Genetic Algorithm as a Tool for the Determination of the Self-Stress States of Tensegrity Domes

Abstract

The aim of the paper is to find the appropriate self-stress state of the tensegrity structures. The first approach provides exact solutions but is suitable for simple structures. In the second approach proposed in this research, it is assumed that the forces of the self-stressed state are a set of randomly selected values, which are then optimized by a genetic algorithm. This procedure is intended for more elaborate structures, for which the spectral analysis identifies many self-stress states that need to be superimposed. Two approaches are used, i.e., the spectral analysis of the compatibility matrix and the genetic algorithm. The solution procedures are presented on the example of a simple two-dimensional truss. Next, three different tensegrity domes are considered, i.e., Geiger, Levy and Kiewitt. The significant difference between these domes lies in the cable system. The obtained results are compared with those documented in the literature. It follows from the considerations that the self-stressed states found in the literature are not always accurate (forces do not balance themselves). The presented results confirm the effectiveness of the genetic algorithm for finding self-balanced forces of the existing structures. The method is relatively simple and provides sufficiently accurate results.

1. Introduction

Tensegrity is a term derived from the English language as a compound of two words: “tension”—stretching, and “integrity”—stability. The idea of tensegrity structures has been around for over 60 years. During this period, many researchers dealt with this unique type of structure, each of them trying to define the essence of this idea in their own way. The first attempts to describe tensegrity structures come from the creators of the idea. The concept of tensegrity systems was conceived in 1962 by Fuller [1], which reflected his idea as “islands of compression in a sea of tension”. In 1964, Emmerich [2] added the condition of the self-stress state. In 1965, Snelson [3] patented a system called “Continuous Tension, Discontinuous Compression Structures”. The definition provided in 1976 by Pugh [4] is the most widely accepted: “A tensegrity system is established when a set of discontinuous compression components interacts with a set of continuous tensile components to define a stable volume in space”. This definition was narrowed in 1992 by Motro [5]: “A tensegrity system is a system in a stable self-equilibrated state comprising a discontinuous set of compressed components inside a continuum of tensioned components”. In general, tensegrity systems are structures whose integrity is based on a balance between tension and compression. Thus far, neither a precise nor a general definition has been agreed upon. Moreover, there are several definitions and classifications that take into account the different features of tensegrity [6]. However, from a mechanical point of view, the most important features are closely related to the advantages of the tensegrity structures. The first step to understand the unique properties is the qualitative analysis, which includes the identification of self-stress state (or states) and infinitesimal mechanism (or mechanisms) [7,8]. The self-stress state is understood as a system of self-equilibrated internal forces, which satisfies homogeneous equations of equilibrium, while the infinitesimal mechanism means the geometrical variability that is stabilized by the self-stress states. Additionally, the modification of self-equilibrated internal forces allows the behavior of the structure to be controlled; i.e., stiffness and frequency can be influenced. Such structures are characterized by stabilization through tension, efficiency, deployability and easy tunability, the possibility of reliable modeling and more precise controllability [9,10].

Tensegrity as a structural system offers many advantages over conventional structural systems. Over the years, this concept has evolved, finding a wider audience in engineering, mathematics and biology [11]. In civil engineering, the idea of tensegrity relates to specific self-balancing trusses consisting of continuous tensioned cables and individual compression struts. The self-equilibrium, called the self-stress state, means that there is an equilibrium stress state between struts and cables under zero external loads. An example of tensegrity systems is a cable dome. A typical cable dome consists of the load-bearing girders (built with ridge cables, diagonal cables and vertical struts) connected by a circumferential cable system. The girders are connected by an outer compression ring. To ensure the structural feasibility, continuous cables are often subjected to tension and individual struts subjected to compression. The rigidity of the dome is a result of self-equilibrium between cables and struts. However, opinions are divided as to whether cable domes can be regarded as tensegrity systems. Those in favor regard the outer compression ring as a curved strut. The others insist that the tensegrity system should be self-supporting, which means that it should not be stabilized with a compression ring. However, most people agree that cable domes are a successful extension of the concept of tensegrity [12,13,14,15,16] due to the mechanical and mathematical properties that distinguish them from conventional systems. The stiffness of the structure is determined not only by the geometry and material characteristics but also by the level of self-stress. This means that the stiffness of these structures can be controlled.

In contrast to traditional steel domes [17,18], tensegrity domes are lightweight structures. This feature is very important, especially in long span space structures [19,20]. In the process of designing such structures, the key issue is how to reduce the weight of the structure and thus the cost of construction. The first tensegrity-inspired dome was proposed and patented in 1988 by Geiger [21]. In the opening paragraph of his tensegrity patent, he claims that “the cable truss dome is adapted for spanning large areas where the cables form a low shallow arch which supports a flexible membrane as a covering”. The Geiger dome was used as the roofs of the Olympic Gymnastics Hall and the Fencing Hall in Seoul [20]. The Seoul gymnastics stadium dome is 118 m in diameter and 15 m high. This type of roof has low-profile configurations that reduce wind lift, uneven snow settling and use less material to cover the roof. One of the main advantages of this structure is that its weight per square meter does not change as the span increases. Additionally, the diameter of Geiger domes can be expanded indefinitely with little additional weight or expense per square meter of roof. The second cable dome was developed in 1994 by Levy [19,22]. Levy’s idea was used to build the largest existing cable dome, the Georgia Dome, designed for the Atlanta Olympics in 1996. The Georgia Dome with its elliptical configuration is covered by a roof extending 193 m along the minor axis and 240 m along the major axis and is referred to in the literature as a hyper-tensegrity dome [12,16]. The main difference between the Geiger dome and the Levy dome is the arrangement of the ridge cables. In a Geiger dome, the ridge cables are arranged radially so that the roof is divided into multiple flat load-bearing girders. In the Levy dome, the ridge cables form a triangular pattern. The roof surface consists mainly of quadrilateral anticlastic membrane panels connected by ridge cables. In addition, it facilitates the application of cable domes on an elliptical plan shape. The Geiger and Levy domes are the most known types of tensegrity domes. They have been and still are an inspiration to create new and hybrid forms of cable domes [14,23,24,25,26,27]. These structures have been repeatedly studied by the researchers throughout the years. There are more than 10,000 articles from the years 1990–2022 that mention the Geiger dome and more than 18,000 that mention the Levy dome (according to the information from Google Scholar). Another cable dome, less popular than Geiger and Levy (appeared only in 500 articles), is the Kiewitt dome form [14]. This structure is inspired by the lamella lattice shell developed in 1960 by Kiewitt [28]. The Kiewitt cable domes, as with the tensegrity structures, have structural rigidity only when the self-equilibrium forces are applied to cables and struts [12,13,14,15,16,29]. Therefore, calculating the self-stress state is the key step for any cable structure.

The search for self-stress states, independent of the load, is called form-finding methods in the literature. In fact, these methods consist of determining such a configuration of elements at which self-balancing forces occur. The review of the literature has shown a great variety of form-finding methods, and these methods are constantly being modified and improved. The overviews of the methods can be found, e.g., in [30,31,32,33]. The commonly used form-finding methods are the force density method [34], dynamic relaxation [35], energy optimization [36], reduced coordinates method [37], iteration method [38], singular value decomposition [7,39] or spectral analysis of the truss matrices [7]. In recent years, in the process of searching for a stable configuration of tensegrity structures, evolutionary optimization techniques have been increasingly used. These methods look for better solutions, modeling their operation on mechanisms occurring during natural evolution. One of the optimization techniques is the genetic algorithm. This algorithm, inspired by the process of natural selection and genetics, was first presented by Holland [40]. In the case of tensegrity, the genetic algorithm is mostly used as a form-finding method for regular [27,41,42,43] and irregular [44,45,46] structures.

The methods mentioned above are mainly used to shape the structure, but they can also be used to search for the self-balanced forces (force finding) of the existing structure [47,48,49,50]. In [49], the genetic algorithm is used to determine the self-stress states. In this paper, flat structures and a three-dimensional double quadruplex module were considered. In the case of cable domes, the literature lacks a study using the genetic algorithm to search for the self-stress states. The authors of this study intend to fill that gap. In this paper, the genetic algorithm and the spectral analysis of compatibility matrix are used. The obtained results are compared with those documented in the literature in order to verify the effectiveness of the present method. Three selected tensegrity domes are considered, i.e., Geiger, Levy and Kiewitt.

The rest of the paper is organized as follows: Section 2 describes two approaches used for the identification of the self-stress states, i.e., the spectral analysis of compatibility matrix and the genetic algorithm. In Section 3, in order to compare the proposed methods, the solution procedures are presented on the example of a simple two-dimensional truss. Furthermore, the results for three tensegrity domes are provided in Section 4. The calculation procedure was written in the Python language. The genetic algorithm was implemented using pyGAD (Python library) [51]. Finally, in Section 5, some conclusions are drawn.

2. Methods of Analysis

The tensegrity structure is described as an $n$-element spatial truss ($e=1,2,\dots ,n$) with $m$ degrees of freedom described by the vector $\mathbf{q}\left(\in {\mathbb{R}}^{m\times 1}\right)$. This system consists of tensioned cables and compressed struts in a self-stress state. The geometry of the tensegrity structure is described by the expansion matrix $\mathbf{B}\left(\in {\mathbb{R}}^{n\times m}\right)$ defined using the finite element formalism [52]:

$$\mathbf{B}=\left[\begin{array}{c}{\mathbf{B}}_1{\mathbf{C}}_1\\ {\mathbf{B}}_2{\mathbf{C}}_2\\ \dots \\ {\mathbf{B}}_n{\mathbf{C}}_n\end{array}\right];{\mathbf{B}}_e=\left[\begin{array}{cccccc}{-\mathrm{c}}_x& {-\mathrm{c}}_y& {-\mathrm{c}}_z& {\mathrm{c}}_x& {\mathrm{c}}_y& {\mathrm{c}}_z\end{array}\right],$$

(1)

where ${\mathrm{c}}_x=\frac{x_j-x_i}{L_e},{\mathrm{c}}_y=\frac{y_j-y_i}{L_e},{\mathrm{c}}_z=\frac{z_j-z_i}{L_e}$, $L_e$ is a length of element, ${\mathbf{B}}_e\left(\in {\mathbb{R}}^{1\times 6}\right)$ is an extension vector of element, ${\mathbf{C}}_e\left(\in {\mathbb{R}}^{6\times m}\right)$ is a Boolean matrix. It is assumed that the number of global degrees of freedom of each element $q_i(i=1,2,\dots ,6)$ corresponds to the number of global nodes of elements $n_1,n_2$, as shown in Figure 1. Consequently, the non-zero elements of ${\mathbf{C}}_e$ can be expressed as ${\mathbf{C}}_{i{q}_i}=1$.

The existence of self-balancing forces depends only on the configuration of elements. It is not necessary to know the cross-sections of the elements and their material properties. The simplest method of determining the self-stress state is to check the static equilibrium of each node. This approach is exact but only applies to simple structures. The second exact method is the spectral analysis of the compatibility matrix. This method is effective if there is only one existing self-stress state. If more than one self-stress state occurs, it is necessary to check the suitability of the self-stress states, i.e., whether the forces correctly identify the type of elements (what is a strut and what is a cable). Usually, only the superposition of all self-stress states provides the expected results; however, it requires additional calculations. In such cases, an optimization technique, such as the genetic algorithm, seems to be an effective method of finding appropriate self-balanced forces. Using this technique, the type of elements (tensed or compressed), is described in the initial conditions.

In the paper, the identification of self-stress state $\mathbf{S}\left(\in {\mathbb{R}}^{n\times 1}\right)$ is carried out through the spectral analysis of the compatibility matrix and the genetic algorithm. In both cases, only the geometry of the structure is required, taking into account support conditions. The obtained self-balanced forces have to ensure the stability of the structure. It means that the stiffness matrix $\mathbf{K}\left(\in {\mathbb{R}}^{m\times m}\right)$:

$${\mathbf{K}=\mathbf{B}}^{\mathit{T}}\mathbf{B}+{\mathbf{K}}_G\left(\mathbf{S}\right),$$

(2)

must be positive definite. The stiffness matrix (2) consists of the linear stiffness matrix with a unitary matrix of elasticity ${\mathbf{B}}^{\mathit{T}}\mathbf{B}$ and the geometric stiffness matrix that depends on the self-stress state ${\mathbf{K}}_G\left(\mathbf{S}\right)$:

$${\mathbf{K}}_G\left(\mathbf{S}\right)=\sum _{i=1}^n{\left({\mathbf{C}}_e\right)}^T{\mathbf{K}}_G^e\left(S_e\right){\mathbf{C}}_e;{\mathbf{K}}_G^e\left(S_e\right)=\frac{S_e}{L_e}\left[\begin{array}{cc}\mathbf{I}& -\mathbf{I}\\ -\mathbf{I}& \mathbf{I}\end{array}\right];\mathbf{I}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],$$

(3)

where $S_e$ is a longitudinal force.

2.1. Spectral Analysis of Compatibility Matrix

The equilibrium equations of the tensegrity structure can be presented, e.g., in the form of symmetrized equilibrium equations:

$$\mathbf{B}{\mathbf{B}}^{\mathit{T}}\mathbf{S}=\mathbf{B}\mathbf{P}$$

(4)

where P$\left(\in {\mathbb{R}}^{n\times 1}\right)$ is an external load vector, $\mathbf{B}{\mathbf{B}}^{\mathit{T}}\left(\in {\mathbb{R}}^{n\times n}\right)$ is a compatibility matrix. The spectral analysis of the matrix $\mathbf{B}{\mathbf{B}}^{\mathit{T}}$, with non-negative eigenvalues, leads to the identification of the self-stress states [6,7,9,10]:

$$\left(\mathbf{B}{\mathbf{B}}^T-\mu \mathbf{I}\right)\mathbf{y}=0$$

(5)

If all eigenvalues $\mu$ are positive definite, there are no self-equivalent systems of longitudinal forces in the structure. The zero eigenvalues $({\mu }_i=0)$ of the matrix (5) are related to the non-zero solution of the homogeneous equations ((4) for $\mathbf{P}=\mathbf{0}$) called the self-stress state, or more precisely to the self-balanced normal forces that satisfy the homogeneous equilibrium equations. The self-stress state can be considered as an eigenvector related to the zero eigenvalue $\mathbf{S}={\mathbf{y}}_i({\mu }_i=0)$. The self-stress state $\mathbf{S}$ is appropriate when the eigenvalues of the matrix (2) are positive definite and the forces correctly identify the type of elements.

2.2. Genetic Algorithm

The genetic algorithm is one of the most popular computational algorithms for searching problems based on the mechanics of natural selection and genetics. In this approach, an initial random population of feasible solutions evolves to create a better solution based on genetic operators, i.e., parent selection, crossover or mutation. The genetic algorithm procedure is shown in Figure 2.

The first part of the algorithm is the selection of appropriate groups of elements. The selection can be completed automatically or manually. The automatic selection usually consists of the length selection [53]. The groups of elements are divided according to their length and type. This method can be less precise for structures with elements of comparable lengths. The manual selection is more complicated and involves creating groups manually. It is helpful for structures with complex shapes or when higher accuracy is required. Due to the specificity of tensegrity structures, the selection is conducted in a mixed way, i.e., partly automatically and partly manually. Two types of the element groups (tensed or compressed) were used. These groups lead to different definitions in the encoding procedure. An automatic selection was then completed within these groups based on the length selection. The set of all elements $\mathsf{𝚪}$ is divided into the sets of elements with the same length ${\mathit{g}}_i$:

$${\mathit{g}}_i=\left\{e_l,\cdots ,e_z\right\}, {\mathit{g}}_i\in \mathsf{𝚪},$$

(6)

where $e_l$ and $e_z$ are, respectively, the first and last element of group. For each group of elements ${\mathit{g}}_i$, the normalized longitudinal force in element $S^e$ is equal to:

$$S_e=\frac{\pm S_i}{S_{max}},$$

(7)

where $\pm S_i$ is a value of the self-stress state (“+” for tensed element, “−” for compressed element) and $S_{max}$ is a maximum value.

Then, the initialization of the population based on the given parameters is required. As in the first step, it can be completed in two ways, i.e., automatically or manually. In this paper, the initial population is prepared automatically by encoding the number of solutions in the populations and a number of searched genes. Obtaining an optimal result requires significant computational effort. Genetic operators are used for the natural selection of valuable solutions. The leading operators in the genetic algorithm are selection, crossover and mutation. The selection operator prefers better solutions (chromosomes) to pass its genes to the algorithm without mutation. The crossover combines the features of the genes of two parents to form two outcomes. The mutation operator is applied on the chromosome generated from the crossover operation with the probability of mutation. There are several methods for genetic operators. In this paper, the following parameters are proposed:

• parent selection methods: steady state selection (in each generation, a few good chromosomes are selected to create a new offspring; then, some of the bad (with low fitness) chromosomes are removed and the new offspring is placed in their place; the rest of the population survives to the new generation),
• crossover type: scattered (it randomly selects the gene from one of two parents),
• mutation type: random (the values of some genes change randomly; the number of genes is specified on the basis of the mutation number of genes or the percentage of genes to mutate; for each gene, a random value is selected according to the range specified by the minimum and maximum value),
• number of genes: the number of the groups of elements,
• percentage of genes to mutate: 10.

For chosen operators, the following parameters must be specified, i.e.,:

• population size,
• number of generations,
• solutions in the population.

The most critical part of the genetic algorithm is the fitness function. This function determines how the obtained solution fits this particular problem. In the case of seeking of the self-stress state, the fitness function proposed in this paper is equal to:

$$F=F_1·F_2,$$

(8)

where the components $F_1$, $F_2$ are described as follows:

$$F_1=\left\{\begin{array}{cc}0,& if\mathrm{the}\mathrm{stiffness}\mathrm{matrix}\left(2\right)\mathrm{is}\mathrm{not}\mathrm{positive}\mathrm{definite}\\ 1,& if\mathrm{the}\mathrm{stiffness}\mathrm{matrix}\left(2\right)\mathrm{is}\mathrm{positive}\mathrm{definite}\end{array}\right\},$$

(9)

$$F_2=\frac{1}{\sqrt{EN}},$$

(10)

where $EN$ is the equilibrium of nodes. There is no physical interpretation of the equilibrium of nodes, which, for the sake of simplicity, was assumed as:

$$EN=\sum _{i=1}^n{[\left(S_{ix}\right)}^2+{\left(S_{iy}\right)}^2+{\left(S_{iz}\right)}^2],$$

(11)

where $S_{ix}$, $S_{iy}$, $S_{iz}$ are force projections.

The feasible solution is obtained by maximizing the fitness function (8). The values of the fitness function should increase with the number of generations. The appropriate solution has to satisfy the stable equilibrium.

3. Results for Two-Dimensional Truss

In order to compare the proposed methods, a simple two-dimensional truss (Figure 3a) is considered. The structure consists of six cables and two struts ($n=8$) and is characterized by eight degrees of freedom $(m=8)$: $\mathbf{q}={\left[\begin{array}{cccccccc}{q}_3& q_4& q_5& q_6& q_9& q_{10}& q_{11}& q_{12}\end{array}\right]}^T$. The expansion matrix (1) takes the form $\mathbf{B}\left(\in {\mathbb{R}}^{8\times 8}\right)$:

$$\mathbf{B}=\left[\begin{array}{cccccccc}0.894& 0.447& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0.894& -0.447& 0& 0\\ 0& 0& -0.894& 0.447& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -0.894& -0.447\\ -1& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 1& 0\\ 0& 1& 0& 0& 0& -1& 0& 0\\ 0& 0& 0& 1& 0& 0& 0& -1\end{array}\right].$$

(12)

In this case, the self-stress state $\mathbf{S}={\left[\begin{array}{cccccccc}{S}_1& S_2& S_3& S_4& S_5& S_6& S_7& S_8\end{array}\right]}^T$ can be simply determined by checking the static equilibrium of each node (Figure 3b):

$$S_2=S_3=S_4=S_1,S_5=S_6=S_1\sin \alpha ,S_7=S_8=-S_1\cos \alpha ;\sin \alpha =\frac{2\sqrt{5}}{5};\cos \alpha =\frac{\sqrt{5}}{5}.$$

(13)

The forces (13) are normalized in such a way that the maximum compressive force in struts is equal to $-1$:

$$\mathbf{S}={\left[\begin{array}{cccccccc}\sqrt{5}& \sqrt{5}& \sqrt{5}& \sqrt{5}& 2.0& 2.0& -1.0& -1.0\end{array}\right]}^T$$

(14)

In turn, the spectral analysis of the compatibility matrix (5) leads to the following eigenvalues:

$$\mathsf{\mu }={\left[\begin{array}{cccccccc}3.0& 2.86015& 2.30623& 2.0& 1.0& 0.69377& 0.13985& 0.0\end{array}\right]}^T$$

(15)

The zero eigenvalue in (15)—${\mu }_8=0$ is responsible for the existence of one self-stress state considered as an eigenvector $\mathbf{S}={\mathbf{y}}_8({\mu }_8=0)$. The normalization of the eigenvector leads to:

$$\mathbf{S}={\left[\begin{array}{cccccccc}2.23607& 2.23607& 2.23607& 2.23607& 2.0& 2.0& -1.0& -1.0\end{array}\right]}^T.$$

(16)

The self-stress state (16) correctly identifies the type of elements (it is the same as (14)). Taking into account the geometric stiffness matrix (3), all eigenvalues of the stiffness matrix (2) are positive:

$$\mathsf{\sigma }={\left[\begin{array}{cccccccc}9.24& 7.26& 6.73& 6.71& 3.16& 2.65& 2.63& 1.18\end{array}\right]}^T,$$

(17)

which means that the stiffness matrix is positive definite and the structure is stable.

Using the genetic algorithm, the selection of elements was completed manually. Two approaches were used. Firstly, elements were divided into three groups (two groups of tensed elements and one of compressed elements), and then into two groups (two groups of tensed elements). For both approaches, two series with different values of population size and number of generations were used (Series 1 and 2 for the approach with three genes; Series 3 and 4 for the approach with two genes). The genetic main parameters and results of the genetic algorithm are presented in

Table 1. Genetic main parameters and results of genetic algorithm for the two-dimensional truss.

	Series 1	Series 2	Series 3	Series 4
Number of genes	3	3	2	2
Population size	18	100	18	100
Number of generations	4	10	4	10
Solutions in the population	9	20	9	20
$\sum {\mathit{S}}_{\mathbf{2}\mathit{x}}$	−0.0576	−0.0028	−0.0706	−0.00002
$\sum {\mathit{S}}_{\mathbf{2}\mathit{y}}$	−0.1360	−0.0623	−0.2141	−0.00009
Equilibrium of nodes$\mathit{E}\mathit{N}$(11)	0.0873	0.0794	0.2034	0
Fitness function (8)	3.38481	3.54881	2.21706	503.083

, whereas the values of self-stress states and relative errors in

Table 2. Values of self-stress states of the two-dimensional truss.

		Genetic Algorithm
Groups of Elements	Exact Solutions	Series 1	Relative Error	Series 2	Relative Error	Series 3	Relative Error	Series 4	Relative Error
1	2.23607	2.54022	13.60%	2.37559	6.24%	2.71499	21.42%	2.23829	0.1%
2	2.00000	2.21443	10.72%	1.99847	0.08%	2.49898	24.95%	2.00197	0.1%
3	−1.00000	−1.00000	0%	−1.00000	0%	−1.00000	0%	−1.00000	0%

.

The accuracy of the results obtained using the genetic algorithm depends on the number of genes, the population size and the number of generations. The larger the population size is, the more diverse the possible solutions are and the larger the chance that there is the solution among those randomly generated by the program. The higher the number of generations is, the more genetic operations are applied and the greater the chance of obtaining the right solution is. In Series 1 and 2, three genes were searched for and the range of the solution was set from 0 to 1. Then, the values were normalized so that the forces in compressed elements were equal to −1. In Series 3 and 4, knowing the approximate solutions from Series 1 and 2, only the forces in tensed elements were searched for, the range of the solution was set from 2 to 3 and the force in strut was assumed as normalized −1. For low level of genetic parameters (Series 1 and 3), the maximum relative error for the series is equal to 14% and 25%, respectively. Increasing the level of the parameters (Series 2 and 4) improves the accuracy of solutions. In the case of Series 2, the maximum relative error is 6.24%, while, in the case of Series 4, the relative error is insignificant—0.01%. It means that the solutions obtained in Series 4 can be considered accurate.

In the considered case, the exact solutions are known and the accuracy of the solutions can be evaluated by comparing them. What if they are not known? For the appropriately selected solution, equilibrium of each node needs to be close to zero. To measure the accuracy of the self-stress states, the equilibrium of each node and the fitness functions are calculated (Table 1).

4. Results for Tensegrity Domes

The paper contains the following considerations, i.e., is the genetic algorithm an effective method of the identification of the self-stress state? Does it lead to the satisfactory results? Do results converge or not compared to other methods? Is it possible to use the same algorithm parameters for different types of domes? To answer the questions, three types of tensegrity domes are considered, i.e., Geiger, Levy and Kiewitt. The load-bearing girders of the considered structures have a similar arrangement of the elements. The significant difference among these domes lies in the cable system. The structures are supported in all external nodes. Using the genetic algorithm, the selection of elements was completed manually according to their length and the type of group selection ($\mathrm{C}$—struts; $\mathrm{J}$, $\mathrm{X}$, $\mathrm{H}$, $\mathrm{N}$—cables). Two series with different values of genetic parameters were applied:

• number of genes: number of groups of elements,
• population size: 1000 (Series 1), 1100 (Series 2),
• number of generations: 100 (Series 1), 150 (Series 2),
• solutions in the population: 200 (Series 1), 250 (Series 2).

Since domes are more elaborate structures than the example in Chapter 3, a correspondingly larger population size and the number of generations were selected for Series 1. In order to investigate how the increase in these parameters improves the accuracy, it was assumed that, in Series 2, they are even higher. To verify the efficiency of the genetic algorithm, the geometry of each dome was taken from the literature. Due to the inclusion of real structures in the literature, the values of the self-stress states (Original) were normalized (Normalized) to compare them with the values obtained from the genetic algorithm.

4.1. Geiger Dome

As the first, the original 12-girder Geiger dome (Figure 4) proposed by Jiang et al. [47] is considered. The structure consists of 120 cables and 36 struts ($n=156)$ and is characterized by 216 degrees of freedom $(m=216)$. The Geiger dome consists of flat load-bearing girders (Figure 4a) connected by perimeter cables. The elements were divided into 13 groups (Figure 4a).

In the case of the original Geiger domes, the formulas for the self-equilibrium forces can be derived from the equilibrium of nodes. These formulas depend on the inclination angle of cables ${\alpha }_i$ (Figure 4a) and on the angle between perimeter cables $2\beta$ (Figure 4c):

$$\begin{gathered} \mathrm{J}3=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t};{\mathrm{C}}_i=-{\mathrm{X}}_i\sin \left({\alpha }_{2i}\right);{\mathrm{X}}_i={\mathrm{J}}_i\frac{\sin \left({\alpha }_{2i-1}\right)}{\sin \left({\alpha }_{2i}\right)};i=1,2,3, \\ {\mathrm{J}}_j=\frac{{\mathrm{J}}_{j+1}\cos \left({\alpha }_{j+2}\right)+{\mathrm{X}}_{j+1}\cos \left({\alpha }_{j+2}\right)}{\cos \left({\alpha }_j\right)};{\mathrm{H}}_j=0.5{\mathrm{X}}_j\frac{\cos \left({\alpha }_{2j}\right)}{\cos \left(\beta \right)};j=\mathrm{1,2}, \\ \mathrm{N}1=0.5\frac{\cos \left({\alpha }_6\right)}{\cos \left(\beta \right)}\mathrm{X}3;\mathrm{N}2=0.5\frac{\cos \left({\alpha }_5\right)}{\cos \left(\beta \right)}\mathrm{J}3. \end{gathered}$$

(18)

The angle between perimeter cables ($2\beta$) is determined by the number of the load-bearing girders. Thus, four cases of domes with 6, 8, 10 and 12 flat load-bearing girders are considered.

The correctness of the Formulas (18) was confirmed by the spectral analysis of the compatibility matrix (5). In the case of the Geiger domes, as in the case of a two-dimensional truss, the self-stress state obtained from the spectral analysis of the compatibility matrix is the same as obtained from the equilibrium of nodes (

Table 3. Values of the self-stress state of the Geiger domes with 6, 8, 10 and 12 load-bearing girders obtained from formulas (18) and from the spectral analysis of compatibility matrix (5).

No. of Girders	Groups of Elements
	J1	J2	J3	X1	X2	X3	H1	H2	N1	N1	C1	C2	C3
6	4.401	2.109	1.284	2.952	2.255	0.814	2.778	2.168	0.796	1.273	−1.000	−0.414	−0.169
8							3.708	2.959	1.062	1.699
10							4.495	3.587	1.288	2.060
12							5.366	4.282	1.537	2.459

). The self-stress states correctly identify the type of elements and ensure the stability of the structure (the stiffness matrix (2) is positive definite). The results show that only the forces in circumferential cables depend on the number of the load-bearing girders of the structure.

The search for the self-stress state using the genetic algorithm was performed for the 12-girder dome proposed by Jiang et al. [47]. The obtained results for Series 1 and Series 2 are shown in

Table 4. Values of self-stress states of the Geiger dome.

Groups of Elements		Jiang et al. [47]			Present Study
						Genetic Algorithm
		Original	Normalized	Relative Error	Exact Solutions	Series 1	Relative Error	Series 2	Relative Error
1	J1	1452.497	4.287	3%	4.401	4.826	10%	4.511	2%
2	J2	701.205	2.069	2%	2.109	2.486	18%	2.235	6%
3	J3	431.514	1.274	1%	1.284	1.315	2%	1.488	16%
4	X1	941.433	2.778	6%	2.952	2.952	0%	2.940	0%
5	X2	751.292	2.217	2%	2.255	2.336	4%	2.240	1%
6	X3	269.694	0.796	2%	0.814	1.163	43%	0.736	10%
7	H1	1818.387	5.367	0%	5.366	5.359	0%	5.329	1%
8	H2	1451.127	4.283	0%	4.282	4.416	3%	4.273	0%
9	N1	520.917	1.537	0%	1.537	2.213	44%	1.372	11%
10	N2	833.467	2.459	0%	2.459	2.527	3%	2.848	16%
11	C1	−338.833	−1.000	0%	−1.000	−1.000	0%	−1.000	0%
12	C2	−140.228	−0.414	0%	−0.414	−0.524	27%	−0.402	3%
13	C3	−57.534	−0.169	0%	−0.169	−0.201	19%	−0.197	17%

. For comparison, the exact solutions and values obtained by Jiang et al. [47] are additionally presented. In [47], the authors used the catenary-equation-based method. It should be noted that the values presented in [47] for groups X and J differ from exact solutions; the maximum relative error is equal to 5.9% and 2.6%, respectively.

The accuracy of the results obtained by the genetic algorithm depends on the values of the population size and the number of generations. Compared to the exact solutions, in the case of Series 1, the maximum relative error is equal to 44%, whereas, in the case of Series 2, 17%. The values of the relative error show how Series 2 improved the accuracy of the solution. To measure the suitability of the self-stress states, the equilibrium of unsupported nodes (2, 3, 4, 5, 6, 7) and the fitness functions (FF) are calculated. The effectiveness of the genetic algorithm should be assessed by the fitness function, the value of which should be as large as possible. The results for the values obtained by Jiang et al. [47] (Normalized [47]) and for the genetic algorithm (Series 1, Series 2) are shown in Figure 5.

In the case of the exact solution, the equilibrium of nodes equals $6.7\times {10}^{-12}$ and the value of fitness function is 385,880. For solutions presented by Jiang et al. [47], the lowest value of the fitness function was obtained. In this case, the equilibrium of nodes ranges from 0.009 to 0.223 and the value of fitness function equals 1.56. In contrast, for the genetic algorithm, the equilibrium of nodes and the fitness function, respectively, equal 0.03–0.085 and 2.32 for Series 1 and 0.001–0.034 and 6.02 for Series 2.

4.2. Levy Dome

The second considered structure is the Levy dome (Figure 6) proposed by Chen and Feng [54]. The structure consists of 168 cables and 36 struts ($n=204)$ and is characterized by 216 degrees of freedom $(m=216)$. The system consists of the 12 repetitive load-bearing girders (Figure 6b). The elements were divided into 15 groups (Figure 6a).

In the case of the Levy dome, the presence of the additional cables complicates the calculation of the self-stress state using the equilibrium of nodes. Using the spectral analysis of the compatibility matrix (5), two self-stress states are obtained. The superposition of them leads to a suitable self-stress state; the equilibrium of nodes equals $5.15\times {10}^{-15}$ and the fitness function—$1.39\times {10}^7$ (

Table 5. Values of self-stress states of the Levy dome.

Groups of Elements			Chen & Feng [54]			Present Study
							Genetic Algorithm
			Original	Normalized	Relative Error	Exact Solutions	Series 1	Relative Error	Series 2	Relative Error
1	J1		1343	4.403	57%	2.809	2.097	25%	2.234	20%
2	J2		644	2.111	105%	1.032	0.525	49%	0.627	39%
3	J3		392	1.285	23%	1.044	0.431	59%	0.675	35%
4	X1	a	901	2.954	276%	0.786	0.190	76%	0.281	64%
		b			12%	2.639	2.805	6%	2.934	11%
5	X2		688	2.256	133%	0.969	0.962	1%	0.963	1%
6	X3		248	0.813	15%	0.704	0.459	35%	0.390	45%
7	H1	a	1637	5.367	47%	3.644	2.664	27%	2.916	20%
		b			8%	4.986	4.542	9%	4.874	2%
8	H2		1307	4.285	37%	3.124	3.067	2%	3.093	1%
9	N1		469	1.538	15%	1.333	0.884	34%	0.727	45%
10	N2		750	2.459	23%	2.000	0.838	58%	1.282	36%
11	C1		−305	−1.000	0%	−1.000	−1.000	0%	−1.000	0%
12	C2		−126	−0.413	20%	−0.345	−0.289	16%	−0.270	22%
13	C3		−52	−0.170	23%	−0.138	−0.074	46%	−0.083	40%

—Exact solution). It should be noted that the division of elements into 13 groups proposed in [54] (Table 4) is not sufficient. The forces in circumferential cables H1 and X1 are not the same for each load-bearing girder; the forces are repeated every other girder (Figure 6c). Comparing the exact solution with the forces obtained in [54], there is a large discrepancy (for six groups of cables, the relative error is above 100%). Due to the fact that the authors [54] did not specify the method used to obtain the self-stress state, it is impossible to determine the reason for such a discrepancy in values. To measure the suitability of the obtained self-stress state, the equilibrium of unsupported nodes (2, 3, 4, 5, 6, 7) and the fitness functions (FF) are calculated. The results are shown in Figure 6(Normalized [54]). In this case, the equilibrium of nodes ranges from 0.001 to 2.3 and the value of fitness function equals 0.08. In turn, the values of self-stress state obtained using the genetic algorithm (Table 4—Series 1 and Series 2) are much better. The equilibrium of nodes and fitness function (Figure 7), respectively, equal 0.013–0.097 and 1.41 for Series 1 and 0.004–0.075 and 1.75 for Series 2.

4.3. Kiewitt Dome

The last considered structure is the Kiewitt dome (Figure 8) proposed by Yuan et al. [27]. The structure consists of 126 cables and 19 struts ($n=145)$ and is characterized by 114 degrees of freedom $(m=114)$. The system consists of six repetitive load-bearing girders (Figure 8b). The elements were divided into 18 groups.

In the case of the Kiewitt dome, as for the Levy dome, the calculation of the self-stress state using the equilibrium of nodes is complicated. In turn, the spectral analysis of compatibility matrix (5) leads to obtaining 31 self-stress states. None of them correctly identify the type of elements, so the superposition of them is required. Because it is complicated in the case of 31 states, the genetic algorithm seems to be a reasonable method of searching for the appropriate self-stress states. The results obtained with the genetic algorithm for Series 1 and Series 2 are shown in

Table 6. Values of self-stress states of the Kiewitt dome.

Groups of Elements		Yuan et al. [27]		Present Study
				Genetic Algorithm
		Original	Normalized	Series 1	Series 2
1	J1	290.24	1.554	1.386	0.888
2	J2	234.96	1.258	1.542	1.260
3	J3	369.24	1.977	1.952	1.748
4	J4	164.34	0.879	0.591	0.653
5	J5	99.06	0.530	0.749	0.911
6	J6	251.74	1.348	1.020	1.059
7	X1	60.42	0.323	1.223	0.673
8	X2	168.39	0.902	0.765	0.758
9	X3	56.54	0.303	0.292	0.291
10	X4	65.59	0.351	0.367	0.339
11	X5	341.31	1.827	0.931	1.274
12	X6	275.31	1.474	1.326	1.268
13	H1	875.31	4.686	4.350	4.143
14	H2	221.04	1.183	1.088	1.060
15	C1	−198.74	−1.064	−0.920	−0.902
16	C2	−125.53	−0.672	−0.633	−0.601
17	C3	−186.78	−1.000	−1.000	−1.000
18	C4	−173.17	−0.927	−0.790	−0.765

. Additionally, the results obtained by Yuan et al. [27] are presented. The authors used the double singular value decomposition method.

In the case of the results obtained in [27], the equilibrium of nodes ranges from 0.001 to 0.09 and the value of the fitness function equals 3.38 (Figure 9). In turn, in the case of the genetic algorithm, the equilibrium of nodes and the fitness function, respectively, equal 0.008–0.057 and 5.47 for Series 1 and 0.01–0.081 and 6.65 for Series 2.

5. Conclusions

The aim of the paper was to investigate the application of the genetic algorithm to find a suitable self-stress state of tensegrity structures. Two approaches are used: the spectral analysis of the compatibility matrix and the genetic algorithm. In response to the questions posed in the introduction, the following conclusions can be drawn:

• the genetic algorithm is an effective method of the identification of the self-stress states, especially in the case of the structures with complex designs. The procedure proposed in this paper is relatively simple in mathematical description, uses an open-source Python library and can be used for structures for which the spectral analysis of compatibility matrix generates many self-stress states. In such situations, the self-stress states must be superimposed to obtain the one that identifies struts and cables properly,
• the genetic algorithm leads to the satisfactory results; i.e., the obtained self-stress forces provide relatively small value of the equilibrium of nodes. The accuracy of solutions highly depends on the values of the population size and the number of generations; increasing these values leads to a more precise solution but requires more computational power. As the comparisons with the self-stress state presented in the literature show, the forces are not always accurate. It should be noted that the appropriate solution has to satisfy the stable equilibrium (the equilibrium of nodes needs to be close to zero).
• the results are convergent compared to the exact methods, as is indicated by the comparison with the solutions obtained with the spectral analysis. As mentioned before, the accuracy of the solutions depends on the selected parameters of the algorithm, and higher convergence could be achieved by increasing these parameters,
• it is possible to use the same algorithm parameters for different types of domes with comparable geometry.

In conclusion, the considerations confirmed the effectiveness of the genetic algorithm for identifying the self-balanced forces of the existing structures. The results of this research can be applied for the considerations of the behavior of tensegrity domes under external load (quantitative analysis). The authors also intend to apply the proposed method to analyze other tensegrity structures, such as bridges or double-layered tensegrity grids.