CK
Christiaan M. Klaij
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The augmented Lagrangian (AL) preconditioner and its variant have been successfully applied to solve saddle point systems arising from the incompressible Navier-Stokes equations discretized by the finite element method. Attractive features are the purely algebraic construction and robustness with respect to the Reynolds number and mesh renement. In this report, we reconsider the application of the AL preconditioner in the context of the stabilized finite volume methods and present the extension to the
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations. ...
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations. ...
The augmented Lagrangian (AL) preconditioner and its variant have been successfully applied to solve saddle point systems arising from the incompressible Navier-Stokes equations discretized by the finite element method. Attractive features are the purely algebraic construction and robustness with respect to the Reynolds number and mesh renement. In this report, we reconsider the application of the AL preconditioner in the context of the stabilized finite volume methods and present the extension to the
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations.
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations.
The modified augmented Lagrangian preconditioner has attracted much attention in solving nondimensional Navier-Stokes equations discretized by the finite element method. In industrial applications the governing equations are often in dimensional form and discretized using the finite volume method. This paper assesses the capability of this preconditioner for dimensional Navier-Stokes equations in the context of the finite volume method. Two main contributions are made. First, this paper introduces a new dimensionless parameter that is involved in the modified augmented Lagrangian preconditioner. Second, with a number of academic test problems this paper reveals that the convergence of both nonlinear and linear iterations depend on this dimensionless parameter. A way to choose the optimal value of the dimensionless parameter is suggested and it is found that the optimal value is dependent of the Reynolds number, instead of the fluid's properties, e.g., density and dynamic viscosity. The outcomes of this paper yield a potential rule to choose the optimal dimensionless parameter in practice, namely, correspondingly increasing with enlarging the Reynolds number.
...
The modified augmented Lagrangian preconditioner has attracted much attention in solving nondimensional Navier-Stokes equations discretized by the finite element method. In industrial applications the governing equations are often in dimensional form and discretized using the finite volume method. This paper assesses the capability of this preconditioner for dimensional Navier-Stokes equations in the context of the finite volume method. Two main contributions are made. First, this paper introduces a new dimensionless parameter that is involved in the modified augmented Lagrangian preconditioner. Second, with a number of academic test problems this paper reveals that the convergence of both nonlinear and linear iterations depend on this dimensionless parameter. A way to choose the optimal value of the dimensionless parameter is suggested and it is found that the optimal value is dependent of the Reynolds number, instead of the fluid's properties, e.g., density and dynamic viscosity. The outcomes of this paper yield a potential rule to choose the optimal dimensionless parameter in practice, namely, correspondingly increasing with enlarging the Reynolds number.
The augmented Lagrangian (AL) preconditioner and its variant have been successfully applied to solve saddle point systems arising from the incompressible Navier-Stokes equations discretized by the finite element method. Attractive features are the purely algebraic construction and robustness with respect to the Reynolds number and mesh renement. In this report, we reconsider the application of the AL preconditioner in the context of the stabilized finite volume methods and present the extension to the
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations. ...
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations. ...
The augmented Lagrangian (AL) preconditioner and its variant have been successfully applied to solve saddle point systems arising from the incompressible Navier-Stokes equations discretized by the finite element method. Attractive features are the purely algebraic construction and robustness with respect to the Reynolds number and mesh renement. In this report, we reconsider the application of the AL preconditioner in the context of the stabilized finite volume methods and present the extension to the
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations.
Reynolds-Averaged Navier-Stokes (RANS) equations, which are used to model turbulent flows in industrial applications. Furthermore, we propose a new variant of the AL preconditioner, obtained by substituting the approximation of the Schur complement from the SIMPLE preconditioner into the inverse of the Schur complement for the AL preconditioner. This new variant is applied to both Navier-Stokes and RANS equations to compute laminar and turbulent boundary-layer flows on grids with large aspect ratios. Spectral analysis shows that the new variant yields a more clustered spectrum of eigenvalues away from zero, which explains why it outperforms the existing variants in terms of the number of the Krylov subspace iterations.
The iterative error can be an important part of the total numerical error of any Com-putational Fluid Dynamics simulation when the iterative convergence stagnates or when loose convergence criteria are used. In the quest for better iterative convergence of CFD simulations, we consider the design of iterative methods for the Reynolds-averaged Navier-Stokes equations, discretized by finite-volume methods with cell-centered, co-located variables. The central point
is the approximation of the Schur complement (pressure matrix) in the block factorization of the discrete system of mass and momentum equations. We show particular approximations of these blocks that yield either segregated solvers or block preconditioners for fully coupled solvers. The performance of these solvers are then demonstrated by computing the flow over a flat plate and around a tanker on both structured and unstructured grids. We find that iterative convergence to machine precision is attainable despite the high Reynolds numbers and mesh aspect ratio’s. Improved approximations of the Schur complement do result in improved convergence rates, but do not seem to pay-off in terms of total cost compared to the basic SIMPLE-type approximation. ...
is the approximation of the Schur complement (pressure matrix) in the block factorization of the discrete system of mass and momentum equations. We show particular approximations of these blocks that yield either segregated solvers or block preconditioners for fully coupled solvers. The performance of these solvers are then demonstrated by computing the flow over a flat plate and around a tanker on both structured and unstructured grids. We find that iterative convergence to machine precision is attainable despite the high Reynolds numbers and mesh aspect ratio’s. Improved approximations of the Schur complement do result in improved convergence rates, but do not seem to pay-off in terms of total cost compared to the basic SIMPLE-type approximation. ...
The iterative error can be an important part of the total numerical error of any Com-putational Fluid Dynamics simulation when the iterative convergence stagnates or when loose convergence criteria are used. In the quest for better iterative convergence of CFD simulations, we consider the design of iterative methods for the Reynolds-averaged Navier-Stokes equations, discretized by finite-volume methods with cell-centered, co-located variables. The central point
is the approximation of the Schur complement (pressure matrix) in the block factorization of the discrete system of mass and momentum equations. We show particular approximations of these blocks that yield either segregated solvers or block preconditioners for fully coupled solvers. The performance of these solvers are then demonstrated by computing the flow over a flat plate and around a tanker on both structured and unstructured grids. We find that iterative convergence to machine precision is attainable despite the high Reynolds numbers and mesh aspect ratio’s. Improved approximations of the Schur complement do result in improved convergence rates, but do not seem to pay-off in terms of total cost compared to the basic SIMPLE-type approximation.
is the approximation of the Schur complement (pressure matrix) in the block factorization of the discrete system of mass and momentum equations. We show particular approximations of these blocks that yield either segregated solvers or block preconditioners for fully coupled solvers. The performance of these solvers are then demonstrated by computing the flow over a flat plate and around a tanker on both structured and unstructured grids. We find that iterative convergence to machine precision is attainable despite the high Reynolds numbers and mesh aspect ratio’s. Improved approximations of the Schur complement do result in improved convergence rates, but do not seem to pay-off in terms of total cost compared to the basic SIMPLE-type approximation.