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Abstract

Free-form reticulated shells should not only satisfy the requirement of a specific geometric configuration but also have rational mechanical properties. This article introduced two form-finding methods of free-form configuration. The first method was a reverse realization method that is based on hanging fabric with a given boundary condition; the second method was a numerical simulation method that is based on the equilibrium characteristics of a natural tension membrane combined with a geometrical characteristics analysis and a mechanical properties optimization. The article then discusses the conducted beneficial comparison. The analytical results demonstrate that both these form-finding methods are feasible for obtaining rational free-form shells, and the numerical simulation method is more convenient for obtaining the surface family.

Keywords

Reticulated shell free-form surface form-finding dynamic relaxation method reverse engineering

Introduction

A free-form architecture is characterized by a free-flowing expression that seeks to simultaneously reflect and reconcile the inevitability of a diversity of forces that influence any architectural design.¹ The expression comprises a synthesis of design modeling, structural analysis, and mathematical optimization.² Over the past several decades, physical and computational models of free-form shell generation have been developed by researchers, engineers, architects, and designers of structures, in conjunction with contractors.³ These two types of form-finding methods exist for free-form shells to generate optimal shapes.

The first method consists of physical experiments, such as soap film analogies and hanging methods, which are two main experimental approaches.⁴ A minimal surface is properly described as soap film, which forms on a closed curve.⁵ The famous Prato experiment demonstrated that soap film can present a minimal surface under the influence of its equilibrium characteristics with the limitation of the boundary conditions.⁶ Thus, the minimal surfaces realized by soap film analogies have been used to generate optimal shapes of membranes. However, they can physically only be implemented on isotropic films.⁴ The goal of hanging models is to perform the transition from a “bending structure” to a “membrane structure.”⁷ The hanging method is a form generation method that has an important role at the beginning of structural morphology research.⁸ They are used for structures meant to work in compression such as arches and shells.⁴ These techniques of generating physical models are basically inspired by Gaudi’s philosophy and developed by Isler.³ More than 1000 concrete shell structures have been designed and constructed by Heinz Isler over a 40-year period. And many shell shapes result from shape-finding experiments, in which the shapes are created automatically by natural laws.⁹ In the practical experiment, the quantity of the shape data of the physical models acquired by a traditional measure technique such as coordinate measuring machine is unable to satisfy the requirement of the free-form surface description. Coordinate measuring machines were standard tools for making three-dimensional (3D) measurements in the past. Due to the deformable nature of these physical models, the touch probes of coordinate measuring machines cannot be used always to digitize their surface.¹⁰ With the development of reverse engineering, the obtainment of more precise point clouds by 3D laser scanner has become an effective tool for surface reconstruction. Balz and Böhm¹⁰ outlined the point clouds of the morphology scanned by a 3D scanner. According to Kloft,¹¹ in the design process of the MARTa Herford museum, a series of physical models were constructed and digitized using the 3D scanner to correct the shape.

The second method involves numerical simulation to generate the optimal shape of soap films or hanging models, which consists of computer aided geometrical design (CAGD) and computational mechanics. CAGD such as B-splines can describe a free-form surface using a flexible approach. This technique has been gradually applied with possible challenges.⁶ The methods of computational mechanics comprise many different solution approaches, for example, the density method,¹² the dynamic relaxation method,¹³ and nonlinear finite element analysis. For minimizing maximum stress of a shell structure, Shimoda et al.¹⁴ applied the shape gradient function to the free-form optimization method for shells. It became possible to simulate the physical evaluation process using advances in computational models to generate and optimize free-form shells.³ However, these numerical methods inherit the limitations of the physical approaches that they simulate.⁴ A special method that combines CAGD and numerical simulation approaches was previously presented by Brew and Lewis.^15,16 They described a new form-finding approach, based on the Laplace–Young equation and cubic spline fitting to give a minimal surface. However, this method can only adjust the surface along just one direction and lie in the creation of a mesh that would conform to the boundary shape of the surface. Based on this method, Li et al.¹⁷ introduced the quasi-uniform B-splines to describe the surface to make the adjustment of the surface more flexibly. Through the case of hyperbolic paraboloid form-finding, compared with the conventional dynamic relaxation method to meet the same precision, the number of nodes of this method could be reduced. Thus, the calculation scale was smaller and the convergence speed was faster.¹⁷ Furthermore, the prediction technology of tangent vectors was adopted to improve this method and satisfy the specific geometric configuration requirements.¹⁸

In this article, the physical models of hanging fabric are derived from the reverse realization applied with the 3D scanning technique, while the computational models are derived from the improved numerical simulation involved in some studies.^15–18 To obtain a reasonable and optimal free-form surface, this article introduced these two form-finding methods that were based on hanging fabric and the findings of a beneficial comparative study.

Reverse realization

Introduction of reverse experiment

To depict a reverse realization clearly, the reverse experiment of a building cover surface from a physics model to a reticulated shell has been completed by the authors. Figure 1 depicts the process of our reverse experiments, which includes four phases, as shown in Figure 2.

Figure 1.

Reverse realization of building cover surface: (a) physics model, (b) cloud data, (c) free-form surface, and (d) reticulated shell.

Figure 2.

Flowchart of reverse experiment.

Preparation for reverse experiment of hanging fabric

A square and hollow PS plate is fabricated to support the hanging fabric, as shown in Figure 3(a). The PS plate should be adjusted by the leveling rod to maintain a horizontal position. The fabric should be cut along the fixed boundary to ensure that it freely overhangs. Two hanging fabrics with different boundary conditions, which are termed HF1 and HF2, are prepared, as shown in Figure 3(b) and (c), respectively. The four sides of HF1 are fixed with the exception of the four corners, which are free. Two sides of HF2 are fixed, whereas two midpoints of the remaining two sides are fixed.

Figure 3.

Preparation: (a) supporting PS plate, (b) HF1, and (c) HF2.

3D scanning for cloud data

The hanging fabric is scanned by a 3D scanner, as shown in Figure 4. Combined with the hardware of the FARO 3D laser scanner and 3D scanning software, the scanning system is established. While scanning, it is important to adjust the distance and angle between the scanner and the object to improve the quality of the cloud data. The appropriate distance and angle are mostly according to the specifications of the 3D scanner, especially with the laser scan arm. In these experiments, the distance between the scanner and the object is about 50 mm. Meanwhile, keep the scanner perpendicular to the object as much as possible. The scanning continues until all characteristic cloud data of the hanging fabric are collected. As shown in Figure 5, the data include 87,950 points in P-HF1 and 96,083 points in P-HF2. Compared with traditional measurement method such as coordinate measuring machines, the 3D scanning technique can describe all characteristics of the physics models, and the quantity of cloud data is sufficient for rebuilding the free-form surface.

Figure 4.

3D scanning.

Figure 5.

Cloud data: (a) P-HF1 and (b) P-HF2.

Surface reconstruction

After disposal, including the redundant data deletion and the coordinate system alignment, the improved cloud data are shown in Figure 6(a) and (c). After the CAGD modeling, the free-form surface can be obtained as shown in Figure 6(b) and (d).

Figure 6.

Surface reconstruction: (a) HF1 (after disposal), (b) S-HF1, (c) HF2 (after disposal), and (d) S-HF2.

Reticulated shells

Based on the reconstructed surface, the reticulated shells can be generated (Figure 7(a) and (c)) with the reflection meshing method.¹⁷ The models of the reticulated shells are constructed with a 3D printer, as shown in Figure 7(b) and (d).

Figure 7.

Models of reticulated shells: (a) M-HF1, (b) M-HF1 (after 3D print), (c) M-HF2, and (d) M-HF2 (after 3D print).

Numerical simulation

Surface description

The free-form surface can be described by quasi-uniform bicubic B-splines, which are conveniently applied to analyze the geometrical properties of the surface, such as boundary conditions and curvatures. Its patch is defined as

$S (u, w) = UB V_{4 \times 4} B^{T} W^{T}$ (1)

where u and w are the independent parameters, u∈[0, 1], w∈[0, 1], U = [u³u²u 1], W = [w³w²w 1], B is the coefficient matrix of the quasi-uniform B-spline basis functions, and V is the control points’ matrix. Patches arranged in sequence compose the entire surface.

Equilibrium characteristics of tension membrane

A minimal surface is more properly described as soap film, which is a type of tension membrane. From observations with the soap film, its interface is distinctly concave or convex because the surface tension causes the pressure difference ΔP that is sustained across the interface between two static fluids. In physics, the Young–Laplace equation can explain the relationship between these two physical quantities^15–18

$Δ P = 2 σ H = σ C$ (2)

where σ is the surface tension coefficient, H is the arithmetic average of the principal curvatures, and C = 2H. Treating the interface of the tension membrane as the quasi-uniform bicubic B-splines, H is the mean curvature of the surface

$H = \frac{NE - 2 MF + LG}{2 (EG - F^{2})}$ (3)

where E, F, G and L, M, N are the first fundamental quantity and second fundamental quantity of the surface respectively. They can be calculated from the basic parameters of the B-splines.

When H = 0, it is a minimal surface. When H ≠ 0, according to equation (2), ΔP ≠ 0, which indicates that the interface of the tension membrane has not attained the equilibrium state. Otherwise, the excess pressure Δp that is applied on the interface may disturb the equilibrium state. These two factors cause a normal unbalanced force T that acts on the unit area ΔS as^15–18

$T = Δ p Δ S - Δ P Δ S = (Δ p - σ C) Δ S$ (4)

Treating the interface of the tension membrane as the quasi-uniform bicubic B-splines, ΔS can be described as

$Δ S = S_{u} Δ u \times S_{w} Δ w = (S_{u} \times S_{w}) Δ u Δ w$ (5)

where S_u denotes ∂S/∂u, and S_w denotes ∂S/∂w

$\begin{array}{l} Δ S = (y_{u} z_{w} - y_{w} z_{u}, . z_{u} x_{w} - z_{w} x_{u}, x_{u} y_{w} - x_{w} y_{u}) Δ u Δ w \\ = (a_{0} Δ u Δ w, b_{0} Δ u Δ w, c_{0} Δ u Δ w) = (Δ A_{1}, Δ A_{2}, Δ A_{3}) \end{array}$ (6)

Substituting equation (6) into equation (4), the i-direction unbalanced forces T_i (i = 1, 2, 3) along the coordinate direction x, y, z assume the following form¹⁷

$T_{i} = (Δ p - σ C) Δ A_{i} (i = 1, 2, 3)$ (7)

Dynamic relaxation method

As a tension membrane, with the given boundary and control points, the initial surface will gradually change its configuration to seek the equilibrium state until the unbalanced force T disappears. This process is performed to seek the optimal surface, which is suitable for achieving a solution by the dynamic relaxation method. To achieve the static equilibrium of the structure, this iterative numerical method simulates a pseudo-dynamic process and originates from the D’alembert principle.¹³ The following is the summary of results given in literature.^13,15–18 The i-direction equation of motion at node j, which belongs to the surface, is expressed as

$T_{ji} = m_{ji} {\overset{\cdot\cdot}{d}}_{ji} + c_{ji} {\overset{\cdot}{d}}_{ji}$ (8)

where T_ji is the unbalanced force, which is obtained by equation (4), m_ji is the nodal mass, c_ji is the damping, ${\overset{\cdot\cdot}{d}}_{ji}$ is the nodal acceleration, and ${\overset{\cdot}{d}}_{ji}$ is the nodal velocity. The kinetic damping is adopted; thus, c_ji is set to 0. The nodal acceleration is obtained

${\overset{\cdot\cdot}{d}}_{ji}^{t} = \frac{T_{ji}^{t}}{m_{ji}}$ (9)

The relationship with the nodal acceleration ${\overset{\cdot\cdot}{d}}_{ji}$ , the nodal velocity ${\overset{\cdot}{d}}_{ji}$ , and the nodal displacement d_ji can be approximated in centered finite difference form as

${\overset{\cdot}{d}}_{ji}^{t + Δ t / 2} = {\overset{\cdot}{d}}_{ji}^{t - Δ t / 2} + {\overset{\cdot\cdot}{d}}_{ji}^{t} Δ t$ (10)

$d_{ji}^{t + Δ t} = d_{ji}^{t} + {\overset{\cdot}{d}}_{ji}^{t + Δ t / 2} Δ t$ (11)

where Δt is the time step. Due to the kinetic damping, at the start or restart of the process, the nodal velocities are set to 0. For the initialization, the nodal velocity is

${\overset{\cdot}{d}}_{ji}^{Δ t / 2} = \frac{Δ t}{2 m_{ji}} T_{ji}^{0}$ (12)

The total kinetic energy (KE) of the entire surface assumes the following form

$KE = \sum_{j = 1}^{η} \sum_{i = 1}^{3} \frac{1}{2} m_{ji} {\overset{\cdot}{d}}_{ji}^{2}$ (13)

where η is the number of all nodes.

According to the presentation of Barnes,¹³ the virtual mass m_j is

$m_{j} = \frac{Δ t^{2}}{2} \times K_{j}$ (14)

where the virtual stiffness K_j is the largest possible stiffness at node j. Li et al.¹⁷ deduced the virtual stiffness along the normal direction as

$K_{j} = \frac{6 (E + G) σ}{{(EG - F^{2})}^{3 / 2}} Δ S_{j}^{2}$ (15)

and the virtual stiffness K_i (i = 1, 2, 3) along the coordinate direction x, y, z, as

${\begin{matrix} K_{1} = \frac{6 (E + G) σ}{{(EG - F^{2})}^{3 / 2}} a_{0} Δ A_{1} \\ K_{2} = \frac{6 (E + G) σ}{{(EG - F^{2})}^{3 / 2}} b_{0} Δ A_{2} \\ K_{3} = \frac{6 (E + G) σ}{{(EG - F^{2})}^{3 / 2}} c_{0} Δ A_{3} \end{matrix}$ (16)

Process of surface construction and optimization

The iterative procedure of the dynamic relaxation method is shown in Figure 8 as follows:

Step 1. Input initialization: the surface tension coefficient σ, the excess pressure Δp, the time step Δt, and the max iterative number ψ.

Step 2. Apply quasi-uniform B-splines to express the initial surface.

Step 3. Compute the mean curvature H of the current surface and get the unbalanced force T_ji by equation (4).

Step 4. Calculate the virtual stiffness K_ji or K_j and nodal mass m_ji.

Step 5. Update the nodal acceleration ${\overset{\cdot\cdot}{d}}_{ji}$ and the nodal velocity ${\overset{\cdot}{d}}_{ji}$ using equations (9) and (10), respectively.

Step 6. Calculate the current total KE of the entire surface.

Step 7. If the current KE is less than the previous KE, apply small corrections to all node coordinates to ensure that their positions correspond with the peak KE. Then, set all nodal velocities to 0, and set KE to 0. Otherwise, compute all nodal displacements and update the nodal coordinates.

Step 8. Return to Step 3 and repeat the process until the convergence criteria are satisfied, when the iterative number ψ > ψ or max(T_ji) < 0.0001.

Figure 8.

Flowchart of dynamic relaxation method.

In the practical experiments, the excess pressure Δp is derived from the self-weight. The surface tension coefficient σ is uncertain. To compare with the results of the practical experiments, the tentative surface tension coefficient σ is used. Then, the precise surface tension coefficient σ will be searched by the line search technique. The procedure combined with the dynamic relaxation method is shown in Figure 9.

Figure 9.

Flowchart of searching for precise σ.

Numerical example

Comparison

The boundary condition is the same as the boundary condition for the reconstructed surface HF1. Four sides are fixed, with the exception of the four corners, which are free. The precise surface tension coefficient σ is searched considering Δp with the self-weight to keep the z coordinate of node 1 the same with S-HF1. In Figure 10(b) and (c), the continuous lines describe S-HF1, and the dashed lines describe the simulated surface that was obtained from the above numerical simulation. Two distribution states of the surface tension coefficient σ are considered. In one state, the surface tension coefficient σ is uniform. In the other state, the surface tension coefficient σ is relaxed in the corner because of the given boundary condition. Table 1 shows the comparison results. The simulated surface in Figure 10(c) is closer to S-HF1.

Figure 10.

Reconstructed surface: (a) S-HF1, (b) uniform, and (c) relaxed.

Table 1.

Comparative results.

Node no.	z (experiment)	Uniform		Relaxed
		z	Relative error (%)	z	Relative error (%)
1	13.4833	13.4928	0.07	13.4929	0.07
2	13.2853	13.2557	−0.22	13.2894	0.03
3	12.5695	12.1162	−3.61	12.3967	−1.37
4	11.6743	10.2848	−11.90	11.1242	−4.71
5	9.9911	8.5414	−14.51	9.6492	−3.42
6	8.3849	9.0146	7.51	8.7675	4.56

Surface family

The reconstructed surface HF2 is shown in Figure 11(a); the two sides are fixed, whereas two midpoints of the remaining two sides are fixed. The excess pressure Δp is 9.8 unit, and the surface tensions σ belong to a tolerance of 200 unit arithmetic progression from −1600 to −600 unit. Figure 11(c) shows the optimal configuration with six different excess pressures. The profile of shell S1 is slightly sharper than the profile of shell S6 as expected—due to the total pressures at significant deflections with the surface tension. To determine the best results, it should compare some evaluation criteria for grid quality such as the mean value of grid quality coefficient, the variance of element length, and the variance of Gaussian curvature.^17,19

Figure 11.

Optimized surface: (a) S-HF2, (b) top view of surface, and (c) surface family (P–Q cross section).

Conclusion

Two form-finding methods of free-form configuration were introduced, including the realization method and the numerical simulation method. The first method is based on hanging fabric with a given boundary condition. The 3D scanning technique is introduced to describe all characteristics of the physics models, and the quantity of cloud data is sufficient for rebuilding the free-form surface. The second method is based on the equilibrium characteristics of a natural tension membrane combined with a geometrical characteristics analysis and a mechanical properties optimization. The detailed procedure of surface construction and optimization based on dynamic relaxation method is proposed.

The analysis of free-form surface morphology indicates that a reconstructed surface and simulated surface were obtained using these two-form finding methods. The reverse realization method offers a visual method for obtaining a free-form surface and structures, and the technique of surface reconstruction is maturely applied in practical engineering. The numerical simulation method is feasible and time-saving. It is also convenient for obtaining a free-form surface family with the given boundary conditions by adjusting the calculating parameters, which is effective for shortening the design period of free-form reticulated shells.

The analytical results demonstrate that both these form-finding methods are feasible for obtaining rational free-form shells, and the numerical simulation method is more convenient for obtaining the surfaces family.

Footnotes

Handling Editor: David R Salgado

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported by the National Natural Science Foundation of China (nos 51208263 and 51778129);Natural Science Fund Project in Jiangsu Province,China (no. BK20161422);the Foundation of Zhejiang Provincial Key Laboratory of Space Structures,China (no. 201701);and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 12KJB560005).

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