Sparse self‐stress matrices for the finite element force method

Abstract

SUMMARYA basic problem in the ÿnite element force method is that of obtaining a sparse and banded selfstress matrix and a sparse and banded structure exibility matrix. Traditionally the self-stress matrix is obtained through the application of algebraic procedures to the equilibrium matrix. The self-stress matrix for an indeterminate structure is not unique, and it is possible to obtain another self-stress matrix from an existing one through algebraic operations and grouping of redundants. The purpose of this paper is to describe and test an algorithm, called REDUC, which combines the vectors of the self-stress matrix obtained from the LU procedure of the force method. The rows of the transpose of this matrix are combined by using a special form of the Gaussian elimination technique. A plane frame example is presented to demonstrate the algorithm at work. The algorithm REDUC is applied to a plane truss and physical interpretation of the resulting self-stress matrix highlights the grouping of redundants, improved sparsity and bandwidth. Improvements in the conditioning and bandwidth of the structure exibility matrix are also observed. The algorithm yields results similar to those of the turn-back LU procedure, but requires less computation time and programming e ort.

Minimal cycle basis of graph products for the force method of frame analysis

“…There are also algebraic methods for the force method of structural analysis developed by Denke [9], Kaneko et al [10], Topçu [11], Soyer and Topçu [12]. The main advantage of these methods is their generality; however, graph theoretical methods are more efficient when applied to frame structures with no singularities.…”

“…It can be shown that the sum of two cycle set vectors of a graph is also a cycle set vector. Thus, the cycle set vectors of a graph form a vector space over the field of integer modulo 2 [12]. The dimension of a cycle space is equal to the first Betti number of the graph b 1 (S).…”

“…One cycle of length 7 as: (1, 2, 3, 4, 5, 6, 7). Twenty-one cycles of length 4 as: (1,8,2,9); (8,15,9,16); (15,22,16,23); (2, 9, 3, 10); (9,16,10,17); (16,23,17,24); (3,10,4,11); (10,17,11,18); (17, 24, 18, 25); (4,11,5,12); (11,18,12,19); (18, 25, 19, 26); (5,12,6,13); (12,19,13,20); (19,26,20,27); (6,13,7,14); (13,20,…”

Efficient graph‐theoretical force method for two‐dimensional rectangular finite element analysis

“…Algebraic methods have been developed by Denke [9], Robinson [10], Topçu [11], Kaneko et al [12] and Soyer and Topçu [13], and mixed algebraic-topological methods have been used by Gilbert and Heath [14], Coleman and Pothen [15,16], and Pothen [17]. Developments for use with finite element models (FEMs) based on hybrid equilibrium elements are due to Maunder [18] and Maunder and Savage [19].…”

An efficient graph‐theoretical force method for three‐dimensional finite element analysis

“…Algebraic methods have been developed by Denke [9], Robinson [10], Topçu [11], Kaneko et al [12], Soyer and Topçu [13], and mixed algebraic-topological methods have been used by Gilbert and Heath [14], Coleman and Pothen [15,16] and Pothen [17]. The integrated force method has been developed by Patnaik [18,19], in which member forces are used as variables, and the equilibrium equations and the compatibility conditions are satisfied simultaneously in terms of these variables.…”