Jacobi Preconditioned Conjugate Gradient Method Research Articles

Research on application of LU decomposition and CG method in nonlinear iteration of transmission line finite element method

PurposeThe purpose of this study is to develop a reliable finite element algorithm based on the transmission line method (TLM) to solve the nonlinear problem in electromagnetic field calculation.Design/methodology/approachIn this paper, the TLM has been researched and applied to solve nonlinear iteration in FEM. LU decomposition method and the Jacobi preconditioned conjugate gradient method have been investigated to solve the equations in transmission line finite element method (FEM-TLM). The algorithms have been developed in C++ language. The algorithm is applied to analyze the magnetic field of a long straight current-carrying wire and a single-phase transformer.FindingsFEM-TLM is more effective than traditional FEM in nonlinear iteration. The results of FEM-TLM have been compared and analyzed under different calculation scales. It is found that the LU decomposition method is more suitable for FEM-TLM because there is no need to repeatedly assemble the global coefficient matrix in the iterative solution process and it is not affected by the disturbance of the right-hand vector.Originality/valueAn effective algorithm is provided for solving nonlinear problems in the electromagnetic field, which can save a lot of computing costs. The efficiency of LU decomposition and CG method in FEM-TLM nonlinear iteration is investigated, which also makes a preliminary exploration for the research of FEM-TLM parallel algorithms.

Numerical and Experimental Investigation of Temperature Distribution for Oil-Immersed Transformer Winding Based on Dimensionless Least-Squares and Upwind Finite Element Method

To accurately analyze the temperature rising and locate the hot-spot of oil-immersed transformer windings, a hybrid method based on the dimensionless least-squares finite element method (DLSFEM) and upwind finite element method (UFEM) is proposed in this paper. To solve the fluid-thermal coupling problem, the sequential iteration method was utilized. After that, the two-side equilibration method and the Jacobi preconditioned conjugate gradient method were used to solve the equations. Moreover, a product-level experimental platform for oil-immersed transformer windings was built. The influence of the turn-to-turn insulation and gravity on the temperature was discussed, and the results show that the influence is significant. To verify the effectiveness of the proposed method, a comparison between the simulation results of the proposed method and experimental results were made, which shows a good agreement. Aiming at verifying the efficiency of the proposed method, the convergence of the proposed method was compared with that of the commercial computational fluid dynamic (CFD) software Fluent. The results indicate that the convergence speed of the proposed method is much faster than that of Fluent, which means a better performance of the new method.

Simulation of Fluid-Thermal Field in Oil-Immersed Transformer Winding Based on Dimensionless Least-Squares and Upwind Finite Element Method

In order to study the coupling fluid and thermal problems of the local winding in oil-immersed power transformers, the least-squares finite element method (LSFEM) and upwind finite element method (UFEM) are adopted, respectively, to calculate the fluid and thermal field in the oil duct. When solving the coupling problem by sequential iterations, the effect of temperature on the material property and the loss density of the windings should be taken into account. In order to improve the computation efficiency for the coupling fields, an algorithm, which adopts two techniques, the dimensionless LSFEM and the combination of Jacobi preconditioned conjugate gradient method (JPCGM) and the two-side equilibration method (TSEM), is proposed in this paper. To validate the efficiency of the proposed algorithm, a local winding model of a transformer is built and the fluid field is computed by the conventional LSFEM, dimensionless LSFEM, and the Fluent software. While the fluid and thermal computation results of the local winding model of a transformer obtained by the two LSFEMs are basically consistent with those of the Fluent software, the stiffness matrix, which is formed by the dimensionless scheme of LSFEM and preconditioned by the JPCGM and TSEM, has a smaller condition number and a faster convergence rate of the equations. Thus, it demonstrates a broader applicability.

A new extrapolation cascadic multigrid method for three dimensional elliptic boundary value problems

In this paper, we develop a new extrapolation cascadic multigrid method, which makes it possible to solve three dimensional elliptic boundary value problems with over 100 million unknowns on a desktop computer in half a minute. First, by combining Richardson extrapolation and quadratic finite element (FE) interpolation for the numerical solutions on two-level of grids (current and previous grids), we provide a quite good initial guess for the iterative solution on the next finer grid, which is a third-order approximation to the FE solution. And the resulting large linear system from the FE discretization is then solved by the Jacobi-preconditioned conjugate gradient (JCG) method with the obtained initial guess. Additionally, instead of performing a fixed number of iterations as used in existing cascadic multigrid methods, a relative residual tolerance is introduced in the JCG solver, which enables us to obtain conveniently the numerical solution with the desired accuracy. Moreover, a simple method based on the midpoint extrapolation formula is proposed to achieve higher-order accuracy on the finest grid cheaply and directly. Test results from four examples including two smooth problems with both constant and variable coefficients, an H3-regular problem as well as an anisotropic problem are reported to show that the proposed method has much better efficiency compared to the classical V-cycle and W-cycle multigrid methods. Finally, we present the reason why our method is highly efficient for solving these elliptic problems.

An evaluation of four reordering algorithms to reduce the computational cost of the Jacobi-preconditioned Conjugate Gradient Method using high-precision arithmetic

In this work, four heuristics for bandwidth and profile reductions are evaluated. Specifically, the results of a recent proposed heuristic for bandwidth and profile reductions of symmetric and asymmetric matrices using a one-dimensional self-organising map is evaluated against the results obtained from the variable neighbourhood search for bandwidth reduction heuristic, the original reverse Cuthill-McKee method, and the reverse Cuthill-McKee method with starting pseudo-peripheral vertex given by the George-Liu algorithm. These four heuristics were applied to three datasets of linear systems composed of sparse symmetric positive-definite matrices arising from discretisations of the heat conduction and Laplace equations by finite volumes. The linear systems are solved by the Jacobi-preconditioned conjugate gradient method when using high-precision numerical computations. The best heuristic in the simulations performed with one of the datasets used was the Cuthill-McKee method with starting pseudo-peripheral vertex given by the George-Liu algorithm. On the other hand, no gain was obtained in relation to the computational cost of the linear system solver when a heuristic for bandwidth and profile reduction is applied to instances contained in two of the datasets used.

An evaluation of four reordering algorithms to reduce the computational cost of the Jacobi-preconditioned conjugate gradient method using high-precision arithmetic

An evaluation of four reordering algorithms to reduce the computational cost of the Jacobi-preconditioned conjugate gradient method using high-precision arithmetic

Simulation of Stokes flow over microelectrodes with least-squares meshfree method

Microelectrode structures in alternating current (AC) electrokinetics can generate high electric field strength to manipulate, characterize and separate particles in suspending medium. It has been widely used in biological, pharmaceutical and medical fields. In this paper, a least-squares meshfree method (LSMFM) based on the first-order velocity–pressure–vorticity formulation for the Stokes flow, electric potential–electric field strength expression for electric field and temperature–heat flux equations for heat transfer problem is presented to study two-dimensional electrothermally induced fluid flow over microelectrodes. Joule heat generated from electric field acts as heat source and gives rise to electric force and buoyancy force acting on the fluid. The discretization of all system of equations is completed by the least-squares method. The equal-order moving least-squares (MLS) approximation is used with Gaussian quadrature in the background cells constructed by the quadtree algorithm. A matrix-free element-by-element Jacobi preconditioned conjugate gradient method is applied to solve the resulting systems. Finally, an example of steady heat transfer problem with analytical solution is devised to analyze the error estimates of the LSMFM, and the examples of electric field of shielded microstrip line and Stokes flow over microelectrode are also solved to investigate the features of the LSMFM.

Parallelization of EMAP3D based on element-by-element Jacobi preconditioned conjugate gradient method

The demand for more accurate analyses of electromechanical systems, such as electromagnetic launchers and pulsed rotating power supplies, requires an increase in the size of the finite-element model of these systems. It is not uncommon for the number of unknowns for such a model to reach a half million. A parallel computing system with multiple processors and distributed memory, such as a PC cluster, makes it possible to obtain solutions for large-scale problems in reasonable times. In order to utilize this parallel hardware architecture, the software needs to be parallelized accordingly. Electromechanical Analysis Program in Three Dimensions (EMAP3D) is parallelized based on the element-by-element Jacobi preconditioned conjugate gradient (EBEJPCG) method because it is easily adopted into a parallel scheme and has low memory requirements because the formation of the global matrix is not necessary. The details of this algorithm are described in this paper. A block armature railgun was used to investigate this parallel algorithm on the Institute for Advanced Technology's (IAT's) eight-node PC-based Beowulf cluster. The performance of the algorithm in terms of speed-up ratio is presented.

A parallel algebraic multigrid solver for finite element method based source localization in the human brain

Time plays an important role in medical and neuropsychological diagnosis and research. In the field of Electro- and MagnetoEncephaloGraphy (EEG/MEG) source localization, a current distribution in the human brain is reconstructed noninvasively by means of measured fields outside the head. High resolution finite element modeling for the field computation leads to a sparse, large scale, linear equation system with many different right hand sides to be solved. The presented solution process is based on a parallel algebraic multigrid method. It is shown that very short computation times can be achieved through the combination of the multigrid technique and the parallelization on distributed memory computers. A solver time comparison to a classical parallel Jacobi preconditioned conjugate gradient method is given.

A least-squares spectral element method for computational aeroacoustics

Because of its relatively small magnitude compared with the carrier flow field variations, an acoustic signal is not easily tracked without distortion and degradation to the phase and amplitude information contained therein. The spurious dispersion and dissipation effects introduced by most numerical methods for computational fluid dynamics are detrimental to an accurate calculation of the acoustic signal. To rectify such deficiency in current codes, a least-squares spectral element method was developed to solve the linearized acoustic field equations derived from the full Navier–Stokes equations. The method solves the first-order partial-differential equations and minimizes the integral of the squares of the residual over the domain of interest. Spatial derivatives are approximated by Legendre polynomials, and temporal discretization is performed by dual-time stepping. The resultant algebraic equations are solved by a Jacobi preconditioned conjugate gradient method. Numerical examples were presented for a Gaussian pulse in a uniform flow of Mach 0.5 above an infinite plate. The computed acoustic pressures on the wall at different time instances were compared against the exact analytical values, and the comparison showed very good agreement. It indicates that the proposed method is indeed suitable for computing the sound generation and propagation in an unsteady flow regime.

Large-scale computation of incompressible viscous flow by least-squares finite element method

The least-square finite element method (LSFEM) based on the velocity-pressure-vorticity formulation is applied to large-scale/three-dimensional steady incompressible Navier-Stokes problems. This method can accommodate equal-order interpolations, and results in a symmetric, positive definite algebraic system which can be solved effectively by simple iterative methods. The first-order velocity-Bernoulli pressure-vorticity formulation for incompressible viscous flows is also tested. The first-order velocity-pressure-stress formulation is not elliptic in the ordinary sense, so we do not recommend its use for Newtonian flows. For three-dimensional flows, a compatibility equation, i.e., zero divergence of vorticity vector, is included to make the first-order system elliptic. As a by-product of proving the ellipticity of first-order systems, a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the Navier-Stokes equations. The simple substitution or Newton's method is employed to linearize the partial differential equations, the LSFEM is used to obtain discretized equations, and the system of algebraic equations is solved using the Jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices (matrix-free) to achieve high efficiency. To show the validity of this method for large-scale computation, we give numerical results for the 2D driven cavity problem at Re = 10000 with 408 × 400 bilinear elements. The flow in a 3D cavity is calculated at Re = 100, 400, and 1000 with 50 × 52 × 50 trilinear elements. The Taylor-Görtler-like vortices are observed for Re = 1000.

The multigrid method in solid mechanics: Part II—Practical applications

AbstractThe performance of the multigrid algorithm is investigated by solving some large, practical, three dimensional solid mechanics problems. The convergence of the method is sensitive to factors such as the amount of bending present and the degree of mesh non‐uniformity, as was also observed in Part I for two dimensional problems. However, in contrast to Part I, no proportionality is observed between the total number of operations to convergence and the problem size. Despite such behaviour, the multigrid algorithm proves to be an effective matrix equation solver for solid mechanics poblems. It is orders of magnitude faster than a direct factorization method, and yields converged solutions several times faster than the Jacobi preconditioned conjugate gradient method.