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Research Cite this article: Song K, Zhou X, Zang S, Wang H, You Z. 2017 Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns. Proc. R. Soc. A 473: 20170016. http://dx.doi.org/10.1098/rspa.2017.0016 Received: 9 January 2017 Accepted: 10 March 2017

Subject Areas: computer modelling and simulation, structural engineering Keywords: doubly curved origami structure, rigid-foldable, folded ring structure, stacked folded structure Author for correspondence: Xiang Zhou e-mail: [email protected]

Design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns Keyao Song1 , Xiang Zhou1 , Shixi Zang1 , Hai Wang1 and Zhong You2 1 School of Aeronautics and Astronautics, Shanghai Jiao Tong

University, No. 800 Dongchuan Road, Shanghai 200240, People’s Republic of China 2 Department of Engineering Science, University of Oxford, Parks Road, Oxford OX3 0PL, UK XZ, 0000-0003-1317-2912; ZY, 0000-0002-5286-7218 This paper presents a mathematical framework for the design of rigid-foldable doubly curved origami tessellations based on trapezoidal crease patterns that can simultaneously fit two target surfaces with rotational symmetry about a common axis. The geometric parameters of the crease pattern and the folding angles of the target folded state are determined through a set of combined geometric and constraint equations. An algorithm to simulate the folding motion of the designed crease pattern is provided. Furthermore, the conditions and procedures to design folded ring structures that are both developable and flat-foldable and stacked folded structures consisting of two layers that can fold independently or compatibly are discussed. The proposed framework has potential applications in designing engineering doubly curved structures such as deployable domes and folded cores for doubly curved sandwich structures on the aircraft.

1. Introduction

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9. figshare.c.3729916.

Origami, an ancient art of folding a two-dimensional paper into a three-dimensional structure, has aroused considerable research interests from scientists and engineers in recent years due to some of the unique properties exhibited by the folded structures and consequently has inspired a wide spectrum of stateof-the-art applications ranging from deployable space 2017 The Author(s) Published by the Royal Society. All rights reserved.

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(a) Folding kinematics of the trapezoidal pattern Figure 1a illustrates a single degree-4 vertex, where the valley and mountain creases are indicated by the blue and red lines, respectively, sector angles AOB and AOD are both equal to ϕ (∈ [0, π/2]), and creases OA and OC are collinear. The interim folded state of the degree-4 vertex is shown in figure 1b. Denote the dihedral angles along creases OA, OB, OC and OD by θ A , θ B , θ C and θ D (∈ [0, π ]) and the angles formed by creases OA and OC and creases OB and OD by ηa and ηb , respectively. The following relationships can be obtained: θA = θC ,

(2.1)

θB = θD ,

(2.2)

...................................................

2. Fundamental theory

2

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solar panels [1] to medical stents [2], foldable electronic devices [3–5], lightweight sandwich folded cores [6] and mechanical metamaterials [7–11]. In this context, a fundamental yet challenging issue is origami geometric design, which addresses the problem of how to create a crease pattern that folds into the desired three-dimensional shape. Deriving parametric equations to describe the folding kinematics for a certain crease pattern is a commonly used approach by many researchers [12–14]. However, this approach suffers from vast mathematical complexity and often lacks flexibility and applicability in real circumstances. Lang [15] proposed an origami design method, known as the tree method, which allows one to design hierarchical flat-folded origami silhouettes based on the concept of universal molecule. Tachi [16] developed the first practical method for the construction of true three-dimensional origami structures using edgeand vertex-tucking molecules. Later on, Tachi [17] introduced a method to produce threedimensional origami structures from generalized Resch patterns that can approximate a given polyhedral surface. Zhou et al. [18] developed a computer-based method, known as the vertex method, which can generate developable three-dimensional origami structures between two singly curved surfaces as well as simulate the unfolding process of the designed folded structure to the corresponding two-dimensional crease pattern. More recently, Dudte et al. [19] presented a numerical algorithm to generate developable origami tessellations that can yield approximations to given surfaces of constant or varying curvature. Despite these developments, designing developable origami tessellations to strictly fit between two doubly curved target surfaces is still intractable. In this paper, we present a mathematical framework for the generation of rigid-foldable three-dimensional origami tessellations based on trapezoid-mesh crease patterns that can simultaneously fit two doubly curved surfaces with rotational symmetry about a common axis. Specifically, the geometric parameters of the trapezoidal crease pattern and the folding angles of the target three-dimensional state whose outer and inner profiles are prescribed independently are determined by solving a set of mixed geometric and constraint equations. To facilitate the design process, an algorithm to simulate the interim folding sequence of the designed origami tessellation is provided. Furthermore, methods of designing developable and flat-foldable folded rings and stacked folded structures consisting of two layers that can fold independently or in a compatible manner are introduced. The potential applications of the proposed framework include but are not limited to creating deployable roofs or shelters in architecture and doubly curved sandwich structures with folded cores on the aircraft. The layout of the paper is as follows. First, the folding kinematics of the trapezoidal crease pattern is established, based on which a folding simulation algorithm and the fundamental inverse design theory are developed. Two design examples are provided. Then, the conditions and procedures to design folded ring structures and stacked folded structures are discussed. Finally, a brief summary concludes the paper.

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(a)

(b)

3

B

C

j O

A

j

qC

qB ha O

qA

hb qD

D

D

Figure 1. (a) A single degree-4 vertex and (b) The interim folded state of the degree-4 vertex. (Online version in colour.)

and

(1 + cos2 ϕ + sin2 ϕ cos θA )(1 + cos2 ϕ − sin2 ϕ cos θB ) = 4cos2 ϕ,

(2.3)

cos ηa = sin2 ϕ cos θB − cos2 ϕ

(2.4)

2

2

cos ηb = sin ϕ cos θA + cos ϕ.

(2.5)

The detailed derivations of equations (2.1)–(2.5) are provided in the electronic supplementary material, section A. Consider now a crease pattern comprising 12 trapezoidal facets shown in figure 2a, where all vertices locate on five auxiliary dashed lines χ c1 , χ f 1 , χ c2 , χ f 2 and χ c3 that intersect at point O, the angle formed by any two adjacent χ lines equals α 0 and the creases and boundary lines (i.e. the grey lines) extending between any two adjacent χ lines are all parallel to each other. For clarity, we refer herein the creases or boundary lines extending along the χ lines as the radial creases or boundary lines and the rest as the circumferential creases or boundary lines. Any two adjacent circumferential creases or boundary lines are symmetrical about the χ line passing through their common point. Denote the length of line segment C1,k C1,k+1 by ack , where k = 1,2,3, the length of line segment F1,k F1,k+1 by afk , where k = 1,2,3, the length of line segment C1,k F1,k by bk , where k = 1, . . . , 4, the acute angle formed by line segment C11 F11 and line χ c1 by ϕ c , and the acute angle formed by line segment C11 F11 and line χ fk by ϕ f . The following simple geometric relationships can be obtained: ϕ c − ϕf = α 0 , ack sin ϕf = , afk sin ϕc and

bk = b1 +

(2.6) k = 1, 2, 3

k−1 sin α0 aci , sin ϕf

(2.7) k = 2, 3, 4.

(2.8)

i=1

The interim folded state of the crease pattern is illustrated in figure 2b. Each internal vertex of the crease pattern is a degree-4 vertex shown in figure 1. According to equations (2.1)–(2.3), all of the dihedral angles along the collinear radial creases and all of those along the circumferential creases are equal. Owing to symmetry, the dihedral angles along the creases on line χ f 1 are equal to those along the creases on line χ f 2 . Hence, we denote the dihedral angles along the radial creases on lines χ f 1 and χ f 2 by θ fa , those along the radial creases on line χ c2 by θ ca and those along the circumferential creases by θ b . Applying equation (2.3) to each internal vertex, the following two equations can be obtained: (1 + cos2 ϕc + sin2 ϕc cos θca )(1 + cos2 ϕc − sin2 ϕc cos θb ) = 4cos2 ϕc

(2.9)

...................................................

C

p–j

A

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B

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cc3

(a)

cf2

F22

C32

C23 C22

jf

F11 jc

b1 C11

ac1

O r h1

2a 2a h2

C21 C11

h3

F14 F13 b4

b3

C13

C14

hfb

hcb

F11

F23

C23 F13

hca

cc1

ac3

ac2

C33 F21

C31

z

F12 b2

C12

(b)

cf1

af3 af2

af1

C21

O

cc2

C24

F21

a0 a0 a0

...................................................

F23

C33

a0

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F24

C34

C31

4

C13

F24

hfa

C12 F14 C14

Figure 2. (a) A trapezoidal crease pattern comprising 12 trapezoidal facets and (b) The interim folded state of the trapezoidal crease pattern. (Online version in colour.) and (1 + cos2 ϕf + sin2 ϕf cos θfa )(1 + cos2 ϕf − sin2 ϕf cos θb ) = 4cos2 ϕf .

(2.10)

Note that there are a total of three independent geometric parameters θ ca , θ fa and θ b governed by two kinematics equations. Therefore, the crease pattern shown in figure 2 has single degreeof-freedom (d.f.) of rigid folding motion. Besides, when θ b becomes zero, both θ ca and θ fa reach zero according to equations (2.9) and (2.10), indicating that the crease pattern is flat-foldable. The same conclusions are applicable to a general case with any number of χ lines and any number of vertices along the χ lines.

(b) Motion simulation of the trapezoidal pattern According to equation (2.4) and the relationships among the dihedral angles discussed above, the angles formed by the radial creases at all internal vertices that locate on the χ f lines (subsequently

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(2.11)

cos ηcb = sin2 ϕc cos θca + cos2 ϕc ,

(2.12)

cos ηfa = sin2 ϕf cos θb − cos2 ϕf

(2.13)

cos ηfb = sin2 ϕf cos θfa + cos2 ϕf .

(2.14)

To derive the positions of the vertices, we define a cylindrical coordinate system (figure 2b) whose z-axis is perpendicular to the plane defined by the circumferential boundary lines passing through Ci,1 and Fj,1 , where i = 1,2,3 and j = 1,2, r-axis passes through C11 and origin O has equal distances to Ci,1 , i = 1,2,3. Denote the projected lengths of line segments C11 C12 , C12 C13 and C13 C14 on the z-axis by h1 , h2 and h3 , respectively. They can be determined as η ca , k = 1, 2, 3. (2.15) hk = ack cos 2 The sector angle α, defined by the angle formed by line segments OFk,1 and OCk,1 or OCk+1,1 , k = 1,2, is given by ηcb − ηfb . (2.16) α= 2 Finally, for a general trapezoidal crease pattern containing m × n C-type vertices and (m − 1) × n) F-type vertices, the coordinates of the C-type vertices are obtained as rci,j =

bj sin (ηfb /2) sin α

c = 2(i − 1)α, θi,j

zci,1 = 0, zci,j =

and

i = 1, . . . , m, j = 1, . . . , n,

,

i = 1, . . . , m, j = 1, . . . , n,

i = 1, . . . , m

j−1

(−1)k hk ,

(2.17) (2.18) (2.19)

i = 1, . . . , m, j = 2, . . . , n,

(2.20)

i = 1, . . . , m − 1, j = 1, . . . , n,

(2.21)

k=1

and those of the F-type vertices as f

ri,j =

bj sin (ηcb /2)

f

sin α

θi,j = (2i − 1)α, and

f

zi,j = zci,j ,

,

i = 1, . . . , m − 1, j = 1, . . . , n

i = 1, . . . , m − 1, j = 1, . . . , n.

(2.22) (2.23)

The algorithm to simulate the folding motion of the trapezoidal pattern is as follows. First, select one of the three dihedral angles as the control parameter. Then, solve the other two dihedral angles through equations (2.9) and (2.10). Next, ηca , ηfa , ηcb and ηfb are determined through equations (2.11)–(2.14). Finally, the coordinates of all the vertices are obtained using equations (2.17)–(2.23), with which the current interim folded state is determined.

(c) Inverse design algorithm The folded structure of the trapezoidal pattern, as illustrated in figure 2b, has rotational symmetry about the z-axis. Hence, projecting all the vertices and creases onto the r–z plane yields a

...................................................

and

cos ηca = sin2 ϕc cos θb − cos2 ϕc ,

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referred as the F-type vertices) are equal. So are those formed by the radial creases or boundary lines at all vertices that locate on the χ c lines (subsequently referred to as the C-type vertices). The similar relationships can be found for the angles formed by the circumferential creases or boundary lines. Hence, we denote the angles formed by the radial creases at the F-type vertices by ηfa , those by the radial creases or boundary lines at the C-type vertices by ηca and the circumferential counterparts by ηfb and ηcb , respectively. According to equations (2.4) and (2.5), they can be determined as

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z

6

Fj,1

h0

outer profile

Fj,3 Ci,5

Hz Fj,2 Ci,2

Fj,4 Ci,4

Ci,6

inner profile O

Fj,6

r

Figure 3. The cross-sectional view in the r–z plane of the folded state in figure 2b, in which all the vertices and creases are projected onto the same r–z plane. (Online version in colour.)

superimposed two-dimensional pattern shown in figure 3. It is noted that the outer and inner cross-sectional profiles of the folded structure are determined by the F-type vertices with odd radial indices, i.e. Fi,j , i = 1, . . . , m − 1, j = odd and the C-type vertices with even radial indices, i.e. Ci,j , i = 1, . . . ,m, j = even, respectively. Given two non-intersecting doubly curved surfaces with rotational symmetry about the z-axis whose cross-section equations in the r–z plane are given by z = f out (r) and z = f in (r), respectively, in order for the folded structure to fit into the interspace between these surfaces, the following constraints need to be satisfied for arbitrary i: n+1 f f (2.24) z˜ i,2j−1 = fout (ri,2j−1 ), j = 1, . . . , 2 and z˜ ci,2j = fin (rci,2j ), j = 1, . . . ,

n 2

,

(2.25) f

where [ζ ] means taking the integer part of a real number ζ , and z˜ ci,j and z˜ i,j are the z-coordinates of the C-type and F-type vertices of the folded structure with an offset distance h0 along the z-axis, respectively, i.e. f

z˜ ci,j = z˜ i,j = zci,j + h0 ,

i = 1, . . . , m, j = 1, . . . , n.

(2.26)

Furthermore, given that the total height of the folded structure in the z-direction is Hz and the circumferential span is θ s (∈ [0, 2π]), the following constraints hold for arbitrary i: z˜ c − z˜ ci,n , if n = even (2.27) Hz = i,1 c c z˜ i,1 − z˜ i,n−1 , if n = odd, and c c − θ1,j = 2(m − 1)α. θs = θm,j

(2.28)

To design a folded structure, α can be determined independently from equation (2.28) where it is assumed that m and n are design inputs. Then, equations (2.24), (2.25) and (2.27) along with equations (2.9)–(2.14) and (2.16) can be solved together for parameters ϕ c , ϕ f , aci , b1 , θ ca , θ fa , θ b , ηca , ηcb , ηfa , ηfb and h0 , where i = 1, . . . , n − 1. Note that the total number of equations and unknowns are counted as n + 8 and n + 10, respectively. Hence, two out of the n + 10 parameters need to be chosen as the control parameters so that the rest can be uniquely determined by the equations. In other words, the design d.f. is two. Once these parameters are solved, the rest of the geometric parameters of the plane crease pattern can be determined using equations (2.6)–(2.8) and the resulting folded structure determined by equations (2.17)–(2.23).

...................................................

Fj,5

Ci,3

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Ci,1

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Table 1. The calculated parameters of the first example.

7

parameter ac1

value (mm) 16.143

parameter ac11

value (mm) 20.160

ϕf

0.895 rad

ac2

10.175

ac12

5.361

θ ca

1.746 rad

ac3

17.031

ac13

20.733

θ fa

1.523 rad

ac4

9.208

ac14

4.418

θb

1.076 rad

ac5

17.889

ac15

21.128

ηca

1.464 rad

ac6

8.241

ac16

3.493

ηcb

1.451 rad

ac7

18.708

ac17

21.227

ηfa

1.673 rad

ac8

7.276

ac18

2.597

ηfb

1.137 rad

ac9

19.473

ac19

20.789

b1

13.665 mm

ac10

6.315

ac20

1.747

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(d) Examples To demonstrate the inverse design process discussed above, two examples are provided below. In the first example, the outer and inner cross-sectional profiles are given by z = fout (r) = 0.0015r2 + 450

(2.29)

z = fin (r) = 0.0021r2 + 440,

(2.30)

and

respectively. The design inputs m, n, Hz and θ s are taken as 16, 21, 100 mm and 3π /2 rad, respectively. h0 and ϕ c are chosen as the control parameters whose values are taken as 445 mm and π /3 rad, respectively. The calculated parameters are listed in table 1. The resulting folded geometry and the corresponding two-dimensional crease pattern are shown in figure 4. An animation of the folding motion of this example is provided in the electronic supplementary material, movie S1. The outer and inner cross-sectional profiles of the second example are determined as

n2 z = fout (r) = n1 − n21 − 12 (r − m1 + 40)2 m1 and

n2 z = fin (r) = n1 − n22 − 22 (r − m1 + 40)2 , m2

(2.31)

(2.32)

respectively, where m1 , n1 , m2 and n2 are 580, 300, 575, 278, respectively. The design inputs m, n, Hz and θ s are the same as the first example. Again, h0 and ϕ c are taken as the control parameters and their values are taken as 140 mm and π /3 rad, respectively. The calculated parameters and resulting folded structure are shown in table 2 and figure 5, respectively. An animation of the folding motion is provided in the electronic supplementary material, movie S2. It is noted that in both examples the designed folded structures strictly fit between the given outer and inner profiles (figures 4b and 5b), indicating the validity of the proposed design theory (figure 6).

...................................................

value π /20 rad

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parameter α

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(a)

(b)

8

z

–200 –100 0 100 200

–100

0

100

200

400 380

350 50

–200

(c)

z = fin(r)

360

100

150 r

200

250

(d)

300 200 100 0 –100 –200 –300 –400 –300 –200 –100 0

100 200 300 400

Figure 4. (a) The resulting target folded structure of the first example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles; (c) the corresponding two-dimensional crease pattern and (d) a paper model of the designed folded structure. (Online version in colour.)

Table 2. The calculated parameters of the second example. parameter α

value π /20 rad

parameter ac1

ϕf

0.901 rad

ac2

θ ca

2.056 rad

θ fa

value (mm) 49.445

parameter ac11

value (mm) 17.827

5.833

ac12

7.440

ac3

27.089

ac13

17.602

1.856 rad

ac4

5.929

ac14

8.078

θb

1.385 rad

ac5

22.030

ac15

17.677

ηca

1.682 rad

ac6

6.159

ac16

8.847

ηcb

1.671 rad

ac7

19.685

ac17

18.004

ηfa

1.846 rad

ac8

6.490

ac18

9.778

ηfb

1.357 rad

ac9

18.452

ac19

18.566

b1

14.719 mm

ac10

6.915

ac20

10.910

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3. Discussion (a) Developable and flat-foldable ring design For some practical applications, it is desirable to design a folded structure that can not only fit between the given surfaces but at the same time form a closed ring by itself. In other words, it is

...................................................

z = fout(r)

420

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400 450 400 350

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(a)

(b)

200 0

200

–200

(c)

200 r

250

300

350

(d)

400 300 200 100 0 –100 –200 –300 –500

0

500

Figure 5. (a) The resulting target folded structure of the second example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles; (c) the corresponding two-dimensional crease pattern and (d) a paper model of the designed folded structure. (Online version in colour.) 15

a = amax

a (°)

10 fc = 70°, ff = 60° fc = 50°, ff = 40° fc = 30°, ff = 20°

5

qb satisfies equation (4.6). 0

20

40

60

80 100 120 140 160 180 qb (°)

Figure 6. Explanation of the developable and flat-foldable condition of the ring design. (Online version in colour.) desired that the opposite edges in the θ -direction of the folded structure meet each other, which can be achieved by making θ s in equation (2.28) equal 2π , i.e. (m − 1)α = π .

(3.1)

If it is further required that the designed folded structure satisfying equation (3.1) is both developable and flat-foldable for the ease of assembly, storage or transportation, the following

...................................................

–200

z = fout(r)

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140 120 100 z 80 60 z = fin(r) 40 50 100 150

150 100 50 0

9

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conditions must be satisfied: ϕc >

10

π π , ϕ c + ϕf > 4 2

(3.2)

sin 2ϕf (1 + cos2 ϕc ) − sin 2ϕc (1 + cos2 ϕf ) sin 2ϕf sin2 ϕc − sin 2ϕc sin2 ϕf

.

(3.3)

The detailed derivations of these conditions are provided in the electronic supplementary material, section B. A physical illustration of how these conditions work is provided in figure 6, where the solid, dashed and dotted lines correspond to three cases in which ϕ c = 70° and ϕ f = 60°, ϕ c = 50° and ϕ f = 40°, and ϕ c = 30° and ϕ f = 20°, respectively. It is shown that there exists a maximum value for α, denoted by α max , only when inequality (3.2) is strictly satisfied, i.e. the solid line. Under this circumstance, when α of the designed folded structure locates either left or right to the peak, the folded structure is either flat-foldable but not developable or the other way round due to the penetration of the opposite edges in the θ-direction during folding or unfolding which is physically prohibited. Only when α of the designed folded structure is equal to α max can it be both developable and flat-foldable. Equation (3.3) ensures that θ b of the designed structure corresponds to the peak position of α on the curve. If inequality (3.2) is not satisfied (e.g. the dashed and dotted lines), α increases monotonically with θ b , indicating that the designed folded structure is always flat-foldable but not developable. The algorithm to design a developable and flat-foldable self-closed folded ring is as follows. First, α is determined through equation (3.1). Then, equations (2.9)–(2.14), (2.16), (2.24), (2.25), (2.27) and (3.3) are solved together to obtain parameters ϕ c , ϕ f , aci , b1 , θ ca , θ fa , θ b , ηca , ηcb , ηfa , ηfb and h0 , where i = 1, . . . , n − 1. Note that the total number of equations and unknowns are n + 9 and n + 10, respectively. Hence, the design d.f. is now reduced to one. Owing to inequality (3.2), it is convenient to choose ϕ c as the control parameter which is set to be larger than π /4 rad. Once the other parameters are solved, one ought to check back if inequality (3.2) is satisfied. Alternatively, one may specify both ϕ c and h0 and solve the remaining parameters without equation (3.3) in the first step. Next, keep h0 unchanged as the sole control parameter and solve the rest of the parameters including ϕ c with equation (3.3) included. In this second step, the specified value for ϕ c and the solutions of the other parameters obtained in the first step can be used as the initial guesses for solutions of the nonlinear equation system. Usually, as long as m is large enough and ϕ c is set to be well above π /4 rad, inequality (3.2) can always be satisfied. A design example is given here in which m = n = 21, Hz = 100 mm, θ s = 2π rad, and the outer and inner cross-sectional profiles of the target surfaces are

r2 z = fout (r) = 450 1 − (3.4) 5802 and

z = fin (r) = 440 1 −

r2 4802

,

(3.5)

respectively. First, we use h0 and ϕ c as the control parameters whose values are taken as 445 mm and π /3 rad, respectively, and the rest of the parameters are solved using the inverse design algorithm discussed in §2c. Then, h0 is taken as the sole control parameter, and the remaining parameters including ϕ c are solved with equation (3.3). The obtained parameters are listed in table 3. The graphic results are shown in figure 7. An animation of the folding motion of the structure is provided in the electronic supplementary material, movie S3. Note that the designed folded structure forms a complete ring (figure 7a) and strictly fits between the inner and outer profiles (figure 7b). Moreover, the total circumferential span of the folded structure is always smaller than 2π during the entire folding range (figure 7d), indicating that the designed folded structure is both developable and flat-foldable.

...................................................

cos θb =

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and

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Table 3. The calculated parameters of the first developable and flat-foldable ring design example.

11

parameter ac1

value (mm) 15.615

parameter ac11

value (mm) 21.695

ϕf

0.997 rad

ac2

11.924

ac12

11.435

θ ca

1.997 rad

ac3

16.403

ac13

24.095

θ fa

1.768 rad

ac4

11.761

ac14

11.450

θb

1.170 rad

ac5

17.343

ac15

27.495

ηca

1.433 rad

ac6

11.629

ac16

11.521

ηcb

1.728 rad

ac7

18.482

ac17

32.838

ηfa

1.591 rad

ac8

11.530

ac18

11.675

ηfb

1.414 rad

ac9

19.893

ac19

43.161

b1

17.738 mm

ac10

11.464

ac20

11.982

.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................

Consider another example in which the prescribed outer and inner surfaces have negative and positive Gaussian curvatures, respectively, whose cross-sectional profiles are determined as

3002 (r − 540)2 (3.6) z = fout (r) = 300 − 3002 − 5802 and z = fin (r) = −0.0038(r + 80)2 + 190,

(3.7)

respectively. The design inputs m, n, Hz and θ s are the same as the previous example, and the control parameter h0 is taken as 160 mm. The obtained parameters are summarized in table 4, the graphic results are illustrated in figure 8, and an animation of the folding motion of the structure is provided in the electronic supplementary material, movie S4. Again, the designed ring structure successfully satisfies both the prescribed geometries (figure 8b) and the developable and flat-foldable requirement (figure 8d).

(b) Stacked design The design algorithm discussed above can be generalized to design stacked folded structures consisting of multiple folded layers. For simplicity, only the two-layer configuration is discussed here, as shown in figure 9. In the sequel, we refer to the bottom and top layers as layer 1 and 2, respectively. In each individual layer, equations (2.6)–(2.23) still hold except that equation (2.20) in layer 2 is modified as zci,j

=

j−1

(−1)k+1 hk ,

i = 1, . . . , m, j = 2, . . . , n,

(3.8)

k=1

due to the reversed mountain-valley assignment for the creases in layer 2. It is noted that the outer and inner profiles of the folded structure are determined by the F-type vertices with even radial indices in the top layer and the C-type vertices with even radial indices in the bottom layer, respectively. Therefore, given two non-intersecting surfaces with rotational symmetry about the z-axis whose cross-section equations in the r–z plane are given by z = f out (r) and z = f in (r), respectively, the following constraints need to be satisfied for arbitrary i: n f ,2 f ,2 z˜ i,2j = fout (ri,2j ), j = 1, . . . , (3.9) 2

...................................................

value π /20 rad

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parameter α

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(a)

(b)

200

0 0

–200

z = fout(r) z = fin(r) 100

–200

(c)

150

200 r

250

300

350

(d)

300

350

200

300 250 qs (°)

100 0

200 qb of the designed state

150 –100

100

–200

qs = 360° qs versus qb designed state

50

–300

0 –400 –300 –200 –100 0

100 200 300 400

0

20

40

60

80 100 120 140 160 180 qb (°)

(e)

Figure 7. (a) The resulting target folded structure of the first developable and flat-foldable ring design example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles; (c) the corresponding two-dimensional crease pattern; (d) the θ s versus θ b curve where the blue circle indicates the designed state and (e) a paper model of the designed folded structure. (Online version in colour.)

and c,1 z˜ c,1 i,2j = fin (ri,2j ),

j = 1, . . . ,

n 2

.

(3.10)

where the superscripts 1 and 2 denote the layer numbers, and f ,1

i = 1, . . . , m, j = 1, . . . , n

(3.11)

f ,2

i = 1, . . . , m, j = 1, . . . , n.

(3.12)

c,1 1 ˜ z˜ c,1 i,j = zi,j = zi,j + h0 ,

and c,2 2 ˜ z˜ c,2 i,j = zi,j = zi,j + h0 ,

On the interface between the two layers, the C- and F-type vertices with odd radial indices in layer 2 should coincide with those in layer 1, respectively. Under this requirement, the following

...................................................

200

440 420 z 400 380 360 340

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400 350

12

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Table 4. The calculated parameters of the second developable and flat-foldable ring design example.

13

parameter ac1

value (mm) 15.215

parameter ac11

value (mm) 11.286

ϕf

1.254 rad

ac2

6.164

ac12

5.687

θ ca

1.795 rad

ac3

12.009

ac13

13.410

θ fa

1.505 rad

ac4

5.193

ac14

6.998

θb

0.567 rad

ac5

10.521

ac15

17.420

ηca

0.732 rad

ac6

4.768

ac16

9.456

ηcb

1.728 rad

ac7

10.011

ac17

25.392

ηfa

0.844 rad

ac8

4.714

ac18

14.464

ηfb

1.414 rad

ac9

10.254

ac19

44.019

b1

5.560 mm

ac10

4.997

ac20

26.790

.......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... .......................................................................................................................................................................................................... ..........................................................................................................................................................................................................

relationships can be obtained according to figure 2b:

and

1 2 = ηcb , ηcb

(3.13)

1 ηfb

2 = ηfb

(3.14)

α1 = α2 .

(3.15)

Owing to equation (2.16), two of the three equations are independent. Equation (3.15) ensures that the position constraints in the θ-direction for both C- and F-type vertices on the interface are satisfied. For the C-type vertices, the position constraints on the r and z-directions are given by n+1 c,2 (3.16) = r , j = 1, . . . , rc,1 i,2j−1 i,2j−1 2 n+1 c,2 ˜ z˜ c,1 . (3.17) = z , j = 1, . . . , i,2j−1 i,2j−1 2 Substituting equations (2.17), (3.14) and (3.15) into equation (3.16) yields b1j = b2j ,

j = odd.

(3.18)

Considering equations (2.21), (3.13), (3.15) and (3.18) together and due to equation (2.23), the constraints in the r and z-directions for the F-type vertices on the interface are automatically satisfied. Furthermore, given the total height of the stacked folded structure in the z-direction Hz and the circumferential span θ s (∈ [0, 2π ]), the following constraints hold for arbitrary i: ⎧ ⎨z˜ c,2 − z˜ c,1 , if n = even i,2 i,n (3.19) Hz = ⎩z˜ c,2 − z˜ c,1 , if n = odd i,1 i,n−1 and θs = 2(m − 1)α 1 = 2(m − 1)α 2 .

(3.20)

To design a stacked folded structure, α 1 and α 2 can be determined directly from equation (3.20). Then, equations (2.9)–(2.14) and (2.16) for both layers along with constraint equations (3.9), (3.10) and (3.13), (3.16), (3.17) and (3.19) can be solved together for parameters ϕck , ϕfk , akci , bk1 , k , θ k , θ k , ηk , ηk , ηk , ηk and hk , where k = 1,2 and i = 1, . . . , n − 1. Note that the total numbers of θca 0 fa b ca cb fa fb equations and unknowns in the equation system are 2(n + 8) and 2(n + 10), respectively. Therefore, the design d.f. for the two-layer configuration is four.

...................................................

value π /20 rad

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parameter α

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(a)

(b)

14

160

z

100 z = fin(r)

80

100 50 0 –50 –100

–100

–50

0

50

100

60 20

(c)

40

60

80 100 120 140 r

(d) 350

350 300 250 200 150 100 50 0 –50

300 qs (°)

250 200 150 100 50 –300 –200 –100

0

100

200

300

0

qb of the designed state qs = 360° qs versus qb designed state

20 40 60 80 100 120 140 160 180 qb (°)

(e)

Figure 8. (a) The resulting target folded structure of the second developable and flat-foldable ring design example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles; (c) the corresponding two-dimensional crease pattern; (d) the θ s versus θ b curve where the blue circle indicates the designed state and (e) a paper model of the designed folded structure. (Online version in colour.)

If a stacked folded ring structure is to be designed, one needs simply to set θ s in equation (3.20) to 2π . If it is further required that each layer in the structure is both flat-foldable and developable, conditions given in inequality (3.2) and equation (3.3) should be satisfied by each layer. Adding equation (3.3) for each layer into the overall equation system of the stacked structure reduces the total design d.f. from four to two. The selection of control parameters is similar to the singlelayered ring design discussed in §3a.

...................................................

z = fout(r)

120

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140 160 140 120 100 80

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z

F2i,1

h10

C1i,1

F1i,1 C1i,3

C2i,6

C2i,5

F1i,3

Hz F1i,2 C1i,2

F2i,6

C2i,4

F2i,3

C2i,3

...................................................

C2i,1

h20

F2i,4

rspa.royalsocietypublishing.org Proc. R. Soc. A 473: 20170016

C2i,2

15

outer profile

F2i,2

C1i,5

F2i,5 F1i,5

F1i,4 C1i,4

C1i,6

inner profile

F1i,6

r

O

Figure 9. The cross-sectional profile of a two-layer configuration viewed in the r–z plane. (Online version in colour.)

For certain applications, it is desirable that both layers of the designed stacked folded structure can fold or unfold together in a compatible manner. To achieve this, the following condition needs to be satisfied: ϕc1 = ϕc2 .

(3.21)

The detailed derivation of the above equation is provided in the electronic supplementary material, section C. In this context, when a developable and flat-foldable stacked ring structure is to be designed, due to the compatible folding condition enforced by equation (3.21), the entire structure is developable and flat-foldable as long as inequality (3.2) and equation (3.3) are satisfied by any one of the constituent layers. Therefore, the total design d.f. remains two. As a rule of thumb, h10 and ϕc1 are the robust choices for the control parameters. We finalize this section with a design example for a developable, flat-foldable and compatibly foldable stacked ring structure, where m = 19, n = 21, Hz = 117 mm, θ s = 2π rad, and the outer and inner cross-sectional profiles of the target surfaces are given by

r2 (3.22) z = fout (r) = 480 1 − 6502 and

z = fin (r) = 460 1 −

r2 5502

,

(3.23)

respectively. The control parameters are h10 and ϕc1 and they are taken as 470 mm and π /3 rad, respectively. The calculated parameters are listed in tables 5 and 6. The graphic results are plotted in figure 10. Animations of the folding motion of the three-dimensional structure and the crosssectional profile change are provided in the electronic supplementary material, movies S5 and movie S6, respectively. The results suggest that the designed stacked ring structure strictly fits between the inner and outer profiles (figure 10b) and is developable and flat-foldable (figure 10e) and the two layers do not separate throughout the folding motion (electronic supplementary material, movie S6).

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Table 5. The calculated parameters of layer 1 of the stacked ring design example. value

parameter

value (mm)

parameter

value (mm)

Table 6. The calculated parameters of layer 2 of the stacked ring design example. parameter α2

value π /18 rad

parameter a2c1

value (mm) 9.870

parameter a2c11

value (mm) 12.922

h2

470 mm

a2

23.519

a2

17.790

.......................................................................................................................................................................................................... . . . . 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c12 .......................................................

ϕ2 π /3 rad a2 16.682 a2 6.678 . . . . . c. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c13 ....................................................... 2 2 2 ϕ 0.884 rad a 13.869 a 34.827 . . . . . f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c14 ....................................................... 2 2 2 θ 2.171 rad a 8.843 a 10.810 . . . . .ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c15 ....................................................... θfa2 1.963 rad a2c6 26.082 a2c16 22.041 .......................................................................................................................................................................................................... θ2 1.518 rad a2 14.839 a2 5.385 . . . . .b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c17 ....................................................... 2 2 2 η 1.783 rad a 15.488 a 48.055 . . . . .ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c18 ....................................................... 2 2 2 η 1.745 rad a 7.795 a 8.089 . . . . .cb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c19 ....................................................... 2 2 2 η 1.950 rad a 29.460 a 41.048 . . . . .fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c20 ....................................................... 2 η 1.396 rad . . . . .fb ..................................................................................................................................................................................................... 2 b 16.422 mm . . . . 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4. Summary and final remarks In this paper, a mathematical framework for the design of rigid-foldable doubly curved origami tessellations that can fit between two doubly curved target surfaces with rotational symmetry about a common axis has been established. Under the framework, an algorithm to simulate the folding motion of the designed origami structure is provided, and specific conditions for the design of doubly curved folded ring structures that are developable and flat-foldable and stacked folded structures whose constituent layers can fold independently or in a compatible manner are identified. The validity and versatility of the proposed framework were demonstrated by several design examples and paper models. This study paves the way towards various potential engineering applications. For example, the developable and flat-foldable ring design with 1-d.f. rigid folding motion are particularly useful for designing retractable domes or portable shelters requiring minimum driving systems. When manufactured with composite materials, such as carbon fibre-reinforced polymer and Kevlar prepregs, the doubly curved folded structures can help guide the development of lightweight folded cores for doubly curved sandwich structures on

...................................................

α π /18 rad a1 23.519 a1 17.790 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c11 ....................................................... 1 1 1 ϕ 0.884 rad a 9.870 a 12.922 . . . . . f. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c12 ....................................................... 1 1 1 θ 2.171 rad a 13.869 a 34.827 . . . . .ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c13 ....................................................... 1 1 1 θ 1.963 rad a 16.682 a 6.678 . . . . .fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c14 ....................................................... 1 1 1 θ 1.518 rad a 26.082 a 22.041 . . . . .b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c15 ....................................................... ηca1 1.783 rad a1c6 8.843 a1c16 10.810 .......................................................................................................................................................................................................... 1 1 1 η 1.745 rad a 15.488 a 48.055 . . . . .cb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c17 ....................................................... 1 1 1 η 1.950 rad a 14.839 a 5.385 . . . . .fa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c18 ....................................................... 1 1 1 η 1.396 rad a 29.460 a 41.048 . . . . .fb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c19 ....................................................... 1 1 1 b 16.422 mm a 7.795 a 8.089 . . . . 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c20 ....................................................... 1

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parameter

16

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(a)

(b)

17

400 200

0 –200 –400

0

z = fin(r)

350 50

–200

100 150 200 250 300 350 400 450 r

–400

(c)

(d)

500

500

400

400

300

300

200

200

100

100

0

0

–100

–100

–200

–200

–300

–300

–400

–400 –500

–500 –600

–400

–200

0

200

400

600

–600

–400

–200

0

200

400

600

(e) 350 300

qs (°)

250 200

qb of the designed state

150 100

qs = 360° qs versus qb designed state

50 0

0

20

40

60

80 100 120 140 160 180 qb (°)

Figure 10. (a) The resulting target folded structure of the stacked ring design example; (b) the cross-sectional view of the target folded structure in which the red lines indicate the given outer and inner profiles and the black and blue lines indicate layers 1 and 2, respectively; (c) the corresponding two-dimensional crease pattern of layer 1; (d) the corresponding two-dimensional crease pattern of layer 2; (e) the θ s versus θ b curve where the blue circle indicates the designed state. (Online version in colour.) the aircraft, such as fairing and fuselage. Moreover, the stacked folded structures with compatibly foldable layers may lead to new doubly curved tunable metamaterial designs with intriguing properties. The main limitation of the current work is that the designed folded structures can only fit between two doubly curved surfaces with rotational symmetry about a common axis. Chopping off the parallel condition for the creases and boundary lines extending between any two adjacent χ lines in figure 2a and introducing proper diagonal creases to the general quadrilateral facets is a potential direction to overcome this limitation, which will be considered in our future work. Data accessibility. The electronic supplementary material supporting this article are available via https://dx.doi. org/10.6084/m9.figshare.c.3729916.

...................................................

z 400 200

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z = fout(r)

450

450 400 350 400

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Authors’ contributions. K.S. derived some of the equations, wrote computer programs for the examples and made

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rspa.royalsocietypublishing.org Proc. R. Soc. A 473: 20170016

the physical models; X.Z. designed the study, derived some of the equations and wrote the paper; S.Z. derived some of the equations; H.W. coordinated the study and commented on the paper; Z.Y. conceived of the study and commented on the paper. All authors gave final approval for publication. Competing interests. We have no competing interests. Funding. X.Z. is funded by National Science Foundation of China (grant no. 51408357).