An efficient graph theoretical method for plate bending finite element analysis via force method

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Abstract
Purpose: This paper seeks to present an efficient algorithm for the formation of null basis for finite element model discretized as rectangular bending elements. The bases obtained by this algorithm correspond to highly sparse and narrowly banded flexibility matrices and such bases can be considered as an efficient tool for optimal analysis of structures.Design/methodology/approach: In the present method, two graphs are associated with finite element mesh consisting of an “interface graph” and an “associate digraph”. The underlying subgraphs of the self‐equilibrating systems (SESs) (null vectors) are obtained by graph theoretical approaches forming a null basis. Application of unit loads (moments) at the end of the generator of each subgraph results in the corresponding null vector.Findings: In the present hybrid method, graph theory is used for the formation of null vectors as far as it is possible and then algebraic method is utilized to find the complementary part of the null basis.Originality/value: This hybrid approach makes the use of pure force method in the finite element analysis feasible. Here, a simplified version of the algorithm is also presented where the SESs for weighted graphs are obtained using an analytical approach. Thus, the formation of null bases is achieved using the least amount of algebraic operations, resulting in substantial saving in computational time and storage.

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