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13
ON CONVERGENCE ACCELERATION TECHNIQUES FOR UNSTRUCTURED MESHES
, 1998
"... A discussion of convergence acceleration techniques as they relate to computational fluid dynamics problems on unstructured meshes is given. Rather than providing a detailed description of particular methods, the various different building blocks of current solution techniques are discussed and exa ..."
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Cited by 17 (1 self)
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A discussion of convergence acceleration techniques as they relate to computational fluid dynamics problems on unstructured meshes is given. Rather than providing a detailed description of particular methods, the various different building blocks of current solution techniques are discussed and examples of solution strategies using one or several of these ideas are given. Issues relating to unstructured grid CFD problems are given additional consideration, including suitability of algorithms to current hardware trends, memory and cpu tradeoffs, treatment of nonlinearities, and the development of efficient strategies for handling anisotropyinduced stiffness. The outlook for future potential improvements is also discussed.
A Linear Multigrid Preconditioner for the solution of the NavierStokes Equations using a Discontinuous Galerkin Discretization
, 2007
"... A NewtonKrylov method is developed for the solution of the steady compressible NavierStokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element LineJacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum couplin ..."
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Cited by 11 (2 self)
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A NewtonKrylov method is developed for the solution of the steady compressible NavierStokes equations using a Discontinuous Galerkin (DG) discretization on unstructured meshes. An element LineJacobi preconditioner is presented which solves a block tridiagonal system along lines of maximum coupling in the flow. An incomplete blockLU factorization (BlockILU(0)) is also presented as a preconditioner, where the factorization is performed using a reordering of elements based upon the lines of maximum coupling used for the element LineJacobi preconditioner. This reordering is shown to be far superior to standard reordering techniques (Nested Dissection, Oneway Dissection, Quotient Minimum Degree, Reverse CuthillMckee) especially for viscous test cases. The BlockILU(0) factorization is performed inplace and a novel algorithm is presented for the application of the linearization which reduces both the memory and CPU time over the traditional dual matrix storage format. A linear pmultigrid algorithm using element LineJacobi, and BlockILU(0) smoothing is presented as a preconditioner to GMRES. The coarse level Jacobians are obtained using a
Parallel Implicit Adaptive Mesh Refinement Scheme for BodyFitted MultiBlock Mesh
, 2005
"... A parallel implicit adaptive mesh refinement (AMR) algorithm is described for the system of partialdifferential equations governing steady twodimensional compressible gaseous flows. The AMR algorithm uses an upwind finitevolume spatial discretization procedure in conjunction with limited linear s ..."
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Cited by 8 (1 self)
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A parallel implicit adaptive mesh refinement (AMR) algorithm is described for the system of partialdifferential equations governing steady twodimensional compressible gaseous flows. The AMR algorithm uses an upwind finitevolume spatial discretization procedure in conjunction with limited linear solution reconstruction and Riemannsolver based flux functions to solve the governing equations on multiblock mesh composed of structured curvilinear blocks with quadrilateral computational cells. A flexible blockbased hierarchical data structure is used to facilitate automatic solutiondirected mesh adaptation according to physicsbased refinement criteria. A matrixfree inexact Newton method is used to solve the system of nonlinear equations arising from this finitevolume spatial discretization procedure and a preconditioned generalized minimal residual (GMRES) method is used to solve the resulting nonsymmetric system of linear equations at each step of the Newton algorithm. Right preconditioning of the linear system is used to improve performance of the Krylov subspace method. An additive Schwarz global preconditioner with variable overlap is used in conjunction with blockfill incomplete lowerupper (BFILU) type preconditioners based on the Jacobian of the firstorder upwind scheme for each subdomain. The Schwarz preconditioning and blockbased data structure readily allow efficient and scalable parallel implementations of the implicit AMR approach on distributedmemory multiprocessor architectures. Numerical results are described for several flow cases, demonstrating both the effectiveness of the mesh adaptation and algorithm parallel performance. The proposed parallel implicit AMR method allows for anisotropic mesh refinement and appears to be well suited for predicting complex flows with disparate spatial and temporal scales in a reliable and efficient fashion. I.
Aerodynamic Computations Using a HigherOrder Algorithm
 AIAA Paper
, 1999
"... Introduction D ESPITE impressive progress in both algorithms and computers, the time required to compute the solution of the steady compressible NavierStokes equations for the flow about aerodynamic configurations remains excessive for routine use in aircraft design. Consequently, numerical solut ..."
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Cited by 6 (3 self)
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Introduction D ESPITE impressive progress in both algorithms and computers, the time required to compute the solution of the steady compressible NavierStokes equations for the flow about aerodynamic configurations remains excessive for routine use in aircraft design. Consequently, numerical solution techniques applicable to simpler physical models, such as panel methods or solvers incorporating the boundarylayer equations, are heavily used in the design process, despite their limitations. Although a substantial portion of the time required in a simulation based on the NavierStokes equations is involved in problem setup, including geometry definition and grid generation, and this needs to be addressed, the solution time is also excessive. One way to reduce the solution time is to improve the iterative method used to achieve a steady state. 1 Alternatively, the accuracy of the spatial discretization can be increased. This allows a reduction in the n
Discontinuous Galerkin solutions of the NavierStokes equations using linear multigrid preconditioning
 in 18th Computational Fluid Dynamics Conference, (Miami, FL), American Institute of Aeronautics and Astronautics
, 2007
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Improvements to a NewtonKrylov Solver for Aerodynamic Flows
, 1998
"... In this paper, we present and justify the strategies and parameters used in PROBE, a NewtonKrylov solver for steady aerodynamic flows. The Krylov solver GMRES is used in matrixfree form. An approximate Jacobian matrix with some modifications to increase the magnitude of the diagonal entries is use ..."
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In this paper, we present and justify the strategies and parameters used in PROBE, a NewtonKrylov solver for steady aerodynamic flows. The Krylov solver GMRES is used in matrixfree form. An approximate Jacobian matrix with some modifications to increase the magnitude of the diagonal entries is used to form a preconditioner based on an incomplete lowerupper factorization with two levels of fill. The resulting solver is very efficient, generally reducing the residual twelve orders of magnitude with a CPU expense equivalent to between 500 and 1000 function evaluations. For all cases studied, PROBE converged significantly faster than an approximatelyfactored multigrid algorithm used for comparison. We propose a convergence rate based on the reduction in the residual obtained in the CPU time required for one function evaluation. With this definition, PROBE achieves a convergence rate per function evaluation between 0.945 and 0.972 for the cases studied, which include inviscid, laminar, and turbulent flows. This is substantially faster than many current algorithms applied to similar flows.
Newton–Krylov Algorithm with a Loosely Coupled Turbulence Model for Aerodynamic Flows
, 2006
"... A fast Newton–Krylov algorithm is presented that solves the turbulent Navier–Stokes equations on unstructured 2D grids. The model of Spalart and Allmaras provides the turbulent viscosity and is loosely coupled to the meanflow equations. It is often assumed that the turbulence model must be fully c ..."
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A fast Newton–Krylov algorithm is presented that solves the turbulent Navier–Stokes equations on unstructured 2D grids. The model of Spalart and Allmaras provides the turbulent viscosity and is loosely coupled to the meanflow equations. It is often assumed that the turbulence model must be fully coupled to obtain the full benefit of an inexact Newton algorithm. We demonstrate that a loosely coupled algorithm is effective and has some advantages, such as reduced storage requirements and smoother transient oscillations. A transonic singleelement case converges to 1 10 12 in 90 s on recent commodity hardware, whereas the lift coefficient is converged to three figures in one quarter of that time. Nomenclature c = reference chord length e = total internal energy F = the inviscid flux tensor G = the stress and heat flux tensor ^i = unit normal aligned with reference chord ^j = unit normal perpendicular to reference chord M1 = freestream Mach number ^n = generic unit normal vector
Domain Decomposition Preconditioners for HigherOrder Discontinuous Galerkin Discretizations
, 2011
"... Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, hi ..."
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Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higherfidelity simulations remains. Present day CFD simulations are limited by the lack of an efficient highfidelity solver able to take advantage of the massively parallel architectures of modern day supercomputers. A higherorder hybridizable
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld Implicit Finite Element Schemes for the Stationary Compressible Euler Equations
"... Semiimplicit and Newtonlike finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a highresolu ..."
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Semiimplicit and Newtonlike finite element methods are developed for the stationary compressible Euler equations. The Galerkin discretization of the inviscid fluxes is potentially oscillatory and unstable. To suppress numerical oscillations, the spatial discretization is performed by a highresolution finite element scheme based on algebraic flux correction. A multidimensional limiter of TVD type is employed. An important goal is the efficient computation of stationary solutions in a wide range of Mach numbers, which is a challenging task due to oscillatory correction factors associated with TVDtype flux limiters. A semiimplicit scheme is derived by a timelagged linearization of the nonlinear residual, and a Newtonlike method is obtained in the limit of infinite CFL numbers. Special emphasis is laid on the numerical treatment of weakly imposed characteristic boundary conditions. A boundary Riemann solver is used to avoid unphysical boundary states. It is shown that the proposed approach offers unconditional stability as well as higher accuracy and better convergence behavior than in the case of algorithms in which the boundary conditions are implemented in a strong sense. The spatial accuracy of the whole scheme and the boundary conditions is analyzed by grid convergence studies. Copyright c © 2010