On the approximate solution of singular integral equations

A method for obtaining the approximate solution of singular integral equations of the first and second kinds is suggested. The solution is represented in the form of power series with undetermined coefficients multiplied by a function in which the essential features of the singularity of the solution are preserved. The method of collocations is used to determine the unknown coefficients. The examples show that the method suggested is more general and gives good results even in the case when the form of solution does not exactly preserve the essential features of singularity. The method is simpler than others which use the properties of orthogonal polynomials, and is applicable for the solution of single equations as well as systems of simultaneous equations.     

Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations

Direct methods for solving Cauchy-type singular integral equations (S.I.E.) are based on Gauss numerical integration rule [1] where the S.I.E. is reduced to a linear system of equations by applying the resulting functional equation at properly selected collocation points. The equivalence of this formulation with the one based on the Lagrange interpolatory approximation of the unknown function was shown in the paper. Indirect methods for the solution of S. I. E. may be obtained after a reduction of it to an equivalent Fredholm integral equation and an application of the same numerical technique to the latter. It was shown in this paper that both methods are equivalent in the sense that they give the same numerical results. Using these results the error estimate and the convergence of the methods was established.     

A recurrence formula for the direct solution of singular integral equations

An iterative method for the solution of singular integral equations is given in this paper by developing a recurrence formula. Discretizing the above formula, by using appropriate quadrature rules, the solution of the singular integral equation is given in an extremely simple form. The number of numerical operations required for such a solution is considerably reduced, when compared to the number of operations required for a classical type of solution. Illustrative examples are given, indicating the efficiency of the method. It is shown that the number of operations in this procedure is only half the number of the operations for a typical numerical method. The convergence of the method is studied in the space of Hölder continuous functions. In the particular case of plane elasticity more efficient bounds are given. In the same case it is proved that the procedure is equivalent to the Schwarz's alternating method and convergence is assured [18].     

Hybrid finite element methods for the von Karman equations

We analyse the «assumed stresses» hybrid approximation of the Von Karman equations; we provide convergence results and optimal error bounds for a large class of finite element discretizations.     

Hierarchical parallelisation for the solution of stochastic finite element equations

As an example application the elliptic partial differential equation for steady groundwater flow is considered. Uncertainties in the conductivity may be quantified with a stochastic model. A discretisation by a Galerkin ansatz with tensor products of finite element functions in space and stochastic ansatz functions leads to a certain type of stochastic finite element system (SFEM). This yields a large system of equations with a particular structure. They can be efficiently solved by Krylov subspace methods, as here the main ingredient is the multiplication with the system matrix and the application of the preconditioner. We have implemented a “hierarchical parallel solver” on a distributed memory architecture for this. The multiplication and the preconditioning uses a—possibly parallel—deterministic solver for the spatial discretisation as a building block in a black-box fashion. This paper is concerned with a coarser grained level of parallelism resulting from the stochastic formulation. These coarser levels are implemented by running different instances of the deterministic solver in parallel. Different possibilities for the distribution of data are investigated, and the efficiencies determined. On up to 128 processors, systems with more than 5 × 10⁷ unknowns are solved.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

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 class of iterative methods for finite element equations

A general method in the form of an accelerated preconditioned iterative refinement method (including some wellknown iterative methods and direct factorization methods) is presented for the solution of symmetric, sparse matrix problems. An analysis of one such approximate factorization, the SSOR method, is given, and some inherently advantageous properties of the conjugate gradient acceleration method are pointed out. A comparison is made of the computational complexity and storage in the SSOR preconditioned method with some direct methods applied to second order discretized boundary value problems. For plane problems of average size the direct methods are somewhat faster if enough right hand sides are present. For large enough problems (large number of nodes) the iterative method is faster. For three-dimensional problems no Cholesky factorization method can compete with the SSOR preconditioned method, not even for average sized problems.     

On the automatic solution of nonlinear finite element equations

An algorithm for the automatic incremental solution of nonlinear finite element equations in static analysis is presented. The procedure is designed to calculate the pre- and post-buckling/collapse response of general structures. Also, eigensolutions for calculating the linearized buckling response are discussed. The algorithms have been implemented and various experiences with the techniques are given.     

Some practical procedures for the solution of nonlinear finite element equations

Procedures for the solution of incremental finite element equations in practical nonlinear analysis are described and evaluated. The methods discussed are employed in static analysis and in dynamic analysis using implicit time integration. The solution procedures are implemented, and practical guidelines for their use are given.     

On the coupling of boundary integral and finite element methods

We prove some error estimates for a procedure obtained by combining the boundary integral method and the usual finite element method. This work was carried out while Franco Brezzi was visiting the Department of Computer Science at Chalmers Institute of Technology during September 1977.     

On finite element methods for elliptic equations on domains with corners

A finite element method for approximating elliptic equations on domains with corners is proposed. The method makes use of the singular functions of the problem in the trial space and the kernel functions of the adjoint problem in the test space. This leads to good approximates of the coefficients of the singular functions. In the numerical computations, the method is compared with the well known Singular Function Method.     

Parallel solution for finite element linear systems of equations on workstation cluster

With the advances in the high speed computers network technologies, a workstation cluster is becoming the main environment for parallel processing. Finite element linear systems of equations are common throughout structural analysis in Civil Engineering. The preconditioned conjugate gradient method （PCGM） is an iterative method used to solve the finite element systems of equations with symmetric positive definite system matrices. In this paper, the algorithm of PCGM is parallelized and implemented on DELL workstation cluster. Optimization techniques for the sparse matrix vector multiplication are adopted in programming. The storage scheme is analyzed in detail. The experiment result shows that the designed parallel algorithm has high speedup and good efficiency on the high performance workstation cluster. This illustrates the power of parallel computing in solving large problems much faster than on a single processor.     

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.     

A study of variable step-length incremental/iterative methods for nonlinear finite element equations

A deep study is carried out on the variable step-length incremental/iterative methods of nonlinear finite element equations in static structural analysis. Simple interpretations are presented for the conventional incremental/iterative method, single characteristic displacement controlling method and arc-length controlling method from their geometric characteristics in the load-deformation space. Based on the concept of structural equilibrium, three constraint equations (i.e. zero incremental displacement norm, zero residual force norm and zero incremental work) are established for controlling the iteration path. The automatic selection method of initial load increment in each increment is discussed. The suitability and efficiency of the aforementioned methods are studied and compared through tracing the snap through and snap back responses of two cylindrical shells.     

Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

Parallel iterative multilevel solution of mixed finite element systems for scalar equations

A combination of several contemporary techniques is used for the efficient parallel solution of the mixed finite element systems on locally refined Grids. Implementation experience and numerical results are reported. Copyright © 2005 John Wiley & Sons, Ltd.     

An iterative penalty method for the finite element solution of the stationary Navier-Stokes equations

The objective of this paper is to analyse an iterative procedure for the finite element solution of the Stokes and Navier-Stokes stationary problems. For the latter case, the usual condition on the viscosity and the data that ensures uniqueness is assumed. The method is based on the iterative imposition of the incompressibility condition via penalization. Theoretical and numerical results show that this constraint can be approximated iteratively within the same iterative loop used to deal with the nonlinear term of the equations. Two particular iterative schemes are analysed, namely those based on the Picard and Newton-Raphson algorithms.