Multigrid Research Articles

A new transient model for motion and variation of gas phase in pipelines applying collocated multi-grid method with the Eulerian-Lagrangian approach

This paper investigates the gas phase, including bubbles' and gas pockets' migration and evolution in gas-liquid flow in the pipelines, where the safe operation is severely influenced by the gas accumulations. To investigate the formation, as well as location and amount of these gas accumulations, a new transient model applied with the Population Balance Model (PBM), gas pockets capturing and tracking model are proposed, which are discretized and calculated in the collocated multi-grid with Eulerian-Lagrangian coupling scheme. The flow field of the pipeline is modeled with the Eulerian scheme, while the bubbles' migration, gas pockets' capturing and tracking can be modeled in the Lagrangian mesh. To couple the description of bubble-dimension and that of pipe-dimension, the Rayleith-Plesset equation and the Smooth Particle Hydrodynamics (SPH) are applied, thus, the changeable characteristics of bubbles along the pipes can be described. Based on that, the bubbles' transition to the gas pocket under specific conditions can be predicted. And the subsequent capturing and tracking process will work with the help of the collocated multi-grid, which provides the accurate and parallel calculation. The relative parameters are extracted from the published experimental data, and the simulation results are in good agreement with the measured results. Meanwhile, the case studies show the model has the prediction ability of the gas phase's migration and evolution, as well as the formation and location of gas accumulations in the pipeline.

Towards a Multigrid Method for the M1 Model for Radiative Transfer

Towards a Multigrid Method for the M1 Model for Radiative Transfer

Hydrodynamic Lubrication of Partially Textured Gas Parallel Slider Bearings with Orientation Ellipse Dimples

The hydrodynamic lubrication performance of partially textured gas parallel slider bearings with orientation ellipse dimples is investigated in this paper. By using the multigrid finite element method, the pressure distribution between a partially textured slider and a smooth slider is obtained. The geometric parameters of the ellipse dimples are optimized to maximize the average pressure under a given sliding speed. The numerical results show that geometric parameters such as orientation angle, depth, slender ratio, and area density have an important impact on hydrodynamic pressure. Besides, the effect of textured fraction on hydrodynamic pressure is investigated under a given sliding speed. It is observed that the optimum textured fraction for maximizing the average pressure is dependent on the sliding speed.

A mass conserving arbitrary Lagrangian–Eulerian formulation for three‐dimensional multiphase fluid flows

AbstractThe arbitrary Lagrangian–Eulerian (ALE) framework presented in [Sahin and Guventurk, An arbitrary Lagrangian–Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem.Int J Numer Methods Eng. 2017;112:2110–2134] has been extended to simulate three‐dimensional immiscible incompressible multiphase fluid flows. The div‐stable side centered unstructured finite volume formulation is used for the discretization of the incompressible Navier–Stokes equations. A novel kinematic boundary condition that satisfies the local discrete geometric conservation law (DGCL) is implemented to ensure the mass conservation of each species at the machine precision. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. Surface tension force is treated as a tangent force and discretized in a semi‐implicit form. Two different approaches for the computation of unit normal vector have been implemented: the least squares biquadratic surface fitting (LSBSF) and the mean weighted by sine and edge length reciprocals (MWSELR). The resulting large system of algebraic equations is solved in a fully coupled manner in order to improve the time step restrictions. As a preconditioner, an approximate matrix factorization similar to that of the projection method is employed and the parallel algebraic multigrid solver BoomerAMG provided by the HYPRE library, which is accessed through the PETSc library, has been utilized for the scaled discrete Laplacian of pressure and the diagonal blocks of mesh deformation equations. The accuracy of the proposed method is initially validated on the static bubble problem, since the surface tension force is highly sensitive to the accurate evaluation of the unit normal vector and the inaccuracies significantly contribute to unphysical velocities, called parasitic currents. The calculations indicate that the parasitic currents can be reduced to machine precision for the MWSELR method. Then, the proposed approach is applied to the single bubble rising in a viscous quiescent liquid for both low and high density ratios. The calculations produce accurate predictions of the bubble shape, center of mass, rise velocity, and so forth. Furthermore, the mass of each species is conserved at machine precision and discontinuous pressure field is obtained in order to avoid errors due to the incompressibility restriction in the vicinity of liquid‐liquid interfaces at large density and viscosity ratios.

Low‐order preconditioning of the Stokes equations

AbstractA well‐known strategy for building effective preconditioners for higher‐order discretizations of some PDEs, such as Poisson's equation, is to leverage effective preconditioners for their low‐order analogs. In this work, we show that high‐quality preconditioners can also be derived for the Taylor–Hood discretization of the Stokes equations in much the same manner. In particular, we investigate the use of geometric multigrid based on the discretization of the Stokes operator as a preconditioner for the discretization of the Stokes system. We utilize local Fourier analysis to optimize the damping parameters for Vanka and Braess–Sarazin relaxation schemes and to achieve robust convergence. These results are then verified and compared against the measured multigrid performance. While geometric multigrid can be applied directly to the system, our ultimate motivation is to apply algebraic multigrid within solvers for systems via the discretization, which will be considered in a companion paper.

Parallel finite volume simulation of the spherical shell dynamo with pseudo-vacuum magnetic boundary conditions

In this paper, we study the parallel simulation of the magnetohydrodynamic (MHD) dynamo in a rapidly rotating spherical shell with pseudo-vacuum magnetic boundary conditions. A second-order finite volume scheme based on a collocated quasi-uniform cubed-sphere grid is applied to the spatial discretization of the MHD dynamo equations. To ensure the solenoidal condition of the magnetic field, we adopt a widely-used approach whereby a pseudo-pressure is introduced into the induction equation. The temporal integration is split by a second-order approximate factorization approach, resulting in two linear algebraic systems both solved by a preconditioned Krylov subspace iterative method. A multi-level restricted additive Schwarz preconditioner based on domain decomposition and multigrid method is then designed to improve the efficiency and scalability. Accurate numerical solutions of two benchmark cases are obtained with our code, comparable to the existing local method results. Several large-scale tests performed on the Sunway TaihuLight supercomputer show good strong and weak scalabilities and a noticeable improvement from the multi-level preconditioner with up to 10368 processor cores.

Homogeneous multigrid for embedded discontinuous Galerkin methods

We introduce a homogeneous multigrid method in the sense that it uses the same embedded discontinuous Galerkin (EDG) discretization scheme for Poisson's equation on all levels. In particular, we use the injection operator developed in [LuRK2020] for HDG and prove optimal convergence of the method under the assumption of elliptic regularity. Numerical experiments underline our analytical findings.

Efficient Simulations of Large Scale Convective Heat Transfer Problems

We describe an approach for efficient solution of large scale convective heat transfer problems, formulated as coupled unsteady heat conduction and incompressible fluid flow equations. The original problem is discretized in time using classical implicit methods, while stabilized finite elements are used for space discretization. The algorithm employed for the discretization of the fluid flow problem uses Picard's iterations to solve the arising nonlinear equations. Both problems, heat transfer and Navier-Stokes quations, give rise to large sparse systems of linear equations. The systems are solved using iterative GMRES solver with suitable preconditioning. For the incompressible flow equations we employ a special preconditioner based on algebraic multigrid (AMG) technique. The paper presents algorithmic and implementation details of the solution procedure, which is suitably tuned, especially for ill conditioned systems arising from discretizations of incompressible Navier-Stokes equations. We describe parallel implementation of the solver using MPI and elements of PETSC library. The scalability of the solver is favourably compared with other methods such as direct solvers and standard GMRES method with ILU preconditioning.

Thermal elastohydrodynamic lubrication characteristics and optimisation of the ball-type tripod universal joint

Purpose This paper aims to study the influence of three-column groove shell radius, ball radius, lubricating oil viscosity and elastic modulus on the thermal elastohydrodynamic lubrication (TEHL) characteristics and optimisation of the ball-type tripod universal joint. Design/methodology/approach The point contact TEHL model of the joint was developed, and the multi-grid method was used to solve it. The influence of three-column groove shell radius, ball radius, lubricating oil viscosity and elastic modulus on the lubrication characteristics was analysed. Further, the optimisation of the joint TEHL performance was carried out by the Kriging approximation model combined with the multi-objective particle swarm optimisation (MOPSO) algorithm. Findings The research results show that increasing groove shell radius and ball radius can effectively increase the oil film thickness, and decrease the oil film pressure, as well as the temperature rise. Decreasing elastic modulus can reduce the oil film temperature rise and pressure, and increasing viscosity can effectively increase the oil film thickness. The optimised minimum oil film thickness increases by 33.23% and the optimised maximum oil film pressure and maximum temperature rise decrease by 11.92% and 28.87%, respectively. Furthermore, the relative error of each response output is less than 10%. Originality/value This study applies TEHL theory to the tribological research of the ball-type tripod universal joint, and the joint’s lubrication performance is improved greatly by the Kriging model and MOPSO algorithm, which provides an effective measure to raise the joint’s working efficiency.

Convergence of multilevel spectral deferred corrections

The spectral deferred correction (SDC) method is class of iterative solvers for ordinary differential equations (ODEs). It can be interpreted as a preconditioned Picard iteration for the collocation problem. The convergence of this method is well-known, for suitable problems it gains one order per iteration up to the order of the quadrature method of the collocation problem provided. This appealing feature enables an easy creation of flexible, high-order accurate methods for ODEs. A variation of SDC are multi-level spectral deferred corrections (MLSDC). Here, iterations are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. While there are several numerical examples which show its capabilities and efficiency, a theoretical convergence proof is still missing. This paper addresses this issue. A proof of the convergence of MLSDC, including the determination of the convergence rate in the time-step size, will be given and the results of the theoretical analysis will be numerically demonstrated. It turns out that there are restrictions for the advantages of this method over SDC regarding the convergence rate.

Partial Reuse AMG Setup Cost Amortization Strategy for the Solution of Non-Steady State Problems

The partial reuse algebraic multigrid (AMG) setup cost amortization strategy is presented for the solution of non-steady state problems. The transfer operators are reused from the previous time steps, and the system matrices and the smoother operators are rebuilt on each of the AMG hierarchy levels. It is shown on the example of modelling a two-fluid dam break scenario that the strategy may decrease the AMG preconditioner setup cost by 40% to 200%. The total compute time is decreased by up to 20%, but the specific outcome depends on the fraction of time that the setup step initially takes.

ANACIN-X: A software framework for studying non-determinism in MPI applications

<h2>Abstract</h2> Non-determinism in MPI applications negatively impacts the debugging and correctness of high performance computing (HPC) and scientific simulations. To help developers and scientists study and understand sources of non-determinism, we present ANACIN-X, a software framework for quantifying point-to-point communication non-determinism in MPI applications as graph kernel distances. ANACIN-X includes three benchmark applications for testing (i.e., message race, algebraic multigrid, and unstructured mesh).

Geometric multigrid with multiplicative Schwarz smoothers for eddy-current and Maxwell’s equations in deal.II

In this work, we consider the design of a geometric multigrid method with multiplicative Schwarz smoothers for the eddy-current problem and the time-harmonic Maxwell equations. The main purpose is to show numerically that a straightforward application works for the former problem, but not for the latter. The well-known key is a special decomposition of the function spaces within the multigrid algorithm. The failures and performance are shown with the help of a numerical test, implemented in the modern finite element library deal.II, including a github link to the eddy-current implementation.

An efficient matrix-free preconditioned conjugate gradient based multigrid method for phase field modeling of fracture in heterogeneous materials from 3D images

In this paper, we present a new and efficient strategy to perform 3D computational fracture modeling from computed tomography (CT) images. The image-based fracture modeling creates a new way to more accurately predict fracture in realistic structures. However, it is currently complex and expensive to perform such simulations. In this work, we propose to use a matrix-free type preconditioned conjugate gradient solver based multigrid method to achieve the fastest numerical performance for phase field modeling of fracture. Several specific methods are investigated to enhance the robustness and to improve the efficiency of the proposed strategy. An automatic load control strategy to control crack propagation is investigated. High performance computing is applied using a hybrid MPI/OpenMP strategy. With all of the proposed procedures, the phase field modeling of fracture can be performed in the real 3D microstructure of heterogeneous materials using tomographic images.

Novel accurate modeling of dust loaded wire-duct precipitators using FDM-FMG method on one fine computational domains

Worldwide, electrostatic precipitators (ESP) have been extensively utilized to separate fine particles for diverse large-scale industrial applications. In this regard, this paper presents a novel approach for modeling the dust-loaded ESP on the fine computational domain where the need for a fast solver arises. Unlike the previously published numerical techniques, the finite difference method (FDM) integrated with a full multi-grid method (FMG), labeled FDM-FMG, is developed to resolve Poisson and continuity equations on one fine computational domain. For clean and dust-loaded ESP, the proposed FMG is checked versus successive over-relaxation (SOR) on fine domains where the proposed one is greatly transcendent in terms of convergence characteristics and hence the computational performance (CPU time). For the first time, two major issues are highlighted and solved: the first concerning issue is the chosen ion mobility as an important factor in the simulation results and the second one is choosing an optimal computational grid for dust loaded precipitators that grantees both low truncation and roundoff errors, results in well-matched with experimental measurements nominated in the previous publishing. The novel idea of working on various grid sizes and tracking the optimal ones gives the FDM-FMG an advantage of predicting a precise picture for the electrical situations in industrial ESP over the other numerical techniques. After all, the impact of changing the spacing between the different wires and the height of the ionized wires on the distributions of current, ion, and particle charge densities on the ground are deeply simulated and presented in dust-loaded ESP. The proposed FDM-FMG can be a promising tool for the designers and manufacturers of precipitators, thanks to its superior computational performance.

A new FV scheme and fast cell-centered multigrid solver for 3D anisotropic diffusion equations with discontinuous coefficients

In this paper, an efficient cell-centered extrapolation cascadic multigrid (CEXCMG) method is proposed for solving large linear system of equations resulting from finite volume (FV) discretizations of three dimensional (3D) anisotropic diffusion equations with discontinuous coefficients. For cell-centered FV schemes, the values at vertex need to be approximated often by a linear combination of neighboring cell-centered values. In the literature, the weighted coefficients are obtained by solving local linear system of equations which is costly in 3D. One of the novelties of this paper is a new approach for obtaining vertex values by interpolating the cell-centered ones, which avoids solving local linear system of equations even with arbitrary diffusion tensors. Another main novelty of this paper is a new cascadic multigrid solver based on a prolongation operator, the newly developed explicit gradient transfer method, and a splitting extrapolation operator for solving 3D anisotropic diffusion equations with discontinuous coefficients. Numerical experiments are presented to demonstrate the efficiency and robustness of the CEXCMG method in terms of the mesh size and the contrast in the coefficients of the anisotropic diffusion tensor.

Patch‐smoother and multigrid for the dual formulation for linear elasticity

AbstractThe dual formulation for linear elasticity, in contrast to the primal formulation, is not affected by locking, as it is based on the stresses as main unknowns. Thus it is quite attractive for nearly incompressible and incompressible materials. Discretization with mixed finite elements will lead to—possibly large—linear saddle point systems. Whereas efficient multigrid methods exist for solving problems in mixed plane elasticity for nearly incompressible materials, we propose a multigrid method that is also stable in the incompressible limit. There are two main challenges in constructing a multigrid method for the dual formulation for linear elasticity. First, in the incompressible limit, the matrix block related to the stress is positive semidefinite. Second, the stress belongs to and standard smoothers, working for regular problems, cannot be applied. We present a novel patch‐based smoother for the dual formulation for linear elasticity. We discuss different types of local boundary conditions for the patch subproblems. Based on our patch‐smoother, we build a multigrid method for the solution of the resulting saddle point problem and investigate its efficiency and robustness. Numerical experiments show that Dirichlet and Robin conditions work best and eventually lead to textbook multigrid performance.

Parallel multilevel domain decomposition preconditioners for monolithic solution of non-isothermal flow in reservoir simulation

In reservoir simulation, the non-isothermal flow adds a conservation of the energy equation with the involvement of a temperature variable to the complicated physics dynamics, which brings a higher nonlinearity of the corresponding PDE system and therefore exhibits significantly additional challenges on the linear preconditioner of the simulator. However, the commonly used block preconditioning techniques mostly focus on the isothermal petroleum model and do not have a specific treatment of the energy conservation equation for the non-isothermal flow. In this study, we propose and develop a family of multilevel restricted additive Schwarz (MRAS) preconditioners on parallel computers for the fully implicit solution of the non-isothermal flow in porous media. The proposed parallel reservoir simulator incorporates the restricted additive Schwarz preconditioner into a general multilevel preconditioning framework with the use of the domain decomposition technique and the multigrid method, which allows us to flexibly construct several types of multilevel Schwarz preconditioners by using the additive or multiplicative strategies. In particular, it follows the fully implicit discretization scheme for the monolithic solution, and therefore is unconditionally stable with the relaxation of the time step size by the stability condition. We numerically show that the proposed approach is highly robust and efficient for solving some non-isothermal flow problems with high nonlinearity, and good parallel scalabilities are obtained on a supercomputer with a large number of processors.

Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization

Abstract We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 {L_{2}} -norm regularization is replaced by the H - 1 {H^{-1}} -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ ⁢ h {u_{\varrho h}} to the state u and the desired state u ¯ {\overline{u}} in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 {\varrho=h^{2}} ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ {\overline{u}} . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

All-at-once multigrid approaches for one-dimensional space-fractional diffusion equations

We focus on a time-dependent one-dimensional space-fractional diffusion equation with constant diffusion coefficients. An all-at-once rephrasing of the discretized problem, obtained by considering the time as an additional dimension, yields a large block linear system and paves the way for parallelization. In particular, in case of uniform space–time meshes, the coefficient matrix shows a two-level Toeplitz structure, and such structure can be leveraged to build ad-hoc iterative solvers that aim at ensuring an overall computational cost independent of time. In this direction, we study the behavior of certain multigrid strategies with both semi- and full-coarsening that properly take into account the sources of anisotropy of the problem caused by the grid choice and the diffusion coefficients. The performances of the aforementioned multigrid methods reveal sensitive to the choice of the time discretization scheme. Many tests show that Crank–Nicolson prevents the multigrid to yield good convergence results, while second-order backward-difference scheme is shown to be unconditionally stable and that it allows good convergence under certain conditions on the grid and the diffusion coefficients. The effectiveness of our proposal is numerically confirmed in the case of variable coefficients too and a two-dimensional example is given.