Algebraic Multigrid Approach Research Articles

Integrated wellbore-reservoir modeling based on 3D Navier–Stokes equations with a coupled CFD solver

The occurrence of fluid flow near a wellhead is the major concern of the petroleum industry, as pressure drop, loss of formation, and other variables of interest are mostly affected in this region. The fluid flows from the hydrocarbon reservoir to the wellbore can be characterized as laminar to turbulent; thus, it is important to model this phenomenon with the integrated wellbore-reservoir model. Using 3D Navier–Stokes equations, an integrated wellbore-reservoir model is created in this study, and it incorporates the formation damage zone. For the porous-porous and porous-fluid interfaces, the General Grid Interface (GGI) approach is applied in conjunction with the conservative mass flux interface model. Model equations are solved using a velocity-pressure coupling solver that is pressure-based. For reliable and quick results, the system of equations is solved using an algebraic multigrid approach. The pressure diffusivity equation’s analytical solution under steady-state flow circumstances is used to validate the model. The integrated wellbore-reservoir model is applied to different reservoir scenarios, for example, different production rates, formation zones, and reservoir formation conditions. The results indicate that the present Computational Fluid Dynamics (CFD) model can be extended to simulate the real field scale model. integrated wellbore-reservoir modeling based on 3D Navier–Stokes equations with efficient computational techniques can lead the field of petroleum industries to advance current knowledge.

A scalable preconditioning framework for stabilized contact mechanics with hydraulically active fractures

A preconditioning framework for the coupled problem of frictional contact mechanics and fluid flow in the fracture network is presented. We focus on a blended finite element/finite volume method, where the porous medium is discretized by low-order continuous finite elements with nodal unknowns, cell-centered Lagrange multipliers are used to prescribe the contact constraints, and the fluid flow in the fractures is described by a classical two-point flux approximation scheme. This formulation is consistent, but is not uniformly inf-sup bounded and requires a stabilization. For the resulting 3×3 block Jacobian matrix, robust and efficient solution methods are not available, so we aim at designing new scalable preconditioning strategies based on the physically-informed block partitioning of the unknowns and state-of-the-art multigrid techniques. The key idea is to restrict the system to a single-physics problem, approximately solve it by an inner algebraic multigrid approach, and finally prolong it back to the fully-coupled problem. Two different techniques are presented, analyzed and compared by changing the ordering of the restrictions. Numerical results illustrate the algorithmic scalability, the impact of the relative number of fracture-based unknowns, and the performance on a benchmark problem. The objective of the analysis is to identify the most promising solution strategy.

Coarsening in algebraic multigrid using Gaussian processes

Multigrid methods have proven to be an invaluable tool to efficiently solve large sparse linear systems arising in the discretization of Partial Differential Equations (PDEs). Algebraic multigrid methods and in particular adaptive algebraic multigrid approaches have shown that multigrid efficiency can be obtained without having to resort to properties of the PDE. Yet the required setup of these methods poses a not negligible overhead cost. Methods from machine learning have attracted attention to streamline processes based on statistical models being trained on the available data. Interpreting algebraically smooth error as an instance of a Gaussian process, we develop a new, data driven approach to construct adaptive algebraic multigrid methods. Based on Gaussian a priori distributions, kriging interpolation minimizes the mean squared error of the a posteriori distribution, given the data on the coarse grid. Going one step further, we exploit the quantification of uncertainty in the Gaussian process model in order to construct efficient variable splittings. Using a semivariogram fit of a suitable covariance model we demonstrate that our approach yields efficient methods using a single algebraically smooth vector.

A Novel Algebraic Multigrid Approach Based on Adaptive Smoothing and Prolongation for Ill-Conditioned Systems

The numerical simulation of modern engineering problems can easily incorporate millions or even billions of unknowns. In several applications, sparse linear systems with symmetric positive definite...

A line-by-line hybrid unstructured finite volume/Monte Carlo method for radiation transfer in 3D non-gray medium

A hybrid method combing the unstructured finite volume method and the Monte Carlo method and incorporating the line-by-line model has been developed to simulate the radiative transfer in highly spectral and inhomogeneous medium. In this method, the unstructured finite volume method is adopted to solve the spectral radiative transfer equation at wave numbers or spectral locations determined by the Monte Carlo method. The Monte Carlo method takes effects by firstly defining the monotonic random number relations corresponding to the spectral emitted power density of every discretized element of the concerning medium, and then by reversing the spectral location through comparison of these relations with predefined random numbers. Through this Monte Carlo method, the actual number of spectral locations on which the spectral radiative transfer equations are solved may be reduced: only the spectral locations that have higher spectral emissive powers would be more possibly selected. To increase the performance of the presented method, the total variation diminishing scheme on unstructured grids is adopted in treating the spectral radiative intensity at interface between control volumes. And, the discretized radiative transfer equation is implicitly and iteratively solved by an algebraic multi-grid solution approach to accelerate the convergence of the equation. The presented method was applied to 3D homogeneous and inhomogeneous cases for the validation and performance studies. Results show that for both cases, the presented method agree well with pure Monte Carlo benchmark solutions with acceptable number of spectral locations and computing time.

A Fluidity-Based First-Order System Least-Squares Method for Ice Sheets

This paper describes two first-order system least-squares (FOSLS) formulations of a nonlinear Stokes flow model for glaciers and ice sheets. In Glen's law, the most commonly used constitutive equation for ice rheology, the ice viscosity becomes infinite as the velocity gradients (strain rates) approach zero, which typically occurs near the ice surface where deformation rates are low, or when the basal slip velocities are high. The computational difficulties associated with the infinite viscosity are often overcome by an arbitrary modification of Glen's law that bounds the maximum viscosity. In this paper, two FOSLS formulations (the viscosity formulation and fluidity formulation) are presented. The viscosity formulation is a FOSLS representation of the standard nonlinear Stokes problem. The new fluidity formulation exploits the fact that only the product of the viscosity and strain rate appears in the nonlinear Stokes problem, a quantity that, in fact, approaches zero as the strain rate goes to zero. The fluidity formulation is expressed in terms of a new set of variables and overcomes the problem of infinite viscosity. The new formulation is well posed, and linearizations are essentially $H^1$-elliptic around approximations for which the viscosity is bounded. A nested iteration (NI) Newton-FOSLS-AMG (algebraic multigrid) approach is used to solve both nonlinear Stokes problems, in which most of the iterations are performed on the coarsest grid. Both formulations demonstrate optimal finite element convergence. However, the fluidity formulation is more accurate. The fluidity formulation results in linear systems that are more efficiently solved by AMG.

Solver preconditioning using the combinatorial multilevel method on reservoir simulation

The purpose of this paper is to report the first preliminary study of the recently introduced combinatorial multilevel (CML) method for solver preconditioning in large-scale reservoir simulation with coupled geomechanics. The CML method is a variant of the popular algebraic multigrid (AMG) method yet with essential differences. The basic idea of this new approach is to construct a hierarchy of matrices using the discrete geometry of the graph, based on support theory for preconditioners. In this way, the CML method combines the merits of both geometric and algebraic multigrid methods. The resulting hybrid approach not only provides a simpler and faster setup phase compared to AMG, but the method can be proven to exhibit strong convergence guarantees for arbitrary symmetric diagonally dominant matrices. In addition, the underlying theoretical soundness of the CML method contrasts to the heuristic AMG approach, which often can show slow convergence for difficult problems. This new approach is implemented in a reservoir simulator for both pressure and poroelastic displacement preconditioners in the multistage preconditioning technique. We present results based on several known benchmark problems and provide a comparison of performance and complexity with the widespread preconditioning schemes used in large-scale reservoir simulation. An adaptation of CML for non-symmetric matrices is shown to exhibit excellent convergence properties for realistic cases.

A new algebraic multigrid approach for Stokes problems

Standard discretizations of Stokes problems lead to linear systems of equations in saddle point form, making difficult the application of algebraic multigrid methods. In this paper, a new approach ...

Algebraic Multigrid Domain and Range Decomposition (AMG-DD/AMG-RD)

In modern large-scale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMG-DD) and algebraic multigrid range decomposition (AMG-RD), that replace traditional AMG V-cycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135--155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224--233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 1411--1443], they differ primarily in that they are purely algebraic: AMG-RD and AMG-DD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMG-RD and AMG-DD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the all-to-all communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. To this end, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.

Aggregation based on graph matching and inexact coarse grid solve for algebraic two grid

A graph matching is used to construct aggregation-based coarsening for an algebraic two-grid method. Effects of inexact coarse grid solve is analysed numerically for a highly discontinuous convection–diffusion coefficient matrix, and for problems from the Florida matrix market collection. The proposed strategy is found to be more robust compared to a classical algebraic multi-grid approach based on strength of connections. Basic properties of two-grid method are outlined.

An algebraic multigrid approach to solve extended finite element method based fracture problems

SUMMARYThis article proposes an algebraic multigrid (AMG) approach to solve linear systems arising from applications where strong discontinuities are modeled by the extended finite element method. The application of AMG methods promises optimal scalability for solving large linear systems. However, the straightforward (or ‘black‐box’) use of existing AMG techniques for extended finite element method problems is often problematic. In this paper, we highlight the reasons for this behavior and propose a relatively simple adaptation that allows one to leverage existing AMG software mostly unchanged. Numerical tests demonstrate that optimal iterative convergence rates can be attained that are comparable with AMG convergence rates associated with linear systems for standard finite element approximations without discontinuities. Published 2012. This article is a US Government work and is in the public domain in the USA.

Efficient iterative solvers for time-harmonic Maxwell equations using domain decomposition and algebraic multigrid

Paper presents a set of parallel iterative solvers and preconditioners for the efficient solution of systems of linear equations arising in the high order finite-element approximations of boundary value problems for 3-D time-harmonic Maxwell equations on unstructured tetrahedral grids. Balancing geometric domain decomposition techniques combined with algebraic multigrid approach and coarse-grid correction using hierarchic basis functions are exploited to achieve high performance of the solvers and small memory load on the supercomputers with shared and distributed memory. Testing results for model and real-life problems show the efficiency and scalability of the presented algorithms.

Robust and adaptive multigrid methods: comparing structured and algebraic approaches

SUMMARYAlthough there have been significant advances in robust algebraic multigrid (AMG) methods in recent years, numerical studies and emerging hardware architectures continue to favor structured‐grid approaches. Specifically, implementations of logically structured robust variational multigrid algorithms, such as the black box multigrid (BoxMG) solver, have been shown to be 10 times faster than AMG for three‐dimensional heterogeneous diffusion problems on structured grids. BoxMG offers important features such as operator‐induced interpolation for robustness, while taking advantage of direct data access and bounded complexity in the Galerkin coarse‐grid operator. Moreover, because BoxMG uses a variational framework, it can be used to explore advances of modern adaptive AMG approaches in a structured setting. In this paper, we show how to extend the adaptive multigrid methodology to the BoxMG setting. This extension not only retains the favorable properties of the adaptive framework but also sheds light on the relationship between BoxMG and AMG. In particular, we show how classical BoxMG can be viewed as a special case of classical AMG and how this viewpoint leads to a richer family of adaptive BoxMG approaches. We present numerical results that explore this family of adaptive methods and compare its robustness and efficiency to the classical BoxMG solver.Copyright © 2012 John Wiley & Sons, Ltd.

Algebraic multigrid and 4th‐order discrete‐difference equations of incompressible fluid flow

AbstractThis paper investigates the effectiveness of two different Algebraic Multigrid (AMG) approaches to the solution of 4th‐order discrete‐difference equations for incompressible fluid flow (in this case for a discrete, scalar, stream‐function field). One is based on a classical, algebraic multigrid, method (C‐AMG) the other is based on a smoothed‐aggregation method for 4th‐order problems (SA‐AMG). In the C‐AMG case, the inter‐grid transfer operators are enhanced using Jacobi relaxation. In the SA‐AMG case, they are improved using a constrained energy optimization of the coarse‐grid basis functions. Both approaches are shown to be effective for discretizations based on uniform, structured and unstructured, meshes. They both give good convergence factors that are largely independent of the mesh size/bandwidth. The SA‐AMG approach, however, is more costly both in storage and operations. The Jacobi‐relaxed C‐AMG approach is faster, by a factor of between 2 and 4 for two‐dimensional problems, even though its reduction factors are inferior to those of SA‐AMG.For non‐uniform meshes, the accuracy of this particular discretization degrades from 2nd to 1st order and the convergence factors for both methods then become mesh dependent. Copyright © 2009 John Wiley & Sons, Ltd.

Multilevel algorithms for linear ordering problems

Linear ordering problems are combinatorial optimization problems that deal with the minimization of different functionals by finding a suitable permutation of the graph vertices. These problems are widely used and studied in many practical and theoretical applications. In this paper, we present a variety of linear--time algorithms for these problems inspired by the Algebraic Multigrid approach, which is based on weighted-edge contraction. The experimental result for four such problems turned out to be better than every known result in almost all cases, while the short (linear) running time of the algorithms enables testing very large graphs.

The Wall Effect on Ventilated Cavitating Flows in Closed Cavitation Tunnels

For ventilated cavitating flows in a closed water tunnel, the wall effect may exert an important influence on cavity shape and hydrodynamics. An isotropic mixture multiphase model was established to study the wall effect based on the RANS equations, coupled with a natural cavitation model and the RNG κ-Σ turbulent model. The governing equations were discretized using the finite volume method and solved by the Gauss-Seidel linear equation solver on the basis of a segregation algorithm. The algebraic multigrid approach was carried through to accelerate the convergence of solution. The steady ventilated cavitating flows in water tunnels of different diameter were simulated for a conceptual underwater vehicle model which had a disk cavitator. It is found that the choked cavitation number derived is close to the approximate solution of natural cavitating flow for a 3-D disk. The critical ventilation rate falls with decreasing diameter of the water tunnel. However, the cavity size and drag coefficient are rising with the decrease in tunnel diameter for the same ventilation rate, and the cavity size will be much different in water tunnels of different diameter even for the same ventilated cavitation number.

Inhomogeneous Multifluid Model for Prediction of Nonequilibrium Phase Transition and Droplet Dynamics

A pressure based Eulerian multifluid model for application to phase transition with droplet dynamics in transonic high-speed flows is described. It is implemented using an element-based finite-volume method, which is implicit in time and solves mass and momentum conservation across all phases via a coupled algebraic multigrid approach. The model emphasizes treatment of the condensed phases, with their respective velocity and thermal fields, in inertial nonequilibrium and metastable gas flow conditions. The droplet energy state is treated either in algebraic form or through transport equations depending on appropriate physical assumptions. Due to the complexity of the two-phase phenomena, the model is presented and validated by exploring phase transition and droplet dynamics in a turbine cascade geometry. The influence of droplet inertia on localized homogeneous nucleation is examined.

An Algebraic Multigrid Approach Based on a Compatible Gauge Reformulation of Maxwell's Equations

With the rise in popularity of compatible finite element, finite difference, and finite volume discretizations for the time domain eddy current equations, there has been a corresponding need for fast solvers of the resulting linear algebraic systems. However, the traits that make compatible discretizations a preferred choice for the Maxwell's equations also render these linear systems essentially intractable by truly black-box techniques. We propose an algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid (AMG) technique for this reformulated problem. The reformulation process takes advantage of a discrete Hodge decomposition on cochains to replace the discrete eddy current equations by an equivalent $2\times2$ block linear system whose diagonal blocks are discrete Hodge–Laplace operators acting on 1-cochains and 0-cochains, respectively. While this new AMG technique requires somewhat specialized treatment on the finest mesh, the coarser meshes can be handled using standard methods for Laplace-type problems. Our new AMG method is applicable to a wide range of compatible methods on structured and unstructured grids, including edge finite elements, mimetic finite differences, covolume methods, and Yee-like schemes. We illustrate the new technique, using edge elements in the context of smoothed aggregation AMG, and present computational results for problems in both two and three dimensions.

A Multilevel Algorithm for the Minimum 2-sum Problem

In this paper we introduce a direct motivation for solving the minimum 2-sum problem, for which we present a linear-time algorithm inspired by the Algebraic Multigrid approach which is based on weighted edge contraction. Our results turned out to be better than previous results, while the short running time of the algorithm enabled experiments with very large graphs. We thus introduce a new benchmark for the minimum 2-sum problem which contains 66 graphs of various characteristics. In addition, we propose the straightforward use of a part of our algorithm as a powerful local reordering method for any other (than multilevel) framework.

Large-scale kinetics modeling of non-LTE plasmas

Improvements continue to be made in the modeling capabilities provided by the Los Alamos National Laboratory (LANL) suite of atomic data codes. Recent work includes the addition of parallel rate matrix solvers when obtaining the atomic populations necessary to model non-LTE plasmas and the use of various interpolation techniques to obtain the vast amounts of atomic cross sections and rates that are required to build the data models. These enhancements have enabled the creation of models that contain more than one million relativistic configurations. The ATOMIC code has recently been used to synthesize spectra from such large models with two completely independent, sparse solvers: a serial, iterative, generalized minimum residual algorithm and a parallel, iterative, algebraic multigrid approach. The populations resulting from these two methods can be compared directly in order to determine how sensitive is the solution to the positioning of the usual boundary condition associated with charge neutrality. This sensitivity is an important concern when such large systems of equations are being considered. Additionally, this enhancement in model size can be used to better determine the convergence of calculated spectra with respect to model size. As an application of these capabilities, comparisons with a recent xenon experiment are discussed.