A parallel finite element solution method

New parallel computer architectures have revolutionized the design of computer algorithms, and promise to have significant influence on algorithms for structural engineering computations. In this paper, a parallel finite element solution method is presented. The solution method proposed does not require the formation of global system equations, but computes directly the element distortions, as opposed to solving a system of nodal equations. An element or substructure is mapped on to a processor of an MIMD multiprocessing system. Each processor stores only the information relevant to the element or substructure for which the processor represents. The finite element computations can be performed in parallel, in that a processor generates the local stiffness, computes the element distortions and determines the stress-strain characteristics for the element or substructure associated with the processor.     

Implementation of an element-by-element solution algorithm for the finite element method on a coarse-grained parallel computer

An element-by-element solution algorithm for systems of equations arising in applying the finite element method in solid mechanics was implemented on the loosely coupled array of processors (lCAP) parallel computer located at IBM Kingston. The element-by-element algorithm has previously been shown to be advantageous over direct solution algorithms for large problems on sequential computers. It also has the advantage that it can be implemented in parallel on machines such as the lCAP in a relatively straightforward manner. The results show that solution speedup efficiencies of approximately 95% can be readily achieved with this method, with no indication that the speed-up efficiency drops off as more processors are added. The implementation used is applicable to other coarse-grained parallel architectures in addition to the lCAP computer.     

A finite difference scheme for the incompressible advection-diffusion equation

A representation of continuum space by a second order grid system is proposed. A new finite difference scheme for the two-dimensional incompressible advection-diffusion equation is derived for the model. The difference scheme is flux conserving and contains no spatial truncation error with respect to the model except through approximation of the local velocity. It contains no false diffusion and preserves the sign of positive definite quantities. It is simple to program and is subject only to the usual diffusion and Courant-Friedrichs-Lewy stability conditions. It is stable at all grid mesh sizes or cell Reynolds numbers.A sample one-dimensional problem is presented and comparison made with the standard central difference and “upwind” difference schemes.     

Two-grid method for the two-dimensional time-dependent Schrödinger equation by the finite element method

In this paper, we construct a backward Euler full-discrete two-grid finite element scheme for the two-dimensional time-dependent Schrödinger equation. With this method, the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two Poisson equations on the same fine grid. We analyze the error estimate of the standard finite element solution and the two-grid solution in the $H^1$ norm. It is shown that the two-grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy $H=O\left(h^{\frac{k}{k+1}}\right)$. Finally, a numerical experiment indicates that our two-grid algorithm is more efficient than the standard finite element method.     

A space-time least-square finite element scheme for advection-diffusion equations

A space-time least-square finite element scheme is presented for the advection-diffusion problems at moderate to high Peclet numbers. This scheme is designed to eliminate spurious oscillations and can be used to define the steady-state solution as the asymptotic transient solution for large time. Numerical results, using linear elements in a 1D space and bilinear elements in a 2D space, demonstrate the accuracy and the stability of the new scheme.     

A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation

To avoid the local oscillations that still remain using the streamline-upwind/Petrov-Galerkin formulation for the scalar convection-diffusion equation, the introduction of a nonlinear crosswind dissipation is proposed. It is shown that the method is less overdiffusive than other discontinuity-capturing techniques and has better numerical behavior. The design of the crosswind diffusion is based on the study of the discrete maximum principle for some simple cases.     

A parallel method for the numerical solution of integro-differential equation with positive memory

An efficient parallel numerical method is proposed for an integro-differential equation with positive memory. Instead of solving the equation in classical time-marching methods which require massive storage of solutions of previous time steps in order to advance to a next time step, the Fourier–Laplace transformation in time is applied to obtain a set of complex-valued, elliptic problems parameterized by points on a contour in the complex plane. Using the independence of an elliptic problem corresponding to one contour point is independent of those elliptic problems corresponding to other contour points, all elliptic problems can be solved in parallel essentially without data communications. Then the time domain solution can be obtained by the Fourier–Laplace inversion formula. An error analysis and the numerical implementation of this parallel method is presented.     

A finite element method for the solution of rotary pumps

We present in this paper a numerical strategy for the simulation of rotary positive displacement pumps, taking as an example a gear pump. While the two gears of the pump are rotating, the intersection between them changes in time. Therefore, the computational domain should be recomputed in some way at each time step. The strategy used here consists in dividing a cycle into a certain number of time steps and obtaining different computational meshes for each of these time steps. The coupling between two consecutive time steps is achieved by interpolating the flow unknowns in a proper way. This geometrical decomposition enables one to have a plain control over the mesh, particularly in the zones of interest, which are the gap between the gears and the casing, and the engagement and disengagement zones of the gears.     

A new parallel domain decomposition method for the adaptive finite element solution of elliptic partial differential equations

We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15 , it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load-balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright © 2001 John Wiley & Sons, Ltd.     

A transient finite element solution method for the Navier-Stokes equations

This paper is concerned with the discrete finite element formulation and numerical solution of transient incompressible viscous flow in terms of the primitive variables. A restricted variational principle is introduced as equivalent to the momentum equations and the Poisson equation for pressure. The latter is introduced to replace the continuity equation, and thus the incompressibility condition is realized only asymptotically; i.e. through the iterative process. An incomplete cubic interpolation function is used for both the velocities and pressure within a triangular finite element. The discrete equations are integrated in time with backward finite differences. We illustrate the similarity between the (ψ,ζ) finite difference method and the ($\text{u}\text{,p}$) finite element method by calculations on the driven square cavity problem.     

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.     

TVD finite point method for two-dimensional conservation equation

A new meshless method, called total variation diminishing (TVD) finite point method (TVDFP), is proposed. The TVDFP method is developed on the least-square procedure which uses a global stencil of grid points and the two-dimensional (2D) TVD procedure for the approximation of fictitious interface directional fluxes. We present the accuracy of the TVDFP method and several 2D test computations.     

二维不可压缩Navier-Stokes方程的并行谱有限元法求解

针对不可压缩Navier-Stokes （N-S）方程求解过程中的有限元法存在计算网格量大、收敛速度慢的缺点,提出了基于面积坐标的三角网格剖分谱有限元法（TSFEM）并进一步给出了利用OpenMP对其并行化的方法。该算法结合谱方法和有限元法思想,选取具有无限光滑特性的指数函数取代传统有限元法中的多项式函数作为基函数,能够有效减少计算网格数量,提高算法的精度和收敛速度;利用面积坐标便于三角形单元计算的特点,选取三角单元作为计算单元,增强了适用性;在顶盖方腔驱动流问题上对该算法进行验证。实验结果表明,TSFEM较传统有限元法（FEM）无论是收敛速度还是计算效率都有了显著提高。     

A Galerkin finite element method for time-fractional stochastic heat equation

In this study, a Galerkin finite element method is presented for time-fractional stochastic heat equation driven by multiplicative noise, which arises from the consideration of heat transport in porous media with thermal memory with random effects. The spatial and temporal regularity properties of mild solution to the given problem under certain sufficient conditions are obtained. Numerical techniques are developed by the standard Galerkin finite element method in spatial direction, and Gorenflo–Mainardi–Moretti–Paradisi scheme is applied in temporal direction. The convergence error estimates for both semi-discrete and fully discrete schemes are established. Finally, numerical example is provided to verify the theoretical results.     

A parallel DSDADI method for solution of the steady state diffusion equation

A parallel diagonally scaled dynamic alternating-direction-implicit (DSDADI) method is shown to be an effective algorithm for solving the 2D and 3D steady-state diffusion equation on large uniform Cartesian grids. Empirical evidence from the parallel solution of large gridsize problems suggests that the computational work done by DSDADI to converge over an N^d grid with continuous diffusivity is of lower order than O(N^d+α) for any fixed α > 0. This is in contrast to the method of diagonally scaled conjugate gradients (DSCG), for which the computational work necessary for convergence is O(N^d+1). Furthermore, the combination of diagonal scaling, spatial domain decomposition (SDD), and distributed tridiagonal system solution gives the DSDADI algorithm reasonable scalability on distributed-memory multiprocessors such as the CRAY T3D. Finally, an approximate parallel tridiagonal system solver with diminished interprocessor communication exhibits additional utility for DSDADI.     

A mapping algorithm for domain decomposition in massively parallel finite element analysis

A new mapping algorithm is presented for domain decomposition for the purpose of allowing researchers to conduct finite element analysis on massively parallel computers. Over the last few years, massively parallel MIMD machines such as the Intel Touchstone Delta and recently the Intel Touchstone Paragon have become increasingly popular for speeding up finite element computations. Most of these applications use domain decomposition as a first step towards conquering the problem. Many different algorithms have been developed by researchers to achieve an effective domain decomposition. Some of these methods use connectivity information only, some use coordinate information only, while others use both of them together. Some algorithms are based on assigning weights to nodes using a particular strategy while others are recursive in nature. As will be discussed in this paper, the logic employed in various algorithms works perfectly well for certain meshes to be decomposed, in certain numbers of subdomains; while it gives far from perfect results for other meshes or for same meshes to be decomposed in a different number of subdomains. The logic used in the proposed algorithm has been developed in a creative way such that it is closer to a human's natural thinking when making decisions. Fairly large meshes can be decomposed in a matter of seconds on a Sun Sparc station by the proposed algorithm. Its execution time remains almost the same for any number of subdomains.     

A solution algorithm for linear constraint equations in finite element analysis

A general solution algorithm is presented for the incorporation of a general set of linear constraint equations into a linear algebraic system; such situations arise in the application of the finite element method to a variety of physical problems. Implementation of the algorithm, without need for pre-arranging the equations, into an equation solver using Gauss elimination is developed. The method is most attractive as compared to other approaches for constrained systems.     

Distributed evolutionary multi-objective mesh-partitioning algorithm for parallel finite element computations

Majority of the mesh-partitioning algorithms attempt to optimise the interprocessor communications, while balancing the computational load among the processors. However, it is desirable to simultaneously optimise the submesh aspect ratios in order to significantly improve the convergence characteristics of the domain decomposition based Preconditioned-conjugate-gradient algorithms, being used extensively in the state-of-the-art parallel finite element codes. Keeping this in view, a new distributed multi-objective mesh-partitioning algorithm using evolutionary computing techniques is proposed in this paper. Effectiveness of the proposed distributed mesh-partitioning algorithm is demonstrated by solving several unstructured meshes of practical-engineering problems and also benchmark problems.