extended force density method for form-finding of tension structures - 2010

JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS

1

EXTENDED FORCE DENSITY METHOD FOR FORM-FINDING OF TENSION STRUCTURES

Masaaki MIKI *1 and Ken’ich KAWAGUCHI *2 *1Graduate Student, Department of Engineering, University of Tokyo, [email protected]

*2Proffessor, Institute of Industrial Science, University of Tokyo, Dr. Eng., [email protected] Bw502, Komaba 4-6-1, Meguro-ku, Tokyo 153-8505, Japan

Editor’s Note: The first author of this paper is one of the four w inners of the 2010 Hangai Prize, awarded for outstanding papers that are submitted for presentat ion and publication at the annual IASS Symposium by younger members of the Association (under 30 years old). It is re-published here with permission of the editor s of the proceedings of the IASS 2010 Symposium: “Spatial Str uctures – Temporary and Permanent” held in November 2010 in Shanghai, China. ABSTRACT

The objective of this study is to propose an extension of the force density method (FDM)[1], a foregoing numerical method for the form-finding of tension structures. FDM has a great advantage for the form-finding of cable-nets. However, some difficulties arise when it is applied to the pre-stressed structures that consist of a combination of both tension and compression members. Therefore, the FDM has scope for extension. In this paper, we identify the existence of a variational principle in the FDM, although, the mathematical forms used by original FDM are different from those related to the variational principle. Thus, a functional that can be thought to be selected by FDM is clarified and it enables us to extend the FDM by considering various functionals as a generalization of it. Such newly introduced functionals enable us to find the forms of complex tension structures that consist of a combination of cables, membranes, and compression members, such as tensegrities, and suspended membranes with bars, etc. Keywords: Force density method, Form-finding, Tensegrity, Membrane, Cable-net,

Variational principle, Principle of virtual work 1. INTRODUCTION

Tension structures such as cable-nets, suspended membranes, and tensegrities are stabilized by introducing prestress. Hence, they require a special process to ensure that they have a self-equilibrium state, so-called prestress state. This process is called form-finding because the existence of the prestress state of a structure is highly dependent on its form. For the form-finding, various numerical methods have already been proposed by many researchers. The force density method (FDM) [1] is one of such method proposed to determine the form of cable-nets. The purpose of this study is to propose an extension of the FDM [1].

In Section 2, we describe the original FDM and its great advantage. In addition, we identify the limitations of the FDM when it is applied to prestressed structures that consists of a combination of both tension and compression members, e.g. tensegrities. Therefore, the FDM has scope for

extension.

As an analytical expression for form-finding, two types of mathematical expression have been mainly adopted. One is a set of equilibrium equations, whereas the other represents a stationary problem of a functional based on the variational principle. In general, the equilibrium equations and the stationary problems of a functional are closely related.

In Section 3, we show the existence of a variational principle in the FDM, although, the mathematical forms used by the original FDM are different from those related to the variational principle. Therefore, we can propose a functional which can be thought to be selected by the FDM by considering the variational principle. This functional enables us to extend the FDM.

In Section4 and 5, we describe the extended FDM and some of its applications in the form of


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numerical examples, by considering various functionals as generalizations of the selected functional. The newly introduced functionals enable us to find the forms of complex tension structures that consist of a combination of cables, membranes, and compression members, such as tensegrities, and suspended membranes with bars.

2. FORCE DENSITY METHOD

2.1 Original Formulation

The FDM was first proposed by Schek and Linkwitz in 1973. The main characteristics of the FDM are divided into two parts. The first part consists of the definition and use of a quantity called force density. The force density qj is defined as

jjj Lnq /= , (1)

where nj and Lj denote the tension and length of the j-th cable, respectively, as shown in Fig. 1(a). In the FDM, each cable is assigned a force density as a known parameter, whereas nj and Lj are unknown (to be determined). Therefore, some trials must be carried out to obtain an appropriate configuration of force densities.

The second part consists of the linear form of the equilibrium equations. When the force densities are assigned and the fixed nodes and their coordinates are prescribed, the self-equilibrium condition for cable-nets is represented by

,

,

,

ff

ff

ff

and

zDzD

yDyD

xDxD

rr

rr

rr

−=

−=

−=

(2)

where D is the equilibrium matrix and x, y, and z are the column vectors containing the nodal coordinates of each node. The terms with the subscript f are related to the fixed nodes, whereas those with no subscript are related to the free nodes.

The right-hand sides of Eq. (2) represent the reaction forces from the fixed nodes. Hence, Eq. (2) can be considered as an analogue of Hook’s law, i.e., fkx = . Here, note that the linear form of Eq. (2) is not approximated.

There are just three unknown variables, i.e., x, y,

and z, therefore, the nodal coordinates of the free nodes can be obtained as follows,

.)(

),(

,)(

1

1

1

ff

ff

ff

and

zDDz

yDDy

xDDx

rr

rr

rr

−

−

−

−=

−=

−=

(3)

Once the nodal coordinates are obtained, the tension in each cable is calculated using Eq. (1).

Eq. (3) represents the standard procedure for solving a set of simultaneous linear equations; hence, it enables the implementation of the FDM very concise. Thus, the FDM has a great advantage over other methods, which require non-linear iterative computation.

Using the FDM, we can determine the form of cable-nets by changing the coordinates of the fixed nodes or the force densities of the cables, as shown in Fig. 1(b).

n

L

q = n /L

(a) Definition of Force Density

(b) Form-finding Analysis using FDM Figure 1. Force Density Method

2.2 Limitation of FDM

In the application of the FDM to the form-finding of self-equilibrium systems that consist of a combination of both tension and compression members, e.g., tensegrites, some difficulties arise when negative force densities are assigned to the compression members and positive force densities are assigned to the tension members [2][3][4][5].

Let us consider the form-finding of a prestressed structure, e.g., an X-Tensegrity, as shown in Fig. 2.


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An X-Tensegrity is a planar prestressed structure that consists of 4 cables (tension) and 2 struts (compression). As in the case of general tensegrities, the cables connect the struts and the struts do not touch each other.

For such self-equilibrium systems that have no fixed nodes, Eq. (2) reduced to a simpler form:

.,, 0zD0yD0xDrrrrrr

=== (4)

When D is a regular matrix, only a trivial solution is obtained, i.e.,

.0zyxrrrr

=== (5)

On the other hand, when D is a non-regular matrix, Eq. (4) has complementary solutions. Such solutions are obtained by analyzing the null space of D. However, even if we analyze it, the FDM would lose its conciseness as follows:

� When the assigned force densities are in the proportion 1:1:1:1:-1:-1 for the 4 cables and the 2 struts respectively, many solutions are obtained. An example of D and the corresponding solutions are shown below.

,

1

1

0

0

0

0

1

1

1

1

1

1

,

1

1

0

0

0

0

1

1

1

1

1

1

,

1

1

0

0

0

0

1

1

1

1

1

1

,

1111

1111

1111

1111

−

+

−+

=

−

+

−+

=

−

+

−+

=

−−−−

−−−−

=

ihg

andfed

cba

z

y

x

D

r

r

r

(6)

where a…i are arbitrary real numbers. This implies, for example, that both Fig. 3(a) and (b) satisfy Eq. (4). The first terms of the right- hand-sides denote the position of the center

point, i.e., [a d g], and the other terms denote some symmetry that all solutions must have.

� When the assigned force densities are not in the proportion 1:1:1:1:-1:-1, the obtained solutions do not denote a form. An example of D and the corresponding solutions are shown below.

.

1

1

1

1

,

1

1

1

1

,

1

1

1

1

,

1322

3122

2231

2213

=

=

=

−−−−

−−−−

=

candba zyx

D

rrr

(7)

This implies that all nodes converge at one point, i.e., [a b c].

Figure 2. X-Tensegrity

(a) (b)

Figure 3. Self Equilibrium Forms

3 VARIATIONAL PRINCIPLE IN FDM

The functional that is thought to be selected by FDM is given by

∑=Πj

jj Lw )()( 2 xx , (8)

where wj and Lj denote an assigned weight coefficient and a function for the length of the j-th cable, respectively. The column vector x contains the x, y, and z coordinates of the free nodes. It is generalized as an unknown variable vector by


4

[ ]Tnxx L1=x . (9)

Note that the coordinates related to the fixed nodes are eliminated from x beforehand and directly substituted in Lj, because they are prescribed.

The stationary condition of Eq. (8) is expressed as

0xx r

=∇=Π∇=∂

Π∂∑

jjjj LLw2

)(, (10)

∇where is the gradient operator defined as

∂∂

∂∂≡

∂∂≡∇

nx

f

x

fff L

1x. (11)

Eq. (11) denotes the direction of steepest increase in n-dimensional space.

In this paper, Lj is given by

( ) ( ) ( )221

221

221 zzyyxxL −+−+−≡ . (12)

In this case, L∇ represents two normalized vectors attached to both ends of the member, as shown in Fig. 4(a). On the other hand, let us consider a linear member resisting two nodal loads applied to both its ends, as shown in Fig. 4(b), and let its axial force be n. Comparing Fig. 4(a) and (b), a general expression for a self-equilibrium state of prestressed cable-nets is obtained as

∑ =∇j

jj Ln 0r

. (13)

On the basis of Eq. (13), the principle of virtual work for pre-stressed cable-nets is expressed as

∑ ==j

jj Lnw 0δδ , (14)

where jLδ is defined as xδδ ⋅∇= jj LL , and δx is

defined as an arbitrary column vector, [ ]Tnxx δδ L1 . They are usually called the variation of Lj and the virtual displacement, respectively.

Substituting Eq. (1) in Eq. (13), an alternative form of Eq. (2) is obtained as

∑ =∇j

jjj LLq 0r

. (15)

Here, because of the mathematical equivalence of Eq. (10) and Eq. (15), Eq. (8) is thought to be the functional that is selected by FDM. In addition, it is assumed that the assigned weight coefficients play the same role as the force densities in the form-finding process for cable-nets.

Although Eq. (2) and Eq. (15) appear rather dissimilar, they are actually identical when each function Lj is represented using Eq. (12). On the other hand, when the unknown variables do not represent Cartesian coordinates, (e.g., when they represent polar coordinates), Eq. (15) remains valid, whereas Eq. (2) becomes invalid. Thus, Eq. (15) is more general expression than Eq. (2).

On the basis of Eq. (15), the principle of virtual work for the FDM is expressed as

.0∑ ==j

jjj LLqw δδ (16)

Similarly, on the basis of Eq. (10), the principle of virtual work is expressed as

∑ ==j

jjj LLww 02 δδ . (17)

Additionally, Eq. (17) is mathematically equivalent to 0=⋅Π∇ xδ . Thus, we obtain a variational principle as

0=Πδ , (18)

where Πδ is defined as xδδ ⋅Π∇=Π and usually called the variation of Π .

In conclusion, it is important to note that in the original paper [1], Eq. (8) is mentioned by the following theorem:

THEOREM 1. Each equilibrium state of an unloaded network structure with force densities qj is identical with the net, whose sum of squared way lengths weighted by qj is minimal.

(a) L∇ (b) Equilibrium State

Figure4. Linear Member


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4. EXTENDED FORCE DENSITY METHOD

4.1 Generalization of Selected Functional

In the previous section, we obtained a functional that is selected by the FDM. It is possible to extend the FDM by generalizing this functional.

Let us reconsider the form-finding of the X-Tensegrity. The same difficulties that is pointed out in subsection 2.2 would also arise from the stationary problem of Eq. (8), when negative weight coefficients are assigned to the struts and positive weight coefficients are assigned to the cables, because of the mathematical equivalence of Eq. (2) and Eq. (15).

In the case of cable-nets, the coordinates of the fixed nodes are given as kinematic conditions, whereas no kinematic conditions are given in the case of the X-Tensegrities. Thus the length of each strut of the X-Tensegrity is assigned as a kinematic condition instead of a weight coefficient. In this case, according to the Lagrange’s multiplier method, a modified functional is obtained as

∑∑ −+=Πk

kkkj

jj LLLw ))(()(),( 2 xxλx λ , (19)

where the second sum is taken for every strut, and, λk and kL are Lagrange’s multiplier and the given length of the k-th strut, respectively.

However, Eq. (19) does not completely eliminate the aforementioned difficulties. If the assigned weight coefficients of the cables are in the proportion 1:1:1:1, and the given lengths of the struts are in the proportion 1:1, both Fig. 3(a) and (b) satisfy the stationary condition of Eq. (19). By using the famous Pythagorean Theorem, i.e., c2=a2+b2, it can be easily verified that the sum of squared lengths of the cables takes the same value for both the shapes.

Here, another functional, e.g.,

∑∑ −+=Πk

kkkj

jj LLLw ))(()(),( 4 xxλx λ , (20)

can be considered because the functionals such as Eq.(8) and Eq. (19) do not represent any physical quantity, such as energy. Thus, it is possible to use higher powers of Lj.

Solving the stationary problem of Eq. (20), Fig. 3(a) becomes the unique solution when the weight coefficients of the cables and the lengths of the struts are assigned in the proportion 1:1:1:1 and 1:1, respectively. On the other hand, when the weight coefficients and the lengths are assigned in the proportion 1:8:1:8 and 1:1, respectively, Fig. 3(b) becomes the unique solution.

Let us investigate the following generalized functional:

∑∑ −+=Πk

kkkj

jj LLL ))(())((),( xxλx λπ . (21)

The stationary condition of Eq. (21) with respect to x is given by

0x

x

r

=∇+∇∂

∂=

∂Π∂

∑∑k

kkj

jj

jj LLL

Lλ

π ))((. (22)

Then, when Eq. (22) is satisfied, comparing Eq. (13) and Eq. (22), the following non-trivial configuration must satisfy Eq. (13):

∂∂

∂∂

= rm

mm

L

L

L

L λλππLL 1

1

11 ,)()(

n . (23)

Eq. (23) represents one of the self-equilibrium modes. Thus, any functional compatible with Eq. (21) can be applied to such a problem. Henceforth in this paper, we call πj the element functional, and apply it to a stationary problem by selecting it. Thus, we propose the following two policies:

� Perform form-finding analysis by solving a stationary problem of a freely selected functional

� When difficulties arise, test the other functionals.

Let us consider the relation between Eq. (23) and the FDM. If wjLj

2 is selected as the element functional, according to Eq. (23), we have the following relation:

jjjjjj LnwLwn 2/2 =∴= . (24)

Therefore, it is verified that the assignment of the force densities is virtually equivalent to the assignment of the weight coefficients. On the other hand, if wjLj

4 is selected, we have


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33 4/4 jjjjjj LnwLwn =∴= . (25)

Therefore, it is verified that defining and adopting a new quantity wj=nj/4Lj

3 is equivalent to selecting wjLj

4 as the element functional. We Call the new quantities, such as wj=nj/4Lj

3, the extended force density.

Without the linear form, the key features of FDM are reconsidered as follows:

� The coordinates are assigned to each fixed node as kinematic conditions.

� The force densities qj=nj/Lj are assigned to each cable as known parameters.

On the other hand, when wjLj4 is selected as the

element functional, the key features of the extended FDM are as follows:

� The lengths kL are assigned to each strut as kinematic conditions.

� The extended force densities wj=nj/4Lj3 are

assigned to each cable as known parameters.

Thus, it is verified that the extended FDM is quit similar to the FDM. Moreover, the general form of the functionals, i.e., Eq. (21), enables us to select various non-linear computational methods when we carry out the extended FDM.

As an alternative to Eq. (13), the following form can be used to express the principle of virtual work for not only the tensegrities but also general pre-stressed structures that consist of a combination of both cables and struts:

.0=+= ∑∑k

kkj

jj LLnw δλδδ (26)

4.2 Additional Analyses

In this subsection, some additional numerical analyses are reported for further comprehension of the extended FDM.

Let us consider an analytical model that consists of 220 cables connecting one another and 5 fixed nodes, as shown in Fig. 5. The coordinates of the fixed nodes are also provided in the figure.

Fig. 6 shows the results of a series of optimization

problems applied to the analytical model. These problems are to minimize the sum of lengths with powers ranging from 1 to 4.

On the other hand, Fig. 7 shows the results of the same optimization problems applied to another model, the Simplex Tensegrity, which is a prestressed structure that consists of 9 cables and 3 struts. In this case, the optimization was only performed for the cables, whereas the lengths of the struts were kept constant at 10.0.

A comparison between Fig. 6(ii) and Fig. 7(ii) implies that different element functionals are required for cable-nets and tensegrities.

Figure 5. Analytical Model

(i) min→∑ jL (ii) min2 →∑ jL

(iii) min3 →∑ jL (iv) min

4 →∑ jL

Figure 6. Optimization Results of Cable-nets

(i) min→∑ jL (ii) min2 →∑ jL

(iii) min3 →∑ jL (iv) min

4 →∑ jL

Figure 7. Optimization Results of Simplex Tensegrity


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5. NUMERICAL EXAMPLES

In this section, we report some numerical examples as applications of the extended FDM.

In general, the unknown variables (to be determined) in a stationary problem of a functional represented by the form of Eq. (21) are [ ]nxx L1=x and [ ]rλλ L1=λ . However, we just minimized

∑ jπ as an objective function by keeping the

lengths of the struts constant at kL . Hence, in this case, only [ ]nxx L1=x are the unknown variables.

General non-linear computations require appropriate initial values for the unknown variables; however, we roughly assign the random numbers to the unknown variables (from -2.5 to 2.5). Regardless of such a rough initial configuration, we always obtained an expected solution.

5.1 Structures Consisting of Cables and Struts

Let us consider 20 struts assigned with sequential nodal numbers at every end, as shown in Fig. 8(a). For example, 1 and 2 are assigned to the 1st strut, 3 and 4 are assigned to the 2nd strut, and so on.

To determine a form of tensegrities with 9 different connections, let N be an arbitrary number from 1 to 9. To make every end being connected to 4 other ends by 4 cables, let the i-th node be connected to the i+2N-th, i+2N+1-th, i+(40-2N)-th, and i+(40-2N-1)-th nodes. If the calculated nodal number is greater than 40, no connection is added. Thus, we obtain 9 different connections for the form-finding of tensegrities which consist of 20 struts and 80 cables. Fig. 8(b) shows 8 cables connected to the 1st and 2nd nodes when N = 6.

Using the structure with obtained 9 connections, we ran many analyses by solving the following problem:

.stationary

))(()(),( 4

→

−+=Π ∑∑k

kkkj

jj LLLw xxλx λ (27)

The corresponding principle of virtual work is expressed as

0.4 3 =+= ∑∑k

kkj

jjj LLLww δλδδ (28)

In this series of analyses, the weight coefficients of half of the cables connecting the i-th node to the i+2N+1-th and i+(40-2N-1)-th nodes are set to 1.0, whereas those of the other half of the cables connecting the i-th node to the i+2N-th and i+(40-2N)-th nodes are set to w, a variable parameter.

Fig.9 shows 9 results obtained when w was set to 2.0. Interestingly, these results are just a fraction of the many possible results when w is kept constant at 2.0. Fig.9 shows the most frequently obtained results. This fact implies that the functional is multimodal.

(a) Indices of Nodes (b) 8 Cables Connected to 1st and 2nd Nodes (N=6)

Figure 8. Instruction for Configuration

Figure 9. Discovered Tensegrities

5.2 Structures Consisting of Cables, Membranes and Struts

For the form-finding of the structures that consist of


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cables, membranes, and struts, Eq. (21) is extended as follows:

,stationary))((

))(('))((),(

→−

++=Π

∑

∑∑

kkkk

jjj

jjj

LL

SL

x

xxλx

λ

ππ (29)

where the first summation is taken for every linear element; the second, for every triangular element; and the third, for every strut. A function Lj and Sj are defined, which gives the length of the j-th linear element and the area of the j-th triangular element, respectively. The forms of the cables are represented by the linear elements, and the forms of the membranes are represented by the triangular elements.

The stationary condition of Eq. (29) with respect to x is given by

.

))(('

))((

0

x

x

x

r

=∇

+∇∂

∂

+∇∂

∂=

∂Π∂

∑

∑

∑

kkk

jj

j

jj

jj

j

jj

L

SS

S

LL

L

λ

π

π

(30)

Replacing the partial differential factors by

j

jjj

j

jjj S

S

L

Ln

∂∂

=∂

∂=

)(',

)( πσ

π, (31)

the general form of the self-equilibrium state is obtained as

0x

r

=∇+∇+∇=∂Π∂

∑∑∑k

kkjj

jjj

j LSLn λσ , (32)

and the principle of virtual work for general self-equilibrium systems is expressed as

0=++= ∑∑∑k

kkj

jjj

jj LSLnw δλδσδδ . (33)

We report the form-finding analysis by using the analytical model shown in Fig. 10 and by solving the following problem:

,stationary))((

)()(),( 24

→−

++=Π

∑

∑∑

kkkk

jjj

jjj

LL

SwLw

x

xxλx

λ(34)

and the principle of virtual work is expressed as

.024 3 =++= ∑∑∑k

kkj

jjjj

jjj LSSwLLww δλδδδ (35)

The model is based on a cuboctahedron, and it consists of 24 cables, 6 membranes, and 6 struts. Each cable is represented by 8 linear elements, and each membrane is represented by 128 triangular elements. Fig. 11 shows a standard result. By varying the parameters wj and kL , various forms are obtained, as shown in Fig. 12.

Figure 10. Analytical Model Figure 11. Result

Figure 12. Variety of Forms

5.3 Structures Consisting of Cables, Membranes, Struts and Fixed Nodes

In this subsection, we report the form-finding of a suspended membrane structure based on the famous Tanzbrunnen. It is located in Cologne (Köln), Germany, and designed by F. Otto (1957). The form-finding was carried out by solving the following problem:

,stationary))((

)()(),( 24

→−

++=Π

∑

∑∑

kkkk

jjj

jjj

LL

SwLw

x

xxλx

λ (36)

where, as in the previous subsection, jL and jS

do not denote the lengths of the cables or the surface areas of the membranes, but the lengths of the linear elements and the element areas of the triangular elements, respectively.

As shown in Fig. 13, the form-finding was carried out effectively and conveniently by changing the


9

weight coefficients and the lengths of the struts. Note that the form has been improved very much via process from Fig. 13(a) to (f).

(a) (b)

(c) (d)

(e) (f)

Figure 13. Form-finding of Suspended Membrane

6. CONCLUSION

We proposed the extended force density method that enables us to carry out form-finding of general pre-stressed structures that consist of a combination of both tension and compression members. We identified the existence of a variational principle in the FDM, and we extended the FDM by generalizing the functional that is assumed to be selected by FDM. Moreover, we found that various functionals can be selected for the form-finding of tension structures. By using the newly introduced functionals, various self-equilibrium forms were obtained.

REFERENCES

[1] Schek, H. J., The force density method for form finding and computation of general networks, Comput. Methods Appl. Mech. Engrg., 3, pp. 115-134, 1974

[2] Connelly, R., and Back, A., Mathematics and tensegrity, American Scientist, 86, pp.142–151, 1998

[3] Tibert, A. G. and Pellegrino, S., Review of form-finding methods for tensegrity structures, Int. J. Space Struct., 18(4), pp. 209-223, 2003

[4] Zhang, JY., and Ohsaki, M., Adaptive force density method for form-finding problem of tensegrity structures, Int. J. Solids Struct., 43, pp. 5658-5673, 2006

[5] Vassart, N., Motro, R., Multiparametered formfinding method: application to tensegrity systems, Int. J. Space Struct., 14(2), pp. 147-154, 1999

[6] Maurin, B., Motro, R., The surface stress density method as a form-finding tool for tensile membranes, Eng. Struct., 20(8), pp.712-719, 1998

[7] Barnes, M. R., Form Finding and Analysis of Tension Structures by Dynamic Relaxation, Int. J. Space Struct., 14, pp.89-104, 1999

[8] Goto, K., Noguchi, H., Form Finding Analysis of Tensegrity Structure Based on Variational Method, Proceedings of The Forth China – Japan - Korea Joint Symposium on Optimization of Structural and Mechanical Systems, pp. 455-460, 2006.

[9] Lagrange, J. L. ( author), Boissonnade, A. C. and Vagliente, V. N. (translator), Analytical mechanics, Kluwer, 1997


10

APPENDIX (A) PHOTOGRAPHS

Handmade model based on the famous Tanzbrunnen.

By changing the weight coefficients, better form was obtained (corresponds to Fig. 13(f)).

This model corresponds to Fig. 8(d).

This model corresponds to Fig. 9(e).

This is another local minimum of the above model, which implies that the functional is multimodal.

This model corresponds to Fig.11.


11

APPENDIX (B) VISUALIZATIONS

By changing the parameters, it is possible to break the usual symmetry of the optimized forms.

Discovered tensegrities consisting of 40 struts and 160 cables.


12

Form-finding of Tanzbrunnen.

Direct integration of three popular light-weight systems: tensegrity, cable-net and tension membrane.

extended force density method for form-finding of tension structures - 2010

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