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Node-based genetic form-finding of irregular tensegrity structures
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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc
Node-based genetic form-finding of irregular tensegrity structures Buntara Sthenly Gan a,⇑, Jingyao Zhang b, Dinh-Kien Nguyen c, Eiji Nouchi a a
Department of Architecture, College of Engineering, Nihon University, Koriyama, Japan Department of Architecture and Urban Design, Nagoya City University, Nagoya, Japan c Department of Solid Mechanics, Institute of Mechanics, VAST, Hanoi, Viet Nam b
a r t i c l e
i n f o
Article history: Received 9 April 2014 Accepted 7 July 2015 Available online 8 August 2015 Keywords: Tensegrity Form-finding Genetic algorithm Self-stress Force density method
a b s t r a c t A novel and versatile numerical technique to solve a self-stress equilibrium state is developed and used in a combination with a genetic algorithm as a form-finding procedure for an irregular tensegrity structure. An innovative direct encoding scheme of a tensegrity structure is proposed by using the connectivity matrix and prototype tension coefficient vector as the source of genes. The proposed method uses the least design variable, i.e. only the number of nodes needed to find a configuration of an irregular tensegrity structure, without any information on the symmetrical geometry or other predefined initial structural conditions. Three test cases are presented to demonstrate the efficiency of the proposed method in the form-finding of an irregular tensegrity structure. The result from the numerical form-finding procedure was used to design a real tensegrity object. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Since the invention of tensegrity structures by Snelson, Fuller and Emmerich [1], the ingenious forms, simplicity of conception, light weight and deployability of tensegrity structures have been brought to a rapid development stage in civil structures [2,3], space structures [4,5], mechanical cells [6–8] and robotics [9]. As a result, many analytical and numerical methods based on mathematical principles and theories have been developed for form-finding problem of tensegrity structures [10,11]. However, most of these methods are still limited to simple and regular tensegrity structures. In this study, we present a novel method for form-finding of complicated and irregular tensegrity structures. Form-finding of a tensegrity structure is a process of finding structural configurations associated with self-stresses in the self-equilibrium state. It is one of the fundamental problems in the design of any statically indeterminate structure including tensegrity parts. A vast amount of research in form-finding of tensegrity structures has resulted in reliable techniques. Most recently, the stochastic procedure and numerical optimization algorithm inspired by both natural selection and natural genetics are being used to solve the form-finding problem of tensegrity structures. Li et al. [15] proposed a form-finding method ⇑ Corresponding author at: Laboratory for Computational Applied Mechanics, Department of Architecture, College of Engineering, Nihon University, 1-Nakagawara, Tokusada, Koriyama City, Fukushima Prefecture 963-8642, Japan. Tel./fax: +81 24 956 8735. E-mail address: [email protected] (B.S. Gan). http://dx.doi.org/10.1016/j.compstruc.2015.07.003 0045-7949/Ó 2015 Elsevier Ltd. All rights reserved.
for large scale regular and irregular tensegrity structures by using a Monte Carlo simulation. Lobo and Vico [16] developed an evolutionary form-finding process for tensegrity structures. Xu and Luo [17] and Koohestani [18] used the genetic algorithm in the form-finding procedure of tensegrity structures. In the above-mentioned form-finding processes, some initial conditions or assumptions on structural configurations, such as a twisting angle, a strut to cable length ratio, or a force to length ratio (force density), are usually necessary. However, there has been relatively little research on design procedures that use fewer assumptions as the initial conditions. This paper presents a similar track to the works described in [16,17] by using the genetic algorithm, but employs a simpler and easier encoding scheme, therefore requiring the fewest variables, i.e., only the number of nodes, compared to the other works in the form-finding of irregular tensegrity structures. In this paper, the iterative numerical form-finding procedure developed by Estrada et al. [13] is adopted as a form-finding tool for tensegrity structures used in the proposed genetic algorithm. The process was based on the force density formulation with fewer design variables, such as connectivity information and the prototype force densities. The work of Schek [12] on the force density method was originally developed for form-finding of network tensile structures. Since the iterative procedure will result in one particular structural configuration of a tensegrity structure in a self-equilibrium state, a non-iterative scheme is adopted herein. The iteration process in the adopted procedure is replaced in the present study
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by a genetic algorithm which is based on the mechanics of natural selection and genetics [14].
Suppose that the structure is composed of nn nodes and nb members. The connectivity of the members by the nodes can be
2. Tensegrity structures
b
A tensegrity structure consists of two types of structural members: the continuous cables in tension and the discontinuous struts in compression. Tensegrity structures are usually associated with (infinitesimal) mechanisms which are stabilized by the introduction of pre-stresses into their structural members. Moreover, tensegrity structures are free-standing without any support while maintaining their self-equilibrium states, which require the calculation of member forces in a particular spatial arrangement. Without the application of external forces at the nodes, the associated mathematical models and numerical algorithms have to represent non-trivial solutions for the member forces. 2.1. Equilibrium equations Fig. 1 is used to illustrate the equations of static equilibrium of a reference node i which is connected to nodes j and k by members i– j and i–k, respectively. Neglecting the self-weight of members and nodes, the equilibrium equations at node i in Fig. 1 are given in the three-dimensional axis directions by ext
ðxi xj Þf ij =lij þ ðxi xk Þf ik =lik ¼ f i;x ; ext
ðyi yj Þf ij =lij þ ðyi yk Þf ik =lik ¼ f i;y ; ðzi zj Þf ij =lij þ ðzi zk Þf ik =lik ¼
ð1Þ
ext f i;z :
The so-called force density [12], or tension coefficient [19], denoted as qk and defined as ratio of member force f k to member length lk of member k, is often used to simplify the equilibrium equations in Eq. (1) as ext
ðxi xj Þqij þ ðxi xk Þqik ¼ f i;x ; ext
ðyi yj Þqij þ ðyi yk Þqik ¼ f i;y ; ðzi zj Þqij þ ðzi zk Þqik ¼
2.2. Connectivity matrix
ð2Þ
ext f i;z ;
or as
ext qij þ qik xi qij xj qik xk ¼ f i;x ; ext qij þ qik yi qij yj qik yk ¼ f i;y ; ext qij þ qik zi qij zj qik zk ¼ f i;z ;
ð3Þ
0
CT diagðCxÞ
1
B C A ¼ @ CT diagðCyÞ A; CT diagðCzÞ where diagðC Þ is the diagonal version of the vector ðÞ and CT denotes transpose of the matrix C. When a tensegrity structure is in the self-equilibrium state, there is no external load applied. Thus, the equilibrium equations in Eq. (2) for all nodes can be summarized in matrix form as follows:
A q ¼ 0;
ð6Þ
where the zero vector on the right-hand side of the equation indicates that there is no external load applied, and q ¼ fq1 ; q2 ; . . . ; qnb gT is the vector of force densities. The self-equilibrium equations can be written in an alternative form using the force density matrix. According to Eq. (3), the self-equilibrium equations without external loads can be summan n rized as follows in terms of a symmetric matrix D 2 Rn n , known as the force density matrix (FDM), associated with the nodal coordinate vectors x, y, z as follows:
Member fi-j li-j
i
Node
f ext i, x
Mem
ber
(xi,yi,zi) li-k fi-k
k (xk,yk,zk ) Node
ð7Þ
where
D ¼ CT diagðqÞC:
(xi,yi,zi )
f ext i, z
z
The simplest triplex tensegrity structure shown in Fig. 2 is used to illustrate the creation of the connectivity matrix C. The structure consists of twelve members (nb ¼ 12, nine cables and three struts) and six nodes (nn ¼ 6Þ. Its connectivity matrix C is shown in Fig. 2. The rows of matrix C specify the connectivity information between two nodes connecting a member; hence the columns of matrix C indicate the sequence of nodal number information. Let A denote the equilibrium matrix defined as follows:
f ext i, y
y
ð5Þ
The (i; jÞ-th component Dði; jÞ of the force density matrix D can be expressed as
where all the notations in Eqs. (1)–(3) are given in Fig. 1.
Node (xj,yj,zj ) j
8 > < þ1 if p ¼ i Cðk; pÞ ¼ 1 if p ¼ j : > : 0 otherwise
D ½x y z ¼ ½0 0 0;
n
denoted by using the connectivity matrix C 2 Rn n : if a member k is connected to two nodes i and j (i < jÞ, then the entries in the k-th row of C is defined as
x
Coordinates of i-node
f ext External force at i-node in x-direction i, x
li-j
Length of i-j member
fi-j
Internal force of i-j member
qi-j
Force density of i-j member which is given as, qi-j=fi-j / li-j
Fig. 1. Equilibrium at the reference node i.
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Node Number
4
1
4
3
nn nn-th node
9 10 11 12
nb-th member
8
nb
7
3
2
6
1
1
2
5
9
Members' Connectivi ty
12
8
6
0 0 0 0 −1 −1 0 0 −1 0 −1 0
4
7
11
10
4
3
5
3
2
6
5
5
1 −1 0 0 0 0 1 −1 0 0 1 0 −1 0 0 0 0 0 1 −1 0 0 0 0 1 0 0 0 1 0 C= 1 0 0 −1 0 0 1 0 0 −1 0 0 1 0 0 1 0 0 0 −1 0 1 0 0 0 0 0 1 −1 0
1
6
2
Connectivity Matrix n b x n n Fig. 2. Connectivity matrix of a triplex tensegrity structure.
8 qk if i – j and member k is connected by nodes i and j; >
: 0 if nodes i and j are not connected: ð8Þ
2.3. Rank conditions of tensegrity structure For a tensegrity structure in the self-equilibrium state, two necessary but not sufficient rank conditions have to be satisfied in a d-dimensional space [1], one is related to the equilibrium matrix and the other is related to the force density matrix. In order to have at least one self-stress mode, the following rank condition for the equilibrium matrix A has to hold
r ¼ rankðAÞ < nb :
ð9Þ
Using rank of A, the number m of the first-order infinitesimal mechanisms and that s of the self-stress modes can be derived as follows [20,21]:
m ¼ d nn r;
ð10Þ
b
s ¼ n r P 1:
ð11Þ
Note that s should be positive such that the structure is statically indeterminate and it can carry self-stresses in its members. Tensegrity structures are not only statically, but also often kinematically, indeterminate structures. Table 1 shows the classification of pin-jointed structures based on the values of s and m (see
Table 1 Classification of pin-jointed structures. Category
s and m values
Type of structure
I
s ¼ 0; m ¼ 0
II
s ¼ 0; m > 0
III
s > 0; m ¼ 0
IV
s > 0; m > 0
Statically determinate and kinematically determinate Statically determinate and kinematically indeterminate Statically indeterminate and kinematically determinate Statically indeterminate and kinematically indeterminate
Refs. [22,23,30] for a comprehensive description of the classification of pin-jointed structures). The second rank condition is for the force density matrix D:
rankðDÞ < nn d:
ð12Þ
The rank deficiency of a tensegrity structure is (d þ 1Þ as the largest possible rank condition of matrix D in order to find a non-degenerate configuration in d-dimensional space [24,25,31,32]. 3. Numerical form-finding method The numerical form-finding method for tensegrity structures presented in this paper is similar to the procedure proposed by Estrada et al. [13]. However, a genetic algorithm is used to find the self-equilibrium configuration of an irregularly shaped tensegrity structure. Form-finding algorithms are recalled in this section for clarification purposes while iterative procedures and formulations are described in literature. 3.1. Form-finding of tensegrity using force density prototype In contrast to most of the existing form-finding procedures [1,24,33] which require initial assumptions on the length of members, geometry or the symmetry of structure, Estrada et al. [13] proposed a procedure using a predefined connectivity matrix C and a prototype of the force density vector q0 for all members. To calculate the rank deficiency requirement of a tensegrity structure, the spatial dimension d of the problem is also necessary. The prototype force density vector q0 is assigned with a value of þ1 or 1 to members that are chosen to be in tension or in compression, respectively, as
2
3T
q ¼ 4|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} þ1 þ 1 þ 1 0
tension
1 1 1 5 : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð13Þ
compression
The procedure guides both matrices D and A to have the proper rank deficiencies, by selecting the appropriate eigenvector(s) in each decomposition which lead to the existence of at least one self-stress equilibrium state.
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3.2. Approximation of coordinates from tension coefficients A Schur decomposition to the D matrix can be expressed by [13]
D ¼ U V UT :
ð14Þ
If the matrix D has the maximal rank [26,31], i.e., rankðDÞ ¼ nn d 1, the first (d þ 1) columns of the unitary matrix U ¼ ½u1 u2 u3 . . . unn contain the basis of the nodal coordinates which solves the homogeneous Eq. (7) (see Meyer [27] for more details); and the diagonal matrix V has (d þ 1 ¼ 4) zero eigenvalues for a three-dimensional problem. However, the nodal coordinates can only be obtained approximately for an arbitrary q. In the method proposed by Estrada et al. [13], the first three eigenvectors u1 , u2 , and u3 in U corresponding to the absolutely smallest eigenvalues of D assembled from the initial force density vector q0 given in Eq. (13) are picked up for computing the approximation of nodal coordinates:
x ¼ u1 ; y ¼ u2 ;
ð15Þ
z ¼ u3 : However, the force density matrix D corresponding to q0 is unlikely to satisfy Eq. (12) and therefore the structure is not in a self-equilibrium state. The static equilibrium can be achieved from the approximation of the column vectors which do not correspond to the zero eigenvalues in Eq. (14) as
D ½x y z ½0 0 0:
ð16Þ
A full explanation of how to handle the non-equilibrium of Eq. (16) can be found in [13]. At this point in the form-finding procedure, an equilibrium configuration that fulfills Eq. (16) can be approximated.
equilibrium matrix and so on until s > 0. Finally, the force density vector q that fulfills Eq. (20) and the nodal coordinates ½x y z that fulfill Eq. (16) are the solutions. 4. Genetic algorithm Genetic algorithms (GA) are among the currently popular metaheuristic algorithms used for addressing problems based on the mechanics of natural selection and genetics [14]. The most crucial effort in form-finding problem of tensegrity structures is how to define the input parameters which have to be encoded into genes to form a chromosome of an individual population. In this study, the input number of the node is determined to be an even number such that the amounts of struts are half of the amounts of nodes in order to ensure that no pair of struts connect to each other. Based on this idea, both the connectivity matrix and the prototype force density vector can be generated based only on the information on the number of nodes. From this, the dimension of the connectivity matrix and prototype force density vector can be calculated easily. During the GA process, an initial guess of the connectivity matrix is created by ensuring that there are no members that have the same connectivity. In the case of the vector of the prototype force densities, an initial guess is created randomly by either a compressive or a tensile type of member information. These connectivity matrix and prototype force density vector are then encoded into two different chromosomes to form an individual population with different genetic information. 4.1. Solution procedure Fig. 3 gives an outline flow of the present form-finding process for an irregular tensegrity structure based on the predefined
3.3. Approximation of force densities from coordinates By using the approximated nodal coordinates computed from Eq. (15), the equilibrium matrix A in Eq. (4) can be decomposed by using the Singular Value Decomposition [23] as follows:
A ¼ G Y WT ;
ð17Þ
Start Encoding
where the matrices G and W have the following null spaces as
G ¼ ½g1 g2 . . . gr j m1 m2 . . . mdnn r
ð18Þ
Initialize Population
ð19Þ
Decoding
and
W ¼ ½w1 w2 . . . wr j q1 q2 . . . qnb r ;
where mi are the vectors of infinitesimal mechanisms. However, if the structure is not in a self-equilibrium state, the null-space of the equilibrium matrix A does not exist. This is usually the case when A is calculated with an approximation of the nodal coordinates. Alternatively, A can be modified to be rank deficient, by applying a matrix operation that uses ½x y z to compute an approximation of the force density vector q. A self-equilibrium state that fulfills Eq. (17) can be approximated as
0
CT diagðCxÞ
Form-finding Analysis
Genetic Operations : • Selection • Crossover • Mutation
Fitness Evaluations: - Self-stress state - Constraints
Encoding
1
B C A q ¼ @ CT diagðCyÞ A q 0:
ð20Þ
CT diagðCzÞ A comprehensive explanation of how to handle the non-equilibrium of the Eq. (20) can be found in [13]. In summary, the form-finding procedure in [13] iterates Eqs. (14) and (17) until the rank condition of Eq. (12) is satisfied. The force densities and nodal coordinates are updated for the next
Termination
N
Y End Fig. 3. Flow chart of the form-finding process for an irregular tensegrity structure.
B.S. Gan et al. / Computers and Structures 159 (2015) 61–73
number of nodes. The process keeps generating a new population until the tensegrity structure is found. In most GA implementations, there is usually a fitness function which has to be optimized during the evolutionary process. In this study, a single self-stress equilibrium state of a tensegrity structure is sought as the main purpose of the GA implementation, so there is no optimization against a particular fitness function in the overall process of the form-finding of a tensegrity structure. Besides the main objective function, several constraints are transformed into penalty functions to ensure the validity of the tensegrity structure being sought. Thus, all the fitness functions are evaluated together during the form-finding process to obtain a single self-stress equilibrium state of a tensegrity structure which satisfies all the applied constraints. 4.2. Encoding scheme for individual population A 6-node triplex tensegrity structure is used to illustrate encoding herewith. Fig. 4 shows the encoding for the triplex tensegrity as an individual population which is presented by two separate chromosomes: a connectivity matrix and a set of prototype force densities as genetic information. Since the binary encoding scheme cannot represent all the row-wise genes of the connectivity matrix, the chromosome of the connectivity matrix is thus encoded by using the alphabet-based chromosome. Fig. 5 shows the genetic operation used in the present study. The connectivity matrix is encoded into an alphabet-based chromosome. The binary-based chromosome is used for encoding the force density vector, in which the value of 1 represents the tensile member and the value of 0 represents the compressive member. Permutation in the reproduction process is used in the connectivity matrix chromosome to ensure that there are no members that have the same connectivity. 4.3. Fitness and penalty functions This study uses one fitness function combined with two constraints which are transformed into penalty functions [29] to evaluate the fitness of each individual in the population. The fitness function is determined from the result of evaluating the self-equilibrium state. The self-equilibrium state of a tensegrity
65
structure is evaluated from the connectivity matrix and vector of force densities, which are decoded from the individual population. The fitness evaluation for an individual population is defined from a summation of the four fitness functions below:
Total Fitness Value; FV ¼ f 1 þ f 2 þ f 3 þ f 4
ð21Þ
which will be defined in detail later. The total fitness value, which is a combination of the four evaluation functions in Eq. (21), will result in a value close to zero as the best fitness objective of the current GA scheme. 4.3.1. Self-equilibrium state evaluation Following the non-iterative numerical form-finding procedure previously explained, a Schur decomposition to the D matrix in Eq. (14) is used to obtain approximated coordinates. The approximated coordinates are then used in the Singular Value Decomposition of the matrix A in Eq. (17), which results in a singular vector of mechanisms and approximated force densities vector, q. The resulting singular values of diagonal matrix Y follow the decreasing sequence diag(y1;1 . . . yndxn b;n Þ. When the amount of zero or close to zero values in the diagonal matrix Y are found, it shows the rank of the matrix, which further implies the amount of the self-equilibrium state which exists for the tensegrity structure. Thus, the rank deficiency of the diagonal matrix Y is used as the fitness value of the individual population. In order to narrow the search space, only one rank deficiency of the Y matrix is sought, which gives one self-equilibrium state of the tensegrity structure. One rank of deficiency of the matrix is found when the value of the right bottom diagonal in the matrix approaches zero,
f 1 ¼ Yðd nn ; nb Þ:
ð22Þ
Here, Eq. (22) will give a zero value as the best fitness if the end diagonal of matrix Y results in a value close to null; otherwise, the fitness value will be larger than zero. It should be noted that any candidate design can be accepted if the fitness value is smaller than 1010 . 4.3.2. Connectivity at a node constraint To eliminate any infeasible solution with very few member connections at a node, a predefined minimum number of members,
Fig. 4. Encoding scheme for a 6-node triplex tensegrity structure.
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Fig. 5. Crossover operation for a 6-node triplex tensegrity structure.
NC, at a node is imposed by the following constraint evaluation function:
! nn nb X X jCði; jÞj f2 ¼ NC j¼1 i¼1
ð23Þ
Eq. (23) will result in a zero value as the best fitness if the number of members at a node is equal to NC; otherwise the fitness value will be larger than zero. 4.3.3. Only a single strut at a node constraint One of the typical characteristics of a tensegrity structure is that all the compressive strut members have to be independent and separate from each other to guarantee that all the strut members are in compression. In order to eliminate the connection between two struts, the following constraint evaluation function being transformed into a fitness value is adopted:
! nn nb X X T f3 ¼ CTði; jÞ where; CT ¼ diagðq0 Þ jCj; NS j¼1 i¼1
ð24Þ
where NS can be obtained from NS = NC 2. The result of 0 T
diagðq Þ jCj can give the total amount of predefined number of members at all nodes constrained by the NC. Because the prototype force density vector q0 consists of 1 value for a strut member having two nodes will reduce the total amount of members by 2. Eq. (24) will give the best fitness value of zero only if there are no
two strut members connected at a node; otherwise it will be larger than zero. The enforcement of the current constraint implies that there needs to be an even number of nodes in order to achieve a configuration where each strut has its two connecting nodes which are separate from the other nodes. 4.3.4. No intersection between two struts constraint If two lines intersect each other, there is a common point of intersection. Consider two members connecting nodes A, B and C, D, respectively. The nodal coordinates of which are denoted by A, B, C, and D. The equations for these two members are expressed as follows by using two parameters g and h varying between 0 and 1:
Member 1 : ð1 gÞA þ gB Member 2 : ð1 hÞC þ hD:
ð25Þ
Equating the two equations in Eq. (25) in each corresponding direction gives
g ðBx Ax Þ þ hðC x Dx Þ ¼ C x Ax g By Ay þ h C y Dy ¼ C y Ay
ð26Þ
g ðBz Az Þ þ hðC z Dz Þ þ p ¼ C z Az ; where a parameter p, which can be physically interpreted as a residual distance between the two lines in the z-direction, is added in the third equation as an indicator for detecting whether the two lines are intersecting each other or not. Moreover, let r denote a
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7RWDO)LWQHVV9DOXH
7RWDO)LWQHVV9DOXH
Fig. 6. Six-nodes irregular tensegrity structure: variation of total fitness for the best individual included in the 10-individuals population.
threshold value which represents the minimum distance between two lines, such as the physical diameter of the strut or cable. Thus, two members do not intersect if and only if p > r. There are three simultaneous equations in Eq. (26) from which we can solve the unknown parameters g; h and p. Rewriting Eq. (26) in a matrix form, the solution gives
(a) Configuration at 1st generation
(c) Configuration at 100th generation
*HQHUDWLRQ
*HQHUDWLRQ
8 9 2 9 31 8 ðBx Ax Þ ðC x Dx Þ 0 > >
Fig. 8. Eight-nodes irregular tensegrity structure: variation of total fitness for the best individual included in the 40-individuals population.
If the coefficient matrix is not invertible, the two lines representing two members are parallel, and we set p = 2r for this case. Using the solution of p in Eq. (27), the fourth fitness value f 4 is defined as follows:
f4 ¼
0; if p > r ðnot intersectÞ 2; if p 6 r ðintersectÞ
:
ð28Þ
5. Test cases and numerical results
ð27Þ The GA algorithm implemented in the present study includes conventional operators such as selection, crossover, mutation and
(b) Configuration at 40th generation
(d) Configuration at 134th generation
Fig. 7. Evolution of the configuration of the six-nodes irregular tensegrity structure.
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(a) Configuration at 1st generation
(b) Configuration at 30th generation
(c) Configuration at 51st generation
(d) Configuration at 207th generation
Fig. 9. Evolution of the configuration of the eight-nodes irregular tensegrity structure.
5.1. Six-nodes irregular tensegrity structure In the first test case a tensegrity structure with 6 nodes must be designed. The population size is set as 10. The connectivity matrix C is randomly generated and the prototype force density vector q is encoded for each individual included in the population. The total fitness value close to zero indicates that a self-equilibrium state is reached during the form-finding process. In this example, the GA process ends after 134 generations. Fig. 6 shows the total fitness values during the evolutionary process. Fig. 7(a–d) show the evolution of the structural configuration during the evolutionary process at the 1st, 40th, 100th and 134th generations, respectively. In Fig. 7, the parameters p1 and p2 indicate the residual distances between adjacent struts that are also compared with the longest element dimension. The three structures in Fig. 7(a)–(c) are not in self-equilibrium states because the fitness functions are not zero and the constraints are violated. At the 134th generation, the proposed algorithm terminates successfully, and the corresponding self-equilibrated structure is shown in Fig. 7(d).
structure after 207 generations. The population size set for this problem is 40. Fig. 8 shows the variation of the total fitness for the best design included in the population during the evolutionary process. Fig. 9(a–d) show the configurations taken by the structure at the 1st, 30th, 51st and 207th (self-equilibrium solution) generations, respectively. In Fig. 9, the parameters p1, p2 and p3 indicate the residual distances between adjacent struts that are also compared with the longest element dimension.
5.3. Ten-nodes irregular tensegrity structure The tensegrity structure designed in the last test case includes 10 nodes. The GA-based form-finding algorithm utilized the same population size as in the previous test case but a much larger number of generations (i.e. 2581). Fig. 10 shows the variation of the total fitness for the best design included in the population during the evolutionary process. Fig. 11(a–d) show the configurations taken by the structure at the 1st, 1000th, 2000th and 2581st
7RWDO)LWQHVV9DOXH
elitism [14]. The GA is run with several individuals in the population, where one individual is reserved as a single elite after each generation. Probability-based mutation is used to randomly change a gene of the alphabet-based chromosome. The GA is run with mutation probability of 0.2. The best combination of GA internal parameters used in this study was obtained from sensitivity analysis. We consider three (d = 3) dimensional example structures consisting of (nn =) 6, 8, and 10 nodes, respectively. In each example, the predefined number of members connected to a node is set as (NC=)4.
5.2. Eight-nodes irregular tensegrity structure Similar to the previous test case, the GA-based form-finding algorithm finds a self-equilibrium configuration for the 8-node
*HQHUDWLRQ Fig. 10. Ten-nodes irregular tensegrity structure: variation of total fitness for the best individual included in the 40-individuals population.
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(a) Configuration at 1st generation
(b) Configuration at 1000th generation
(c) Configuration at 2000th generation
(d) Configuration at 2581st generation
Fig. 11. Evolution of the configuration of the ten-nodes irregular tensegrity structure.
(a) Six-nodes
(b) Eight-nodes
(c) Ten-nodes
Fig. 12. Several irregular tensegrity structure generated with the present GA algorithm.
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Table 2 Computational cost of the form-finding process for different population sizes.
a
Numbers shown in the ( ) are products of the number of generations and the population size during the evolutionary process. The numbers show the evaluations of GA processes (penalty functions, crossover, selection and mutation) that can be interpreted as the total computational cost.
Fig. 14. The result of form-finding of an eight-nodes irregular tensegrity structure.
Table 3 Nodal coordinates for the real tensegrity structure to be designed. Node number
x
y
z
1 2 3 4 5 6 7 8
0.046997 0.100338 0.462185 0.502951 0.638767 0.334700 0.029177 0.015839
0.612765 0.267531 0.155165 0.101945 0.233056 0.331275 0.470376 0.364918
0.392801 0.369638 0.368403 0.184040 0.026372 0.480091 0.063288 0.551630
Fig. 13. The connectivity matrix C and prototype force density vector q0 . Table 4 Force densities for the real tensegrity structure to be designed.
(self-equilibrium solution) generations, respectively. In Fig. 11, the parameters p1 and p2 indicate the residual distances between adjacent struts that are also compared with the longest element dimension. Finally, several configurations of self-equilibrium state of the six, eight and ten-nodes irregular tensegrity structures generated with the present GA algorithm are shown in Fig. 12. Table 2 reports data on the computational cost of the form-finding process for different population sizes. As expected, the number of generations increases with the number of nodes included in the structure. Increasing population size leads to reducing the required number of generations and structural analyses. 6. Design of a real eight-nodes irregular tensegrity structure The results from the numerical form-finding procedure described in the previous sections can be used to design a real tensegrity structure. For that purpose, the unique configuration of the eight-node irregular tensegrity structure is chosen as an initial design. The real tensegrity object is then sized via finite element analysis to satisfy safety and stability requirements. It should be noted that the form-finding algorithm presented in this study is based on the self-equilibrium state of a structure not subject to external forces at the nodes. This leads to search for non-zero solutions of a homogeneous equation. Therefore, it is
Member connectivity
Force density
Member type
1–2 2–3 3–4 1–4 5–6 6–7 7–8 5–8 1–5 2–6 3–7 4–8 1–6 2–7 3–8 4–5
1.5331 3.2162 4.9010 1.6192 2.8351 1.8807 1.7724 1.3201 3.4248 2.7533 1.6502 5.0517 3.1641 2.0067 3.7128 3.2504
Cable Cable Cable Cable Cable Cable Cable Cable Cable Cable Cable Cable Strut Strut Strut Strut
not necessary to include stress/buckling conditions into the form-finding process, as these constraints can then be scaled up to meet design requirements. 6.1. Form-finding Based on the numerical form-finding process algorithm, an eight-nodes irregular tensegrity configuration was found by using
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Table 5 Scaled nodal coordinates for the tensegrity structure. Node number
X (mm)
Y (mm)
Z (mm)
1 2 3 4 5 6 7 8
0.00 1437.34 2529.21 2015.31 2261.04 2062.49 2517.07 66.22
0.00 689.33 0.00 1808.41 1106.38 902.53 1352.97 1142.40
0.00 1688.40 0.00 0.00 1220.02 2809.20 1185.58 1839.52
Table 6 Sectional properties of members. Member
Type
Size
2–7 4–5 1–6 3–8 Others
Strut Strut Strut Strut Cable
/ ¼ 60:5 mm, t = 2.0 mm / ¼ 114:3 mm, t = 2.0 mm / ¼ 165:2 mm, t = 3.0 mm / ¼ 216:3 mm, t = 3.0 mm / ¼ 4:0 mm, / strand = 0.44 mm
Fig. 15. Finite element model of the eight-nodes tensegrity object.
the following connectivity matrix C and prototype force density vector q0 as for the initial trial configuration. As shown in Fig. 13, the prototype force density vector consists of positive value of ones which indicates tension members, and different negative values which indicate the variation of compressive members. These negative varying values are intended to design the compressive members with different cross sectional properties later in design process. The eight-nodes irregular tensegrity structure obtained from the form-finding process is shown in Fig. 14. The resulting nodal coordinates and force densities t are shown in Tables 3 and 4. The coordinate values and force density vector listed in Tables 3 and 4 are scalable in length and member forces. In order to design the real structure, nodal coordinates resulted from form-finding procedure are rotated, translated and scaled up as given in Table 5. Nodes 1, 3 and 4 belonging to the three largest struts are rested on the ground. Fig. 16. Static equivalent seismic loadings in 12 directions (plane view).
6.2. Sizing of members A 4-mm stainless wire cable made of SUS-304 with 7 7 strands is used for the tensile members. Polished hollow stainless pipes made of SUS-304 with two types of thicknesses and four types of diameters are used for struts. Table 6 shows the detail of structural members. 6.3. Structural analysis The designed tensegrity structure is then analyzed with the finite element method to evaluate member strengths. The tensegrity object is put on the ground without any fixed supports. In the finite element analysis, very small spring constants in three coordinate directions are provided in order to avoid rigid body motion. Fig. 15 shows the finite element model used for the eight-nodes irregular tensegrity with the chosen configuration where the struts with the three largest dimensions are rested on the ground while the pipe with the smallest dimensions is suspended in the air supported by the surrounding cables connected to the others pipes.
6.4. Loads and stress/stability response evaluation Beside the self-weight of the structure, wind loading and static equivalent seismic loadings in 12 directions, as illustrated in Fig. 16, are considered to ensure the safety of the structure. Cable safety is evaluated with respect to the limit strength of SUS-304. A minimum safety factor of 6.0 is achieved for all the cables with the maximum tensile strength under various loading conditions. Struts are verified against hollow pipe member’s elastic buckling strength. The minimum safety factor beyond 50.0 is assured for the strut with the maximum compressive strength under different loading conditions. The configured tensegrity object is constructed inside the College of Engineering, Nihon University, Japan in front of the library building. Fig. 17 shows different views of the installed structure. Since the structure is built as a monumental object, it is expected to give the impression of a modern look inside the campus.
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Fig. 17. Views of the eight-nodes irregular tensegrity structure.
7. Summary and conclusions
References
The object of this study was to integrate an innovative GA encoding scheme in the tensegrity structure form-finding algorithm developed by Estrada et al. [13]. The new approach thus developed allows to find self-equilibrium configurations of irregular tensegrity structures of generic shape by giving in input only the number of nodes. The creation of an irregular tensegrity structure hence becomes a very straightforward process. The efficiency of the proposed method has been proven in three design problems of structures including 6, 8 and 10 nodes, respectively. It was found that the number of generations increases with the complexity of the structure but the total computational effort decreases with the size of population. However, it should be noted that the present GA search does not converge to a unique self-equilibrium configuration but finds different configurations which are theoretically correct although dependent on the first initial configuration generated from a random seed. The form-finding process is the starting basis for the design of a real tensegrity structure. The unique configuration obtained from the form-finding process is chosen as initial design to size the real structure. Finite element analysis must be performed to evaluate stress and stability requirements for struts and cables. It should be noted that it is not necessary to directly include stress/buckling conditions into the form-finding process, as these constraints can then be scaled up to meet design requirements. Overall, the proposed GA-based procedure needs further improvements to become an effective design tool. Moreover, by imposing additional symmetric or predefined geometric constraints, a regular tensegrity structure can be designed. Lastly, by adding some practical constraints, criteria or fitness functions to the present formulation will allow various types of tensegrity structures to be studied.
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Acknowlegments The authors would like to appreciate the valuable comments and discussions made by the reviewers.
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