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59
Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation
, 1997
"... A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a secondorder accurate discretization method is described. The relationship between the continuous formulation and a discret ..."
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Cited by 101 (3 self)
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A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a secondorder accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finitedifference gradients and several design examples are presented.
Grid adaptation for functional outputs: application to twodimensional inviscid flows
 J. Comput. Phys
"... www.elsevier.com/locate/jcp ..."
Implicit/Multigrid Algorithms for Incompressible Turbulent Flows on Unstructured Grids
, 1996
"... this paper use a of 10. ..."
Achieving high sustained performance in an unstructured mesh CFD application
 In Proceedings of SC’99, 1999. Gordon Bell Prize Award Paper in Special Category
, 1999
"... Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA Scientific and Technical Information (STI) Program Office plays a key part in helping NASA maintain this important role. The NASA STI Program Office is operated by Langley Research Center, the l ..."
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Cited by 38 (13 self)
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Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA Scientific and Technical Information (STI) Program Office plays a key part in helping NASA maintain this important role. The NASA STI Program Office is operated by Langley Research Center, the lead ccntcr for NASA's scientific and technical information. The NASA STI Program Office provides access to the NASA STI Database, the largest collection of aeronautical and space science STI in the world. The Program Office is also NASA's institutional mechanism for disseminating the results of its research and development activities. These results are published by NASA in the NASA STI Report Series, which includes the following report types: TECHNICAL PUBLICATION. Reports of completed research or a major significant phase of research that present the results of NASA programs and include extensive data or theoretical analysis. Includes compilations of significant scientific and technical data and information deemed to be of continuing reference value. NASA counterpart or peerreviewed formal professional papers, but having less stringent limitations on manuscript length and extent of graphic presentations. TECHNICAL MEMORANDUM. Scientific and technical findings that are preliminary or of specialized interest, e.g., quick release reports, working papers, and bibliographies that contain minimal annotation. Does not contain extensive analysis.
Multigrid Solution for HighOrder Discontinuous Galerkin . . .
, 2004
"... A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. Thi ..."
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Cited by 26 (12 self)
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A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the pmultigrid solver, which relies on an elementline Jacobi smoother. The elementline Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2D scalar convectiondiffusion shows that the elementline Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higherorder is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.
Toward realistic performance bounds for implicit CFD codes
 Proceedings of Parallel CFD’99
, 1999
"... The performance of scientific computing applications often achieves a small fraction of peak performance [7,17]. In this paper, we discuss two causes of performance problems— insufficient memory bandwidth and a suboptimal instruction mix—in the context of a complete, parallel, unstructured mesh impl ..."
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Cited by 25 (10 self)
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The performance of scientific computing applications often achieves a small fraction of peak performance [7,17]. In this paper, we discuss two causes of performance problems— insufficient memory bandwidth and a suboptimal instruction mix—in the context of a complete, parallel, unstructured mesh implicit CFD code. These results show that the performance of our code and of similar implicit codes is limited by the memory bandwidth of RISCbased processor nodes to as little as 10 % of peak performance for some critical computational kernels. Limits on the number of basic operations that can be performed in a single clock cycle also limit the performance of “cachefriendly ” parts of the code.
Multigrid techniques for unstructured meshes
 in VKI Lecture Series VKILS
, 1995
"... An overview of current multigrid techniques for unstructured meshes is given. The basic principles of the multigrid approach are first outlined. Application of these principles to unstructured mesh problems is then described, illustrating various different approaches, and giving examples of practica ..."
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Cited by 20 (3 self)
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An overview of current multigrid techniques for unstructured meshes is given. The basic principles of the multigrid approach are first outlined. Application of these principles to unstructured mesh problems is then described, illustrating various different approaches, and giving examples of practical applications. Advanced multigrid topics, such as the use of algebraic multigrid methods, and the combination of multigrid techniques with adaptive meshing strategies are dealt with in subsequent sections. These represent current areas of research, and the unresolved issues are discussed. The presentation is organized in an educational manner, for readers familiar with computational fluid dynamics, wishing to
Aerodynamic Applications of NewtonKrylovSchwarz Solvers
 in Proceedings of the 14th International Conference on Numerical Methods in Fluid Dynamics
, 1995
"... : Parallel implicit solution methods are increasingly important in aerodynamics, since reliable lowresidual solutions at elevated CFL number are prerequisite to such largescale applications of aerodynamic analysis codes as aeroelasticity and optimization. In this chapter, a class of nonlinear impl ..."
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Cited by 19 (5 self)
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: Parallel implicit solution methods are increasingly important in aerodynamics, since reliable lowresidual solutions at elevated CFL number are prerequisite to such largescale applications of aerodynamic analysis codes as aeroelasticity and optimization. In this chapter, a class of nonlinear implicit methods and a class of linear implicit methods are defined and illustrated. Their composition forms a class of methods with strong potential for parallel implicit solution of aerodynamics problems. NewtonKrylov methods are suited for nonlinear problems in which it is unreasonable to compute or store a true Jacobian, given a strong enough preconditioner for the inner linear system that needs to be solved for each Newton correction. In turn, KrylovSchwarz iterative methods are suited for the parallel implicit solution of multidimensional systems of linearized boundary value problems. Schwarztype domain decomposition preconditioning provides good data locality for parallel implementatio...
Recent Improvements in Aerodynamic Design Optimization On Unstructured Meshes
 AIAA JOURNAL
, 2002
"... Recent improvements in an unstructuredgrid method for largescale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were difficult to realize, par ..."
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Cited by 18 (1 self)
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Recent improvements in an unstructuredgrid method for largescale aerodynamic design are presented. Previous work had shown such computations to be prohibitively long in a sequential processing environment. Also, robust adjoint solutions and mesh movement procedures were difficult to realize, particularly for viscous flows. To overcome these limiting factors, a set of design codes based on a discrete adjoint method is extended to a multiprocessor environment using a shared memory approach. A nearly linear speedup is demonstrated, and the consistency of the linearizations is shown to remain valid. The full linearization of the residual is used to precondition the adjoint system, and a significantly improved convergence rate is obtained. A new mesh movement algorithm is implemented and several advantages over an existing technique are presented. Several design cases are shown for turbulent flows in two and three dimensions.
Performance modeling and tuning of an unstructured mesh CFD application
 IN PROCEEDINGS OF SC2000
, 2000
"... This paper describes performance tuning experiences with a threedimensional unstructured grid Euler flow code from NASA, which we have reimplemented in the PETSc framework and ported to several largescale machines, including the ASCI Red and Blue Pacific machines, the SGI Origin, the Cray T3E, and ..."
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Cited by 16 (5 self)
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This paper describes performance tuning experiences with a threedimensional unstructured grid Euler flow code from NASA, which we have reimplemented in the PETSc framework and ported to several largescale machines, including the ASCI Red and Blue Pacific machines, the SGI Origin, the Cray T3E, and Beowulf clusters. The code achieves a respectable level of performance for sparse problems, typical of scientific and engineering codes based on partial differential equations, and scales well up to thousands of processors. Since the gap between CPU speed and memory access rate is widening, the code is analyzed from a memorycentric perspective (in contrast to traditional floporientation) to understand its sequential and parallel performance. Performance tuning is approached on three fronts: data layouts to enhance locality of reference, algorithmic parameters, and parallel programming model. This effort was guided partly by some simple performance models developed for the sparse matrixvector product operation.