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A Parallel Three-Dimensional Incompressible Navier-Stokes Solver with a Parallel Multigrid Kernel
The development and applications of a parallel, time-dependent, three-dimensional incompressible Navier-Stokes flow solver and a parallel multigrid elliptic kernel are described.
Evaluation of a multigrid-based Navier-Stokes solver for aerothermodynamic computations
A multigrid acceleration technique developed for solving the three-dimensional Navier-Stokes equations is used for computing high Mach number flows over configurations of practical interest. An explicit multistage Runge-Kutta type of time-stepping scheme is used as the basic algorithm. Solutions are presented for a spherically blunted cone at Mach 10 and a modified shuttle orbiter at Mach 6. The computed surface heat-transfer distributions are shown to compare favorably with the experimental data. Effect of grid-refinement on computed heat-transfer distributions is also examined to assess the numerical accuracy of the computed solutions. The rapid convergence rate associated with multigrid schemes in previous applications at transonic speeds is observed at the higher Mach number flows investigated here.
Multigrid Strategies for Viscous Flow Solvers on Anisotropic Unstructured Meshes
Unstructured multigrid techniques for relieving the stiffness associated with high-Reynolds number viscous flow simulations on extremely stretched grids are investigated. One approach consists of employing a semi-coarsening or directional-coarsening technique, based on the directions of strong coupling within the mesh, in order to construct more optimal coarse grid levels. An alternate approach is developed which employs directional implicit smoothing with regular fully coarsened multigrid levels. The directional implicit smoothing is obtained by constructing implicit lines in the unstructured mesh based on the directions of strong coupling. Both approaches yield large increases in convergence rates over the traditional explicit full-coarsening multigrid algorithm. However, maximum benefits are achieved by combining the two approaches in a coupled manner into a single algorithm. An order of magnitude increase in convergence rate over the traditional explicit full-coarsening algorithm is demonstrated, and convergence rates for high-Reynolds number viscous flows which are independent of the grid aspect ratio are obtained. Further acceleration is provided by incorporating low-Mach-number preconditioning techniques, and a Newton-GMRES strategy which employs the multigrid scheme as a preconditioner. The compounding effects of these various techniques on speed of convergence is documented through several example test cases.
Parallelization of a Multigrid Incompressible Viscous Cavity Flow Solver Using OpenMP
We describe a multigrid scheme for solving the viscous incompressible driven cavity problem that has been parallelized using OpenMP. The incremental parallelization allowed by OpenMP was of great help during the parallelization process. Results show good parallel efficiencies for reasonable problem sizes on an SGI Origin 2000. Since OpenMP allowed us to specify the number of threads (and in turn processors) at runtime, we were able to improve performance when solving on smaller/coarser meshes. This was accomplished by giving each processor a more reasonable amount of work rather than having many processors work on very small segments of the data (and thereby adding significant overhead).
Multigrid acceleration of a fractional-step solver in generalized curvilinear coordinate systems
The efficiency of the INS3D-fractional step (FS) code has been significantly enhanced by accelerating the Poisson solver with a multigrid (MG) method. This is apparently the first implementation of the MG method for solving the 3D discrete Poisson-like equation obtained in the derivation of FS solution methods for the incompressible Navier-Stokes equations in generalized nonorthogonal coordinate systems using a staggered arrangement of the variables. The MG solver is insensitive to the geometry, even in cases of highly nonorthogonal clustered meshes, as well as to the free parameters of MG methods.
An overlapped grid method for multigrid, finite volume/difference flow solvers: MaGGiE
The objective is to develop a domain decomposition method via overlapping/embedding the component grids, which is to be used by upwind, multi-grid, finite volume solution algorithms. A computer code, given the name MaGGiE (Multi-Geometry Grid Embedder) is developed to meet this objective. MaGGiE takes independently generated component grids as input, and automatically constructs the composite mesh and interpolation data, which can be used by the finite volume solution methods with or without multigrid convergence acceleration. Six demonstrative examples showing various aspects of the overlap technique are presented and discussed. These cases are used for developing the procedure for overlapping grids of different topologies, and to evaluate the grid connection and interpolation data for finite volume calculations on a composite mesh. Time fluxes are transferred between mesh interfaces using a trilinear interpolation procedure. Conservation losses are minimal at the interfaces using this method. The multi-grid solution algorithm, using the coaser grid connections, improves the convergence time history as compared to the solution on composite mesh without multi-gridding.
Computations of permeability of large rock images by dual grid domain decomposition
Not Available
Progress report on MPI+X computational kernels for algebraic multigrid and iterative linear solvers.
Abstract not provided.
A fully implicit, scalable, conservative nonlinear relativistic Fokker–Planck 0D-2P solver for runaway electrons
Upon application of a sufficiently strong electric field, electrons break away from thermal equilibrium and approach relativistic speeds. These highly energetic ‘runaway’ electrons (~ MeV) play a significant role in tokamak disruption physics, and therefore their accurate understanding is essential to develop reliable mitigation strategies. As such, we have developed a fully implicit solver for the 0D-2P (i.e., including two momenta coordinates) relativistic nonlinear Fokker–Planck equation (rFP). As in earlier implicit rFP studies (NORSE, CQL3D), electron–ion interactions are modeled using the Lorentz operator, and synchrotron damping using the Abraham–Lorentz–Dirac reaction term. However, our implementation improves on these earlier studies by (1) ensuring exact conservation properties for electron collisions, (2) strictly preserving positivity, and (3) being scalable algorithmically and in parallel. Key to our proposed approach is an efficient multigrid preconditioner for the linearized rFP equation, a multigrid elliptic solver for the Braams–Karney potentials, and a novel adaptive technique to determine the associated boundary values. We verify the accuracy and efficiency of the proposed scheme with numerical results ranging from small electric-field electrical conductivity measurements to the accurate reproduction of runaway tail dynamics when strong electric fields are applied.
Multiscale Simulation Framework for Coupled Fluid Flow and Mechanical Deformation
A multiscale linear-solver framework for the pressure equation associated with flow in highly heterogeneous porous formations was developed. The multiscale based approach is cast in a general algebraic form, which facilitates integration of the new scalable linear solver in existing flow simulators. The Algebraic Multiscale Solver (AMS) is employed as a preconditioner within a multi-stage strategy. The formulations investigated include the standard MultiScale Finite-Element (MSFE) andMultiScale Finite-Volume (MSFV) methods. The local-stage solvers include incomplete factorization and the so-called Correction Functions (CF) associated with the MSFV approach. Extensive testing of AMS, as an iterative linear solver, indicate excellent convergence rates and computational scalability. AMS compares favorably with advanced Algebraic MultiGrid (AMG) solvers for highly detailed three-dimensional heterogeneous models. Moreover, AMS is expected to be especially beneficial in solving time-dependent problems of coupled multiphase flow and transport in large-scale subsurface formations.
A parallel finite-difference method for computational aerodynamics
A finite-difference scheme for solving complex three-dimensional aerodynamic flow on parallel-processing supercomputers is presented. The method consists of a basic flow solver with multigrid convergence acceleration, embedded grid refinements, and a zonal equation scheme. Multitasking and vectorization have been incorporated into the algorithm. Results obtained include multiprocessed flow simulations from the Cray X-MP and Cray-2. Speedups as high as 3.3 for the two-dimensional case and 3.5 for segments of the three-dimensional case have been achieved on the Cray-2. The entire solver attained a factor of 2.7 improvement over its unitasked version on the Cray-2. The performance of the parallel algorithm on each machine is analyzed.
A Scalable Interior‐Point Gauss–Newton Method for PDE‐Constrained Optimization With Bound Constraints
Here, we present a scalable approach to solve a class of partial differential equation (PDE)‐constrained optimization problems with bound constraints. This approach utilizes a robust full‐space interior‐point (IP)‐Gauss–Newton optimization method. To cope with the poorly‐conditioned IP‐Gauss–Newton saddle‐point linear systems that need to be solved approximately, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE‐constrained optimization problems. A block Gauss–Seidel preconditioner is proposed for the GMRES‐based solution of the IP‐Gauss–Newton linear systems. It is shown, for a large‐class of PDE‐ and bound‐constrained optimization problems, that the spectrum of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix is asymptotically independent of discretization and is not impacted by the ill‐conditioning that notoriously plagues interior‐point methods. We exploit symmetry of the IP‐Gauss–Newton linear systems and propose a regularization and log‐barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)‐based solution of the equivalent IP‐Gauss–Newton–Schur complement linear systems. The eigenvalues of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix, that are not equal to one, are identical to the eigenvalues of the regularization and log‐barrier Hessian preconditioned Schur complement matrix. The scalability of the approach is demonstrated on two example problems. The numerical solution of these optimization problems is shown to require a discretization independent number of IP‐Gauss–Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill‐conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of the preconditioner, achieved via algebraic multigrid component solvers when applicable, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE‐ and bound‐constrained problems.
Low-order preconditioning of the Stokes equations
A well-known strategy for building effective preconditioners for higher-order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low-order analogs. In this work, we show that high-quality preconditioners can also be derived for the Taylor–Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the Q 1 iso Q 2 /Q 1 discretization of the Stokes operator as a preconditioner for the Q 2 /Q 1 discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess–Sarazin relaxation schemes and to achieve robust convergence. Furthermore, these results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the Q 2 /Q 1 system, our ultimate motivation is to apply algebraic multigrid within solvers for Q 2 /Q 1 systems via the Q 1 iso Q 2 /Q 1 discretization, which will be considered in a companion paper.
A multigrid and upwind viscous flow solver on 3-D embedded and overlapped grids
A numerically efficient method is presented for solving the three-dimensional governing equations of the viscous compressible flow about complex configurations with topologically different components. The physical domain is decomposed into regions for which the grid generation is relatively simple and virtually with no significant restrictions. The Navier-Stokes equations are solved by an implicit, approximately factored, upwind, finite-volume scheme. The block inversions and the diagonalized scalar inversions of the coefficient matrices are modified to allow the holes created in the computational domain by the embedded and overlapped grids. The convergence is accelerated by a multigrid algorithm despite the existence of such holes. The solution for s supersonic flow past a blunt-nose-cylinder at high angle-of-attack is obtained using a C-O grid embedded in a global Cartesian grid.
Parallel Element-based Algebraic Multigrid for H (c url ) and H (div) Problems Using the ParELAG Library
This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the ParELAG (Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers) library, to produce multilevel preconditioners and solvers for H (c url ) and H (div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and AMS (Auxiliary-space Maxwell Solver) or ADS (Auxiliary-space Divergence Solver) on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library.
Compiler-based code generation and autotuning for geometric multigrid on GPU-accelerated supercomputers
GPUs, with their high bandwidths and computational capabilities are an increasingly popular target for scientific computing. Unfortunately, to date, harnessing the power of the GPU has required use of a GPU-specific programming model like CUDA, OpenCL, or OpenACC. Thus, in order to deliver portability across CPU-based and GPU-accelerated supercomputers, programmers are forced to write and maintain two versions of their applications or frameworks. In this paper, we explore the use of a compiler-based autotuning framework based on CUDA-CHiLL to deliver not only portability, but also performance portability across CPU- and GPU-accelerated platforms for the geometric multigrid linear solvers found in many scientific applications. We also show that with autotuning we can attain near Roofline (a performance bound for a computation and target architecture) performance across the key operations in the miniGMG benchmark for both CPU- and GPU-based architectures as well as for a multiple stencil discretizations and smoothers. We show that our technology is readily interoperable with MPI resulting in performance at scale equal to that obtained via hand-optimized MPI+CUDA implementation.
Manycore Parallel Computing for a Hybridizable Discontinuous Galerkin Nested Multigrid Method
In this work, we present a parallel computing strategy for a hybridizable discontinuous Galerkin (HDG) nested geometric multigrid (GMG) solver. Parallel GMG solvers require a combination of coarse-grain and fine-grain parallelism to improve time-to-solution performance. In this work we focus on fine-grain parallelism. We use Intel's second generation Xeon Phi (Knights Landing) manycore processor. The GMG method achieves ideal convergence rates of 0.2 or less, for high polynomial orders. A matrix free (assembly free) technique is exploited to save considerable memory usage and increase arithmetic intensity. HDG enables static condensation, and due to the discontinuous nature of the discretization, we developed a matrix vector multiply routine that does not require any costly synchronizations or barriers. Our algorithm is able to attain 80% of peak bandwidth performance for higher order polynomials. This is possible due to the data locality inherent in the HDG method. Very high performance is realized for high order schemes, due to good arithmetic intensity, which declines as the order is reduced.