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On computational procedures for the force method
Author(s) -
Kaneko I.,
Lawo M.,
Thierauf G.
Publication year - 1982
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620181004
Subject(s) - factorization , matrix (chemical analysis) , property (philosophy) , algebraic number , unification , algorithm , mathematics , sparse matrix , mathematical optimization , gaussian elimination , flexibility (engineering) , computer science , mathematical analysis , philosophy , statistics , materials science , physics , epistemology , quantum mechanics , composite material , gaussian , programming language
It is known that the matrix force method has certain advantages over the displacement method for a class of structural problems. It is also known that the force method, when carried out by the conventional Gauss‐Jordan procedure, tends to fill in the problem data, making the method unattractive for large size, sparse problems. This poor fill‐in property, however, is not necessarily inherent to the method, and the sparsity may be maintained if one uses what we call the Turn‐Back LU Procedure. The purpose of this paper is two‐fold. First, it is shown that there exist some close relationships between the force method and the least squares problem, and that many existing algebraic procedures to perform the force method can be regarded as applications/extensions of certain well‐known matrix factorization schemes for the least squares problem. Secondly, it is demonstrated that these algebraic procedures for the force method can be unified form the matrix factorization viewpoint. Included in this unification is the Turn‐Back LU Procedure, which was originally proposed by Topçu in his thesis. 8 It is explained why this procedure tends to produce sparse and banded ‘self‐stress’ and flexibility matrices with small band width. Some computational results are presented to demonstrate the superiority of the Turn‐Back LU Procedure over the other schemes considered in this paper.