Geometric Multigrid Research Articles

Performance of Geometric Multigrid Method forTwo-Dimensional Burgers’ Equations with Non-Orthogonal, Structured Curvilinear Grids

This paper seeks to develop an efficient multigrid algorithm for solving the Burgers problem with the use of non-orthogonal structured curvilinear grids in L-shaped geometry. For this, the differential equations were discretized by Finite Volume Method (FVM) with second-order approximation scheme and deferred correction. Moreover, the algebraic method and the differential method were used to generate the non-orthogonal structured curvilinear grids. Furthermore, the influence of some parameters of geometric multigrid method, as well as lexicographical Gauss–Seidel (Lex-GS), η-line Gauss–Seidel (η-line-GS), Modified Strongly Implicit (MSI) and modified incomplete LU decomposition (MILU) solvers on the Central Processing Unit (CPU) time was investigated. Therefore, several parameters of multigrid method and solvers were tested for the problem, with the use of nonorthogonal structured curvilinear grids and multigrid method, resulting in an algorithm with the combination that achieved the best results and CPU time. The geometric multigrid method with Full Approximation Scheme (FAS), V-cycle and standard coarsening ratio for this problem were utilized. This article shows how to calculate the coordinates transformation metrics in the coarser grids. Results show that the MSI and MILU solvers are the most efficient. Moreover, the MSI solver is faster than MILU for both grids generators; and the solutions are more accurate for the Burgers problem with grids generated using elliptic equations.

A comprehensive finite element examination of Carreau Yasuda fluid model in a lid driven cavity and channel with obstacle by way of kinetic energy and drag and lift coefficient measurements

In current pagination the flow problems involving shear rate dependent nonlinear viscosity have been treated successfully in the sense of space discretization and solvers. The stable finite element pair Q2/P1disc is employed to approximate the velocity and pressure spaces independently. Discretized form of non-linear expressions is linearized by implementing Newton’s procedure and the resulting systems are solved using a geometric multigrid approach. The flow generated by way of driven cavity and by an obstacle are very important benchmarks of computational fluid dynamics. In current pagination shear rate reliant viscosity model renowned as Carreau Yasuda fluid is capitalized. The obtained results are demonstrated and analyzed with the help of velocity and viscosity plots. In addition, we have produced new reference data for kinetic energy (K.E) for driven cavity problem and drag and lift coefficients for circular obstacle problem. The obtained results for driven cavity problem reveal the fact that K.E is an increasing function of the relaxation parameter (λ) and power law exponent (n) whereas a decreasing function of the model parameter (a). For the case of flow around obstacle the drag coefficient (CD) and the lift coefficient (CL) show strong dependence on (λ) and (n), however a weak dependence on the parameter (a).

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

Fast matrix splitting preconditioners for higher dimensional spatial fractional diffusion equations

The discretizations of two- and three-dimensional spatial fractional diffusion equations with the shifted finite-difference formulas of the Grünwald-Letnikov type can result in discrete linear systems whose coefficient matrices are of the form D+T, where D is a nonnegative diagonal matrix and T is a block-Toeplitz with Toeplitz-block matrix or a block-Toeplitz with each block being block-Toeplitz with Toeplitz-block matrix. For these discrete spatial fractional diffusion matrices, we construct diagonal and block-circulant with circulant-block splitting preconditioner for the two-dimensional case, and diagonal and block-circulant with each block being block-circulant with circulant-block splitting preconditioner for the three-dimensional case, to further accelerate the convergence rates of Krylov subspace iteration methods, and we analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1, and numerical experiments demonstrate that these structured preconditioners can significantly improve the convergence behavior of the Krylov subspace iteration methods. Moreover, this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant-plus-diagonal preconditioners in both iteration counts and computing times.

A geometric multigrid accelerated incompressible flow simulation over complex geometries

Two algorithms for a fast Cartesian mesh-based flow simulation over complex geometries are presented. As a first category, a geometric multigrid (MG) method using direct restriction of the Heaviside function, which is 0 in the body and 1 in the fluid, is presented. This method is easy to implement, readily parallelizable, and can be applied to any irregular domain. The validity and optimal performance of the current MG method are demonstrated by solving an analytically defined Poisson problem on an irregular domain. Another part of this paper is to introduce a novel efficient method for computing the signed distance function (SDF), which is a typical level-set value to represent bodies on Cartesian meshes. In the Cartesian mesh-based simulation, only the SDF near the body interface is accurately required. Based on this fact, the number of operations for computing the SDF can be significantly reduced by focusing on interface cells. In this study, the adaptive mesh refinement (AMR) strategy is employed to effectively trace the interface cells. By using these two algorithms, an in-house Cartesian mesh-based incompressible flow solver is developed. To demonstrate the applicability of the current approaches, well-known benchmark problems in 2D and 3D are simulated. Finally, as the most challenging case, simulations for flow over an underwater robot, namely Crabster, are presented.

An efficient and accurate MPI-based parallel simulator for streamer discharges in three dimensions

In this paper, we propose an efficient and accurate message-passing interface (MPI)-based parallel simulator for streamer discharges in three dimensions using the fluid model. First, we propose a new second-order semi-implicit scheme for the temporal discretization of the model that relaxes the dielectric relaxation time restriction. Moreover, it requires solving the Poisson-type equation only once at each time step, while the classical second-order explicit schemes typically need to do twice. Second, we introduce a geometric multigrid preconditioned FGMRES solver that dramatically improves the efficiency of solving the Poisson-type equation with either constant or variable coefficients. We show numerically that no more than 4 iterations are required for the Poisson solver to converge to a relative residual of 10−8 during streamer simulations; the FGMRES solver is much faster than R&B SOR and other Krylov subspace solvers. Last but not least, all the methods are implemented using MPI. The parallel efficiency of the code and the fast algorithmic performances are demonstrated by a series of numerical experiments using up to 2560 cores on the Tianhe2-JK clusters. For applications, we study a double-headed streamer discharge as well as the interaction between two streamers, using up to 10.7 billion mesh cells.

Black-box learning of multigrid parameters

This paper studies the optimality of the restriction and prolongation operators in the geometric multigrid method (GMG). GMG is used in solving discretized partial differential equation (PDE) and it relies greatly on the restriction and prolongation operators. Many methods to find these operators were proposed, but most of them have limited optimality proofs. To study their optimality we introduce stochastic convergence functional, which estimates the spectral radius of the iteration matrix for given GMG parameters. We implement the GMG method in a modern machine learning framework that can automatically compute the gradients of the introduced convergence functional with respect to restriction and prolongation operators. Therefore, we can minimize the proposed functional starting from some initial parameters and get better ones after some iterations of stochastic gradient descent. To illustrate the performance of the proposed approach, we carry out experiments on the discretized Poisson equation, Helmholtz equation and singularly perturbed convection–diffusion equation and demonstrate that proposed approach gives operators, which lead to faster convergence.

Hydrodynamic forces and heat transfer of nanofluid forced convection flow around a rotating cylinder using finite element method: The impact of nanoparticles

A computational approach is utilized to study the nanofluid forced convection flow around a rotating circular cylinder in a horizontal channel. Different types of nanoparticles in the base fluid are considered and investigated such as Silica (SiO2), Copper (Cu) and Alumina (Al2O3). Average Nusselt number (Nuavg) are compared for various conditions of the cylinder, namely adiabatic and isothermal cylinders respectively for two locations of cylinder. A dimensionless system of partial differential equations over a complex computational domain is discretized by applying the higher order stable finite element method. The discretized nonlinear system of algebraic equations is linearized using the Newton method. A geometric multigrid method is implemented for the computation of linearized subproblems in each nonlinear sweep. Impact of the pertinent parameters is investigated such as the Reynolds number (10 ≤ Re ≤ 200), angular velocity (−75 ≤ Ω ≤ 75) and the nanoparticles volume fraction (0.0 ≤ ϕ ≤ 0.04). Numerical results are demonstrated in the form of isotherms, streamlines and suitable graphs for various quantities of interest. It is inferred that the clockwise rotation of the cylinder makes the fluid move over the cylinder while the fluid moves below the cylinder in the anticlockwise case. Moreover, an increase in the nanoparticles volume fraction enhances the average Nusselt number and decreases both of the lift and drag coefficients.

An Efficient Geometric Multigrid Solver for Viscous Liquids

We present an efficient geometric Multigrid solver for simulating viscous liquids based on the variational approach of Batty and Bridson [2008]. Although the governing equations for viscosity are elliptic, the strong coupling between different velocity components in the discrete stencils mandates the use of more exotic smoothing techniques to achieve textbook Multigrid efficiency. Our key contribution is the design of a novel box smoother involving small and sparse systems (at most 9 x 9 in 2D and 15 x 15 in 3D), which yields excellent convergence rates and performance improvements of 3.5x - 13.8x over a naïve Multigrid approach. We employ a hybrid approach by using a direct solver only inside the box smoother and keeping the remaining pipeline assembly-free, allowing our solver to efficiently accommodate more than 194 million degrees of freedom, while occupying a memory footprint of less than 16 GB. To reduce the computational overhead of using the box smoother, we precompute the Cholesky factorization of the subdomain system matrix for all interior degrees of freedom. We describe how the variational formulation, which requires volume weights computed at the centers of cells, edges, and faces, can be naturally accommodated in the Multigrid hierarchy to properly enforce boundary conditions. Our proposed Multigrid solver serves as an excellent preconditioner for Conjugate Gradients, outperforming existing state-of-the-art alternatives. We demonstrate the efficacy of our method on several high resolution examples of viscous liquid motion including two-way coupled interactions with rigid bodies.

Composite matrix construction for structured grid adaptive mesh refinement

Structured-grid adaptive mesh refinement (SAMR) is an approach to mesh generation that supports structured access to data and adaptive mesh refinement for discretized partial differential equations (PDEs). Solution algorithms often require that an inverse of an operator be applied, a system of algebraic equations must be solved, and this process is often the primary computational cost in an application. SAMR is well suited to geometric multigrid solvers, which can be effective, but often do not adapt well to complex geometry including material coefficients. Algebraic multigrid (AMG) is more robust in the face of complex geometry, in both boundary conditions and internal material interfaces. AMG requires a stored matrix linearization of the operator. We discuss an approach, and an implementation in the Chombo block-structured AMR framework, for constructing composite grid matrices from a SAMR hierarchy of grids for use in linear solvers in the PETSc numerical library. We consider a case study with the Chombo-based BISICLES ice sheet modeling application.

An unstructured CFD mini‐application for the performance prediction of a production CFD code

SummaryMaintaining the performance of large scientific codes is a difficult task. To aid in this task, a number of mini‐applications have been developed that are more tractable to analyze than large‐scale production codes while retaining the performance characteristics of them. These “mini‐apps” also enable faster hardware evaluation and, for sensitive commercial codes, allow evaluation of code and system changes outside of access approval processes. In this paper, we develop MG‐CFD, a mini‐application that represents a geometric multigrid, unstructured computational fluid dynamics (CFD) code, designed to exhibit similar performance characteristics without sharing commercially sensitive code. We detail our experiences of developing this application using guidelines detailed in existing research and contributing further to these. Our application is validated against the inviscid flux routine of HYDRA, a CFD code developed by Rolls‐Royce plc for turbomachinery design. This paper (1) documents the development of MG‐CFD, (2) introduces an associated performance model with which it is possible to assess the performance of HYDRA on new HPC architectures, and (3) demonstrates that it is possible to use MG‐CFD and the performance models to predict the performance of HYDRA with a mean error of 9.2% for strong‐scaling studies.

Sharp error estimates for spline approximation: Explicit constants, n-widths, and eigenfunction convergence

In this paper, we provide a priori error estimates in standard Sobolev (semi-)norms for approximation in spline spaces of maximal smoothness on arbitrary grids. The error estimates are expressed in terms of a power of the maximal grid spacing, an appropriate derivative of the function to be approximated, and an explicit constant which is, in many cases, sharp. Some of these error estimates also hold in proper spline subspaces, which additionally enjoy inverse inequalities. Furthermore, we address spline approximation of eigenfunctions of a large class of differential operators, with a particular focus on the special case of periodic splines. The results of this paper can be used to theoretically explain the benefits of spline approximation under [Formula: see text]-refinement by isogeometric discretization methods. They also form a theoretical foundation for the outperformance of smooth spline discretizations of eigenvalue problems that has been numerically observed in the literature, and for optimality of geometric multigrid solvers in the isogeometric analysis context.

Analysis of Current Density Inside Human Body Using Geometric Multi-Grid Method

This paper describes the effectiveness of a geometric multi-grid (GMG) method in current density analysis using numerical human body models. The scalar potential finite difference (SPFD) method is used as a current analysis method inside a human body in the low-frequency domain, and studies have been conducted to solve faster large-scale linear equations made by the SPFD method. In the previous research, the block incomplete Cholesky conjugate gradients (ICCG) method is proposed as an effective method to solve linear equations faster. However, even though the block ICCG method is applied, many iterations are still needed. Therefore, in this research, the GMG method is considered as an effective solver for the problem. GMG method is developed and evaluated performances comparing with the block ICCG method with multi-color (MC) ordering in terms of computation time and the number of iterations. The results show that the number of iterations needed for GMG method is much smaller than that for the block ICCG and the ICCG with MC ordering. In addition, computation times are much shorter, depending on the number of threads and the number of coarse grids. Also, by using MC ordering, the scalability of the GMG method can be greatly improved.

A geometric full multigrid method and fourth-order compact scheme for the 3D Helmholtz equation on nonuniform grid discretization

A full multigrid method and fourth-order compact difference scheme is designed to solve the 3D Helmholtz equation on unequal mesh size. Three dimensional restriction and prolongation operators of the multigrid method on unequal grids could be constructed based on volume law. Two numerical experiments are implemented, and the results show the computational efficiency and accuracy of the full multigrid method. The study also illustrates that the full multigrid method with fourth-order compact difference scheme has great advantages in computation, which has been time consuming, and in iterative convergence efficiency.

On the Validity of the Local Fourier Analysis

Local Fourier analysis (LFA) is a useful tool in predicting the convergence factors of geometric multigrid methods (GMG). As is well known, on rectangular domains with periodic boundary conditions this analysis gives the exact convergence factors of such methods. In this work, using the Fourier method, we extend these results by proving that such analysis yields the exact convergence factors for a wider class of problems.

Geometric multigrid methods for Darcy–Forchheimer flow in fractured porous media

In this paper, we present a monolithic multigrid method for the efficient solution of flow problems in fractured porous media. Specifically, we consider a mixed-dimensional model which couples Darcy flow in the porous matrix with Forchheimer flow within the fractures. A suitable finite volume discretization permits to reduce the coupled problem to a system of nonlinear equations with a saddle point structure. In order to solve this system, we propose a full approximation scheme (FAS) multigrid solver that appropriately deals with the mixed-dimensional nature of the problem by using mixed-dimensional smoothing and inter-grid transfer operators. Numerical experiments show that the proposed multigrid method is robust with respect to the fracture permeability, the Forchheimer coefficient and the mesh size. The case of several possibly intersecting fractures in a heterogeneous porous medium is also discussed.

Mixed convective magnetonanofluid flow over a backward facing step and entropy generation using extended Darcy–Brinkman–Forchheimer model

This article focuses on a numerical study for the effect of porosity and internal heat generation/absorption parameters on the mixed convective flow of nanofluid over a backward facing step along with the entropy generation. The channel downstream bottom wall is isothermally heated while the remaining walls of the channel are thermally insulated. The dimensionless governing equations are discretized utilizing the Galerkin-based finite element method. The discrete systems of nonlinear algebraic equations are computed using the Newton method and the corresponding linearized systems are treated using the monolithic geometric multigrid solver. Effect of different physical parameters in the specified ranges such as \((10 \le Re \le 200)\), \((0.01 \le Ri \le 20)\), \((0\le Ha \le 100)\), \((-10 \le q \le 10)\), \((10^{-6} \le Da \le 10^{-3})\), \((0.2 \le \epsilon \le 0.8)\) and \((0^\circ \le \gamma \le 180^\circ )\) on the fluid flow is analyzed with the help of the streamlines, isotherm patterns, and various graphs. It is noticed that the average heat transfer increases with the porosity parameter and reduces with the internal heat generation parameter. Furthermore, the entropy generation produced by the heat transfer and fluid friction is found to amplify with the porosity parameter, whereas a decline is recorded in it due to the magnetic field.

Solving a trillion unknowns per second with HPGMG on Sunway TaihuLight

Benchmarks for supercomputers are important tools, not only for evaluating and ranking modern supercomputers, but also for providing hints for future architecture design. As a new benchmark, HPGMG (high performance geometric multigrid) solves a linear equation set with a full geometric multi-grid algorithm. It involves computation on different scales, data movement with various volumes, global communication and neighbor communication with both large and small messages, etc., and is more correlated to real world applications than traditional benchmarks such as LINPACK. Therefore, it is desirable to examine how well HPGMG can perform on leadership supercomputers such as Sunway Taihulight. Sunway Taihulight, the No. 1 supercomputer in the Top 500 list from June 2016 to June 2018, which uses a specially designed many-core architecture SW26010, is of great interest to the community of high performance computing. With careful analysis and code design, we came up with an efficient implementation of HPGMG on SW26010 processors. We not only employed traditional optimization techniques such as 2.5D partitioning, double buffering, and collective data load, but also introduced a micro-benchmark to help with the choice of optimization direction and parameter tuning. Another contribution is that we proposed a new procedure for the major operations, by granulating and reordering the smooth function and the ghost exchange operation, leading to reduced memory copy and accelerated communication process. Our optimized implementation of HPGMG on Sunway TaihuLight achieved a ground-breaking performance of $$1.036\times 10^{12}$$ Degrees of Freedom per second at the finest level, which is No. 1 on the HPGMG list of Nov 2017.

Multigrid for Matrix-Free High-Order Finite Element Computations on Graphics Processors

This article presents matrix-free finite-element techniques for efficiently solving partial differential equations on modern many-core processors, such as graphics cards. We develop a GPU parallelization of a matrix-free geometric multigrid iterative solver targeting moderate and high polynomial degrees, with support for general curved and adaptively refined hexahedral meshes with hanging nodes. The central algorithmic component is the matrix-free operator evaluation with sum factorization. We compare the node-level performance of our implementation running on an Nvidia Pascal P100 GPU to a highly optimized multicore implementation running on comparable Intel Broadwell CPUs and an Intel Xeon Phi. Our experiments show that the GPU implementation is approximately 1.5 to 2 times faster across four different scenarios of the Poisson equation and a variety of element degrees in 2D and 3D. The lowest time to solution per degree of freedom is recorded for moderate polynomial degrees between 3 and 5. A detailed performance analysis highlights the capabilities of the GPU architecture and the chosen execution model with threading within the element, particularly with respect to the evaluation of the matrix-vector product. Atomic intrinsics are shown to provide a fast way for avoiding the possible race conditions in summing the elemental residuals into the global vector associated to shared vertices, edges, and surfaces. In addition, the solver infrastructure allows for using mixed-precision arithmetic that performs the multigrid V-cycle in single precision with an outer correction in double precision, increasing throughput by up to 83%.

Camellia: A Rapid Development Framework for Finite Element Solvers

Abstract The discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan guarantees the optimality of the finite element solution in a user-controllable energy norm, and provides several features supporting adaptive schemes. The approach provides stability automatically; there is no need for carefully derived numerical fluxes (as in DG schemes) or for mesh-dependent stabilization terms (as in stabilized methods). In this paper, we focus on features of Camellia that facilitate implementation of new DPG formulations; chief among these is a rich set of features in support of symbolic manipulation, which allow, e.g., bilinear formulations in the code to appear much as they would on paper. Many of these features are general in the sense that they can also be used in the implementation of other finite element formulations. In fact, because DPG’s requirements are essentially a superset of those of other finite element methods, Camellia provides built-in support for most common methods. We believe, however, that the combination of an essentially “hands-free” finite element methodology as found in DPG with the rapid development features of Camellia are particularly winsome, so we focus on use cases in this class. In addition to the symbolic manipulation features mentioned above, Camellia offers support for one-irregular adaptive meshes in 1D, 2D, 3D, and space-time. It provides a geometric multigrid preconditioner particularly suited for DPG problems, and supports distributed parallel execution using MPI. For its load balancing and distributed data structures, Camellia relies on packages from the Trilinos project, which simplifies interfacing with other computational science packages. Camellia also allows loading of standard mesh formats through an interface with the MOAB package. Camellia includes support for static condensation to eliminate element-interior degrees of freedom locally, usually resulting in substantial reduction of the cost of the global problem. We include a discussion of the variational formulations built into Camellia, with references to those formulations in the literature, as well as an MPI performance study.