A Fast Finite Element Analysis of Millimeter Wave Waveguide Filters

In this letter, an effective symmetric successive overrelaxation (SSOR) preconditioning scheme is applied to the conjugate-gradient method for solving a large system of linear equations resulting from the use of the edge-based finite-element method (FEM). With SSOR as the preconditioner as well as its efficient implementation in the conjugate-gradient (CG) algorithm, the PCG method converges five times as fast as the CG method. This result demonstrates that SSOR is a good preconditioner for the CG iterative method when the FEM is applied to solve large-scale electromagnetic problems. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 235–238, 2000.

Gauss-Seidel preconditioned conjugate gradient (GS-PCG) method based on element-by-element (EBE) is derived and the calculation process of finite element method (FEM) is presented in this paper. The corresponding FEM program are developed and implemented on the magnetic field calculations which produced by straight conductor carrying current and 3-phases induction motor respectively. The validity and accuracy of the method have been verified by comparing the result with the analytical solution. Compared with Jacobi preconditioned conjugate gradient (J-PCG) method based on EBE-FEM, less computational time and cost are needed for the proposed method with same accuracy. The proposed method can be applied to electromagnetic field analysis of transformer and motor, which may refer large scale numerical computation.

Although most finite element programs have quite effective iterative solvers such as an incomplete Cholesky (IC) or symmetric successive overrelaxation (SSOR) preconditioned conjugate gradient (CG) method, the solution time may still become unacceptably long for very large systems. Convergence and thus total solution time can be shortened by using better preconditioners such as geometric multigrid methods. Algebraic multigrid methods have the supplementary advantage that no geometric information is needed and can thus be used as black box equation solvers. In the case of a finite element solution of a non-linear magnetostatic problem, the algebraic multigrid method reduces the overall computation time by a factor of 6 compared to a SSOR-CG solver.

A partial preconditioned conjugate gradient method for large eigenproblems

In this paper finite element method for 3D DC resistivity modeling accelerated using multi-GPU (Graphics Processing Unit). Solution of the large system of linear equations is the most expensive computation in finite element method performed in GPUs to reduce the computational time. Conjugate gradient solver used to solve large system of linear equations. We developed kernel for conjugate gradient solver that exploit data vectorization and written in PTX assembly form. We perform test on GTX 750Ti GPU and Tesla C2050 GPU. Our kernel have better sparse matrix-vector performance than CUSPARSE library in the first GPU, but lower performance in the latter GPU. The performance comparisons to the library are about 1.4 times and 0.7 times respectively. Our multi-GPU implementation achieved about 1.9 times performance of single GPU by using 2 identical GPUs. In comparison to the serial CPU implementation, about 10 times speedup could be achieved by using 2 GPUs.

The focus of this work is the numerical approximation of differential problems involving partial differential equations (PDEs) of elliptic nature, in the context of modelling and simulation in uid dynamics in porous media. The dissertation aims to contribute with an implementation of a multiscale multigrid algorithm, recently introduced in the literature, designed for solving ill-conditioned large linear systems of equations derived from those classes of PDEs, typically associated with Poisson problems of pressure-velocity with boundary conditions typical of ow in porous media. The Poisson problem discussed here is identied from the coupled convectiondiffusion transport system counterpart of PDEs, with dominated convection, and by a linearization by means the use of an operator splitting approach based on the physics of the underlying model. The methodology used for the discretization of the Poisson elliptic problem is mixed hybrid nite elements. The numerical solution of the resulting linear system will be addressed by a multiscale multigrid preconditioned conjugate gradient (PCG) method. We combine both methodologies in order to capture the distinct wavelengths associated with the different wavelengths from the assosiated self-adjoint Poisson operator, strongly inuenced by the heterogeneity of the geological properties of the porous media, in particular to the absolute permeability, which in turn might exhibit very large uctuations of orders of magnitude. Numerical experiments in applications of uid dynamics problems in porous media are presented and discussed for a verication of the results obtained by direct numerical simulations with the multiscale multigrid algorithm under consideration.

The temperature stress due to temperature changes is often more than the concrete tensile strength during construction period and operation period in massive concrete structures in hydraulic engineering, so the analysis of temperature stress, temperature control and measures to prevent the cracks become very important. The solution of large sparse linear equations is essential to finite element simulation in simulation analysis of temperature field. However, the traditional direct method will cost too much memory and calculation time, and the nodes is limited. The symmetric successive over relaxation (SSOR)-preconditioned conjugate gradient (PCG) method is a very efficient iterative method for solving large sparse linear equations. Construction process simulation and temperature filed calculation during construction period about Caohe river landing rectangle aqueduct of Beijing-Shijiazhuang emergency water supply engineering of middle-route of south-to-north water transfer project are carried out by FEM code using SSOR-PCG method, then the calculation result and monitoring value are compared. The comparison result shows that temperature field during construction can be accurately simulated by simulation calculation, which is helpful for guiding construction and choosing construction scheme, and worthy to popularize to other similar projects.

Parallel preconditioned conjugate gradient algorithm on GPU

Computer Solution of Large Linear Systems

The perfectly matched layer (PML) is very efficient and popular for grid truncation of open-region electromagnetic problems. In the context of the finite element method (FEM), the PML has been widely used for frequency-domain computations. On the other hand, the time-domain FEM formulation of the PML has been less investigated. Some of the first attempts involved approximations of the dispersive characteristics of the PML such that good absorption could only be achieved for a limited frequency band. We present a PML formulation for the time-domain FEM applied to three-dimensional problems. The proposed PML can be used for both scattered and total field formulations and we test it on radiating structures. Specifically, we demonstrate the accuracy and stability for the proposed algorithm given a short line current source.

The perfectly matched layer (PML) is very popular and efficient for grid truncation of open-region problems. The concept of PML was introduced by J. P. Berenger (see J. Comput. Phys., vol.114, p.185-200, 1994) together with a numerical implementation based on the finite-difference time-domain (FDTD) scheme. However, the basic FDTD scheme is formulated on Cartesian grids and in many cases an unnecessarily large free-space region must be discretized between the PML and the object under investigation. F.L. Teixeira et al. (see IEEE Trans. Antennas Propag., vol.49, p.902-7, 2001) developed a conformal PML for the FDTD scheme. We continue this effort by presenting a new conformal PML formulation for the finite element method (FEM) in the time domain. It is based on the anisotropic material which has been used for the time domain FEM by D. Jiao et al. (see IEEE Antennas and Propag. Soc. Int. Symp., vol.2, p.158-61, 2002). We present the 2D-version of the conformal PML, which is a special case of our 3D-formulation. Furthermore, we consider only radar cross section (RCS) computations based on the scattered field formulation and emphasize that our PML is applicable to the total field formulation as well as radiating structures.

The focus of this study is on the efficient solution for three-dimensional (3D) time-harmonic eddy current problems discretised by the finite-element method (FEM) with thin elements. The systems of equations arising from the finite-element formulations are solved by the geometric multigrid preconditioned conjugate gradient (MGCG) method. Numerical examples show that the MGCG method has stable convergence, which does not deteriorate for elements with high aspect ratios and its computation time is much less than that of the conjugate gradient method with incomplete Cholesky factorisation as the preconditioner (ICCG).

This paper presents a symmetric coupling between the finite element and boundary element methods for analysing open microwave devices. The proposed finite element boundary element (FEBE) method is based on the E-field vector Helmholtz equation. Second-order hierarchical Curl-conforming vector finite elements over tetrahedrons are used to discretize the interior region. The boundary element (BE) truncation surface is discretized with the second-order divergence-conforming surface elements with triangular support. The symmetry of the BE part is restored through the application of the Calderon-projector. Moreover, the BE computations are accelerated using a single level QR algorithm. This reduces both memory and computational time. The resulted symmetric system of equations is solved with a very efficient preconditioned conjugate gradient (PCCG) method with a p-multiplicative Schwarz (pMUS) preconditioner. Copyright © 2005 John Wiley & Sons, Ltd.

Quark propagator on the Connection Machine

Problems involving the diffusion and transport (angular dependence considered) of neutron radiation are frequently encountered in radiation physics and nuclear engineering. Neutron diffusion problems require substantial computer storage arising from the energy and spatial dependence, while in the case of transport problems the angular dependence of the neutron flux and of the neutron scattering process gives rise to a further considerable increase in storage. In the present work, a method has been developed which greatly eases the bandwidth problem in the solution of large systems of linear equations and algorithms have been developed to assemble and solve such systems of equations in one and two dimensions. The formalism used is the variational method and, for simplicity, linear interpolating functions have been used to derive a symmetric banded system of equations. Finite element algorithms have been implemented in the codes FEED1 and FEED2, which treat one and two space dimensions, respectively. Three benchmark problems are analysed in this paper and results are compared with finite difference discrete ordinate method solutions. It is shown that the finite element method provides fast and accurate solutions to neutron diffusion and transport problems.