Parallel Solution Methods Research Articles

On parallel solution of linear elasticity problems. Part III: higher order finite elements

SUMMARYThis is the third part of a trilogy on parallel solution of the linear elasticity problem. We consider the separate displacement ordering for a plain isotropic problem with full Dirichlet boundary conditions. The parallel solution methods presented in the first two parts of the trilogy are here generalised to higher order by using hierarchical finite elements. We discuss node numberings on regular grids for high degree of parallelism and even processor load as well as the problem of stability of the modified incomplete Cholesky factorisations used. Several preconditioning techniques for the conjugate gradient method are studied and compared for quadratic finite elements. Bounds for the condition numbers of the corresponding preconditioning methods are derived, and computer experiments are performed in order to confirm the theory and give recommendations on the choice of method. The parallel implementation is performed by message passing interface. Copyright © 2012 John Wiley & Sons, Ltd.

Parallel solution methods for Poisson-like equations in two-phase flows

A parallel CFD code is being modified to improve performance for multiphase flows. The code uses a hybrid front-tracking/front-capturing technique to solve for two immiscible fluids simultaneously on a regular 3D Cartesian grid. The pressure solver uses a multigrid method to solve a Poisson-like equations with variable coefficients. A Schwarz–Aitken acceleration technique is applied to the solver, and the results analysed.

Parallel solution of model predictive control using the alternating direction multiplier method

In this work a parallel solution method for model predictive control is presented based on the alternating direction multiplier method. Our approach solves the overall problem by decomposing it along the time-axis into smaller subproblem. The resulting subproblems can be solved efficiently in parallel and are coupled by the multiplier update. We specifically considers box constraints, since this allows to use tailored methods to solve the subproblems. We outline the applicability of the approach and the performance using a multi-core systems.

An operator-splitting finite element method for the efficient parallel solution of multidimensional population balance systems

A finite element method for solving multidimensional population balance systems is proposed where the balance of fluid velocity, temperature and solute partial density is considered as a two-dimensional system and the balance of particle size distribution as a three-dimensional one. The method is based on a dimensional splitting into physical space and internal property variables. In addition, the operator splitting allows to decouple the equations for temperature, solute partial density and particle size distribution. Further, a nodal point based parallel finite element algorithm for multi-dimensional population balance systems is presented. The method is applied to study a crystallization process assuming, for simplicity, a size independent growth rate and neglecting agglomeration and breakage of particles. Simulations for different wall temperatures are performed to show the effect of cooling on the crystal growth. Although the method is described in detail only for the case of d=2 space and s=1 internal property variables it has the potential to be extendable to d+ s variables, d=2, 3 and s ≥ 1 .

Progressive hedging‐based metaheuristics for stochastic network design

AbstractWe consider the stochastic fixed‐charge capacitated multicommodity network design (S‐CMND) problem with uncertain demand. We propose a two‐stage stochastic programming formulation, where design decisions make up the first stage, while recourse decisions are made in the second stage to distribute the commodities according to observed demands. The overall objective is to optimize the cost of the first‐stage design decisions plus the total expected distribution cost incurred in the second stage. To solve this formulation, we propose a metaheuristic framework inspired by the progressive hedging algorithm of Rockafellar and Wets. Following this strategy, scenario decomposition is used to separate the stochastic problem following the possible outcomes, scenarios, of the random event. Each scenario subproblem then becomes a deterministic CMND problem to be solved, which may be addressed by efficient specialized methods. We also propose and compare different strategies to gradually guide scenario subproblems to agree on the status of design arcs and aim for a good global design. These strategies are embedded into a parallel solution method, which is numerically shown to be computationally efficient and to yield high‐quality solutions under various problem characteristics and demand correlations. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011.

Parallel simulation of an infinitesimal elasto‐plastic Cosserat model

AbstractWe review the continuous and fully‐discrete problem setting of an elasto‐plastic Cosserat model, which is an enhanced continuum model with independent rotational degrees of freedom. The continuous model is defined by a variational inequality, and for the discrete model a corresponding non‐smooth nonlinear variational problem is derived which can be solved with a generalized Newton method. We present numerical results for two representative geometries, where we test the convergence for different finite element discretizations, we study the parameter dependency, and we demonstrate the efficiency of the parallel solution method for very large problem sizes (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Workload Distribution Framework for the Parallel Solution of Large Structural Models on Heterogeneous PC Clusters

One of the main problems of substructure-based parallel solution methods is the imbalances in the condensation times of substructures when direct solvers are used. Such imbalances usually decrease the performance of the parallel solution. Thus, in this study, a workload distribution framework for such methods at heterogeneous computing environment is presented. The main idea behind this framework is to iteratively adjust the shapes of substructures so that the imbalance in their condensation times is minimized. Both generated and actual structural models were solved to illustrate the applicability and the efficiency of this approach. In these examples, a PC cluster having eight different computers was used.

マルチコアクラスタにおける電力系統シミュレーションの階層的並列化

Multi-core CPU is widely used for PC now. PC cluster, which is composed of many PC's, can easily get its high performance using multi-core CPU's. It is named as a multi-core cluster. In this paper, we propose hierarchical parallel solution methods of power system transient stability analysis on a multi-core cluster.In a multi-core cluster, there are two level layers in communication system. One is the communication among cores in a CPU, and the other is the communication among CPU's in the multi-core cluster. To communicate between cores in a CPU, there are two ways. One is the communication through cache, and the other is the communication through bus in a CPU. The speed of the former one through cache is higher than that through bus. In addition, to get maximum speed of communication through cache, the size of data to process should be kept within the size of each cache used in each calculation step. A long development term is necessary to optimize software for two communication ways. We developed an optimization technique of matrix operations and a hierarchical parallel processing method for a multi-core cluster, and applied them to solve differential equations and simultaneous equations in power system simulation. We obtained 1.5 times speedup using two cores in a PC, which communicate through shared memory and 1.39 times speedup using two PCs with single core which communicate through bus.

マルチコアＰＣクラスタにおける電力系統シミュレーションの並列化手法の検討

This letter describes a parallel solution method of power system transient stability analysis on a multi-core PC cluster system. We use a cash blocking method for solving differential equations and simultaneous equations. This method can use multi-core PC cluster efficiently by dividing matrix calculation of differential equations and simultaneous equations of power system simulation into cash size of each core.

On a high-order compact scheme and its utilization in parallel solution of a time-dependent system on a distributed memory processor

The focus of this study is the design of a parallel solution method that utilizes a fourth-order compact scheme. The applicability of the method is demonstrated on a time-dependent parabolic system with Neumann boundaries. The core of the parallel computing facilities used in the study is a 2-head-node, 224-compute-node Apple Xserve G5 multiprocessor. The system is first discretized in both time and space such that it remains in its stability regimes, before being solved with the method. The solution requires time marching in which every time step, h t , calls for a single parallel solve of the intermediary subsystems generated. The solution uses p processors ranging in numbers from 3 to 63. The speedups, s p , approach their limiting value of p only when p is small. The solution produces good computational results at large p, but poor results as p becomes progressively small. Also, the parallel solution produces accurate results yielding good speedups and efficiencies only when p is within some reasonable range of values. The intermediary systems generated by this method are linear and fine-grained, therefore, they are best suited for solution on massively-parallel processors. The solution method proposed in this study is, therefore, expected to yield more impressive results if applied in a massively-parallel computing environment.

Development and application of a parallel multigrid solver for the simulation of spreading droplets

AbstractWe consider the parallel application of an efficient solver developed for the accurate solution of a range of droplet spreading flows modelled as a coupled set of nonlinear lubrication equations. The underlying numerical scheme is based upon a second‐order finite difference discretization in space and a second‐order, fully implicit, adaptive scheme in time. At each time step, this leads to the need to solve a large system of nonlinear algebraic equations, for which the full approximation storage multigrid algorithm is employed. The motion of the contact line between the three phases (liquid, air and the solid substrate) is based upon the assumption of a thin precursor film, with a corresponding disjoining pressure term in the governing equations. It is the inclusion of this precursor film in the model that motivates the need for a parallel solution method. This is because the thickness of such a film must be very small in order to yield realistic predictions, while the finite difference grid must be correspondingly fine in order to obtain accurate numerical solutions. Results are presented which demonstrate that the parallel implementation is sufficiently efficient and robust to allow reliable numerical solutions to be obtained for a level of mesh resolution that is an order of magnitude finer than is possible using a single processor. Copyright © 2008 John Wiley & Sons, Ltd.

Numerical performance of a parallel solution method for a heterogeneous 2D Helmholtz equation

The parallel performance of a numerical solution method for the scalar 2D Helmholtz equation written for inhomogeneous media is studied. The numerical solution is obtained by an iterative method applied to the preconditioned linear system which has been derived from a finite difference discretization. The preconditioner is approximately inverted using multigrid iterations. Parallel execution is implemented using the MPI library. Only a few iterations are required to solve numerically the so-called full Marmousi problem [Bourgeois, A., et al. in The Marmousi Experience, Proceedings of the 1990 EAEG Workshop on Practical Aspects of Seismic Data Inversion: Eur. Assoc. Expl. Geophys., pp. 5–16 (1991)] for the high frequency range.

Parallel Solution Methods for Porous Media Models in Biomechanics

AbstractWe present a biomechanical application of our parallel finite element model for coupled problems in solid mechanics. This programming framework provides a very lean and flexible interface, which allows to realize time‐dependent nonlinear simulations. In this context, a special variant of a stabilized local Gauß‐Seidel preconditioner is introduced, which can be successfully employed to large scale computations. Finally, the efficiency of the implemented algorithm is shown by a numerical example considering the axial compression of a L4‐L5 motion segment of the spine. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Parallel iterative solution for the Helmholtz equation with exact non-reflecting boundary conditions

Efficient and scalable parallel solution methods are presented for the Helmholtz equation with global non-reflecting DtN boundary conditions. The symmetric outer-product structure of the DtN operator is exploited to significantly reduce inter-processor communication required by the non-locality of the DtN to one collective communication per iteration with a vector size equal to the number of harmonics included in the DtN series expansion, independent of the grain size. Numerical studies show that in the context of iterative equation solvers, and for the same accuracy, the global DtN applied to a tightly fitting spheroidal boundary and implemented as a low-rank update with the multiplicative split is more cost-effective (both in wall-clock times and memory) compared to local approximate boundary conditions.

混合型有限要素を用いた液体構造強連成解析における連立一次方程式に対する並列化手法(OS20c 流体構造連成解析)

混合型有限要素を用いた液体構造強連成解析における連立一次方程式に対する並列化手法(OS20c 流体構造連成解析)

A New Wz Factorization For Parallel Solution Of Tridiagonal Systems

Motivated by the structure of a matrix factorization introduced recently by Evans (1999), we introduce a new WZ factorization for use with the partition method for parallel solution of tridiagonal systems. The factorization helps us to uncouple partitioned subsystems for parallel processing of their solution. A crucial question for the validity of the partition method is the existence and stability of the whole solution across the partitioning blocks . We show that if the given system is nonsingular and diagonally dominant, then within each block the WZ factorization exists and is (numerically) strongly stable, and the solution across the partitioning blocks exists (does not terminate prematurely).

Effective multidimensional resistivity inversion using finite-element techniques

SUMMARY This paper describes the development of a multidimensional resistivity inversion method that is validated using two- and three-dimensional synthetic pole‐pole data. We use a finite-element basis to represent both the electric potentials of each source problem and the conductivities describing the model. A least-squares method is used to solve the inverse problem. Using a leastsquares method rather than a lower-order method such as non-linear conjugate gradients, has the advantage that quadratic terms in the functional to be optimized are treated implicitly allowing for a near minimum to be found after a single iteration in problems where quadratic terms dominate. Both the source problem for a potential field and the least-squares problem are solved using (linear) pre-conditioned conjugate gradients. Coupled with the use of parallel domain decomposition solution methods, this provides the numerical tools necessary for efficient inversion of multidimensional problems. Since the electrical inverse problem is ill-conditioned, special attention is given to the use of model-covariance matrices and data weighting to assist the inversion process to arrive at a physically plausible result. The model-covariance used allows for preferential model regularization in arbitrary directions and the application of spatially varying regularization. We demonstrate, using two previously published synthetic models, two methods of improving model resolution away from sources and receivers. The first method explores the possibilities of using depth-dependent and directionally varying smoothness constraints. The second method preferentially applies additional weights to data known to contain information concerning poorly resolved areas. In the given examples, both methods improve the inversion model and encourage the reconstruction algorithm to create model structure at depth.

ChemInform Abstract: Preparation of Small Molecule Libraries for Agrochemical Discovery. Development of Parallel Solution and Solid Phase Synthesis Methods

AbstractChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 100 leading journals. To access a ChemInform Abstract of an article which was published elsewhere, please select a “Full Text” option. The original article is trackable via the “References” option.

A parallel preconditioned conjugate gradient solution method for finite element problems with coarse–fine mesh formulation

The object of this paper is a parallel preconditioned conjugate gradient iterative solver for finite element problems with coarse-mesh/fine-mesh formulation. An efficient preconditioner is easily derived from the multigrid stiffness matrix. The method has been implemented, for the sake of comparison, both on a IBM-RISC590 and on a Quadrics-QH1, a massive parallel SIMD machine with 128 processors. Examples of solutions of simple linear elastic problems on rectangular grids are presented and convergence and parallel performance are discussed.

Parallel generation of triangular and quadrilateral meshes

We describe a parallel solution method to perform mesh partitioning and mesh generation completely in parallel without the preceding serial mesh-generation process. The proposed approach avoids this serial bottleneck. The finite element data are generated exactly at the memory location where they are processed in the subsequent analysis process. The geometric description of the computational domain consists of vertices, edges and faces, boundary conditions, loads and mesh density parameters. The geometric description is recursively partitioned, whereby the domain interfaces are minimised with respect to the number of interface nodes. Load balance is guaranteed for uniform and locally refined meshes by a specially adapted estimation procedure. Applications for two-dimensional all triangular and all quadrilateral meshes in structural mechanics are demonstrated.