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The efficiency of the finite element analysis via force method depends on the overall flexibility matrix of the structure, while this matrix is directly affected from null bases vectors. As the null bases for an indeterminate structure are not unique, for an optimal analysis, the selected null bases should be sparse and banded corresponding to sparse, banded and well-conditioned flexibility matrix. This paper aims to present an efficient method for the formation of optimal flexibility matrix of finite element models comprising tetrahedron elements via mathematical optimization technique.

For this purpose, a linear mixed integer programming model is presented for finding sparse solution of underdetermined linear system, which is correspond to sparse null vector. The charged system search algorithm is improved and used to find the best generator for formation of null bases.

The efficiency of the present method is illustrated through some examples. The proposed method leads to highly sparse, banded and accurate null basis matrices. It makes an efficient force method feasible for the analysis of finite element model comprising tetrahedron elements.

The force method, in which the member forces are used as unknowns, can be appealing to engineers. The main problem in the application of the force method is the formation of a self-stress matrix corresponding to a sparse flexibility matrix. In this paper, the highly sparse, banded and accurate null basis matrices gains by using mathematical optimization technique.

The interest in the ability to detect damage at the earliest possible stage is pervasive throughout the civil engineering over the last two decades. In general, the experimental techniques for damage detection are expensive and require that the vicinity of the damage is known and readily accessible; therefore several methods intend to detect damage based on numerical model and by means of minimum experimental data about dynamic properties or response of damaged structures. The paper aims to discuss these issues.

In this paper, the damage detection problem is formulated as an optimization problem such as to obtain the minimum difference between the numerical and experimental variables, and then a modified ant colony optimization (ACO) algorithm is proposed for solving this optimization problem. In the proposed algorithm, the structural damage is detected by using dynamically measured flexibility matrix, since the flexibility matrix of the structure can be estimated from only the first few modes. The continuous version of ACO is employed as a probabilistic technique for solving this computational problem.

Compared to classical methods, one of the main strengths of this meta-heuristic method is the generally better robustness in achieving global optimum. The efficiency of the proposed algorithm is illustrated by numerical examples. The proposed method enables the deduction of the extent and location of structural damage, while using short computational time and resulting good accuracy.

Finding accurate results by means of minimum experimental data, while using short computational time is the final goal of all researches in the structural damage detection methods. In this paper, it gains by applying flexibility matrix in the definition of objective function, and also via using continuous ant colony algorithm as a powerful meta-heuristic techniques in the constrained nonlinear optimization problem.