Sideroff, Chris; Stephens, Darrin W; Jemcov, Aleksandar A Projection Method Based Fast Transient Solver for Incompressible Turbulent Flows Conference Annual Meeting of CFD Society of Canada (CFDSC2015), Waterloo, Canada, 2015. Abstract  Links  BibTeX  Tags: Algebraic Multigrid Solver, Caelus, Fractional Step Algorithm, High Performance Computing, PISO, Projection Method, SIMPLE, SLIM, Transient solutions, Validation @conference{sideroffprojection,
title = {A Projection Method Based Fast Transient Solver for Incompressible Turbulent Flows},
author = {Chris Sideroff and Darrin W Stephens and Aleksandar Jemcov},
doi = {10.13140/RG.2.1.4094.2242},
year = {2015},
date = {20150608},
booktitle = {Annual Meeting of CFD Society of Canada (CFDSC2015), Waterloo, Canada},
abstract = {This paper presents a fast transient solver suitable for the simulation of transient turbulent flows. The main characteristic of the solver is that it is based on the projection method and requires only one pressure and the momentum solve per time step. Furthermore, using the projection method has the additional advantage that the formulation of the pressure equation is particularly efficient because the Laplacian term depends only on geometric quantities. This improves the parallel scalability of the Algebraic MultiGrid (AMG) method because the coarse agglomeration levels can be cached if the grid is not changing and are consistent throughout each time step. The fractional step error near the boundaries is removed by utilizing the incremental version of the algorithm. The solver is implemented using v5.04 of the openly available Caelus computational mechanics library. Previous work by Stephens et al. [12] demonstrated the accuracy of SLIM on several validation cases. In this work, performance of the solver was investigated through several test cases. The results indicate that SLIM has superior serial and parallel performance over the PISO and transient SIMPLE solvers. Transient solutions of incompressible, turbulent flows occupy an increasing portion of engineering computations. The majority of solvers that use the finite volume method on unstructured meshes with arbitrary number of faces per cell use either a transient SIMPLE [8] or the PISO algorithm [5]. While these algorithms are known to produce satisfactory spatial and temporal accuracy, they are not particularly efficient due to algorithmic constraints. Typically they require multiple solutions of the pressure equation (PISO algorithm) or multiple solutions of both momentum and pressure per time step (SIMPLE algorithm). In these algorithms, multiple solutions of the pressure and/or momentum equation per time step are required to remove the fractional step error due to splitting of equations and recover the time accuracy. Unlike the PISO and SIMPLE algorithms, the projection algorithm introduced originally by Chorin [2], does not require multiple pressure and momentum solves per time step. In this paper we describe an efficient implementation of the projection algorithm that can efficiently utilize the Algebraic MultiGrid method for the pressure equation.
},
keywords = {Algebraic Multigrid Solver, Caelus, Fractional Step Algorithm, High Performance Computing, PISO, Projection Method, SIMPLE, SLIM, Transient solutions, Validation},
pubstate = {published},
tppubtype = {conference}
}
This paper presents a fast transient solver suitable for the simulation of transient turbulent flows. The main characteristic of the solver is that it is based on the projection method and requires only one pressure and the momentum solve per time step. Furthermore, using the projection method has the additional advantage that the formulation of the pressure equation is particularly efficient because the Laplacian term depends only on geometric quantities. This improves the parallel scalability of the Algebraic MultiGrid (AMG) method because the coarse agglomeration levels can be cached if the grid is not changing and are consistent throughout each time step. The fractional step error near the boundaries is removed by utilizing the incremental version of the algorithm. The solver is implemented using v5.04 of the openly available Caelus computational mechanics library. Previous work by Stephens et al. [12] demonstrated the accuracy of SLIM on several validation cases. In this work, performance of the solver was investigated through several test cases. The results indicate that SLIM has superior serial and parallel performance over the PISO and transient SIMPLE solvers. Transient solutions of incompressible, turbulent flows occupy an increasing portion of engineering computations. The majority of solvers that use the finite volume method on unstructured meshes with arbitrary number of faces per cell use either a transient SIMPLE [8] or the PISO algorithm [5]. While these algorithms are known to produce satisfactory spatial and temporal accuracy, they are not particularly efficient due to algorithmic constraints. Typically they require multiple solutions of the pressure equation (PISO algorithm) or multiple solutions of both momentum and pressure per time step (SIMPLE algorithm). In these algorithms, multiple solutions of the pressure and/or momentum equation per time step are required to remove the fractional step error due to splitting of equations and recover the time accuracy. Unlike the PISO and SIMPLE algorithms, the projection algorithm introduced originally by Chorin [2], does not require multiple pressure and momentum solves per time step. In this paper we describe an efficient implementation of the projection algorithm that can efficiently utilize the Algebraic MultiGrid method for the pressure equation.

Sideroff, Chris; Stephens, Darrin W; Jemcov, Aleksandar A Projection Method Based Fast Transient Solver for Incompressible Turbulent Flows Presentation 08.06.2015. Links  BibTeX  Tags: Algebraic Multigrid Solver, Caelus, Fractional Step Algorithm, High Performance Computing, PISO, Projection Method, SIMPLE, SLIM, Transient solutions, Validation @misc{Sideroff2015b,
title = {A Projection Method Based Fast Transient Solver for Incompressible Turbulent Flows},
author = {Chris Sideroff and Darrin W Stephens and Aleksandar Jemcov },
url = {http://www.appliedccm.com/wpcontent/uploads/2015/08/cfdsc2015presentation.pdf},
year = {2015},
date = {20150608},
keywords = {Algebraic Multigrid Solver, Caelus, Fractional Step Algorithm, High Performance Computing, PISO, Projection Method, SIMPLE, SLIM, Transient solutions, Validation},
pubstate = {published},
tppubtype = {presentation}
}

Jemcov, Aleksandar; Maruszewski, Joseph P Nonlinear Flow Solver Acceleration by Reduced Rank Extrapolation Incollection 46th AIAA Aerospace Sciences Meeting and Exhibit, pp. 609, 2008. Abstract  Links  BibTeX  Tags: Acceleration, Algorithm, Extrapolation, GMRES, Nonlinear, RRE, SIMPLE, Solver @incollection{jemcov2008nonlinear,
title = {Nonlinear Flow Solver Acceleration by Reduced Rank Extrapolation},
author = { Aleksandar Jemcov and Joseph P Maruszewski},
url = {https://www.researchgate.net/profile/Aleksandar_Jemcov/publication/275275199_Nonlinear_Flow_Solver_Acceleration_by_Reduced_Rank_Extrapolation/links/5536ca190cf2058efdea9401.pdf?origin=publication_detail_rebranded&ev=pub_int_prw_xdl&msrp=mHcz4Kt3tcM0fKKvzoZAfJu5CZ7bJAZTebm7svkdI8HyTV2DqiloQBxWK3ZC9HHtc1VaKywIo53Gq4ACRkkbmQ%3D%3D_mZOVHfxww2SuKoJv0YEv1yJ19XXhbwqiKJ%2FuZ%2FsAeTe%2BZ2RdfWruA0KDkoFLyPO0ebx%2BKC9jrGwRj%2B%2F%2BKjsf7A%3D%3D},
year = {2008},
date = {20080101},
booktitle = {46th AIAA Aerospace Sciences Meeting and Exhibit},
pages = {609},
abstract = {Convergence acceleration of nonlinear flow solvers through use of vector sequence extrapolation techniques is presented. In particular, suitability of the Reduced Rank Extrapolation (RRE) algorithm for the use of convergence acceleration of nonlinear flow solvers is examined. In the RRE algorithm, the solution is obtained through a linear combination of Krylov vectors with weighting coefficients obtained by minimizing L2 norm of error in this space with properly chosen constraint conditions. This process effectively defines vector sequence extrapolation process in Krylov subspace that corresponds to the GMRES method applied to nonlinear problems. Moreover, when the RRE algorithm is used to solve nonlinear problems, the flow solver plays the role of the preconditioner for the nonlinear GMRES method. Benefits of the application of the RRE algorithm include better convergence rates, removal of residual stalling and improved coupling between equations in numerical models. Proposed algorithm is independent of the type of flow solver and it is equally applicable to explicit, implicit, pressure and density based algorithms. Nomenclature Q Vector of conserved variables R Residual vector F c Vector of convective fluxes F v Vector of viscous fluxes S Source vector H Total enthalpy, J/m 3 p pressure, Pa τ ij viscous tensor ρ Density, Kg/m 3 u Velocity vector, m/s u Xcomponent of velocity vector, m/s v Ycomponent of velocity vector, m/s w Zcomponent of velocity vector, m/s V Contravariant velocity, m/s f e,i Vector of external forces Φ Generic transport variable M Nonlinear preconditioning operator F Fixedpoint function ∂ Q (·) Jacobian with respect to Q α ν Extrapolation coefficients c ν o t Time, s ∂Ω Boundary of computational domain Ω Computational domain},
keywords = {Acceleration, Algorithm, Extrapolation, GMRES, Nonlinear, RRE, SIMPLE, Solver},
pubstate = {published},
tppubtype = {incollection}
}
Convergence acceleration of nonlinear flow solvers through use of vector sequence extrapolation techniques is presented. In particular, suitability of the Reduced Rank Extrapolation (RRE) algorithm for the use of convergence acceleration of nonlinear flow solvers is examined. In the RRE algorithm, the solution is obtained through a linear combination of Krylov vectors with weighting coefficients obtained by minimizing L2 norm of error in this space with properly chosen constraint conditions. This process effectively defines vector sequence extrapolation process in Krylov subspace that corresponds to the GMRES method applied to nonlinear problems. Moreover, when the RRE algorithm is used to solve nonlinear problems, the flow solver plays the role of the preconditioner for the nonlinear GMRES method. Benefits of the application of the RRE algorithm include better convergence rates, removal of residual stalling and improved coupling between equations in numerical models. Proposed algorithm is independent of the type of flow solver and it is equally applicable to explicit, implicit, pressure and density based algorithms. Nomenclature Q Vector of conserved variables R Residual vector F c Vector of convective fluxes F v Vector of viscous fluxes S Source vector H Total enthalpy, J/m 3 p pressure, Pa τ ij viscous tensor ρ Density, Kg/m 3 u Velocity vector, m/s u Xcomponent of velocity vector, m/s v Ycomponent of velocity vector, m/s w Zcomponent of velocity vector, m/s V Contravariant velocity, m/s f e,i Vector of external forces Φ Generic transport variable M Nonlinear preconditioning operator F Fixedpoint function ∂ Q (·) Jacobian with respect to Q α ν Extrapolation coefficients c ν o t Time, s ∂Ω Boundary of computational domain Ω Computational domain 