Full Multigrid Method Research Articles

Improved prediction of ion mobility for precipitators using FDM-FMG on fine computational domains

• FDM integrated with FMG method is a very fast convergent technique for solving the precipitator problem. • Gauss-Siedel is a perfect smoothing solver for FMG in terms of time performance. • Global optimal computational grid size is tracked and perfectly matched the experimental results. • Effective ion mobility is tracked at different applied voltages in precipitator with clean air. • FDM integrated with FMG is an effective technique in the early design stage of precipitators. This paper presents an improved approach for modelling electrostatic precipitators on one fine computational domain in clean air. Contrary to the previously published numerical techniques, the Finite Difference Method (FDM) integrated with Full Multi-grid method (FMG) is used to predict the optimal value of ion mobility at a certain applied voltage in precipitators. Three smoothing solvers for FMG are implemented including Gauss–Siedel, Successive Over Relaxation (SOR) and Successive Under Relaxation (SUR). Gauss–Siedel is examined against SOR and SUR on finer domains and it is found that Gauss–Siedel is greatly superior in terms of the timing performance. Two main ideas are discussed in this paper for the first time; the first idea is that the value of ion mobility is not constant for all applied voltages as published previously in the precipitators. The second one is predicting only one global optimal grid for solving Poisson and continuity equations that grantees both low truncation and round off errors that well matched the previously published experimental measurements more than that of the previously published numerical techniques. Moreover, the idea of tracking the optimal ion mobility and grid size gives the FDM-FMG the advantage of predicting an accurate picture for the electrical conditions in the early design of precipitators over the other numerical techniques. Finally, the FDM-FMG gives a more confident results for current density and voltage characteristics for precipitator which is proved by experiments.

A Cyclic Multi-Stage Implementation of the Full-Waveform Inversion for the Identification of Anomalies in Dams

For the safe and efficient operation of dams, frequent monitoring and maintenance are required. These are usually expensive, time consuming, and cumbersome. To alleviate these issues, we propose applying a wave-based scheme for the location and quantification of damages in dams. To obtain high-resolution “interpretable” images of the damaged regions, we drew inspiration from non-linear full-multigrid methods for inverse problems and applied a new cyclic multi-stage full-waveform inversion (FWI) scheme. Our approach is less susceptible to the stability issues faced by the standard FWI scheme when dealing with ill-posed problems. In this paper, we first selected an optimal acquisition setup and then applied synthetic data to demonstrate the capability of our approach in identifying a series of anomalies in dams by a mixture of reflection and transmission tomography. The results had sufficient robustness, showing the prospects of application in the field of non-destructive testing of dams.

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.

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.

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.

A new type of full multigrid method for the elasticity eigenvalue problem

AbstractThis paper introduces a new type of full multigrid method for the elasticity eigenvalue problem. The main idea is to avoid solving large scale elasticity eigenvalue problem directly by transforming the solution of the elasticity eigenvalue problem into a series of solutions of linear boundary value problems defined on a multilevel finite element space sequence and some small scale elasticity eigenvalue problems defined on the coarsest correction space. The involved linear boundary value problems will be solved by performing some multigrid iterations. Besides, some efficient techniques such as parallel computing and adaptive mesh refinement can also be absorbed in our algorithm. The efficiency and validity of the multigrid methods are verified by several numerical experiments.

Optimal grid size for precipitators using finite difference method based on full multi-grid method

The paper presents a novel approach for tracking the optimal grid size of the computational domain for modeling the corona problem within the wire-duct precipitators, which in turn helps in decreasing the experimental efforts. The Finite Difference Method (FDM) is used to model the corona problem using the full multi-grid method (FMG) as a powerful convergent iterative solution for Poisson equation particularly on finer computational domains. The full multi-grid method is examined against successive over relaxation (SOR) strategy and the latter is effectively transcendent in terms of timing performance. Indeed, using finer grids is a double ended weapon; on one hand it reduces the truncation error of the Finite Difference Method which reflects in getting more accurate view for the corona problem in precipitators. While on the other hand, the round off error will be increased which might give un-accurate results. Accordingly, the issue of choosing the optimal grid size arises. The full multi-grid method tracked the optimal grid size that gives the appropriate results for the potential and current density that well matched the previous published experimental measurements.

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 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.

Fast corona discharge solver for precipitators using multi‐grid methods on fine grids

A novel approach for modelling the corona problem in the wire-duct precipitators using the finite difference technique integrated with multi-grid methods is adopted in this study. The multi-grid method is applied as a fast convergent iterative solution for the finite difference technique to solve Poisson equation especially on finer grids. Two schemes of the multi-grid method, mainly the V-cycle and the full multi-grid method, are adopted in the present study. Compared with Gauss–Seidel method, the above-mentioned schemes are successfully transcendent owing to timing performance. By using finer grids, the proposed algorithm allows to get a more accurate picture about the performance of the precipitators in the design stage without suffering from the excessive computational time. Accurate results for the potential and current density computations, closed to the previous published experimental measurements, are obtained in comparison with earlier numerical techniques for several design parameters of the precipitators. Finally, the effect of changing the effective ion mobility and the surface roughness factor on the voltage–current density is considered.

A full multigrid method for the Steklov eigenvalue problem

ABSTRACTThis paper introduces a kind of parallel multigrid method for solving Steklov eigenvalue problem based on the multilevel correction method. Instead of the common costly way of directly solving the Steklov eigenvalue problem on some fine space, the new method contains some boundary value problems on a series of multilevel finite element spaces and some steps of solving Steklov eigenvalue problems on a very low dimensional space. The linear boundary value problems are solved by some multigrid iteration steps. We will prove that the computational work of this new scheme is truly optimal, the same as solving the corresponding linear boundary value problem. Besides, this multigrid scheme has a good scalability by using parallel computing technique. Some numerical experiments are presented to validate our theoretical analysis.

Parallel Multigrid for Nonlinear Cloth Simulation

AbstractAccurate high‐resolution simulation of cloth is a highly desired computational tool in graphics applications. As single‐resolution simulation starts to reach the limit of computational power, we believe the future of cloth simulation is in multi‐resolution simulation. In this paper, we explore nonlinearity, adaptive smoothing, and parallelization under a full multigrid (FMG) framework. The foundation of this research is a novel nonlinear FMG method for unstructured meshes. To introduce nonlinearity into FMG, we propose to formulate the smoothing process at each resolution level as the computation of a search direction for the original high‐resolution nonlinear optimization problem. We prove that our nonlinear FMG is guaranteed to converge under various conditions and we investigate the improvements to its performance. We present an adaptive smoother which is used to reduce the computational cost in the regions with low residuals already. Compared to normal iterative solvers, our nonlinear FMG method provides faster convergence and better performance for both Newton's method and Projective Dynamics. Our experiment shows our method is efficient, accurate, stable against large time steps, and friendly with GPU parallelization. The performance of the method has a good scalability to the mesh resolution, and the method has good potential to be combined with multi‐resolution collision handling for real‐time simulation in the future.

Development of a structured overset Navier–Stokes solver with a moving grid and full multigrid method

An unsteady Reynolds averaged Navier–Stokes solver with a structured overset grid method has been developed. Velocity–pressure coupling is achieved using an artificial compressibility approach, spatial discretization is based on a finite-volume method, and convective fluxes are evaluated by a third-order upwind scheme based on flux-difference splitting. Body motions are considered using the grid deformation technique and grid velocities in the convective term. Viscous fluxes are evaluated via second-order centered differencing. The full multigrid (FMG) method is applied to obtain fast convergence. The cell flag on a coarse grid level is determined using the cell flag on a fine grid level. In the coarse grid-level calculations at the FMG stage, the data are interpolated until the finest grid level is achieved at an overset update interval. Then, the data are updated based on the overset relations at the finest grid level and then transferred to a coarser grid level. For free surface treatments, a single-phase localized level-set method is employed. The computations for flows around a hull form, including an unsteady simulation with regular waves, are demonstrated.

An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection\u2013diffusion equations

In this paper, we propose an extrapolation full multigrid (EXFMG) algorithm to solve the large linear system arising from a fourth-order compact difference discretization of two-dimensional (2D) convection diffusion equations. A bi-quartic Lagrange interpolation for the solution on previous coarser grid is used to construct a good initial guess on the next finer grid for V- or W-cycles multigrid solver, which greatly reduces the number of relaxation sweeps. Instead of performing a fixed number of multigrid cycles as used in classical full multigrid methods, a series of grid level dependent relative residual tolerances is introduced to control the number of the multigrid cycles. Once the fourth-order accurate numerical solutions are obtained, a simple method based on the midpoint extrapolation is employed for the fourth-order difference solutions on two-level grids to construct a sixth-order accurate solution on the entire fine grid cheaply and directly. Numerical experiments are conducted to verify that the proposed method has much better efficiency compared to classical multigrid methods. The proposed EXFMG method can also be extended to solve other kinds of partial differential equations.

A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations

AbstractA full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is presented. An optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary conditions. The optimality system that results from a Lagrange multiplier framework, form a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods with finite difference discretization on staggered grids. A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators. A distributive-Gauss-Seidel smoothing scheme is employed to update the state- and adjoint-variables and a gradient update step is used to update the control variables. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

English

A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and semilinear problems on a very low dimensional space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient numerical methods can also serve as the linear solver for solving boundary value problems. The optimality of the computational work is also proved. Compared with the existing multigrid methods which need the bounded second order derivatives of the nonlinear term, the proposed method only needs the Lipschitz continuation in some sense of the nonlinear term.

An efficient multigrid strategy for large-scale molecular mechanics optimization

Static mechanical properties of materials require large-scale nonlinear optimization of the molecular mechanics model under various controls. This paper presents an efficient multigrid strategy to solve such problems. This strategy approximates solutions on grids in a quasi-atomistic and inexact manner, transfers solutions on grids following a coarse-to-fine (oneway) schedule, and finds physically relevant minimizers with linear scaling complexity. Compared to the full multigrid method which has the same complexity, the prefactor of this strategy is orders of magnitude smaller. Consequently, the required CPU time of this strategy is orders of magnitude smaller than that of the full multigrid method, and is smaller than that of the brute-force optimization for systems with more than 200,000 atoms. Considerable savings are found if the number of atoms becomes even larger due to the super-linear scaling complexity of the brute-force optimization. For systems with 1,000,000 atoms (over three million degrees of freedom), on average a more than 70% reduction of CPU time is observed regardless of the type of defects, including vacancies, dislocations, and cracks. In addition, linear scalability of the proposed strategy is tested in the presence of a dislocation pair for systems with more than 100 million atoms (over 400 million degrees of freedom).