Full Multigrid Research Articles

AN APPROACH TO THE IMPLEMENTATION OF THE MULTIGRID METHOD WITH FULL APPROXIMATION FOR CFD PROBLEMS

This paper is devoted to the use of the multigrid method to accelerate calculations of compressible flows in a stationary formulation on unstructured grids. The multigrid method is used with the construction of a full approximation for each grid level (FAS MG—Full Approximation Scheme Multigrid). In the case of an unstructured grid, such a method can cause difficulties associated both with the construction of grid levels and transition operators between them, and with software implementation in the existing simulation code. The program needs to deal with several different discretizations at once. If the entire data structure, including arrays with grid data, topology, and time integration data, was designed to work on a single grid, then the implementation of the FAS MG can turn into a disaster involving rewriting the entire code. The purpose of this work is to achieve multiple acceleration of calculations at the cost of minimal effort. The problem of implementing the multigrid method on the basis of an existing software package that was not designed to work with several grid levels is solved. The implementation of the multigrid method in an MPI parallel code is carried out in such a way that there is no need to rewrite the program to work with multiple grids at all. Also, difficulties with constructing grid levels for an unstructured grid are avoided; agglomeration of cells is not used, and the number of faces per cell at coarse levels is not increased. In fact, this paper describes how to deploy a FAS MG accelerator in literally a week, even in code that is outdated from the point of view of software architecture.

Fundamental data structures for matrix-free finite elements on hybrid tetrahedral grids

This paper presents efficient data structures for the implementation of matrix-free finite element methods on block-structured, hybrid tetrahedral grids. It provides a complete categorization of all geometric sub-objects that emerge from the regular refinement of the unstructured, tetrahedral coarse grid and describes efficient iteration patterns and analytical linearization functions for the mapping of coefficients to memory addresses. This foundation enables the implementation of fast, extreme-scalable, matrix-free, iterative solvers, and in particular geometric multigrid methods by design. Their application to the variable-coefficient Stokes system subject to an enriched Galerkin discretization and to the curl-curl problem discretized with Nédélec edge elements showcases the flexibility of the implementation. Finally, the solution of a curl-curl problem with 1.6 ⋅ 10 11 (more than one hundred billion) unknowns on more than 32,000 processes with a matrix-free full multigrid solver demonstrates its extreme-scalability.

SUHMO: an adaptive mesh refinement SUbglacial Hydrology MOdel v1.0

Abstract. Water flowing under ice sheets and glaciers can have a strong influence on ice dynamics, particularly through pressure changes, suggesting that a comprehensive ice sheet model should include the effect of basal hydrology. Modeling subglacial hydrology remains a challenge, however, mainly due to the range of spatial and temporal scales involved – from subglacial channels to vast subglacial lakes. Additionally, networks of subglacial drainage channels dynamically evolve over time. To address some of these challenges, we have developed an adaptive mesh refinement (AMR) model based on the Chombo software framework. We extend the model proposed by Sommers et al. (2018) with a small but significant change to accommodate the transition from unresolved to resolved flow features. We handle the strong nonlinearities present in the equations by resorting to an efficient nonlinear full approximation scheme multigrid (FAS-MG) algorithm. We outline the details of the algorithm and present convergence analysis results demonstrating its good performance. Additionally, we present results validating our approach, using test cases from the Subglacial Hydrology Model Intercomparison Project (SHMIP) (de Fleurian et al., 2018). We finish by presenting a more complex, 100 km-by-100 km synthetic test case with peaks and valleys that we use to investigate the effective pressure distribution as the number of AMR levels increases. These preliminary results suggest that a minimum spatial resolution is needed to properly capture channel features, but additional work is required to precisely quantify this and its impact on accurately modeling the coupled ice sheet–hydrology system. The efficiency of our approach, relying on localized refinement, is also demonstrated. Future work will include coupling the SUbglacial Hydrology MOdel (SUHMO) with the BISICLES AMR ice sheet model (Cornford et al., 2013), both built on the same numerical framework.

A Bootstrap Multigrid Eigensolver

This paper introduces bootstrap multigrid methods for solving eigenvalue problems arising from the discretization of partial differential equations. Inspired by the full bootstrap algebraic multigrid (BAMG) setup algorithm that includes an AMG eigensolver, it is illustrated how the algorithm can be simplified for the case of a discretized partial differential equation (PDE), thereby developing a bootstrap geometric multigrid (BMG) approach. We illustrate numerically the efficacy of the BMG method for: (1) recovering eigenvalues having large multiplicity, (2) computing interior eigenvalues, and (3) approximating shifted indefinite eigenvalue problems. Numerical experiments are presented to illustrate the basic components and ideas behind the success of the overall bootstrap multigrid approach. For completeness, we present a simplified error analysis of a two-grid bootstrap algorithm for the Laplace-Beltrami eigenvalue problem.

Textbook Efficiency: Massively Parallel Matrix-Free Multigrid for the Stokes System

We employ textbook multigrid efficiency (TME), as introduced by Achi Brandt, to construct an asymptotically optimal monolithic multigrid solver for the Stokes system. The geometric multigrid solver builds upon the concept of hierarchical hybrid grids (HHG), which is extended to higher-order finite-element discretizations, and a corresponding matrix-free implementation. The computational cost of the full multigrid (FMG) iteration is quantified, and the solver is applied to multiple benchmark problems. Through a parameter study, we suggest configurations that achieve TME for both, stabilized equal-order, and Taylor-Hood discretizations. The excellent node-level performance of the relevant compute kernels is presented via a roofline analysis. Finally, we demonstrate the weak and strong scalability to up to $147,456$ parallel processes and solve Stokes systems with more than $3.6 \times 10^{12}$ (trillion) unknowns.

Study and Simulation of the Thrust Vectoring in Supersonic Nozzles

The present study aims to numerically simulate the thrust vectoring by secondary injection. The analysis of flow for this study was done using ANSYS Fluent. The kω-sst turbulence model was used. The baseline solver was selected as a double-precision Density-based coupled solve with implicit time integration. Least squares cell-based spatial discretization in which the solution was assumed to vary linearly was used and a second-order upwind scheme is used for interpolating the values of pressure, momentum, turbulent kinetic energy, specific dissipation rate, and energy. The computational analysis was conducted under study conditions. The initialization for the steady-state problem was done using full multigrid (FMG) initialization to get the initial solution and the boundary was provided to get the reference value. The results of the study with and without fluid injection are presented in terms of pressure, angle of deflection, and efficiency. These results are then compared with those obtained numerically or experimentally by other authors. This comparison was very interesting and the results were very close, the error was only 5%.

A neural network multigrid solver for the Navier-Stokes equations

We present the deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. DNN-MG uses the multi-grid method to classically solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed there. This results in a reduction in computation time through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a “guide” for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on different ones. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.

On a multigrid solver for stationary Navier–Stokes velocity–pressure tracking-type control problems

In this article, a multigrid solver for distributed optimal control problems governed by time-independent Navier–Stokes equations is presented. A mixed (velocity–pressure) tracking-type control problem is considered and first-order optimality conditions are discussed. We investigate a full multigrid method with coarsening by a factor-of-three strategy to stationary Navier–Stokes control problems. The potential advantage of multigrid with coarsening by a factor-of-three strategy is that it results in nested hierarchy of staggered grids and thus simplifies the inter-grid transfer operators, reduces the number of levels, and hence the CPU time. The construction of the multigrid algorithm for Stokes control problems of our earlier work gives us a natural extension but still significant challenges are rooted in the nonlinear part of the Navier–Stokes equations (constraints) and mixed (velocity–pressure) tracking-type control formulation. Numerical experiments are reported to show the behavior and efficiency of the proposed multigrid algorithm for small Reynolds numbers and moderate values of regularization parameter.

A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering

In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

Effects of joule heating on shock train structure, mathematical modeling with modification of k-ω shear stress transport

In this study, the two dimensional shock train behavior in a straight isolator was evaluated using a modified k-ω shear stress transport turbulence model. Therefore, this work focused on assessing the constant coefficient (β*) of k-ω shear stress turbulence model to predict the location of the first lambda shock, number of shocks, and changes of principle parameters related to shock train, as well as a comprehensive numerical investigation of a shock train system. In addition, the full multigrid initialization method was applied for the reduction of computational cost. The results revealed that when β* was 0.075, the exact shock structure was detectable. After obtaining the appropriate coefficient for the turbulence model, the effects of Joule heating were investigated on shock train structure, using five electric fields. The obtained results showed that applying Joule heating led to the movement of the first lambda shock, changes in its height, and enhancement of the distance between the primary shocks.

Measurement and assessment of corona current density for HVDC bundle conductors by FDM integrated with full multigrid technique

This paper presents an intensive measurement and analysis of monopolar ionized fields in bundled high voltage direct current (HVDC) conductors using the finite difference method based on the full multigrid technique. The positive feature of this study is that it considers the comprehensive representation of the bundle conductor, unlike the existing studies that approximate the bundle conductor with an equivalent conductor radius. Firstly, the proposed method is compared with previous experimental results. Secondly, a flexible laboratory model for the bundled HVDC conductors is constructed. Thirdly, the laboratory model is exploited to validate the numerically computed current density distribution on the ground plane and corona current for different bundles’ numbers and different distances between bundles. Bundles of one, two, and four conductors are adopted in the experimental setup. For the same applied voltage, the results verified that the corona current decreases by increasing the bundles’ number and/or minimizing the spacing between bundles. Consequently, the obtained results confirmed that corona power losses can be minimized, without needing the traditional procedures that involve increasing either the conductor radius or its height above the ground. The results of the proposed numerical approach concurred well with the present and past laboratory results.

Numerical simulations of capsule deformation using a dual time-stepping lattice Boltzmann method.

In this work a quasisteady, dual time-stepping lattice Boltzmann method is proposed for simulation of capsule deformation. At each time step the steady-state lattice Boltzmann equation is solved using the full approximation storage multigrid scheme for nonlinear equations. The capsule membrane is modeled as an infinitely thin shell suspended in an ambient fluid domain with the fluid structure interaction computed using the immersed boundary method. A finite element method is used to compute the elastic forces exerted by the capsule membrane. Results for a wide range of parameters and initial configurations are presented. The proposed method is found to reduce the computational time by a factor of ten.

Geometric Multigrid Method for Isogeometric Analysis

The isogeometric analysis method (IGA) is a new type of numerical method solving partial differential equations. Compared with the traditional finite element method, IGA based on geometric spline can keep the model consistency between geometry and analysis, and provide higher precision with less freedom. However, huge stiffness matrix from the subdivision progress still leads to the solution efficiency problems. This paper presents a multigrid method based on geometric multigrid (GMG) to solve the matrix system of IGA. This method extracts the required computational data for multigrid method from the IGA process, which also can be used to improve the traditional algebraic multigrid method (AGM). Based on this, a full multigrid method (FMG) based on GMG is proposed. In order to verify the validity and reliability of these methods, this paper did some test on Poisson’s equation and Reynolds’ equation and compared the methods on different subdivision methods, different grid degrees of freedom, different cyclic structure degrees, and studied the convergence rate under different subdivision strategies. The results show that the proposed method is superior to the conventional algebraic multigrid method, and for the standard relaxed V-cycle iteration, the method still has a convergence speed independent of the grid size at the same degrees.

An efficient multigrid-DEIM semi-reduced-order model for simulation of single-phase compressible flow in porous media

In this paper, an efficient multigrid-DEIM semi-reduced-order model is developed to accelerate the simulation of unsteady single-phase compressible flow in porous media. The cornerstone of the proposed model is that the full approximate storage multigrid method is used to accelerate the solution of flow equation in original full-order space, and the discrete empirical interpolation method (DEIM) is applied to speed up the solution of Peng–Robinson equation of state in reduced-order subspace. The multigrid-DEIM semi-reduced-order model combines the computation both in full-order space and in reduced-order subspace, which not only preserves good prediction accuracy of full-order model, but also gains dramatic computational acceleration by multigrid and DEIM. Numerical performances including accuracy and acceleration of the proposed model are carefully evaluated by comparing with that of the standard semi-implicit method. In addition, the selection of interpolation points for constructing the low-dimensional subspace for solving the Peng–Robinson equation of state is demonstrated and carried out in detail. Comparison results indicate that the multigrid-DEIM semi-reduced-order model can speed up the simulation substantially at the same time preserve good computational accuracy with negligible errors. The general acceleration is up to 50–60 times faster than that of standard semi-implicit method in two-dimensional simulations, but the average relative errors of numerical results between these two methods only have the order of magnitude 10−4–10−6%.

An efficient extrapolation full multigrid method for elliptic problems in two and three dimensions

An extrapolation full multigrid (EXFMG) method is proposed for solving the large linear systems arising from linear finite element discretization of two- (2D) and three-dimensional (3D) elliptic boundary value problems. A good initial guess on the next finer grid is constructed through combining Richardson extrapolation and quadratic FE interpolation for the numerical solutions on two-level of grids (current and previous grids). And the FE linear system on the next finer grid is solved to obtain the EXFMG solution with discretization-level accuracy by performing one or two multigrid cycles with Gauss–Seidel smoother. Moreover, Richardson extrapolation is further used to achieve higher-order accuracy. The numerical results of five typical elliptic problems including one 2D problem and four 3D problems are conducted to show the effectiveness and efficiency of the EXFMG algorithm.

Accelerating CFD solver computation time with reduced‐order modeling in a multigrid environment

SummaryReduced‐order models (ROMs) are more and more considered for use in aerodynamic applications. Benefits of these methods can be expected for optimization problems or predicting aerodynamic loads for the entire flight envelope. For these applications it is often possible to perform computations for various parameter combinations before any ROM evaluations are needed. The order reduction of the CFD solutions in this article is done using proper orthogonal decomposition. Coupled with an interpolation method predictions for unknown parameter combinations can be made. The CFD solutions are computed using a discontinuous Galerkin finite element method combined with a nonlinear multigrid scheme. The nonlinear multigrid solver algorithms depend on a good initial guess of the flow field to be computed. Typically, the initial guess on a fine mesh is obtained by solving the problem on a agglomerated mesh. Alternatively, an initial guess could be obtained from a ROM prediction, if available. The main objective is to identify the benefits of using a ROM in a higher‐order multigrid environment. Integral as well as distributed surface quantities of the ROM predictions originating from several multigrid levels will be compared with fully converged flow solutions on the top level of the multigrid algorithm. Furthermore, initializing the flow solver with predictions from several multigrid levels will be analyzed in comparison to full multigrid computations as well as to a scenario where already converged solutions for similar parameter combinations are used as initial flow solutions on the top multigrid level.

Predicting the unsteady states of 2D and 3D lid-driven cavities induced by a centrally located circle and sphere

This investigation addresses a systematic numerical method based on the finite volume method and a full multigrid technique to study two-dimensional and three-dimensional flow of an incompressible fluid inside a cavity driven by the motion of the upper lid. Quantitative aspects of two and three dimensional flows in lid-driven cavities are analyzed by encompassing descriptive Reynolds numbers Re bounded by 1000 and 8000 for 2D case, 100 and 1000 for 3D case. Furthermore, the effects of inner spherical shape on fluid flow characteristics have been also investigated. An analysis of the flow evolution demonstrates that, by further increasing Reynolds beyond a certain critical value, the steady flow becomes unstable and bifurcates to the unsteady flow. Results show that the transition to the unsteady regime follows the classical scheme of a Hopf bifurcation, giving rise to a perfectly periodic state. Flow periodicity has been verified through time history plots for the velocity component and phase-space trajectories as a function of Reynolds number for both cases 2D and 3D. It is also concluded that for 2D and 3D configurations, whether the cavity is obstructed or non-obstructed, the refinement of the grid accelerates the appearance of the unsteady regime and delays it in the opposite case. A one-to-one comparison results between a cavity induced by a centrally located inner shape (a sphere for the 3D case and a circle for the 2D case) and a cavity without an inner shape confirms that by using a grid size 642 for the 2D case or 803 for the 3D case, critical Reynolds number for a cavity with an inner shape are less than that without an inner shape. Whilst, for the grid sizes 1282 and 2562 (2D case) the opposite phenomena occurs and it can be inferred that a bidimensional cavity with an inner circle at the center has a critical Reynolds number higher than that without an inner circle. Hence, the unsteadiness is delayed in this case.

Flow instability of cylinder embedded within two-parallel moving walls of cuboidal cavity at different radii sizes

A computational analysis has been performed to study the flow instability of two-parallel wall motions in a Cuboidal enclosure incorporated by a cylinder under different radii sizes. A numerical methodology based on the Finite Volume Method (FVM) and a full Multigrid acceleration is utilized in this paper. Left and right parallel walls of the cavity are maintained driven and all the other walls completing the domain are motionless. Different radii sizes ([Formula: see text], 0.1, 0.125, 0.15 and 0.175) are employed encompassing descriptive Reynolds numbers that range three orders of magnitude 100, 400 and 800 for the steady state. The obtained results show positions [Formula: see text] and [Formula: see text] of the inner cylinder promote cell distortion. Also, when the radius equates to [Formula: see text], it may lead to the birth of tertiary cells at [Formula: see text] which are more developed for [Formula: see text]. Thereafter, analysis of the flow evolution shows that with increasing Re beyond a certain critical value, the flow becomes unstable and undergoes a Hopf bifurcation. A nonuniform variation with the radius size of the inner cylinder is observed. Otherwise said, elongating the radius of the cylinder leads to decrease in the critical Reynolds number. Hence, the acceleration of the unsteadiness is realized. On the other hand, by further increasing Reynolds number more than the critical value from 1200 to 2100, we note that the kinetic energy is monotonously increasing with Reynolds number and a stronger motion in the velocity near the rear wall of the enclosure is observed. Furthermore, the symmetry of flow patterns observed in the steady state has been lost. Therefore, a systematic description of various effects illuminating the optimum geometrical parameters to achieve effective flow behavior in those systems has been successfully established through this paper.

Fast Corona Discharge Assessment Using FDM integrated With Full Multigrid Method in HVDC Transmission Lines Considering Wind Impact

A novel approach for solving the monopolar corona in high voltage direct current (HVDC) transmission line systems is proposed by the finite difference method (FDM) and a full multigrid method (FMG). Specifically, the FMG is implemented as a fast solver with respect to existing iterative solutions for the FDM to solve the Poisson equation, particularly on fine grids. The advantage features of the proposed approach are that it avoids the hypothesis of a constant electric field around the conductor’s surface. Further, it considers the influence of space charge on both the magnitude and the direction of the electrical field. The proposed approach is employed for computing the electric field and current density on the ground plane with and without wind effect. Considering the impact of wind in the present study, the findings confirm that both corona current density and electric field on the ground plane are influenced by the transverse wave. Eventually, the effect of changing the wind speed on the electric field profiles and the current density is deeply studied in HVDC transmission line systems. To prove the efficacy of the proposed approach, it is compared with previous experimental results where a better agreement is reached rather than other numerical techniques.

A Multigrid Method for Nonlocal Problems: Non--Diagonally Dominant or Toeplitz-Plus-Tridiagonal Systems

The nonlocal problems have been used to model very different applied scientific phenomena, which involve the fractional Laplacian when one looks at the L\'{e}vy processes and stochastic interfaces. This paper deals with the nonlocal problems on a bounded domain, where the stiffness matrices of the resulting systems are Toeplitz-plus-tridiagonal and far from being diagonally dominant, as it occurs when dealing with linear finite element approximations. By exploiting a weakly diagonally dominant Toeplitz property of the stiffness matrices, the optimal convergence of the two-grid method is well established [Fiorentino and Serra-Capizzano, {\em SIAM J. Sci. Comput.}, {17} (1996), pp. 1068--1081; Chen and Deng, {\em SIAM J. Matrix Anal. Appl.}, {38} (2017), pp. 869--890]; and there are still questions about best ways to define coarsening and interpolation operator when the stiffness matrix is far from being weakly diagonally dominant [St\"{u}ben, {\em J. Comput. Appl. Math.}, {128} (2001), pp. 281--309]. In this work, using spectral indications from our analysis of the involved matrices, the simple (traditional) restriction operator and prolongation operator are employed in order to handle general algebraic systems which are {\em neither Toeplitz nor weakly diagonally dominant} corresponding to the fractional Laplacian kernel and the constant kernel, respectively. We focus our efforts on providing the detailed proof of the convergence of the two-grid method for such situations. Moreover, the convergence of the full multigrid is also discussed with the constant kernel. The numerical experiments are performed to verify the convergence with only $\mathcal{O}(N \mbox{log} N)$ complexity by the fast Fourier transform, where $N$ is the number of the grid points.