Multigrid Solution Research Articles

A One-Cell Local Multigrid Method for Solving Unsteady Incompressible Multiphase Flows

An original local multigrid method for solving incompressible two-phase flow with surface tension is described. The dynamics of the interface are resolved on a hierarchy of structured and uniform grids (orthogonal Cartesian meshes). A new type of composite boundary condition is proposed to solve the dynamics of the multigrid calculation domains. The interface tracking is described by a TVD VOF algorithm and the equations of motion are solved using an augmented Lagrangian method. The surface tension is calculated using a continuous surface force method. The one-cell local multigrid method is compared to relevant analytical scalar advection tests. Several classical two-phase flow problems, including nonlinear drop oscillations, Rayleigh–Taylor instabilities, and the drop impact on liquid film, have also been considered. The local character of the method and the differences between a single-grid and a multigrid solution are discussed. For unsteady problems, such as the Rayleigh–Taylor instability, the memory costs and the computational time have been reduced by up to 50%.

High accuracy multigrid solution of the 3D convection–diffusion equation

We present an explicit fourth-order compact finite difference scheme for approximating the three-dimensional (3D) convection–diffusion equation with variable coefficients. This 19-point formula is defined on a uniform cubic grid. Fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1. We design a parallelization-oriented multigrid method for fast solution of the resulting linear system using a four-color Gauss–Seidel relaxation technique for robustness and efficiency, and a scaled residual injection operator to reduce the cost of multigrid inter-grid transfer operator. Numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method. Several test problems are solved and highly accurate solutions of the 3D convection–diffusion equations are obtained for small to medium values of the grid Reynolds number. Effects of using different residual projection operators are compared on both vector and serial computers.

Fast multigrid solution method for nested edge-based finite element meshes

In this paper a fast multigrid solution method for edge-based finite element magnetostatic field computation with nested meshes is introduced and its efficiency is investigated. Special prolongation and restriction matrices were constructed according to the nature of the edge based field approximation. The comparison of the computation speed between the multigrid method and the ICCG method is also presented, showing that the multigrid method is very promising as a fast solution method for large system of equations.

Adaptive multigrid solution of the 2D-Euler equations on an unstructured grid

The AUSM+ (Advection Upstream Splitting Method) scheme which was developed on a structured grid has been extended to be used on an unstructured grid. The monotone linear reconstruction procedure using Green-Gauss theory is utilized to obtain second-order spatial accuracy. Barth's limiter is employed to prevent the solution from spurious oscillations. An adaptive multigrid strategy is used to accelerate the convergence. Numerical results for several test cases are presented.

A note on an accelerated high-accuracy multigrid solution of the convection-diffusion equation with high Reynolds number

A note on an accelerated high-accuracy multigrid solution of the convection-diffusion equation with high Reynolds number

A fast nested multi-grid viscous flow solver for adaptive Cartesian/Quad grids

A nested multi-grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body-fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub-division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell-cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell-centered finite volume discretization, Roe's flux splitting, a least-squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell-cutting. Several cycling strategies, such as the saw-tooth, W- and V-cycles, have been studies. The V-cycle has been found to be the most efficient. In general, the multi-grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.

Optimal injection operator and high order schemes for multigrid solution of 3D poisson equation

We present a multigrid solution of the three dimensional Poisson equation with a fourth order 19-point compact finite difference scheme. Using a red–black ordering of the grid points and some geometric considerations, we derive an optimal scaled injection operator for the multigrid algorithm. Numerical computations show that this operator yields not only the smallest overall CPU time, but also the best convergence rate compared to other more traditional projection operators. In addition, we present a family of 19-point compact schemes and numerically show that each one has a different optimal scaled injection operator.

First-Order System Least Squares for the Stokes and Linear Elasticity Equations: Further Results

First-order system least squares (FOSLS) was developed in [ SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741; SIAM J. Numer. Anal., 35 (1998), pp. 320--335] for Stokes and elasticity equations. Several new results for these methods are obtained here. First, the inverse-norm FOSLS scheme that was introduced but not analyzed in [ SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741] is shown to be continuous and coercive in the L2 norm. This result is shown to hold for pure displacement or pure traction boundary conditions in two or three dimensions, and for mixed boundary conditions in two dimensions. Next, the FOSLS schemes developed in [ SIAM J. Numer. Anal., 35 (1998), pp. 320--335] are applied to the pure displacement problem in planar and spatial linear elasticity byeliminating the pressure variable in the FOSLS formulations of [SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741]. The idea of two-dimensional variable rotation is then extended to three dimensions to make the intervariable coupling subdominant (uniformly so in the Poisson ratio for elasticity). This decoupling ensures optimal (uniform) performance of finite element discretization and multigrid solution methods. It also allows special treatment of the new trace variable, which corresponds to the divergence of velocity in the case of Stokes, so that conservation can be easily imposed, for example. Numerical results for various boundary value problems of planar linear elasticity are studied in a companion paper [SIAM J. Sci. Comput., 21 (2000), pp. 1706--1727].

A note on an accelerated high‐accuracy multigrid solution of the convection‐diffusion equation with high Reynolds number

A note on an accelerated high‐accuracy multigrid solution of the convection‐diffusion equation with high Reynolds number

First-Order System Least Squares For Linear Elasticity: Numerical Results

Two first-order system least squares (FOSLS) methods based on L2 norms are applied to various boundary value problems of planar linear elasticity. Both use finite element discretization and multigrid solution methods. They are two-stage algorithms that solve first for the displacement flux variable (the gradient of displacement, which easily yields the deformation and stress variables), then for the displacement variable itself. As a complement to a companion theoretical paper, this paper focuses on numerical results, including finite element accuracy and multigrid convergence estimates that confirm uniform optimal performance---even as the material tends to the incompressible limit.

Least Squares for the Perturbed Stokes Equations and the Reissner--Mindlin Plate

In this paper, we develop two least-squares approaches for the solution of the Stokes equations perturbed by a Laplacian term. (Such perturbed Stokes equations arise from finite element approximations of the Reissner--Mindlin plate.) Both are two-stage algorithms that solve first for the curls of the rotation of the fibers and the solenoidal part of the shear strain, then for the rotation itself (if desired). One approach uses L2 norms and the other approach uses H-1 norms to define the least-squares functionals. It is shown that the H-1 norm approach, under general assumptions, and the L2 norm approach, under certain H2 regularity assumptions, admit optimal performance for standard finite element discretization and either standard multigrid solution methods or preconditioners. These methods do not degrade when the perturbed parameter (the plate thickness) approaches zero. We also develop a three-stage least-squares method for the Reissner--Mindlin plate, which first solves for the curls of the rotation and the shear strain, next for the rotation itself, and then for the transverse displacement.

Efficient fully-coupled solution techniques for two-phase flow in porous media: Parallel multigrid solution and large scale computations

This paper is concerned with the fast resolution of nonlinear and linear algebraic equations arising from a fully implicit finite volume discretization of two-phase flow in porous media. We employ a Newton-multigrid algorithm on unstructured meshes in two and three space dimensions. The discretized operator is used for the coarse grid systems in the multigrid method. Problems with discontinuous coefficients are avoided by using a newly truncated restriction operator and an outer Krylov-space method. We show an optimal order of convergence for a wide range of two-phase flow problems including heterogeneous media and vanishing capillary pressure in an experimental way. Furthermore, we present a data parallel implementation of the algorithm with speedup results.

Multigrid solution of the incompressible Navier-Stokes equations for three-dimensional recirculating flow: coupled and decoupled smoothers compared

A comparison of multigrid methods for solving the incompressible Navier-Stokes equations in three dimensions is presented. The continuous equations are discretised on staggered grids using a second-order monotonic scheme for the convective terms and implemented in defect correction form. The convergence characteristics of a decoupled method (SIMPLE) are compared with those of the cellwise coupled method (SCGS). The convergence rates obtained for computations of the three-dimensional lid-driven cavity problem are found to be very similar to those obtained for computations of the corresponding two-dimensional problem with comparable grid density. Although the convergence rate of SCGS is thus superior to that of SIMPLE, the decoupled method is found to be more efficient computationally and requires less computing time for a given level of convergence. The linewise implementation of the coupled method (CLGS) is also investigated and shown to be more efficient than SCGS, although the convergence rate and computing time required per cycle are both found to depend on the direction of sweep

Multigrid solution of convection problems with strongly variable viscosity

Summary We investigate the robustness and efficiency of various multigrid (MG) algorithms used for simulation of thermal convection with strongly variable viscosity. We solve the hydrodynamic equations in the Boussinesq approximation, with infinite Prandtl number and temperature- and depth-dependent viscosity in two dimensions. A full approximation storage (FAS) MG method with a symmetric coupled Gauss–Seidel (SCGS) smoother on a staggered grid is used to solve the continuity and Stokes equations. Time stepping of the temperature equation is done by an alternating direction implicit (ADI) method. A systematic investigation of different variants of the algorithm shows that modifications in the MG cycle type, the viscosity restriction, the smoother and the number of smoothing operations are significant. A comparison with a well-established finite element code, utilizing direct solvers, demonstrates the potentials of our method for solving very large equation systems. We further investigate the influence of the lateral boundary conditions on the geometrical structure of convective flow. Although a strong influence exists, even in the case of very wide boxes, a systematic difference between periodic and symmetric boundary conditions, regarding the preferred width of convection cells, has not been found.

Multigrid solutions of the Euler and Navier-Stokes equations on unstructured grids

A computationally efficient multigrid algorithm for upwind edge-based finite element schemes is developed for the solution of the two-dimensional Euler and Navier–Stokes equations on unstructured triangular grids. The basic smoother is based upon a Galerkin approximation employing an edge-based formulation with the explicit addition of an upwind-type local extremum diminishing (LED) method. An explicit time stepping method is used to advance the solution towards the steady state. Fully unstructured grids are employed to increase the flexibility of the proposed algorithm. A full approximation storage (FAS) algorithm is used as the basic multigrid acceleration procedure. Copyright © 1999 John Wiley & Sons, Ltd.

Multigrid Preconditioner for Unstructured Nonlinear 3D FE Models

Multigrid and multigrid-preconditioned conjugate-gradient solution techniques applicable for unstructured 3D finite-element models that may involve sharp discontinuities in material properties, multiple element types, and contact nonlinearities are developed. Their development is driven by the desire to efficiently solve models of rigid pavement systems that require explicit modeling of spatially varying and discontinuous material properties, bending elements meshed with solid elements, and separation between the slab and subgrade. General definitions for restriction and interpolation operators applicable to models composed of multiple, displacement-based isoparametric finite-element types are proposed. Related operations are used to generate coarse mesh element properties at integration points, allowing coarse-level coefficient matrices to be computed by a simple assembly of element stiffness matrices. The proposed strategy is shown to be effective on problems involving spatially varying material properties, even in the presence of large variations within coarse mesh elements. Techniques for solving problems with nodal contact nonlinearities using the proposed multigrid methods are also described. The performance of the multigrid methods is assessed for model problems incorporating irregular meshes and spatially varying material properties, and for a model of two rigid pavement slabs subjected to thermal and axle loading that incorporates nodal contact conditions and both solid and bending elements.

Investigation of the efficiency of the multigrid method for finite element electromagnetic field computations using nested meshes

Investigation of the efficiency of the multigrid solution methods for electromagnetic field computations using nested finite element meshes is presented. Two type of multigrid algorithms, the V-cycle and the W-cycle multigrids are investigated and the results for the convergence rate of the iterative process and the computation time are compared with those of the ICCG solution method which is the commonly used solution method in finite element analysis. It is proven that the efficiency of the multigrid methods is better than that of the ICCG method especially for the solution of large systems of simultaneous algebraic equations.

Multigrid Solution of an Elliptic Boundary-Value Problem with Integral Constraints

An efficient multigrid approach for solving a discretized elliptic equation whose boundary values are determined in part by integral relations is developed, analyzed, and tested. The algorithm is motivated by a problem that is solved during integration of the 3D quasigeostrophic (QG) equations, which model large-scale rotating stratifiedflows, where the integral constraints represent mass conservation.

Comparison between experiments and numerical solutions for isothermal elastohydrodynamic point contacts

This paper presents multi-level, multi-grid solutions for isothermal elastohydrodynamic circular contacts under pure entraining motion. The solutions are valid for quasi-static conditions and Newtonian lubricants. Multi-grid solutions allow rapid evaluation of contact pressure distributions and corresponding elastic film shapes. This allows the deployment of a large number of elements, thus improving the accuracy of the numerical solution compared with that of conventional finite difference schemes. The numerical results are compared with experimental measurements and reasonable agreement has been obtained under a wide range of operating conditions.

Multigrid Solution of the Potential Field in Modeling Electrical Nerve Stimulation

In this paper, multilevel techniques are introduced as a fast numerical method to compute 3-D potential field in nerve stimulation configurations. It is shown that with these techniques the computing time is reduced significantly compared to conventional methods. Consequently, these techniques greatly enhance the possibilities for parameter studies and electrode design. Following a general description of the model of nerve stimulation configurations, the basic principles of multilevel solvers for the numerical solution of partial differential equations are briefly summarized. Subsequently, some essential elements for successful application are discussed. Finally, results are presented for the potential field in a nerve bundle induced by tripolar stimulation with a cuff electrode surrounding part of the nerve.