Efficient force method for the analysis of finite element models comprising of triangular elements using ant colony optimization

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Abstract

An efficient algorithm is presented for the formation of null basis of triangular plane stress and plane strain finite element models, corresponding to highly sparse flexibility matrices. This is achieved by applying a modified ant colony system (ACS). An integer linear programming formulation is also presented to evaluate the quality of the results obtained by the proposed ant colony system algorithm. The efficiency of the present algorithm is illustrated through some examples.

Introduction

The force method of structural analysis, in which the member forces are used as unknowns, is appealing to engineers, since the properties of members of a structure most often depend on the member forces rather than joint displacements.

Four different approaches are adopted for the force method of structural analysis, which can be classified as: (1) topological force methods, (2) algebraic force methods, (3) mixed algebraic–topological force methods, and (4) integrated force method.

Topological methods have been developed by Henderson and Maunder [1] and Maunder [2] for rigid-jointed skeletal structures using manual selection of cycle bases. Methods suitable for computer programming are due to Kaveh [3], [4]. Algebraic methods have been developed by Denke [5], Robinson [6], Topçu [7], Kaneko et al. [8], and mixed algebraic–topological methods have been used by Gilbert and Heath [9], Coleman and Pothen [10], [11]. The integrated force method has been developed by Patnaik [12] and Patnaik et al. [13], in which the equilibrium equations and compatibility conditions are satisfied simultaneously in terms of the force variables.

The force method of structural analysis requires the formation of a maximal set of independent self-equilibrating stress systems (SESs), known as a null basis [14], [15]. The elements of this basis form the columns of an m×γ(S) matrix, B1, known as the self-stress matrix.

The main problem in the application of the force method is the formation of a self-stress matrix corresponding to a sparse flexibility matrix G=B1tFmB1, where Fm contains the flexibility matrices of the individual members of the structure in a block diagonal form.

The combinatorial methods for the force method are very efficient for skeletal structures and, in particular, for rigid-jointed frames. For a general structure, the underlying graph or hypergraph of a SES has not yet been properly defined, and further research is needed. Algebraic methods, on the other hand, can be formulated in a more general form to cover different types of structures such as skeletal structures and finite element models (FEM). The main drawbacks of these methods are the large storage requirements and the higher number of operations.

Heuristic algorithms, such as ant colony algorithms, have found many applications in optimization problems in the last decade. The essence of these algorithms lies in the fact that their capability to converge to a good solution does not depend on the specific search space to which they are applied. In this paper, the ant colony system (ACS) which is a variation of the ant colony optimization (ACO) is applied to the formation of null bases of triangular plane stress and plane strain finite element models corresponding to highly sparse and banded flexibility matrices. An integer linear programming formulation is presented to evaluate the quality of the results obtained by the proposed ACS algorithm. The efficiency of the present method is illustrated through simple examples.

Section snippets

Formulation of the force method

Consider a structure S, which is γ(S) times statically indeterminate. γ(S) independent unknown forces are selected as redundants. These unknown forces can be selected from external reactions and or internal forces of the structure. These redundants are denoted by a vector as q={q1,q2,…,qγ(S)}t. In order to obtain a statically determinate structure, known as the basic (released or primary) structure of S, the constraints corresponding to redundants are removed. Consider the joint loads vector as

Constant stress triangular element

For this element, the element forces, Fi={Fαi, Fβi, Fγi}, are taken as the natural forces acting along the side of the triangles, as shown in Fig. 1. The corresponding displacements are denoted by υi={υαi, υβi, υγi}.

In a global coordinate system, the nodal forces for each element have six components and the nodal forces and element forces can be related by projection.

The simple flexibility matrix is as fi=1tAlφctlwhere l={Lα, Lβ, Lγ}, A is the area of the element, t is the thickness, and φct=12G

Mathematical modeling for optimization problem

Since the overall flexibility matrix of the structure G is B1tFmB1, for the sparsity of G one should select a null basis corresponding to sparse B1 matrix, which is often referred to as the sparse null basis problem. The main objective of this paper is to find sparse self-stress matrices to simplify the solution and to ensure the formation of well-conditioned flexibility matrices.

For a SES (null vector), no applied load is required, thus the equilibrium conditions can be expressed asAB1=0

This

Optimization by ant colony systems

A meta-heuristic algorithm based on the ants behavior was developed in early 1990s by Dorigo and Gambardella [16]. This algorithm was called ant colony optimization because it was motivated by social behavior of ants. Ant colony system is a variation of the ACO which has proven to behave more robustly and provide far better results for certain problems. In this work, ACS is chosen as a suitable tool for finding sparse null vectors. A brief description of ACO is given in the next section, when

The effect of generator sequence and edge ordering on the sparsity of null basis

According to the proposed mathematical modeling, the numbering the members affect the results of the selected null vectors. This can be found out by considering the additional equations which are used in the process of normalization and orthogonalisation. Taking e1tS1=1, it is required to have a force equal to unity in the first entry of S1 vector. This entry is called the generator of S1. Since e1tS2=0 and e2tS2=1, therefore the first entry in S2 vector, which is the generator of S1, must be

ACS algorithm for the formation of sparse null basis

In order to apply the ACO algorithm to a specific problem, it is necessary to represent it as a set of different paths for ants to travel. In the problem of finding sparse null basis, different sequence of generators is considered as a tour for an ant to travel, therefore the cooperative ant agents search to find the best generator sequence resulting in a sparse null basis.

Since both the edge numbering, and its order in the generator sequence are important, therefore the pheromone amount is

A lower bound for optimal null basis selection

Heuristic optimization algorithms like ant colony optimization, seek good feasible solutions to optimization problems in circumstances where the complexities of the problem or the limited time available for solution do not allow exact solution, although these algorithms are often capable of leading to the optimal solutions.

In this section, an integer linear programming formulation is presented to evaluate the efficiency of the suboptimal solution which is obtained by the proposed ACS algorithm.

Numerical results

In this section, four examples are presented to illustrate the performance of the proposed ACS algorithm, and to provide a measure of its efficiency. This algorithm is coded by MATLAB, and is run on a personal computer Pentium® 4 CPU 3.40 GHz, 1.00 GB of RAM.

Example 1

Consider a finite element model as shown in Fig. 10(a). The numbering of the edges for the elements of the model is shown on the disconnected elements in Fig. 10(b). This model is divided into 24 elements, and its degree of static

Conclusions

In this paper, an ant colony system is developed for the formation of sparse null basis leading to sparse self-stress matrices, and correspondingly highly sparse flexibility matrices for triangular plane stress and plane strain finite element models. An integer linear programming formulation is also presented to evaluate the quality of the solutions obtained by the proposed ACS algorithm. The method presented in this paper is not confined to plane stress and plane strain triangular elements and

Acknowledgement

The first author is grateful to Iran National Science Foundation for the support.

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