The use and efficiency of the theorems of structural variation for finite element analysis

Abstract

SUMMARYThis paper describes how the theorems of structural variation mag be used in finitc clement analysis. The efficiency of the theorems is examined and their application to structural and geotechnical problems illustrated. 1NTRODUCTIONEfficient reanalysis tcchniques must be used during a non-linear analysis or optimizition design procedure if numerical methods are to be used effectively. This is particularly true when the structure to be analysed is large and repeated application of the matrix displacement method would require a large number of simultaneous equations to be reformed and solved. In an attempt to avoid repeated analysis a number of reanalysis algorithms4 have been developed 'to find the response of a modified structure using the original analysis of the structure such that the computational effort is less than that required for a fresh analysis'. These reanalysis algorithms are obviously applicable to any numerical procedure in which the stiffness of the structure changes, such as in non-linear analysis and optimization design. One such technique, based on the theorems of structural variation, was presented for trusses by Majid and Elliott *-I0 and dekeloped for frames by Majid, Saka and Celik.'1,'2 The usc of the theorems for finite elemcnt analysis was proposed in References 14 and 15.In this paper a general formulation of the theorems of structural variation for finite element problems is presented. These theorems of structural variation were previously presented for variations in structural properties of one element.'4,15 In this paper they are extended for the simultaneous modification of more than one element. This is important if the theorems are to be used as an efficient rcanalysis technique. The theorems are then presented in matrix form.5 The approach enables an efficient computer implementation of the technique to be identificd.In addition, the efficiency of the theorems of structural variation as a reanalysis technique is investigated. The efficiency tests have been undertaken on some simple finite element structures where the number of modified elements is progressively increased. In each case, the reanalysis time was compared with the time taken for a complete or fresh analysis. The measure of efficiency was

The Theorems of Structural and Geometric Variation: A Review

“…The design studies and efficiency tests presented in Refs [l, 20,21,22, 231 indicate that the Theorems of Structural and Geometric Variation may be used efficiently for the reanalysis of structures. Potential applications include nonlinear analysis, interactive design studies, optimization and other structural sensitivity calculations.…”

Fast exact linear and non‐linear structural reanalysis and the Sherman–Morrison–Woodbury formulas

“…In general, non-linear fast reanalysis requires solution of m simultaneous non-linear equations (Equation (39) or (40)). Such a system may be solved with Newton's method, which involves the solution of a linear system of size m at each iteration step.…”

“…Once ' is computed, it is substituted into the non-linear coupled equations in Equation (39) to compute the remaining m n unknown weights i for the non-linear changes. For an elasto-plastic member whose cross-sectional area has been modiÿed and which, consequently, has itself gone into the plastic range in the example given in the section 'Non-linear Reanalysis', the scalar function in Equation (42) …”

A method to derive approximate explicit solutions for structural mechanics problems

“…exact methods) and approximate methods. Direct methods provide exact explicit solutions which are efficient if the number of modified elements is small; approximate methods are more general but usually in order to achieve accuracy they resort to iterative algorithms which may considerably reduce, even nullify, convenience over a fresh analysis (Kavlie and Powell, 1971;Topping and Abu Kassim, 1987).…”