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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.

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.

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.

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.

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 determination of feasible self-stress modes and grouping of elements for tensegrities with predefined geometry and multiple self-stress modes is very important, though difficult, in the design of these structures. The purpose of this paper is to present a novel approach to the automated element grouping and self-stress identification of tensegrities.

A set of feasible solutions conforming to the unilateral behaviour of elements is obtained through an optimisation process, which is solved using a genetic algorithm. Each chromosome in the population having a negative fitness is a distinctive feasible solution with its own grouping characteristic, which is automatically determined throughout the evolution process.

The self-stress identification is formulated through an unconstrained minimisation problem. The objective function of this minimisation problem is defined in such a way that takes into account both the feasibility of a solution and grouping of elements. The method generates a set of feasible self-stress modes rather than a single one and automatically and simultaneously suggests a grouping of elements for every feasible self-stress mode. A self-stress mode with a minimal/subminimal grouping of elements is also obtained.

The method can efficiently generate sets of feasible solutions rather than a single one. The authors also address one of the challenging issues related to this identification, i.e., automated grouping of elements. These features makes the method very efficient since most of the state-of-the-art methods address the self-stress identification of tensegrities based on predefined groupings of elements whilst providing only a single corresponding solution.

This paper aims to propose a study on the static behavior of prismatic tensegrity structures and an innovative form for determining the effect of mechanical properties and geometric parameters on the minimal mass design of these structures.

The minimal mass design in this paper considers a stable class-two tensegrity tower built through stable models. Using the proposed structures, comprehensive parametric studies are performed to examine the mass (in which the masses of joints are ignored), the mass ratio between a class-two tensegrity tower and a single element, both having the same diameter and length and afterward determine a reliable mass saving structure under various circumstances.

The simulations show that the mass ratio versus the number of units is a nonlinear regressive curve and predicts that the proposed model outperforms the standard model when the variation parameter considered is a vertical force. The difference in mass between these structures is visible when the gap gradually decreases while the number of units increases. On the geometrical aspect, the gap between the masses is not significant.

This paper helps to understand the influences of geometric parameters and the mechanical properties on the design of cylinder tensegrity structures dealing with a compressive force.

The aim of this article is to obtain a stable tensegrity structure by using the minimum knowledge of the structure.

Three methods have been formulated based on the eigen value decomposition (EVD) and singular value decomposition theorems. These two theorems are being implemented on the matrices, which are computed from the minimal data of the structure. The required minimum data for the structure is the dimension of the structure, the connectivity matrix of the structure and the initial force density matrix computed from the type of elements. The stability of the structure is analyzed based on the rank deficiency of the force density matrix and equilibrium matrix.

The main purpose of this article is to use the defined methods to find (1) the nodal coordinates of the structure, (2) the final force density values of the structure, (3) single self-stress from multiple self-stresses and (4) the stable structure.

By using the defined approaches, one can understand the difference of each method, which includes, (1) the selection of eigenvalues, (2) the selection of nodal coordinates from the first decomposition theorem, (3) the selection of mechanism mode and force density values further and (4) the solution of single feasible self-stress from multiple self-stresses.

Graph products are extensively used in the analysis and design of regular structures. It is often thought that these products are only applicable to regular graphs. The main aim of this paper is develop new products which are applicable to regular as well and non‐regular structural models.

New graph products are defined with specified domains. In these products the logical operations of the graph products are only performable in specified domains, and therefore these products can produce configurations which do not need to be regular.

New graph products are defined and a general theorem is proved for the formation of their adjacency matrices.

The presented graph products overcome the difficulty of employing graph products in structural mechanics, and in particular in space structures. The general theorem of this paper can efficiently be used in the formation of adjacency matrices of the structural models.

This paper aims to present a reconfiguration strategy for actuated tensegrity structures. The main idea is to use the infinitesimal mechanisms of the structure to generate a path along which the tensegrity can change its shape while maintaining the equilibrium.

Combining the force density method with a marching procedure, the solution to the equilibrium problem is given by a set of differential equations. Beginning from an initial stable position, the algorithm calculates a small displacement until a new stable configuration is reached, and recurrently repeats the process during a given interval of time.

By means of three numerical simulations and their respective experimental example, the efficacy of this algorithm for reconfiguring the well-known three-bar tensegrity prism along different directions is shown. The proposed method shows efficiency only for small changes of string length. Further work should consider the application of this method to more complex tensegrity structures.

The advantage of this reconfiguration method is its simplicity for finding new stable positions for tensegrity structures, and the fact that it doesn’t need the information of the material of the structure for the computations.

There are many structures that have a repetitive pattern. If a relationship can be established between a repetitive structure and a circulant structure, then the repetitive structure can be analyzed by using the properties of the corresponding circulant structure. The purpose of this paper is to develop such a transformation.

A circulant matrix has certain properties that can be used to reduce the complexity of the analysis. In this paper, repetitive and near-repetitive structures are transformed to circulant structures by adding and/or eliminating some elements of the structure. Numerical examples are provided to show the efficiency of the present method.

A transformation is established between a repetitive structure and a circulant structure, and the analysis of the repetitive structure is performed by using the properties of the corresponding circulant structure.

Repetitive and near-repetitive structures are transformed to circulant structures, and the complexity of the analysis of the former structures is reduced by analyzing the latter structures.

The purpose of this paper is to propose an accurate and efficient numerical method for determining the dynamic responses of a tensegrity structure consisting of bars, which can work under both compression and tension, and cables, which cannot work under compression.

An accurate time-domain solution is obtained by using the precise integration method when there is no cable slackening or tightening, and the Newton–Raphson scheme is used to determine the time at which the cables tighten or slacken.

Responses of a tensegrity structure under harmonic excitations are given to demonstrate the efficiency and accuracy of the proposed method. The validation shows that the proposed method has higher accuracy and computational efficiency than the Runge–Kutta method. Because the cables of the tensegrity structure might be tense or slack, its dynamic behaviors will exhibit stable periodicity, multi-periodicity, quasi-periodicity and chaos under different amplitudes and frequencies of excitation.

The steady state response of a tensegrity structure can be obtained efficiently and accurately by the proposed method. Based on bifurcation theory, the Poincaré section and phase space trajectory, multi-periodic vibration, quasi-periodic vibration and chaotic vibration of the tensegrity structures are predicted accurately.

The purpose is to reduce round-off errors in numerical simulations. In the numerical simulation, different kinds of errors may be created during analysis. Round-off error is one of the sources of errors. In numerical analysis, sometimes handling numerical errors is challenging. However, by applying appropriate algorithms, these errors are manageable and can be reduced. In this study, five novel topological algorithms were proposed in setting up a structural flexibility matrix, and five different examples were used in applying the proposed algorithms. In doing so round-off errors were reduced remarkably.

Five new algorithms were proposed in order to optimize the conditioning of structural matrices. Along with decreasing the size and duration of analyses, minimizing analytical errors is a critical factor in the optimal computer analysis of skeletal structures. Appropriate matrices with a greater number of zeros (sparse), a well structure and a well condition are advantageous for this objective. As a result, a problem of optimization with various goals will be addressed. This study seeks to minimize analytical errors such as rounding errors in skeletal structural flexibility matrixes via the use of more consistent and appropriate mathematical methods. These errors become more pronounced in particular designs with ill-suited flexibility matrixes; structures with varying stiffness are a frequent example of this. Due to the usage of weak elements, the flexibility matrix has a large number of non-diagonal terms, resulting in analytical errors. In numerical analysis, the ill-condition of a matrix may be resolved by moving or substituting rows; this study examined the definition and execution of these modifications prior to creating the flexibility matrix. Simple topological and algebraic features have been mostly utilized in this study to find fundamental cycle bases with particular characteristics. In conclusion, appropriately conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.

(1) Five new algorithms were proposed in order to optimize the conditioning of structural flexibility matrices. (2) A JAVA programming language was written for all five algorithms and a friendly GUI software tool is developed to visualize sub-optimal cycle bases. (3) Topological and algebraic features of the structures were utilized in this study.

This is a multi-objective optimization problem which means that sparsity and well conditioning of a matrix cannot be optimized simultaneously. In conclusion, well-conditioned flexibility matrices are obtained, and analytical errors are reduced accordingly.

Engineers always finding mathematical modeling of real-world problems and make them as simple as possible. In doing so, lots of errors will be created and these errors could cause the mathematical models useless. Applying decent algorithms could make the mathematical model as precise as possible.

Errors in numerical simulations should reduce due to the fact that they are toxic for real-world applications and problems.

This is an original research. This paper proposes five novel topological mathematical algorithms in order to optimize the structural flexibility matrix.