Block Gauss-Seidel Algorithm Research Articles

A unified and modular coupling of particle methods with fem for civil engineering problems

In this work, a modular coupling approach for particle methods with the FEM (finite element method) is presented. The proposed coupled strategy takes advantage from the ability of particle methods of dealing with large displacements and deformations, especially when solving complex fluid–structure and solid–structure interaction problems. The coupling between the FEM and particle methods is done using a co-simulation approach implemented in the open-source Kratos Multiphysics framework. The particle methods considered in this work are the DEM (discrete element method) and the PFEM (particle finite element method). The Lagrangian description of the PFEM is well suited for modeling fluids undergoing large deformations and free-surface motions, and the DEM can be used to simulate rocks, debris and other solid objects. To accelerate the convergence of the coupled strategy, a block Gauss–Seidel algorithm with Aitken relaxation is used. Several numerical examples, with an emphasis on natural hazards, are presented to test and validate the proposed coupled method.

An Efficient Scheme for Coupling OpenMC and FLUENT with Adaptive Load Balancing

This paper develops a multi-physics interface code MC-FLUENT to couple the Monte Carlo code OpenMC with the commercial computational fluid dynamics code ANSYS FLUENT. The implementations and parallel performances of block Gauss–Seidel-type and block Jacobi-type Picard iterative algorithms have been investigated. In addition, this paper introduces two adaptive load-balancing algorithms into the neutronics and thermal-hydraulics coupled simulation to reduce the time cost of computation. Considering that the different scalability of OpenMC and FLUENT limits the performance of block Gauss–Seidel algorithm, an adaptive load-balancing algorithm that can increase the number of nodes dynamically is proposed to improve its efficiency. Moreover, with the natural parallelism of block Jacobi algorithm, another adaptive load-balancing algorithm is proposed to improve its performance. A 3 x 3 PWR fuel pin model and a 1000 MWt ABR metallic benchmark core were used to compare the performances of the two algorithms and verify the effectiveness of the two adaptive load-balancing algorithms. The results show that the adaptive load-balancing algorithms proposed in this paper can greatly improve the computing efficiency of block Jacobi algorithm and improve the performance of block Gauss–Seidel algorithm when the number of nodes is large. In addition, the adaptive load-balancing algorithms are especially effective when a case demands different computational power of OpenMC and FLUENT.

A doubly stochastic block Gauss–Seidel algorithm for solving linear equations

We propose a doubly stochastic block Gauss–Seidel algorithm for solving linear systems of equations. By varying the row partition parameter and the column partition parameter for the coefficient matrix, we recover the Landweber algorithm, the randomized Kaczmarz algorithm, the randomized coordinate descent algorithm, and the doubly stochastic Gauss–Seidel algorithm. For arbitrary (consistent or inconsistent, full column rank or rank-deficient) linear systems, we prove the exponential convergence of the norm of the expected error via exact formulas. We also prove the exponential convergence of the expected norm of the error for consistent linear systems, and the exponential convergence of the expected norm of the residual for arbitrary linear systems. Numerical experiments for linear systems with synthetic and real-world coefficient matrices are given to demonstrate the efficiency of our algorithm.

Partitioned water hammer modeling using the block Gauss–Seidel algorithm

The monolithic classical and extended fluid-structural interaction water hammer models are based on the simplified ring hypothesis to characterize the structural radial response of the pipe under pressure. Recent work has shown that this approach predicts an unrealistic discontinuity in the pipe wall in the neighborhood of the pressure wave front. In order to remedy this limitation, the present study adopts a partitioned approach which couples a continuous shell-based finite element model with a water hammer fluid model based on the method of characteristics, within the context of the partitioned block Gauss–Seidel iterative algorithm. In order to demonstrate the validity of the block Gauss–Seidel implementation, the classical and extended water hammer models, normally solved using a monolithic approach, are solved under the present partitioned model and shown to predict pressure histories identical to those based on the monolithic solution. Comparisons are then conducted for the pressure histories as predicted by the classical, extended, and shell-based approaches. The study shows that the accuracy is optimized for integer values of the Courant number. By adopting either an optimal constant relaxation factor or the Aitken relaxation factor, the number of iterations needed for convergence was significantly reduced. The accuracy and computational efficiency are shown to highly depend on the number of subdivisions in the fluid model. In contrast, the number of subdivisions in the structural model, while influencing the computational efficiency, is shown not to influence the accuracy of the predictions. When applying the partitioned approach to the classical and shell-based models, the predicted pressure histories are found to be independent of the specified tolerance. In contrast, when applied to the extended model, the chosen tolerance has implications on the stability and accuracy of the solution.

On maximum residual block and two-step Gauss–Seidel algorithms for linear least-squares problems

The block Gauss–Seidel algorithm can significantly outperform the simple randomized Gauss–Seidel algorithm for solving overdetermined least-squares problems since it moves a large block of columns rather than a single column into working memory. Here, with the help of the maximum residual rule, we construct a two-step Gauss–Seidel (2SGS) algorithm, which selects two different columns simultaneously at each iteration. As a natural extension of the 2SGS algorithm, we further propose a multi-step Gauss–Seidel algorithm, that is, the maximum residual block Gauss–Seidel (MRBGS) algorithm for solving overdetermined least-squares problems. We prove that these two different algorithms converge to the unique solution of the overdetermined least-squares problem when its coefficient matrix is of full column rank. Numerical experiments on Gaussian models as well as on 2D image reconstruction problems, show that 2SGS is more effective than the greedy randomized Gauss–Seidel algorithm, and MRBGS apparently outperforms both the greedy and randomized block Gauss–Seidel algorithms in terms of numerical performance.

Finite element analysis of a thin piezoelectric bimorph with a metal shim using solid direct-piezoelectric and shell inverse-piezoelectric coupling with pseudo direct-piezoelectric evaluation

A thin triple-layer piezoelectric bender, also known as a piezoelectric bimorph with a metal shim, consists of an elastic layer (metal shim) sandwiched between two piezoelectric layers. It is one of the most commonly used piezoelectric composite structures in both sensing and actuation applications. Therefore, this study deals with the coupled piezoelectric-structure interaction analysis of a thin piezoelectric bimorph with a metal shim. In this work, the best features of the solid and shell elements were combined to analyze the piezoelectric and structural fields, respectively, in a thin piezoelectric bimorph with a metal shim. This approach addresses the shortcomings of using a single finite element mesh for both the piezoelectric and structural fields. A transformation method using the block Gauss-Seidel algorithm is employed to exchange the variables between solid direct-piezoelectric and shell inverse-piezoelectric analyses. The metal layer can be analyzed as an elastic body with electrical boundary conditions. A pseudo direct-piezoelectric evaluation method for the metal shim in the piezoelectric analysis is proposed. This method allows us to reuse the existing piezoelectric analysis program, where the metal shim is analyzed as the pseudo direct-piezoelectric material in a single analysis procedure. The homogenization method for the evaluation of bending rigidity and mass is used for modeling the single shell structure in the inverse-piezoelectric analysis of a thin piezoelectric bimorph with a metal layer. Numerical results are given for various electrical configurations of actuators and sensors to validate the present method. Comparison with an exact solution illustrates the accuracy, efficiency, and capability of the developed solid direct and shell inverse-piezoelectric analysis coupled with a pseudo direct-piezoelectric evaluation method to capture the sensor and actuator response of a thin piezoelectric bimorph with a metal shim.

The LIR space partitioning system applied to the Stokes equations

We introduce a novel approach to solve the stationary Stokes equations on very large voxel geometries. One main idea is to coarsen a voxel geometry in areas where velocity does not vary much while keeping the original resolution near the solid surfaces. For spatial partitioning a simplified LIR-tree is used which is a generalization of the Octree and KD-tree. The other main idea is to arrange variables in a way such that each cell is able to satisfy the Stokes equations independently. Pressure and velocity are discretized on staggered grids. But instead of using one velocity variable on the cell surface we introduce two variables. The discretization of momentum and mass conservation yields a small linear system (block) per cell that allows to use the block Gauss–Seidel algorithm as iterative solver. We compare our method to other solvers and conclude superior performance in runtime and memory for high porosity geometries.

Geometric Mesh Three-Point Discretization for Fourth-Order Nonlinear Singular Differential Equations in Polar System

Numerical method based on three geometric stencils has been proposed for the numerical solution of nonlinear singular fourth-order ordinary differential equations. The method can be easily extended to the sixth-order differential equations. Convergence analysis proves the third-order convergence of the proposed scheme. The resulting difference equations lead to block tridiagonal matrices and can be easily solved using block Gauss-Seidel algorithm. The computational results are provided to justify the usefulness and reliability of the proposed method.

A complex boundary integral method for multiple circular holes in an infinite plane

A complex boundary integral equation method, combined with series expansion technique, is presented for the problem of an infinite, isotropic elastic plane containing multiple circular holes. Loading is applied at infinity or on the boundaries of the holes. The sizes and locations of the holes are arbitrary provided they do not overlap. The analysis procedure is based on the use of a complex hypersingular integral equation that expresses a direct relationship between all the boundary tractions and displacements. The unknown displacements on each circular boundary are represented by truncated complex Fourier series, and all of the integrals involved in the complex integral equation are evaluated analytically. A system of linear algebraic equations is obtained by using a Taylor series expansion, and the block Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the accuracy, versatility, and efficiency of the approach.

NECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL GEOMETRIC CONVERGENCE OF BLOCK GAUSS-SEIDEL ITERATION ALGORITHM APPLIED TO MARKOV CHAINS

Convergence properties of the block Gauss-Seidel algorithm applied to ergodic Markov chains are discussed in this paper. This algorithm is one of the most prevalent methods for computing ergodic probability vectors of large-scale Markov chains. We will provide necessary and sufficient conditions for global geometric convergence of this algorithm. To apply this algorithm, the state space of a Markov process is decomposed into mutually exclusive and exhaustive lumps. The convergence properties depend on this lumping. It is also shown that, there exists at least one set of lumps, for any ergodic stochastic matrix, which assures geometric convergence of the algorithm.

An iterative method for adaptive finite element computation

An iterative method designed for solving systems of equations arising from adaptive finite element computation is presented. The method is based on the block Gauss-Seidel algorithm. The convergence theory presented in the paper gives very sharp convergence bounds, which can be easily evaluated. The method applies for h, p-extensions and various types of elements.

Solution of three-dimensional boundary layer equations by a discontinuous finite element method, part I: Numerical analysis of a linear model problem

We present, in this paper, a new numerical method designed to solve three-dimensional boundary layer equations for incompressible stationary fluid flows. The method consists in using finite elements piecewise constant in the directions parallel to the surface and continuous piecewise linear in the normal direction. We consider a linear model problem describing the heat transfer process in a three-dimensional boundary layer. For this problem, we derive an error estimate and construct a block Gauss-Seidel algorithm whose convergence is proved. This algorithm consists in solving the problem by progressing on the surface at each iteration. A numerical test is given to confirm the theoretical results. In a subsequent paper, the case of nonlinear boundary layer equations is handled in detail.

Convergence rate analysis of the state increment dynamic programming method

It is shown that if the discretization of the state space is sufficiently fine and if the limiting trajectory is an interior point of the admissible policies then the state increment dynamic programming technique (SIDP) has linear convergence, and the coefficient of convergence can be explicitly related to the control problem components. In the course of this analysis, it is seen that the SIDP method is, in essence, a nonlinear generalization of the block Gauss-Seidel algorithm for solving linear equations. Some consequences of this insight are suggested, and a computational example comparing SIDP with differential dynamic programming is offered.