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Coarse Grid (coarse + grid)
Selected AbstractsA dual mesh multigrid preconditioner for the efficient solution of hydraulically driven fracture problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 13 2005A. P. Peirce Abstract We present a novel multigrid (MG) procedure for the efficient solution of the large non-symmetric system of algebraic equations used to model the evolution of a hydraulically driven fracture in a multi-layered elastic medium. The governing equations involve a highly non-linear coupled system of integro-partial differential equations along with the fracture front free boundary problem. The conditioning of the algebraic equations typically degrades as O(N3). A number of characteristics of this problem present significant new challenges for designing an effective MG strategy. Large changes in the coefficients of the PDE are dealt with by taking the appropriate harmonic averages of the discrete coefficients. Coarse level Green's functions for multiple elastic layers are constructed using a single dual mesh and superposition. Coarse grids that are sub-sets of the finest grid are used to treat mixed variable problems associated with ,pinch points.' Localized approximations to the Jacobian at each MG level are used to devise efficient Gauss,Seidel smoothers and preferential line iterations are used to eliminate grid anisotropy caused by large aspect ratio elements. The performance of the MG preconditioner is demonstrated in a number of numerical experiments. Copyright © 2005 John Wiley & Sons, Ltd. [source] Improving Kirchhoff migration with repeated local plane-wave imaging?GEOPHYSICAL PROSPECTING, Issue 6 2005A SAR-inspired signal-processing approach in prestack depth imaging ABSTRACT A local plane-wave approach of generalized diffraction tomography in heterogeneous backgrounds, equivalent to Kirchhoff summation techniques when applied in seismic reflection, is re-programmed to act as repeated synthetic aperture radar (SAR) imaging for seismic prestack depth migration. Spotlight-mode SAR imaging quickly provides good images of the electromagnetic reflectivity of the ground via fast Fourier transform (FFT)-based signal processing. By calculating only the Green's functions connecting the aircraft to the centre of the illuminated patch, scattering structures around that centre are also recovered. SAR technology requires us to examine seismic imaging from the local point of view, where the quantity and quality of the available information at each image point are what are important, regardless of the survey geometry. When adapted to seismics, a local image of arbitrary size and sampling is obtained by FFT of seismic energy maps in the scattering wavenumber domain around each node of a pre-calculated grid of Green's functions. These local images can be used to generate a classic prestack depth-migrated section by collecting only their centres. However, the local images also provide valuable information around the centre, as in SAR. They can therefore help to pre-analyse prestack depth migration efficiently, and to perform velocity analysis at a very low cost. The FFT-based signal-processing approach allows local, efficient and automatic control of anti-aliasing, noise and resolution, including optimized Jacobian weights. Repeated local imaging could also be used to speed up migration, with interpolation between local images associated with a coarse grid of Green's functions, as an alternative to interpolation of Green's functions. The local images may, however, show distortions due to the local plane-wave approximation, and the velocity variations across their frame. Such effects, which are not necessarily a problem in SAR, should be controlled and corrected to further enhance seismic imaging. Applications to realistic models and to real data show that, despite the distortion effects, the local images can yield similar information to prestack depth migration, including common-image-point gathers for velocity analyses and AVO/AVA effects, at a much lower cost when a small target is considered. [source] A two-grid method for expanded mixed finite-element solution of semilinear reaction,diffusion equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2003Yanping Chen Abstract We present a scheme for solving two-dimensional semilinear reaction,diffusion equations using an expanded mixed finite element method. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method. The solution of a non-linear system on the fine space is reduced to the solution of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H=O(h1/3). As a result, solving such a large class of non-linear equation will not be much more difficult than solving one single linearized equation. Copyright © 2003 John Wiley & Sons, Ltd. [source] Chebyshev super spectral viscosity solution of a two-dimensional fluidized-bed modelINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 3 2003Scott A. SarraArticle first published online: 13 MAY 200 Abstract The numerical solution of a model describing a two-dimensional fluidized bed by a Chebyshev super spectral viscosity (SSV) method is considered. The model is in the form of a hyperbolic system of conservation laws with a source term, coupled with an elliptic equation for determining a stream function. The coupled elliptic equation is solved by a finite-difference method. The mixed SSV/finite-difference method produces physically shaped bubbles, on a very coarse grid. Fine scale details, which were not present in previous finite-difference solutions, are present in the solution. Copyright © 2003 John Wiley & Sons, Ltd. [source] Resolution errors associated with gridded precipitation fieldsINTERNATIONAL JOURNAL OF CLIMATOLOGY, Issue 15 2005C. J. Willmott Abstract Spatial-resolution errors are inherent in gridded precipitation (P) fields,such as those produced by climate models and from satellite observations,and they can be sizeable when P is averaged spatially onto a coarse grid. They can also vary dramatically over space and time. In this paper, we illustrate the importance of evaluating resolution errors associated with gridded P fields by investigating the relationships between grid resolution and resolution error for monthly P within the Amazon Basin. Spatial-resolution errors within gridded-monthly and average-monthly P fields over the Amazon Basin are evaluated for grid resolutions ranging from 0.1° to 5.0°. A resolution error occurs when P is estimated for a location of interest within a grid-cell from the unbiased, grid-cell average P. Graphs of January, July and annual resolution errors versus resolution show that, at the higher resolutions (<3° ), aggregation quickly increases resolution error. Resolution error then begins to level off as the grid becomes coarser. Within the Amazon Basin, the largest resolution errors occur during January (summer), but the largest percentage errors appear in July (winter). In January of 1980, e.g., resolution errors of 29, 52 and 65 mm,or 11, 19 and 24% of the grid-cell means,were estimated at resolutions of 1.0°, 3.0° and 5.0°. In July of 1980, however, the percentage errors at these three resolutions were considerably larger, that is, 15%, 27% and 33% of the grid-cell means. Copyright © 2005 Royal Meteorological Society [source] On the efficient evaluation of Fourier patterns for nanoparticles and clustersJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 9 2006Antonio Cervellino Abstract Samples made of an isotropically oriented ensemble of atomic clusters or structures that are not large crystals (i.e. extended less than 10 periods in each direction) are at the frontier of today's material science and chemistry. Examples are nanoparticles, nanotubes, amorphous matter, polymers, and macromolecules in suspension. For such systems the computation of powder diffraction patterns (which may provide an efficient characterization) is to be performed the hard way, by summing contributions from each atom pair. This work deals with performing such computation in the most practical and efficient way. Three main points are developed: how to encode the enormous array of interatomic distances (which increase as the square or higher powers of the cluster diameter) to a much smaller array of equispaced values on a coarse grid (whose size increases linearly with the diameter); how to perform a fast computation of the diffraction pattern from this equispaced grid; how to optimize the grid step to obtain an arbitrarily small error on the computed diffraction pattern. Theory and examples are jointly developed and presented. © 2006 Wiley Periodicals, Inc. J Comput Chem 27: 995,1008, 2006 [source] A fast hybrid algorithm for exoplanetary transit searchesMONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, Issue 2 2006A. Collier Cameron ABSTRACT We present a fast and efficient hybrid algorithm for selecting exoplanetary candidates from wide-field transit surveys. Our method is based on the widely used SysRem and Box Least-Squares (BLS) algorithms. Patterns of systematic error that are common to all stars on the frame are mapped and eliminated using the SysRem algorithm. The remaining systematic errors caused by spatially localized flat-fielding and other errors are quantified using a boxcar-smoothing method. We show that the dimensions of the search-parameter space can be reduced greatly by carrying out an initial BLS search on a coarse grid of reduced dimensions, followed by Newton,Raphson refinement of the transit parameters in the vicinity of the most significant solutions. We illustrate the method's operation by applying it to data from one field of the SuperWASP survey, comprising 2300 observations of 7840 stars brighter than V= 13.0. We identify 11 likely transit candidates. We reject stars that exhibit significant ellipsoidal variations caused indicative of a stellar-mass companion. We use colours and proper motions from the Two Micron All Sky Survey and USNO-B1.0 surveys to estimate the stellar parameters and the companion radius. We find that two stars showing unambiguous transit signals pass all these tests, and so qualify for detailed high-resolution spectroscopic follow-up. [source] A time-independent approach for computing wave functions of the Schrödinger,Poisson systemNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2008C.-S. Chien Abstract We describe a two-grid finite element discretization scheme for computing wave functions of the Schrödinger,Poisson (SP) system. To begin with, we compute the first k eigenpairs of the Schrödinger,Poisson eigenvalue (ESP) problem on the coarse grid using a continuation algorithm, where the nonlinear Poisson equation is solved iteratively. We use the k eigenpairs obtained on the coarse grid as initial guesses for computing their counterparts of the ESP on the fine grid. The wave functions of the SP system can be easily obtained using the formula of separation of variables. The proposed algorithm has the following advantages. (i) The initial approximate eigenpairs used in the fine grid can be obtained with low computational cost. (ii) It is unnecessary to discretize the partial derivative of the wave function with respect to the time variable in the SP system. (iii) The major computational difficulties such as closely clustered eigenvalues that occur in the SP system can be effectively computed. Numerical results on the ESP and the SP system are reported. In particular, the rate of convergence of the proposed algorithm is O(h4). Copyright © 2007 John Wiley & Sons, Ltd. [source] Parallel coarse-grid selectionNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 8 2007David M. Alber Abstract Algebraic multigrid (AMG) is a powerful linear solver with attractive parallel properties. A parallel AMG method depends on efficient, parallel implementations of the coarse-grid selection algorithms and the restriction and prolongation operator construction algorithms. In the effort to effectively and quickly select the coarse grid, a number of parallel coarsening algorithms have been developed. This paper examines the behaviour of these algorithms in depth by studying the results of several numerical experiments. In addition, new parallel coarse-grid selection algorithms are introduced and tested. Copyright © 2007 John Wiley & Sons, Ltd. [source] Total FETI for contact problems with additional nonlinearitiesPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008í Dobiá The paper is concerned with application of a new variant of the Finite Element Tearing and Interconnecting (FETI) method, referred to as the Total FETI (TFETI), to the solution to contact problems with additional nonlinearities. While the standard FETI methods assume that the prescribed Dirichlet conditions are inherited by subdomains, TFETI enforces both the compatibility between subdomains and the prescribed displacements by the Lagrange multipliers. If applied to the contact problems, this approach not only transforms the general nonpenetration constraints to the bound constraints, but it also generates an enriched natural coarse grid defined by the a priori known kernels of the stiffness matrices of the subdomains exhibiting rigid body modes. We combine our in a sense optimal algorithms for the solution to bound and equality constrained problems with geometric and material nonlinearities. The section on numerical experiments presents results of solution to bolt and nut contact problem with additional geometric and material nonlinear effects. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Comparing mass-consistent atmospheric moisture budgets on an irregular grid: An Arctic exampleTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 592 2003M. Göber Abstract We present a method to minimize the effects of different resolution and mass imbalance when comparing atmospheric energy and water budgets from different datasets. Sizeable differences between re-analysis- and radiosonde-based atmospheric budgets had been found in earlier studies and it had been suspected that the different resolutions of the datasets strongly contributes to these discrepancies. Furthermore, most studies so far had used mass-imbalanced wind fields, which can lead to serious errors. We balance the wind field by using a variational modification algorithm combined with a finite-element discretization which allows the use of data on a grid defined by the radiosonde network. This method permits the computation of flux divergences in integral form and gives a consistent numerical method to get a mass-balanced wind field with minimum modifications. Applying this method to Arctic radiosonde and re-analysis data on the same grid leads to a better agreement with respect to the horizontal distribution and the mean annual cycle of the moisture flux convergence. The constraint of mass balance on the wind field leads to a greatly reduced and more realistic variability in space and time. However, a systematic difference of about 20% remains between the estimate based on a re-analysis dataset sampled only on the coarse grid of the radiosonde network and an estimate based on the use of the full, fine grid of the re-analysis. These systematic differences can be significantly reduced by creating a simulated radiosonde dataset from the re-analysis with doubled resolution. We undertake an extensive analysis of the uncertainty of the estimates originating from the choices made in the specification of the algorithm. Based solely on radiosonde data, which are likely to result in a low bias, we estimate the net water gain of the Arctic atmosphere as 164 ± 10 mm yr,1 (0.45 ± 0.03 mm d,1) for 1979,93. Copyright © 2003 Royal Meteorological Society. [source] A Stable and Efficient Numerical Algorithm for Unconfined Aquifer AnalysisGROUND WATER, Issue 4 2009Elizabeth Keating The nonlinearity of equations governing flow in unconfined aquifers poses challenges for numerical models, particularly in field-scale applications. Existing methods are often unstable, do not converge, or require extremely fine grids and small time steps. Standard modeling procedures such as automated model calibration and Monte Carlo uncertainty analysis typically require thousands of model runs. Stable and efficient model performance is essential to these analyses. We propose a new method that offers improvements in stability and efficiency and is relatively tolerant of coarse grids. It applies a strategy similar to that in the MODFLOW code to the solution of Richard's equation with a grid-dependent pressure/saturation relationship. The method imposes a contrast between horizontal and vertical permeability in gridblocks containing the water table, does not require "dry" cells to convert to inactive cells, and allows recharge to flow through relatively dry cells to the water table. We establish the accuracy of the method by comparison to an analytical solution for radial flow to a well in an unconfined aquifer with delayed yield. Using a suite of test problems, we demonstrate the efficiencies gained in speed and accuracy over two-phase simulations, and improved stability when compared to MODFLOW. The advantages for applications to transient unconfined aquifer analysis are clearly demonstrated by our examples. We also demonstrate applicability to mixed vadose zone/saturated zone applications, including transport, and find that the method shows great promise for these types of problem as well. [source] A node-based agglomeration AMG solver for linear elasticity in thin bodiesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 3 2009Prasad S. Sumant Abstract This paper describes the development of an efficient and accurate algebraic multigrid finite element solver for analysis of linear elasticity problems in two-dimensional thin body elasticity. Such problems are commonly encountered during the analysis of thin film devices in micro-electro-mechanical systems. An algebraic multigrid based on element interpolation is adopted and streamlined for the development of the proposed solver. A new node-based agglomeration scheme is proposed for computationally efficient, aggressive and yet effective generation of coarse grids. It is demonstrated that the use of appropriate finite element discretization along with the proposed algebraic multigrid process preserves the rigid body modes that are essential for good convergence of the multigrid solution. Several case studies are taken up to validate the approach. The proposed node-based agglomeration scheme is shown to lead to development of sparse and efficient intergrid transfer operators making the overall multigrid solution process very efficient. The proposed solver is found to work very well even for Poisson's ratio >0.4. Finally, an application of the proposed solver is demonstrated through a simulation of a micro-electro-mechanical switch. Copyright © 2008 John Wiley & Sons, Ltd. [source] Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics,INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2002Mark Adams Abstract Multigrid has been a popular solver method for finite element and finite difference problems with regular grids for over 20 years. The application of multigrid to unstructured grid problems, in which it is often difficult or impossible for the application to provide coarse grids, is not as well understood. In particular, methods that are designed to require only data that are easily available in most finite element applications (i.e. fine grid data), constructing the grid transfer operators and coarse grid operators internally, are of practical interest. We investigate three unstructured multigrid methods that show promise for challenging problems in 3D elasticity: (1) non-nested geometric multigrid, (2) smoothed aggregation, and (3) plain aggregation algebraic multigrid. This paper evaluates the effectiveness of these three methods on several unstructured grid problems in 3D elasticity with up to 76 million degrees of freedom. Published in 2002 by John Wiley & Sons, Ltd. [source] Simulations of the turbulent channel flow at Re, = 180 with projection-based finite element variational multiscale methodsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 5 2007Volker John Abstract Projection-based variational multiscale (VMS) methods, within the framework of an inf,sup stable second order finite element method for the Navier,Stokes equations, are studied in simulations of the turbulent channel flow problem at Re, = 180. For comparison, the Smagorinsky large eddy simulation (LES) model with van Driest damping is included into the study. The simulations are performed on very coarse grids. The VMS methods give often considerably better results. For second order statistics, however, the differences to the reference values are sometimes rather large. The dependency of the results on parameters in the eddy viscosity model is much weaker for the VMS methods than for the Smagorinsky LES model with van Driest damping. It is shown that one uniform refinement of the coarse grids allows an underresolved direct numerical simulations (DNS). Copyright © 2007 John Wiley & Sons, Ltd. [source] Algebraic multigrid, mixed-order interpolation, and incompressible fluid flowNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2010R. Webster Abstract This paper presents the results of numerical experiments on the use of equal-order and mixed-order interpolations in algebraic multigrid (AMG) solvers for the fully coupled equations of incompressible fluid flow. Several standard test problems are addressed for Reynolds numbers spanning the laminar range. The range of unstructured meshes spans over two orders of problem size (over one order of mesh bandwidth). Deficiencies in performance are identified for AMG based on equal-order interpolations (both zero-order and first-order). They take the form of poor, fragile, mesh-dependent convergence rates. The evidence suggests that a degraded representation of the inter-field coupling in the coarse-grid approximation is the cause. Mixed-order interpolation (first-order for the vectors, zero-order for the scalars) is shown to address these deficiencies. Convergence is then robust, independent of the number of coarse grids and (almost) of the mesh bandwidth. The AMG algorithms used are reviewed. Copyright © 2009 John Wiley & Sons, Ltd. [source] Distance-two interpolation for parallel algebraic multigridNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2008Hans De Sterck Abstract Algebraic multigrid (AMG) is one of the most efficient and scalable parallel algorithms for solving sparse linear systems on unstructured grids. However, for large 3D problems, the coarse grids that are normally used in AMG often lead to growing complexity in terms of memory use and execution time per AMG V-cycle. Sparser coarse grids, such as those obtained by the parallel modified independent set (PMIS) coarsening algorithm, remedy this complexity growth but lead to nonscalable AMG convergence factors when traditional distance-one interpolation methods are used. In this paper, we study the scalability of AMG methods that combine PMIS coarse grids with long-distance interpolation methods. AMG performance and scalability are compared for previously introduced interpolation methods as well as new variants of them for a variety of relevant test problems on parallel computers. It is shown that the increased interpolation accuracy largely restores the scalability of AMG convergence factors for PMIS-coarsened grids, and in combination with complexity reducing methods, such as interpolation truncation, one obtains a class of parallel AMG methods that enjoy excellent scalability properties on large parallel computers. Copyright © 2007 John Wiley & Sons, Ltd. [source] Modifying CLJP to select grid hierarchies with lower operator complexities and better performanceNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2006David M. Alber Abstract Algebraic multigrid (AMG) is an efficient algorithm for solving certain types of large, sparse linear systems. For solving very large problems with AMG it becomes necessary to use parallel algorithms. Coarse grid selection algorithms such as CLJP were created to parallelize the setup phase of AMG. For some problems, such as those discretized on structured meshes, CLJP tends to select coarse grids with more nodes than alternative coarsening algorithms. In this paper, the cause for the selection of too many coarse nodes by CLJP is examined, and a new technique which lowers the operator complexities generated by CLJP is introduced. To validate the new method, the modified CLJP is compared to other coarsening algorithms for large-scale problems. Copyright © 2006 John Wiley & Sons, Ltd. 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