Cholesky Method Research Articles

Design aspects and detailed study of medium tonnage mechanical press brakes

The fabricated stands, bed and ram of a medium tonnage mechanical press brake under load are studied. The machine analyzed is capable of handling a variety of bending operations on sheets of up to 6.5 mm thickness. FEM is used to study the static response of the stands when the machine is in operation, three-noded triangular plate elements being used for this purpose. The response is computed for the worst case of loading. Cholesky's method in conjunction with the constant bandwidth approach is used to study the behavior of the stand, the solution of which results in the computing of a large number of unknowns. The objective of the study is to evaluate the performance of the machine stands, the bed and the ram, when the machine is subjected to various loading conditions. Isostress lines are plotted to enable the stress pattern to be studied, based on which study various design and structural modifications to arrive at a satisfactory design have been suggested.

The Cholesky method for interval data

We apply the well-known Cholesky method to bound the solutions of linear systems with symmetric matrices and right-hand sides both of which are varying within given intervals. We derive criteria to guarantee the feasibility and the optimality of the method. Furthermore, we discuss some general properties.

Block Sparse Cholesky Algorithms on Advanced Uniprocessor Computers

As with many other linear algebra algorithms, devising a portable implementation of sparse Cholesky factorization that performs well on the broad range of computer architectures currently available is a formidable challenge. Even after limiting the attention to machines with only one processor, as has been done in this paper, there are still several interesting issues to consider. For dense matrices, it is well known that block factorization algorithms are the best means of achieving this goal. This approach is taken for sparse factorization as well.This paper has two primary goals. First, two sparse Cholesky factorization algorithms, the multifrontal method and a blocked left-looking sparse Cholesky method, are examined in a systematic and consistent fashion, both to illustrate the strengths of the blocking techniques in general and to obtain a fair evaluation of the two approaches, Second, the impact of various implementation techniques on time and storage efficiency is assessed, paying particularly clo...

Design aspects and parametric study of 3-roll heavy-duty plate-bending machines

The fabricated stands of a three-roll heavy duty plate bending machine under load have been studied and a parametric study has been performed on them. The two machines analyzed are capable of cold rolling 14 and 40 mm thick plates. The FEM was used to study the static response of the stands when the machine is in operation, and three-noded triangular plate elements were used for this purpose. The response was computed for the worst case of loading. Cholesky's method in conjunction with the skyline approach was used to study the behaviour of the stand by solving the large-size problem. The objective of the parametric study was to evaluate the performance of the machine stands in relation to various design- or structural-modification and arrive at a satisfactory design.

Impact of mapping and sparsity on parallelized finite element method modules

The efficient parallelization of a flexible h-version finite element method (FEM) algorithm is investigated. The actual implementation and benchmarking are performed on a distributed memory-multiple instruction multiple data (MIMD) machine (NCUBE). After an initial modularization and benchmarking, three specific parameters that could influence the efficiency of a concurrent finite element analysis (FEA) solver are identified: (i) sparsity of matrices, (ii) storage format for stiffness matrix, and (iii) mapping of domain onto processors. These factors are interdependent and they heavily influence the overall efficiency of the implemented FE code. This study includes a performance evaluation of the FE modules using the techniques outlined below. The results demonstrate the importance of these parameters, and the advantages of exploiting different levels of concurrency in a parallel FEA code. Three direct methods of solving systems of linear equations are analyzed, a Block Cholesky method exhibiting a coarse grain concurrency, and Gauss Elimination and Cholesky Decomposition methods with inherit medium grain parallelism. Additionally, work on two Assembly/Storage Modules (One-Way Dissection and Shifted Banded), and their corresponding simple Domain Decomposers (Vertical Strip and Wrap Mapping) is performed.

Nonlinear Analysis of Steel Space Structures

A second‐order nonlinear analysis of steel space structures has been presented. Of the two types of nonlinearities, material and geometric, only geometric nonlinearity has been considered. The material of the structure steel has been assumed to be linearly elastic. In geometric nonlinearity, the effects of instability produced by axial forces, the bowing of the deformed members, and finite deflections have all been included. For this purpose, the secant stiffness matrix in the deformed state and the modified kinematic matrices along with the geometric matrix necessary for formulating the tangent stiffness matrix, have been developed. These matrices are used in the analysis, which is carried out by the displacement method through an iterative‐incremental procedure based on Newton‐Raphson technique. The iterations that take into account the latest geometry are repeated until the unbalanced loads become negligible and equilibrium is obtained. The equilibrium equations are solved by Cholesky's method. Results...

The accuracy of least squares calculations with the Cholesky algorithm

In a detailed study of the accuracy of the Cholesky method for solving least squares equations, points in the algorithm at which accuracy is lost are identified, methods for improving accuracy are discussed, and simple formulas for predicting accuracy from the maximum variance inflation factor (as well as from the condition number) are evaluated. The central role of cancellation in determining accuracy is emphasized.

A historical overview of iterative methods

The object of this paper is to present a historical overview of the development of iterative methods for the solution of large sparse systems of linear equations. The emphasis is on methods which are applicable to linear systems arising in the numerical solution of partial differential equations. Aspects to be covered including the methods of L.F. Richardson and of Liebmann as well as relaxation methods used by Southwell and others; the SOR method and extensions such as block SOR methods, and p-cyclic matrices; Chebyshev polynomial methods; alternating direction implicit methods; the SSOR method; approximate matrix factorization methods including the strongly implicit method (SIP) and the incomplete Cholesky method (ICC); fast direct methods; conjugate gradient methods; adaptive methods for the automatic determination of iteration parameters; multigrid methods; methods for nonsymmetric systems; and iterative software. Future developments will be discussed with emphasis on the use of vector and parallel processors.

AN ADAPTED CHOLESKY DECOMPOSITION METHOD FOR THE SOLUTION OF COUPLED MAGNETIC‐ELECTRIC EQUATIONS

A model coupling magnetic and electric equations is developed in this paper. The unknowns in this model are the magnetic vector potential and the mesh currents. The nature of the resulting matrix system does not permit the use of the classical Cholesky algorithm. A new decomposition, based on the Cholesky method, is introduced to solve it. To illustrate the efficiency of the method, an example of an iron core coil is modelled. Both calculation time and memory storage are compared with classical methods.

Triangularization of a positive definite matrix on a parallel computer

The problem of computing the triangular factors of a square, real, symmetric, and positive definite matrix by using the facilities of a multiprocessor MIMD-type computer is considered. The parallel algorithms based on Cholesky decomposition and Gaussian elimination are derived and analyzed in terms of their speedup and efficiency, when the available number of processors is O( n), where n is the size of the matrix. It is shown that the parallel elimination method can achieve the same speedup as the parallel Cholesky method while using only half the number of processors required by the Cholesky method.

Auxiliary Storage Methods for Solving Finite Element Systems

In this paper we consider the direct solution of the system of linear equations $Ax = b$, where A is a large, sparse, symmetric and positive definite matrix. This system is solved using the Cholesky method by factoring A into $R^T R$, where R is an upper triangular matrix, and then solving $R^T y = b$ and $Rx = y$. We are particularly interested in the case where A is so large that auxiliary storage must be used to store the Cholesky factor R.The approach we use, which fully exploits the sparsity of A, is based on partitioning the given system into subsystems, each of which is sufficiently small that it can be processed in a relatively small amount of main memory. We give a detailed analysis of the input/output (I/O) traffic generated when our method is applied to a model n by n grid problem. The grid is partitioned using an incomplete nested dissection, which is a minor modification of the standard nested dissection ordering. Our analysis shows that if we have $O(n^2 )$ main memory, then the total I/O traffic is $O(n^2 \log n)$. This result implies that the $O(n^3 )$ numerical computations dominate the I/O traffic. A widely used method for solving such large linear systems is the band method, in which zeros outside the band of A are exploited. The I/O traffic and arithmetic operations in this case are given by $O(n^3 )$ and $O(n^4 )$, respectively.We also consider an improvement for our basic strategy which reduces the I/O traffic to the extent that it is dominated by writing the Cholesky factor alone. This enhancement reduces the I/O traffic associated with storing intermediate results from $O(n^2 \log n)$ to $O(n^2 \log \log n)$. Numerical experiments are also provided to illustrate the performance of our algorithms.

Skyline algorithms for multilevel substructure analysis

AbstractA matrix derivation of the method of partial decomposition for substructure equilibrium equations is given and algorithms and subroutines necessary for implementing a multi‐level substructuring scheme are presented. The algorithms are based upon partial decomposition of skyline matrices by the Cholesky method. Transformation of co‐ordinates and prediction of the skyline for higher level substructures are also discussed.

The cholesky method in substructuring with an application to fracture mechanics

The substructuring procedure proposed involves the modification of the Cholesky solution scheme. The required boundary stiffness matrix and equivalent load vector are obtained during the factorisation and forward substitution stage, respectively. The required alterations to the original algorithms are clearly shown, illustrating the minimal programming changes required. The implementation and ease of use is demonstrated in the finite element package, FINEL. An application to fracture mechanics is also provided, with the Virtual Crack Extension Method adapted to the Cholesky scheme and, in particular, the use in a substructuring environment. An example is presented demonstrating the use of the techniques proposed.

Numerical accuracy of block adjustments

A successful application of aerial triangulation presupposes adequate numerical accuracy of the adjusted data. The numerical accuracy is limited by rounding-off errors of the arithmetic operations and their propagation in the block adjustment algorithm. The present paper investigates the effect of block size, control distribution and overlap on the numerical accuracy of the adjusted terrain coordinates in a purely empirical way. For that, a large number of simulated block adjustments by independent models was performed at the CDC 6600 computer of Stuttgart University, using the computer program PAT-M43. The reduced normal equations are solved by a generalized Cholesky method. The simulations confirm, that a word length of 60 bit is sufficient to guarantee adequate numerical accuracy of the adjusted coordinates. Even with a free block adjustment of 1250 models 8 correct digits were obtained. Moreover, the results have shown that the numerical accuracy in x and y can be well approximated by a constant multiplied by n · i 3, with n denoting the block length and i the planimetric control spacing along the block perimeter, expressed in units of the base length. In first approximation the obtained results are also valid for bundle block adjustments with a Gaussian or a Cholesky solution.

On systems of linear equations of the form [formula omitted] error analysis and certain consequences for structural applications

An error analysis of the solution of linear systems of the form A tAx = b is presented. The analysis is based on the backward technique of J.H. Wilkinson applied to the QR method on A, which involves the use of elementary Hermitians. The accuracy of the method is compared to that of the classical Cholesky approach on the formed product K = A tA . Although the initial analysis shows no improved error bound for the QR approach compared to the Cholesky method, a more refined treatment reveals that the relative solution error using QR methods is usually considerably smaller than that using the Cholesky procedure, particularly for commonly occurring right-hand side vectors b. Practical error estimates are provided, both based on the spectral conditioning number measure as well as on the first correction in a special iterative improvement technique. As systems of equations similar to the above arise in the new natural factor technique of J.H. Argyris and O.E. Brönlund for matrix structural analysis, some discussion of structural applications in both statics and dynamics is presented. It is shown that the only case where the accuracy of the QR method is not considerably better than that for Cholesky reduction corresponds to an unusual physical situation. In addition, a natural factor formulation for the mixed method in finite element analysis is presented, extending the first writer's previous work on this topic. Finally, several numerical examples are given, based on randomly generated data as well as on data from real structures, to demonstrate the validity and applicability of the method and the corresponding error analysis.

Stress Analysis of Axisymmetric Solids with Asymmetric Properties

The finite element method for the determination of stresses and displacements in arbitrary axisymmetric solids is extended to problems involving asymmetric mechanical and thermal loads and materials with asymmetric temperature dependent mechanical properties. All dependent variables, including the mechanical properties, are represented by a Fourier series expansion of the circumferential coordinate. Due to the mechanical property variation in the circumferential direction, the resulting equation set becomes fully coupled and must be solved simultaneously. By grouping all Fourier coefficients together at each nodal point, a banded stiffness matrix is achieved the efficient solution of which is feasible by the use of a blocked square root Cholesky method. A useroriented computer program based on the method of analysis is described and applied to several example problems to verify accuracy and demonstrate versatility of the program.

Direct Methods for Solving Symmetric Indefinite Systems of Linear Equations

Methods for solving symmetric indefinite systems are surveyed including a new one which is stable and almost as fast as the Cholesky method.

Algoritmy. 21. PCHOLES. Řešení $(2m+1)$ diagonálového systému lineárních algebraických rovnic s $p$ pravými stranami Choleského metodou

Algoritmy. 21. PCHOLES. Řešení $(2m+1)$ diagonálového systému lineárních algebraických rovnic s $p$ pravými stranami Choleského metodou

Algoritmy. 22. PNCHOLES. Řešení systému lineárních algebraických rovnic Choleskeho metodou

Algoritmy. 22. PNCHOLES. Řešení systému lineárních algebraických rovnic Choleskeho metodou

COMPUTATIONS OF NETWORK MODELS OF POROUS MEDIA*

ABSTRACTIn a paper presented at last year's Amsterdam meeting (viz. Geophysical Prospecting 14 (1966), 3, 301–341), J. R. Schopper derived formulas relating formation factor, permeability and porosity, by means of a statistical‐network approach and by treating the electric and hydraulic resistance analogously.The model of the porous medium consists of a network of branch resistors, their values being statistically distributed about a mean Ro with a relative standard deviation (variation coefficient) s. A properly defined total resistance R of the network can be expressed by the relationship: image Here α is a geometrical factor dependent only on the shape of the network (i.e. the number of meshes in the longitudinal and transversal direction), ε is a characteristical constant dependent only on the individual mesh shape (i.e. the number of nodes and branches within a mesh).This network constant ε enters the equations relating formation factor, permeability and porosity, ε had been found to be in the range zero to one by calculating algebraically two special limiting network cases. However, for a better understanding of which value exactly this constant will have in actual porous media, networks with various mesh shapes have to be treated generally.Because of the basically statistical approach, the networks have to be large so that a general algebraic treatment is precluded. Hence numerical methods using digital computers must be applied.The determination of the total resistance R of any resistance network leads to the problem of solving a system of linear, inhomogeneous equations; i.e. Ohm's law written in matrix form: image (R) is the matrix of the coefficients, composed of the individual branch resistances. (I) is the column vector, its components being fictitious circular mesh currents. (U) is the inhomogeneity column, its components being source voltages within the individual meshes. The matrix (R) has characteristic properties that depend on the mesh shape on the one hand and on the number and arrangement of the meshes on the other hand. With the regular arrangement of identical meshes investigated here, the matrix always has a banded structure and is symmetrical with respect to the main diagonal, positive definite, and non‐singular.For the numerical determination of the wanted constant ε the coefficients matrix is provided with values having a known distribution. Here, in particular, a computer‐generated pseudorandom homogeneous distribution is used. The system, of equations is solved for R by a modified Cholesky method. Equation (1′) can then be solved fore. The main features of an ALGOL program written for this purpose and optimized with respect to storage space requirement and computer time are discussed.Networks of triangular, square and hexagonal meshes have been investigated. The results are discussed.