynomial orders. First, we focus the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the Truncation Error. Then, we show the validity of the analysis for the incompressible Navier---Stokes equations. First on the Kovasznay flow, where an analytical solution is known, and finally in the lid driven cavity (LDC). We demonstrate that this approach yields a highly accurate estimation of the Truncation Error if the precision of the approximations increases with the polynomial order.
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The Estimation of Truncation Error by $$\tau $$-Estimation for Chebyshev Spectral Collocation Method
Journal of Scientific Computing, 2013Co-Authors: Gonzalo Rubio, François Fraysse, J. De Vicente, Eusebio ValeroAbstract:In this paper we show how to accurately estimate the local Truncation Error of the Chebyshev spectral collocation method using $$\tau $$ -estimation. This method compares the residuals on a sequence of approximations with different polynomial orders. First, we focus the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the Truncation Error. Then, we show the validity of the analysis for the incompressible Navier---Stokes equations. First on the Kovasznay flow, where an analytical solution is known, and finally in the lid driven cavity (LDC). We demonstrate that this approach yields a highly accurate estimation of the Truncation Error if the precision of the approximations increases with the polynomial order.
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comparison of mesh adaptation using the adjoint methodology and Truncation Error estimates
AIAA Journal, 2012Co-Authors: François Fraysse, Eusebio Valero, Jorge PonsinAbstract:Mesh adaptation based on Error estimation has become a key technique to improve th eaccuracy o fcomputational-fluid-dynamics computations. The adjoint-based approach for Error estimation is one of the most promising techniques for computational-fluid-dynamics applications. Nevertheless, the level of implementation of this technique in the aeronautical industrial environment is still low because it is a computationally expensive method. In the present investigation, a new mesh refinement method based on estimation of Truncation Error is presented in the context of finite-volume discretization. The estimation method uses auxiliary coarser meshes to estimate the local Truncation Error, which can be used for driving an adaptation algorithm. The method is demonstrated in the context of two-dimensional NACA0012 and three-dimensional ONERA M6 wing inviscid flows, and the results are compared against the adjoint-based approach and physical sensors based on features of the flow field.
Joelle Caro - One of the best experts on this subject based on the ideXlab platform.
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sph Truncation Error in estimating a 3d derivative
International Journal for Numerical Methods in Engineering, 2011Co-Authors: Andrea Amicarelli, Jeanchristophe Marongiu, Francis Leboeuf, Julien Leduc, Magdalena Neuhauser, Le Fang, Joelle CaroAbstract:Following the procedure proposed in Quinlan et al. (Int. J. Numer. Meth. Engng. 2006; 66:2064–2085) for a 1D generic derivative, a 3D formulation of the Smoothed Particle Hydrodynamics (SPH) Truncation Error (eT) has been derived and validated. We have then underlined the differences between traditional SPH simulations, which are not consistent, and estimations using renormalization, a first-order consistency technique. The consistency order is here defined as the highest degree of a generic polynomial function, which can be exactly reproduced by an SPH approximation. Under the homogeneous conditions assumed in our analyses renormalization generally reduces the relative Truncation Error by 1 or 2 orders of magnitude, both at inner points and boundary locations. Due to renormalization the Error tends to a lowest constant value as the kernel support size (h) goes to zero, while in general with no consistency the Error behaves like 1/h. In contrast to formulations without any consistency estimations, using renormalization there is a weak dependence of the Error on the absolute value of the displacement of the particles from their volume barycentre (δ). In addition, for simulations with renormalization, the best choice for the kernel function seems to be the closest to Dirac's delta, while for the ones with no consistency, the preferences are altered. Furthermore, we observe that renormalization reduces the number of neighbors that are necessary to obtain a discretization Error that is negligible with respect to the integral Error. Copyright © 2011 John Wiley & Sons, Ltd.
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sph Truncation Error in estimating a 3d function
Computers & Fluids, 2011Co-Authors: Andrea Amicarelli, Jeanchristophe Marongiu, Francis Leboeuf, Julien Leduc, Joelle CaroAbstract:Abstract The SPH Truncation Error (eT) can be defined as the sum of the integral kernel and the particle approximation Error in Smoothed Particle Hydrodynamics modelling. Following the procedure proposed by Quinlan et al. [16] for a 1D generic derivative, we have derived an approximated 3D formulation of eT in reproducing a generic function. This kind of estimation is implemented in some SPH models in order to reproduce density or some transported scalars. Then a corresponding sensitivity analysis of eT has been performed adopting regular and irregular distributions of particles, arranged within a cube, delimited by lateral walls at each side. The evolution of eT has been analyzed and compared to the proposed formulation, which has been numerically estimated under some simple conditions. The SPH Truncation Error has then been investigated on a simple free surface test case: a supercritical flow over a channel sill. We have developed some conclusions about the dependence of eT on the position of the particles (inner or boundary), the shape of the function to be reproduced (f), the kernel support size (h), the particle volumes (ω), the kernel function (W), a non-dimensional distance between the volume barycentre and the particle location ( δ ), and a geometric anisotropy index of the particle volumes ( I ). We have finally underlined the difference between non-consistent simulations and estimations using Shepard’s correction.
Tyrone S. Phillips - One of the best experts on this subject based on the ideXlab platform.
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A Truncation Error Based Mesh Adaption Metric for CFD
54th AIAA Aerospace Sciences Meeting, 2016Co-Authors: Tyrone S. Phillips, Carl F. Ollivier GoochAbstract:Computational fluid dynamics has enourmous potential to influence the design and optimization of engineering systems; however, the Error due to the computational mesh (discretization Error) is often the largest source of numerical Error. Automatic mesh adaption can be used to generate an optimal mesh given a smooth indicator of Error. Truncation Error is the local source of discretization Error and has been shown to be a good adaption driver for structured grids [Roy, 2009]; however, the Truncation Error for general unstructured meshes is too noisy. A new method is developed that removes the excessive noise by interpolating the numerical solution to a smooth mesh matching only the control volume physical location and size. The smooth mesh Truncation Error is used to drive a mesh adaption process for two different solutions to Poisson’s equation and shows good initial results resolving peak Truncation Errors.
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adjoint and Truncation Error based adaptation for finite volume schemes with Error estimates
53rd AIAA Aerospace Sciences Meeting, 2015Co-Authors: Joseph M. Derlaga, Christopher J. Roy, Tyrone S. Phillips, Jeff BorggaardAbstract:In addition to design, control, and optimization applications, adjoint methods can be used to provide discretization Error estimation and solution adaptation for solution functionals. In this paper, adaptation based on estimates of Truncation Error is coupled with adjoint-based Error estimation to provide improved estimates of discretization Error in solution functionals. Comparisons between different types of adaptation indicators and Error estimation techniques are made for the two-dimensional Euler equations.
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Finite Volume Solution Reconstruction Methods For Truncation Error Estimation
21st AIAA Computational Fluid Dynamics Conference, 2013Co-Authors: Tyrone S. Phillips, Christopher J. Roy, Joseph M. Derlaga, Jeff BorggaardAbstract:The numerical solution to differential equations results in a discrete solution space for the finite volume and finite difference discretization methods. For various reasons, it can be necessary to prolong the solution from a discrete space to a continuous space. The prolongation to a continuous space can be done using various curve-fitting methods which adds an additional level of approximation to the solution. The allowable Error of a prolongation operation depends on the specific task required by the user. In this paper we investigate various prolongation methods and identify the minimal requirements specifically for the purpose of Truncation Error estimation (the difference between the discrete and integral governing equations) for finite volume methods. The reconstruction methods investigated will include k-exact and ENO methods. Truncation Error estimation for 1D Burgers’ equation and the k-exact method suggest that the minimum polynomial order is dependent on the highest derivatives in the Truncation Error expression and, therefore, the discretization scheme. The effect of different reconstruction methods on Truncation Error estimation is investigated and the minimum polynomial order for accurate Truncation Error estimation is identified for the Euler equations and is found to be second-order for the weak formulation and third-order for the strong formulation.
François Fraysse - One of the best experts on this subject based on the ideXlab platform.
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Quasi-A Priori Truncation Error Estimation in the DGSEM
Journal of Scientific Computing, 2014Co-Authors: Gonzalo Rubio, François Fraysse, David A. Kopriva, Eusebio ValeroAbstract:In this paper we show how to accurately perform a quasi-a priori estimation of the Truncation Error of steady-state solutions computed by a discontinuous Galerkin spectral element method. We estimate the spatial Truncation Error using the $$\tau $$?-estimation procedure. While most works in the literature rely on fully time-converged solutions on grids with different spacing to perform the estimation, we use non time-converged solutions on one grid with different polynomial orders. The quasi-a priori approach estimates the Error while the residual of the time-iterative method is not negligible. Furthermore, the method permits one to decouple the surface and the volume contributions of the Truncation Error, and provides information about the anisotropy of the solution as well as its rate of convergence in polynomial order. First, we focus on the analysis of one dimensional scalar conservation laws to examine the accuracy of the estimate. Then, we extend the analysis to two dimensional problems. We demonstrate that this quasi-a priori approach yields a spectrally accurate estimate of the Truncation Error.
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the estimation of Truncation Error by tau estimation for chebyshev spectral collocation method
Journal of Scientific Computing, 2013Co-Authors: Gonzalo Rubio, François Fraysse, J. De Vicente, Eusebio ValeroAbstract:In this paper we show how to accurately estimate the local Truncation Error of the Chebyshev spectral collocation method using $$\tau $$ -estimation. This method compares the residuals on a sequence of approximations with different polynomial orders. First, we focus the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the Truncation Error. Then, we show the validity of the analysis for the incompressible Navier---Stokes equations. First on the Kovasznay flow, where an analytical solution is known, and finally in the lid driven cavity (LDC). We demonstrate that this approach yields a highly accurate estimation of the Truncation Error if the precision of the approximations increases with the polynomial order.
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The Estimation of Truncation Error by $$\tau $$-Estimation for Chebyshev Spectral Collocation Method
Journal of Scientific Computing, 2013Co-Authors: Gonzalo Rubio, François Fraysse, J. De Vicente, Eusebio ValeroAbstract:In this paper we show how to accurately estimate the local Truncation Error of the Chebyshev spectral collocation method using $$\tau $$ -estimation. This method compares the residuals on a sequence of approximations with different polynomial orders. First, we focus the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the Truncation Error. Then, we show the validity of the analysis for the incompressible Navier---Stokes equations. First on the Kovasznay flow, where an analytical solution is known, and finally in the lid driven cavity (LDC). We demonstrate that this approach yields a highly accurate estimation of the Truncation Error if the precision of the approximations increases with the polynomial order.
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comparison of mesh adaptation using the adjoint methodology and Truncation Error estimates
AIAA Journal, 2012Co-Authors: François Fraysse, Eusebio Valero, Jorge PonsinAbstract:Mesh adaptation based on Error estimation has become a key technique to improve th eaccuracy o fcomputational-fluid-dynamics computations. The adjoint-based approach for Error estimation is one of the most promising techniques for computational-fluid-dynamics applications. Nevertheless, the level of implementation of this technique in the aeronautical industrial environment is still low because it is a computationally expensive method. In the present investigation, a new mesh refinement method based on estimation of Truncation Error is presented in the context of finite-volume discretization. The estimation method uses auxiliary coarser meshes to estimate the local Truncation Error, which can be used for driving an adaptation algorithm. The method is demonstrated in the context of two-dimensional NACA0012 and three-dimensional ONERA M6 wing inviscid flows, and the results are compared against the adjoint-based approach and physical sensors based on features of the flow field.
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The estimation of Truncation Error by τ -estimation revisited
Journal of Computational Physics, 2012Co-Authors: François Fraysse, J. De Vicente, Eusebio ValeroAbstract:The aim of this paper was to accurately estimate the local Truncation Error of partial differential equations, that are numerically solved using a finite difference or finite volume approach on structured and unstructured meshes. In this work, we approximated the local Truncation Error using the @t-estimation procedure, which aims to compare the residuals on a sequence of grids with different spacing. First, we focused the analysis on one-dimensional scalar linear and non-linear test cases to examine the accuracy of the estimation of the Truncation Error for both finite difference and finite volume approaches on different grid topologies. Then, we extended the analysis to two-dimensional problems: first on linear and non-linear scalar equations and finally on the Euler equations. We demonstrated that this approach yields a highly accurate estimation of the Truncation Error if some conditions are fulfilled. These conditions are related to the accuracy of the restriction operators, the choice of the boundary conditions, the distortion of the grids and the magnitude of the iteration Error.
