Multigrid Research Articles

The Numerical Solution of Poisson Equations using Wavelet based Multigrid Method

The potential energy field produced by a given charge or mass density distribution is described using the Poisson equation. This study presents a numerical solution to the Poisson equation using a wavelet-based multi-grid approach. The proposed method is superior to the existing method in terms of fast convergence and reduced processing time. Some of the problems are used to show the method's applicability and validity.

Numerical Analysis Of a Falling Circular Particle Passing through a Fluid Channel having Diamond Shaped Obstacles

It has been analyzed that the particle motion inside a vertical channel while passing across diamond shaped obstacles produces severe effects on the fluid. Particle interaction with outer boundary, internal obstacles and with the fluid is inspected. An Eulerian based approach using a computational mesh is used in which solid particles are allowed to move freely in fluid domain. Fluid and particle interaction inside the whole domain is carried using Fictitious boundary method (FBM). A multigrid finite element method combined with the fictitious boundary method (FEM-FBM) is used for the simulation of in-compressible fluid flow along with rigid particle falling and colliding inside a fluid domain. A collision model to treat the Particle-obstacle and particle-wall interactions is used to avoid particle overlapping. The particulate flow is evaluated using an open source multigrid finite element solver FEATFLOW. Numerical investigations are executed in view of different particle positions and different alignment of diamond shaped obstacles. Effects on the movement of the particle and on the interaction of the fluid-particle system due to particle-wall, particle-Obstacle, particle-fluid interactivity has been analyzed.

An efficient multigrid method for semilinear interface problems

In this paper, we study an efficient multigrid method to solve the semilinear interface problems. We first give an optimal finite element error estimate for the semilinear interface problems under a weak condition for the nonlinear term compared with the existing conclusions. Then next based on the finite element error estimate, we design a novel multigrid method for semilinear elliptic problems. The proposed multigrid method only requires to solve a linear interface problem in each level of the multilevel space sequence and a small-scale semilinear interface problem in a correction space. The involved linear interface problem can be solved efficiently by the multigrid iteration. The dimension of the correction space is small and fixed, which is independent from the fine spaces. Thus the computational time of the correction step is negligible compared with that of the linear interface problems in the fine spaces. On the whole, the efficiency of the presented multigrid method is nearly the same as that of the multigrid method for linear interface problems. Additionally, unlike the existing finite element error estimates and the multigrid methods for semilinear interface problems, which always require the bounded second order derivatives of the nonlinear terms, all the analysis in our paper only requires a Lipschitz continuous condition.

Hybrid multi-grid parallelisation of WAVEWATCH III model on spherical multiple-cell grids

Spherical Multiple-Cell (SMC) grid is an unstructured grid, supporting flexible domain shapes and multi-resolutions. It retains the quadrilateral cells as in the latitude-longitude (lat-lon) grid so that simple finite-difference schemes can be used. Sub-timesteps are applied on refined cells and grid cells are merged at high latitudes to relax the CFL restriction. A fixed reference direction is used in polar regions to solve vector polar problems. The SMC grid was implemented in the WAVEWATCH III (WW3) wave model in 2012 as an alternative for the lat-lon grid and updated in the latest WW3 V6.07. The WW3 model is parallelised by wave spectral component decomposition (CD) in MPI mode, which has a limit on number of MPI ranks. Hybrid or combined MPI-OpenMP parallelisation may extend the node usage but the OpenMP scalability flattens out beyond a few threads. Another parallelisation method that combines CD with domain decomposition (DD) is enabled in WW3 model by a multi-grid framework for further extension of node usage. Those regular lat-lon grid parallelisation options are gradually added to the SMC grid and this article reports the recent extension of the SMC grid into the multi-grid framework with hybrid parallelisation. The flexible domain shape of the SMC grid allows optimised domain splitting and minimised boundary exchanges. The combined CD-DD method is tested on SMC sub-grids with various hybrid node-thread combinations. Results indicate that switching from MPI to hybrid MPI-OpenMP mode can halve the global model elapsed time and using hybrid CD-DD on 3 SMC sub-grids may reduce it further by 30%. Elapsed time for one model day run is reduced from about 3 min on 12 nodes to less than 1 min on 90 or 180 nodes. Besides, the hybrid multi-grid method reduces memory demand on one computing node and allows future model updates for higher resolutions.

A linearly implicit scheme and fast multigrid solver for 3D Fitzhugh-Nagumo equation

In this work, a second-order finite difference (FD) scheme for three-dimensional (3D) nonlinear Fitzhugh-Nagumo (FN) equation with the nonlinear term treated with semi-implicitly technique is proposed. The existence and uniqueness of the difference scheme is proved, and the stability and convergence of numerical solution in L∞-norm are also shown. Then, we employ an efficient extrapolation cascadic multigrid (EXCMG) method to solve the large linear system arising from the proposed second-order FD discretization for the 3D FN equation. Numerical results are presented to verify our theoretical findings of the difference scheme and the efficiency of the EXCMG method. The EXCMG method can also be extended to solve other kinds of time-dependent nonlinear partial differential equations.

Lubrication characteristics of double involute gears under the tribo-dynamic condition

PurposeThe purpose of this paper is to study the lubrication characteristics of double involute gear (DIG), compare its lubrication differences under quasi-static and tribo-dynamic conditions and study the influence of different factors on its lubrication characteristics under the tribo-dynamic condition.Design/methodology/approachAccording to the meshing characteristics of DIG and elastohydrodynamic lubrication (EHL) theory, a tribo-dynamic model of DIG is established based on the “subsection method.” The Runge-Kutta method and the multigrid method are integrated to solve the model, and the dynamic analytical model of lubricating oil is established in the iterative solution.FindingsThe load and the transmission error fluctuate obviously under the tribo-dynamic condition, which is not conducive to the lubrication of DIG. The influence of rotational speeds and torque on the lubrication properties of DIG has obvious differences under tribo-dynamic and quasi-static conditions.Originality/valueThis research can provide a theoretical basis for improving the lubrication performance, reducing the loss of mechanical efficiency and improving the bearing capacity and service life of DIG.

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.

Parameter estimation with the multigrid-homotopy method for a nonlinear diffusion equation

This paper is concerned with the numerical identification of the piecewise constant permeability function for a nonlinear diffusion equation within the two-phase porous media flow. A novel iteration scheme based on the multigrid idea and homotopy method, named as the multigrid-homotopy method, has been developed for solving this problem. Numerical results indicate that this method can overcome the numerical instability and be robust to data noise. In addition, compared with the multigrid method, the multigrid-homotopy method expands the convergence domain; compared with the homotopy method, this method improves the convergence rate. The results are promising.

Schwarz methods by domain truncation

Schwarz methods use a decomposition of the computational domain into subdomains and need to impose boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and must also put boundary conditions on the computational domain boundaries. In both fields there are vast bodies of literature and research is very active and ongoing. It turns out to be fruitful to think of the domain decomposition in Schwarz methods as a truncation of the domain onto subdomains. Seminal precursors of this fundamental idea are papers by Hagstrom, Tewarson and Jazcilevich (1988), Després (1990) and Lions (1990). The first truly optimal Schwarz method that converges in a finite number of steps was proposed by Nataf (1993), and used precisely transparent boundary conditions as transmission conditions between subdomains. Approximating these transparent boundary conditions for fast convergence of Schwarz methods led to the development of optimized Schwarz methods – a name that has become common for Schwarz methods based on domain truncation. Compared to classical Schwarz methods, which use simple Dirichlet transmission conditions and have been successfully used in a wide range of applications, optimized Schwarz methods are much less well understood, mainly due to their more sophisticated transmission conditions.A key application of Schwarz methods with such sophisticated transmission conditions turned out to be time-harmonic wave propagation problems, because classical Schwarz methods simply do not work in this case. The past decade has given us many new Schwarz methods based on domain truncation. One review from an algorithmic perspective (Gander and Zhang 2019) showed the equivalence of many of these new methods to optimized Schwarz methods. The analysis of optimized Schwarz methods, however, is lagging behind their algorithmic development. The general abstract Schwarz framework cannot be used for the analysis of these methods, and thus there are many open theoretical questions about their convergence. Just as for practical multigrid methods, Fourier analysis has been instrumental for understanding the convergence of optimized Schwarz methods and for tuning their transmission conditions. Similar to local Fourier mode analysis in multigrid, the unbounded two-subdomain case is used as a model for Fourier analysis of optimized Schwarz methods due to its simplicity. Many aspects of the actual situation,e.g.boundary conditions of the original problem and the number of subdomains, were thus neglected in the unbounded two-subdomain analysis. While this gave important insight, new phenomena beyond the unbounded two-subdomain models were discovered.This present situation is the motivation for our survey: to give a comprehensive review and precise exploration of convergence behaviours of optimized Schwarz methods based on Fourier analysis, taking into account the original boundary conditions, many-subdomain decompositions and layered media. We consider as our model problem the operator$-\Delta + \eta $in the diffusive case$\eta>0$(screened Laplace equation) or the oscillatory case$\eta <0$(Helmholtz equation), in order to show the fundamental difference in behaviour of Schwarz solvers for these problems. The transmission conditions we study include the lowest-order absorbing conditions (Robin), and also more advanced perfectly matched layers (PMLs), both developed first for domain truncation. Our intensive work over the last two years on this review has led to several new results presented here for the first time: in the bounded two-subdomain analysis for the Helmholtz equation, we see strong influence of the original boundary conditions imposed on the global problem on the convergence factor of the Schwarz methods, and the asymptotic convergence factors with small overlap can differ from the unbounded two-subdomain analysis. In the many-subdomain analysis, we find the scaling with the number of subdomains,e.g.when the subdomain size is fixed, robust convergence of the double-sweep Schwarz method for the free-space wave problem, either with fixed overlap and zeroth-order Taylor conditions or with a logarithmically growing PML, and we find that Schwarz methods with PMLs work like smoothers that converge faster for higher Fourier frequencies; in particular, for the free-space wave problem, plane waves (in the error) passing through interfaces at a right angle converge more slowly. In addition to our main focus on analysis in Sections 2 and 3, we start in Section 1 with an expository historical introduction to Schwarz methods, and in Section 4 we give a brief interpretation of the recently proposed optimal Schwarz methods for decompositions with cross-points from the viewpoint of transmission conditions. We conclude in Section 5 with a summary of open research problems. In Appendix A we provide a Matlab program for a block LU form of an optimal Schwarz method with cross-points, and in Appendix B we give the Maple program for the two-subdomain Fourier analysis.

Local grid refinement in multigrid method for point contact problems of polycrystalline anisotropic material under dry and lubricated conditions

Predicting rolling bearing fatigue life requires knowledge of the three-dimensional (3D) stress fields in the roller and raceway near the lubricated contact. Owing to the increasingly severe operating conditions, the effect of localized features such as surface roughness, subsurface inclusions, and even the crystallographic structure of the material becomes important. Achieving such detail requires (locally) extremely dense gridding in simulations, which in 3D is a major challenge. Multigrid techniques have been demonstrated to be capable of solving such problems. In this study, multigrid techniques are shown to further increase the efficiency of the solution by exploiting local grid refinement while maintaining the simplicity of a uniform discretization. This is achieved by employing increasingly finer grids only locally, where the highest resolution is required. Results are presented for dry contact and elastohydrodynamically lubricated contact cases, circular as well as elliptic, with varying crystallographic structure, and with surface roughness. The results show that the developed algorithm is very well suited for detailed analysis, with also excellent prospects for computational diagnostics involving actual material crystallographic structure from electron backscatter diffraction measurements.

An efficient cascadic multigrid solver for 3-D magnetotelluric forward modelling problems using potentials

SUMMARY The fast and accurate 3-D magnetotelluric (MT) forward modelling is core engine of the interpretation and inversion of MT data. In this study, we develop an improved extrapolation cascadic multigrid method (EXCMG) to solve the large sparse complex linear system arising from the finite-element (FE) discretization on non-uniform orthogonal grids of the Maxwell’s equations using potentials. First, the vector Helmholtz equation and the scalar auxiliary equation are derived from the Maxwell’s equations using Coulomb-gauged potentials. The weighted residual method is adopted to discretize the weak formulation and assemble the FE equation. Secondly, carefully choosing the preconditioned complex stable bi-conjugate gradient method (BiCGStab) as multigrid smoother, we develop an improved EXCMG method on non-uniform grids to solve the resulting large sparse complex non-Hermitian linear systems. Finally, several examples including three standard testing models (COMMEMI3D-1, COMMEMI3D-2 and DTM1.0) and a topographic model are used to validate the accuracy and efficiency of the proposed multigrid solver. Numerical results show that the proposed EXCMG algorithm greatly improves the efficiency of 3-D MT forward modelling, is more efficient than some existing solvers, such as Pardiso, incomplete LU factorization preconditioned biconjugate gradients stabilized method (ILU-BiCGStab) and flexible generalized minimum residual method with auxiliary space Maxwell preconditioner (FGMRES-AMS), and capable to simulate large-scale problems with more than 100 million unknowns.

Smoothed aggregation for difficult stretched mesh and coefficient variation problems

AbstractFour adaptations of the smoothed aggregation algebraic multigrid (SA‐AMG) method are proposed with an eye toward improving the convergence and robustness of the solver in situations when the discretization matrix contains many weak connections. These weak connections can cause higher than expected levels of fill‐in within the coarse discretization matrices and can also give rise to suboptimal smoothing within the prolongator smoothing phase. These smoothing drawbacks are due to the relatively small size of some diagonal entries within the filtered matrix that one obtains after dropping the weak connections. The new algorithms consider modifications to the Jacobi‐like step that defines the prolongator smoother, modifications to the filtered matrix, and also direct modifications to the resulting grid transfer operators. Numerical results are given illustrating the potential benefits of the proposed adaptations.

Numerical simulations of capsule deformation in a combined shear flow and DC electric field

In this work a numerical method for fully three dimensional simulations of capsules in the electrohydrodynamic regime is proposed. A quasi-steady dual time-stepping scheme allows for iterative computation of the capsule's fluid velocity using the multigrid lattice Boltzmann method at each time step. The capsule's elasticity is computed using a linear finite element method and the membrane's bending resistance is computed from the Helfrich bending energy. The immersed interface method (IIM) is used to compute the electric field arising due to the electrical properties of the interior fluid, exterior fluid, and the membrane. The fluid structure interaction is facilitated through the immersed boundary method (IBM), which is coupled to the IIM through least squares interpolation between the control points of the IIM and the Lagrangian nodes of the IBM. The method is validated by comparing the numerical results to analytical solutions and previously published studies. The method is then used to study the deformation of capsules in a combined shear flow and DC electric field for various membrane conductances, membrane capacitances, and conductivity ratios. For nonconducting membranes the interaction between the distribution of the electric forces and the capsule inclination angle due to the shear flow result in complex equilibrium dynamics not reported for neutral capsules.

Robust Multigrid Techniques for Augmented Lagrangian Preconditioning of Incompressible Stokes Equations with Extreme Viscosity Variations

We present augmented Lagrangian Schur complement preconditioners and robust multigrid methods for incompressible Stokes problems with extreme viscosity variations. Such Stokes systems arise, for instance, upon linearization of nonlinear viscous flow problems, and they can have severely inhomogeneous and anisotropic coefficients. Using an augmented Lagrangian formulation for the incompressibility constraint makes the Schur complement easier to approximate, but results in a nearly singular (1,1)-block in the Stokes system. We present eigenvalue estimates for the quality of the Schur complement approximation. To cope with the near-singularity of the (1,1)-block, we extend a multigrid scheme with a discretization-dependent smoother and transfer operators from triangular/tetrahedral to the quadrilateral/hexahedral finite element discretizations $[\mathbb{Q}_k]^d\times \mathbb{P}_{k-1}^{\text{disc}}$, $k\geq 2$, $d=2,3$. Using numerical examples with scalar and with anisotropic fourth-order tensor viscosity arising from linearization of a viscoplastic constitutive relation, we confirm the robustness of the multigrid scheme and the overall efficiency of the solver. We present scalability results using up to 28,672 parallel tasks for problems with up to 1.6 billion unknowns and a viscosity contrast up to ten orders of magnitude.

Multiscale coupling of FFT-based simulations with the LDC approach

FFT based solvers have proven to be efficient computational methods for the numerical simulations of composite materials with complex microstructures and sophisticated material behavior. Nevertheless, these solvers require a regular discretization of the investigated domain and periodic boundary conditions.Local multi-grid methods appear as an appropriate framework to perform simulations involving local refinement with FFT solvers. A coarse (global) discretization is improved by one or several local refined discretizations defined in subdomains overlapping the global grid. The Local Defect Correction (LDC) method is adapted to couple the resolution on the global coarse grid problem with the resolution of the local grids problems. Numerical tests demonstrate a significant reduction of CPU time and memory consumption with a relatively low loss in accuracy in comparison to the computation on one fully refined global grid.This approach also provides a baseline for extensions to the coupling of FFT codes with other solvers in a multi-scale and multi-physics framework.

Two Methods to Improve the Efficiency of Supersonic Flow Simulation on Unstructured Grids

The paper presents two methods to improve the efficiency of supersonic flow simulation using arbitrarily shaped unstructured grids. The first method promotes increasing the numerical solution convergence rate and is based on the geometric multigrid method for initialization of the flow field. The method is used to obtain the initial field of distributed physical quantity values, which maximally corresponds to the converged solution. For this purpose, the problem simulation is performed on a series of coarse grids beginning from the coarsest one in this series. Upon completion of simulations, the solution obtained is interpolated to a finer grid and used for initialization of simulations on this grid. The second method allows increasing the numerical solution accuracy and is based on statically adapting the computational grid to the flow specifics. The static adaptation algorithm provides automatic refinement of the computational grid in the region of specific features of flow, such as shock waves typical for supersonic flows. This algorithm provides a better description of the shock-wave front owing to the local grid refinement, with the local refinement region being automatically selected. Results of using these methods are demonstrated for the two supersonic aerodynamics problems: the simulation of the bow shock strength at a given distance under axially symmetric body Seeb-ALR and a mock-up aircraft Lockheed Martin 1021. It is shown that in both cases, the numerical solution convergence rate is increased owing to the use of the geometric multigrid method for initialization and a higher quality and a higher accuracy of solution is gained owing to the local grid refinement (using static adaptation means) near the shock-wave front.

A Comparison of Linear Solvers for Resolving Flow in Three‐Dimensional Discrete Fracture Networks

AbstractWe compare various methods for resolving steady flow within three‐dimensional discrete fracture networks, including direct methods, Krylov subspace methods with and without preconditioning, and multi‐grid methods. We compared the performance of the methods based on compute times and scaling of the solution as a function of the number of grid nodes and log‐variance of the hydraulic aperture. The methods are applied to three test cases: (a) variable density of networks with a truncated power‐law distribution of fracture lengths, (b) a fixed network composed of monodisperse fracture sizes but varied permeability/aperture heterogeneity, (c) and a network based on field site in Nevada, US. We chose these cases to allow us to study the impact of the mesh size and flow properties, as well as to demonstrate our conclusions on a large‐scale, realistic problem (more than 40 million mesh nodes). A direct solution using Cholesky factorization outperformed other methods for every example but was closely followed in performance by some algebraic multigrid (AMG) preconditioned Krylov subspace methods. Among the Krylov methods, conjugate gradients (CG) with an AMG preconditioner performs the best. Generally, Cholesky factorization is recommended, but CG with an AMG preconditioner may be suitable for very large problems beyond 40 million nodes where the entire linear system cannot reside in memory.

Efficient Solution of 3D Elasticity Problems with Smoothed Aggregation Algebraic Multigrid and Block Arithmetics

Efficient Solution of 3D Elasticity Problems with Smoothed Aggregation Algebraic Multigrid and Block Arithmetics

An aggregation-based nonlinear multigrid solver for two-phase flow and transport in porous media

A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. The resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newton's method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newton's method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.

Scalable preconditioning for the stabilized contact mechanics problem

We present a family of preconditioning strategies for the contact problem in fractured and faulted porous media. We combine low-order continuous finite elements to simulate the bulk deformation with piecewise constant Lagrange multipliers to impose the frictional contact constraints. This formulation is not uniformly inf-sup stable and requires stabilization. We improve previous work by Franceschini et al. (2020) by introducing a novel jump stabilization technique that requires only local geometrical and mechanical properties. We then design scalable preconditioning strategies that take advantage of the block structure of the Jacobian matrix using a physics-based partitioning of the unknowns by field type, namely displacement and Lagrange multipliers. The key to the success of the proposed preconditioners is a pseudo-Schur complement obtained by eliminating the Lagrange multiplier degrees of freedom, which can then be efficiently solved using an optimal multigrid method. Numerical results, including complex real-world problems, are presented to illustrate theoretical properties, scalability and robustness of the preconditioner. A comparison with other approaches available in the literature is also provided.