Frontal solution program for unsymmetric matrices

A frontal solution program is presented which may be used for the solution of unsymmetric matrix equations arising in certain applications of the finite element method to boundary value problems. Based on the Gaussian elimination algorithm, it has advantages over band matrix methods in that core requirements and computation times may be considerably reduced; furthermore numbering of the finite element mesh may be completed in an arbitrary manner. The program is written in FORTRAN and a glossary of terms is provided.     

A frontal solution program for finite element analysis

The program given here assembles and solves symmetric positive–definite equations as met in finite element applications. The technique is more involved than the standard band–matrix algorithms, but it is more efficient in the important case when two-dimensional or three-dimensional elements have other than corner nodes. Artifices are included to improve efficiency when there are many right hand sides, as in automated design. The organization of the program is described with reference to diagrams, full notation, specimen input data and supplementary comments on the ASA FORTRAN print-out.     

Computer program for solution of large,sparse, unsymmetric systems of linear equations

A computer program is presented for in-core solution of a large, sparse, unsymmetric, unbanded system of linear equations. The program employs two partially packed arrays (one for storing non-zero elements, and one for column identifications). Used as the pivotal row is the row with the minimum number of non-zero elements. To avoid instability, the pivot is the largest absolute element in the pivotal row. The method was tested on a system of equations encountered in field application of the three-dimensional Galerkin finite element solution of flow and mass transport through porous media. Performance is compared with that of available alternatives.     

Computer program for solution of large,sparse, unsymmetric systems of linear equations

A computer program for the in-core solution of large, sparse, unsymmetric systems of linear equations is presented in this paper. The program employs elimination techniques for solution of systems of linear equations. A limited number of zeros is stored and trivial arithmetic is by-passed to preserve computer storage and to reduce the time required for solution. Several techniques for selecting the pivotal elements are discussed and their effect on accuracy and computational time are examined.     

Band algorithm for unsymmetric matrices in finite element and semi-discrete methods

A compact FORTRAN subroutine for reducing unsymmetric band matrices (in-core solution) is presented. As usual, the zeros inside the band are skipped during the Gauss elimination. However, the theme of the paper is centred on a new approach in applying the above-mentioned band solver to transient problems. The approach results in considerable saving of computational time while also achieving reduction in memory space. This double advantage makes it particularly useful in handling dynamic real world problems. Extension of the use of the algorithm to moving boundary problems is indicated.     

A block equation solver for large unsymmetric linear equation systems with dense coefficient matrices

The presented solver uses the Crout L – U decomposition method. Disk memory space requirements are reduced to an absolute minimum and disk input–output is spectacularly limited. Very extensive pivoting improves numerical accuracy and stability. Experimental test runs prove the overall efficiency of the solver.     

A block equation solver for large unsymmetric matrices arising in the boundary integral equation method

An equation solver for the large unsymmetric systems of linear equations arising in the application of the boundary integral equation method to problems of linear elasticity in homogeneous or piecewise homogeneous solids is presented. The solver uses Gaussian elimination. The advantages of the solver over many existing eliminational or iterational methods are that it ignores many of the large groups of zeros that occur in piecewise homogeneous work, and also restrains growth of the number of non-zero matrix entries. Memory requirements and computation times therefore are reduced for many piecewise homogeneous problems.     

A block solver for large,unsymmetric, sparse,banded matrices with symmetric profiles

A block equation solver for the solution of large, sparse, banded unsymmetric system of linear equations is presented in this paper. The method employs Crout variation of Gauss elimination technique for the solution. The solver ensures the efficient use of the available memory by doing block factorization and storage. It uses a skyline storage scheme which will avoid unnecessary operations on zero elements above the skyline which has found widespread use in banded symmetric solvers. A FORTRAN code with ample comments is provided. © 1997 John Wiley & Sons, Ltd.     

On the mechanical inconsistency of symmetrization of unsymmetric coupling matrices for BEFEM discretizations of solids

For several technical problems discretization of the boundary of a part of a solid by the BEM and of the remaining part of the solid by the FEM is useful in order to exploit the complementary advantages of the BEM and of the FEM optimally. A characteristic feature of the employed local FEM approach for the coupling of such discretizations are coupling matrices representing load-stiffness matrices which are associated with displacement-dependent node forces acting on the interface of the two parts of the solid. Because of the nature of the employed BEM these matrices are unsymmetric. It is proved theoretically that symmetrization of such coupling matrices is mechanically inconsistent. It is also demonstrated that this symmetrization may lead to a significant deterioration of the quality of numerical results.     

An algorithm for the solution of very large banded unsymmetric linear equation systems

This paper presents an algorithm and corresponding FORTRAN program for the solution of unsymmetric banded linear equation systems. The algorithm is based on the Crout method. A special technique, called double windowing, enables the solution of very large equation systems with a total equation number reaching 30,000 and a full bandwidth in excess of 1,000. Special attention was devoted to minimization of peripheral processor time (communication with backing disc memory). An appendix lists the complete program for Cyber series computers (CDC).     

Eigenproblem for Jacobi matrices: hypergeometric series solution

We study the perturbative power series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d. The (small) expansion parameters are the entries of the two diagonals of length d-1 sandwiching the principal diagonal that gives the unperturbed spectrum.The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series in 3d-5 variables in the generic case. To derive the result, we first rewrite the spectral problem for the Jacobi matrix as an equivalent system of algebraic equations, which are then solved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.     

Automatic element reordering for finite element analysis with frontal solution schemes

This paper describes an element reordering algorithm which is suitable for use with a frontal solution package. The procedure is shown to generate efficient element numberings for a wide variety of test examples. In an effort to obtain an optimum elimination order, the algorithm first renumbers the nodes, and then uses this result to resequence the elements. This intermediate step is necessary because of the nature of the frontal solution procedure, which assembles variables on an element-by-element basis but eliminates them node by node. To renumber the nodes, a modified version of the King¹ algorithm is used. In order to minimize the number of nodal numbering schemes that need to be considered, the starting nodes are selected automatically by using some concepts from graph theory. Once the optimum numbering sequence has been ascertained, the elements are then reordered in an ascending sequence of their lowest-numbered nodes. This ensures that the new elimination order is preserved as closely as possible. For meshes that are composed of a single type of high-order element, it is only necessary to consider the vertex nodes in the renumbering process. This follows from the fact that mesh numberings which are optimal for low-order elements are also optimal for high-order elements. Significant economies in the reordering strategy may thus be achieved. A computer implementation of the algorithm, written in FORTRAN IV, is given.     

A finite element solution program for large structures

A straightforward and general computer program for assembling and solving (using Gauss elimination technique) widely sparsed finite element matrix equations with very large bandwidth and capable of handling different degrees-of-freedom and variable bandwidth at different nodes, is described herein. The program assembles any type of finite elements having arbitrary number of nodes and each node may have differnt degrees-of-freedom. It requires only a small core memory in the computer, although a fast random access device is also needed. The two very important features of this program are (i) it does not store any zero submatrices within the band and (ii) during the solution of equations all operations dealing with zero submatrices within the band are automatically skipped and thus the savings of a considerable amount of disc storage space and computer time can be effected in many cases. Another feature is that many right hand sides can be handled simultaneously. Hence the program is very economical for structures having widely sparsed matrix equations. A listing of the computer program written in FORTRAN IV for CDC 6400 computer is readily available from the authors, but unfortunately could not be given here because of lack of space. The program is so general that it can be used to solve a wide class of finite element problems without actually having to understand fully the techniques behind it.     

Note on an exact solution for the pipe flow of a third-grade fluid

Summary We obtain a new exact power law solution for the pipe flow of a third-grade fluid. Moreover, we provide general analytical expressions from which all previous known solutions can be constructed.     

Direct-iterative solution of ill-conditioned finite element stiffness matrices

In this paper we discuss and expand the direct-iterative method proposed originally by Wilson.¹ First we introduce several simple numerical examples to illustrate the basic idea of this method before we proceed to prove the convergence of the direct-iterative method. We then discuss the methods for selecting the transformative matrix (Q) to be used in transforming an ill-conditioned matrix into a well-conditioned matrix in the direct-iterative method. There are two methods used to choose the matrix Q, namely the rigid body movement method and the imaginary element method. From examples 1-3 we can see that the imaginary element mesh is optional, and the finite element mesh is not necessary. The imaginary element method is a generalization of the mesh refinement method development in Reference 3. Because instead of local rotation angle we only choose displacements of nodes to represent rigid body movement, the rigid body movement method is an improvement of the method in Reference 2. The advantage of these two methods is that, in order to obtain well-conditioned matrices, only a few changes in the stiffness matrices are required even with general ill-conditioned stiffness matrices, and then convergency is achieved rapidly under SOR iteration. Finally, the examples for computing each type of the ill-conditioned matrix in three-dimensional finite element analysis are presented to demonstrate the effectiveness of the direct-iterative method in solving the large bandwidth problems.