Geometry optimization for geodesic dome with symmetric grid pattern

Abstract

The geometries, loading conditions, joint conditions, and geometrical imperfections of lattice domes affect their mechanical performance. Various optimization methods for lattice domes have been proposed. Some methods have been applied in actual buildings; however, they often create a new external dome shape, which may lose the aesthetic inner space intended by designers. This study conducted geometry optimization for geodesic domes using a precise mathematical expression. The symmetry of the grid pattern and the external shape of the initial dome were maintained during optimization. The optimization considered the member lengths on the meridian as design variables to reduce the number of iterations for the optimal solution. The proposed optimization scheme was applied to 64 geodesic domes using six objective functions. The optimization results were compared with those of the authors’ previous study on optimal domes, which considered the nodal coordinates on a spherical surface as the design variables. The previous and present optimal domes exhibited significant mechanical and geometric similarities when the objective function was considered as the standard deviation of the member length and minimum buckling safety factor. In addition, the computational complexity of the present optimization scheme for reaching the optimal solution was approximately 10–100 times lower than that of the previous optimization scheme. The proposed scheme can be used to optimize lattice domes with less computational complexity while ensuring their mechanical effectivity and retaining their intended inner space and external shape.

Keywords

Single-layer lattice dome geodesic dome geometry optimization symmetric grid pattern low computational complexity

Introduction

Lattice structures are widely used in many industrial fields (civil engineering, bioengineering, aerospace, and automobiles) owing to their outstanding properties, including light weight, efficient use of materials, high strength, and high energy absorption.^1
–4 Lattice structures can be classified according to their external shape, grid patterns, joint type, and number of layers.⁵ Lattice structures can create large spaces without intermediate support and can be utilized as pavilions, sports, exhibitions, and shelter facilities in civil engineering. Many lattice structures have simple forms, such as domes and cylinders, whereas free-form lattice structures with complex overall shapes have become increasingly popular using form-finding methods.^6
–8

The seismic performances of lattice domes have been investigated experimentally and numerically.^9
–15 Takeuchi et al.¹⁰ numerically investigated the seismic response properties of lattice domes with various span-to-depth ratios and substructural stiffness values. Yan et al.¹⁵ demonstrated a non-linear finite element analysis of single-layer lattice domes with various grid patterns to reveal the instability mechanism. Zhou et al.¹¹ proposed a constitutive model for a fiber beam element to reveal the progressive collapse mechanism of single-layer lattice domes. They conducted a parametric study with span, span-to-depth ratio, and live load parameters. Fan et al.¹⁴ investigated the reduction rate of the buckling load caused by geometrical imperfections for two types of lattice domes. López et al.⁹ considered the effect of joint rigidity on the critical load of a diamatic dome and modeled the ORTZ joint. These results demonstrated that the structural geometry (span, depth, and grid patterns), loading conditions, joint conditions, and geometrical imperfections are the key factors affecting the seismic performance of lattice domes.

Many optimization methods have been proposed for lattice domes.^16
–24 Kaveh et al.^18,20
–22 conducted extensive research on lattice dome optimization. They proposed many novel optimization algorithms based on the big bang–big crunch, colliding body, and firefly algorithms, among others. In addition, they applied the algorithms to the sizing and topology optimization of lattice domes with various grid patterns and validated them by comparing with conventional algorithms. Lattice dome optimization can be categorized into geometry, topology, and sizing optimizations.^19,23 Geometry optimization considers nodal coordinates as design variables. Topology optimization optimizes the number of members and connectivity between them. Sizing optimization optimizes the cross-sectional properties of members. Optimal domes are often obtained by changing the initial external shape, which may differ from the aesthetic space intended by designers. The sizing optimization of members can automatically maintain the initial shape of the lattice domes, whereas geometry and topology optimizations require appropriate geometrical constraints. To meet this demand, the authors geometrically optimized geodesic domes in which the nodal coordinates, which could only move on a spherical surface, were considered as design variables.²⁵ However, the computational complexity of the optimization is high owing to the design variables.

This study proposes a geometric optimization method for geodesic domes with low computational complexity using the precise mathematical expression of the geodesic dome proposed by Saka.¹⁶ The proposed optimization scheme was applied to 64 geodesic domes. Six objective functions were employed in the optimization, and their interrelationships were investigated. The mechanical and geometrical characteristics and convergence toward optimization in the present optimal domes were compared with those of a previous study.²⁵

Mathematical expression of geodesic domes

This section provides an overview of the precise mathematical expressions for geodesic domes proposed by Saka.¹⁶ Figure 1 shows a geodesic dome with m = 3 and n = 4, where m and n are the number of meridian members and sectors, respectively. Note that this study refers to a grid pattern with at least one symmetric axis on a plan view as a symmetric grid pattern, as shown in Figure 1. Figure 2 shows the xy and xz planes of the geodesic dome, where S, H, R, and θ are the span, apex height, dome radius, and half-open angle, respectively. The radius of the k-th ring is r_k. Sequential node numbering started from the apex of the dome and increased toward the periphery. The number of nodes on each ring was counted counterclockwise from the x-axis. Sequential ring numbering was performed from the apex to the periphery of the dome. The number of nodes on the k-th ring can be calculated as

p_{k} = n k .

(1)

Figure 1.

Node numbering rule of geodesic dome.

Figure 2.

Vertical and horizontal cross-sections of geodesic dome.

The first node number on the k-th ring can be calculated as

f_{k} = 2 + \frac{p_{k} (k - 1)}{2} .

(2)

The sequential node numbering for the i-th node on the k-th ring can be obtained as follows:

s (i, k) = f_{k} + (i - 1) .

(3)

The angle of the i-th node on the k-th ring from the x-axis is defined as follows:

α_{s (i, k)} = \frac{2 π}{p_{k}} (i - f_{k}) .

(4)

Using the above equation derives the x and z coordinates of the i-th node on the k-th ring as

x_{s (i, k)} = r_{k} \cos α_{s (i, k)},

(5)

z_{s (i, k)} = - r_{k} \sin α_{s (i, k)} .

(6)

From the geometric relationship, the coordinate value of y on the k-th ring can be calculated as follows:

y_{k} = \sqrt{R^{2} - {r_{k}}^{2}} - (R - H) .

(7)

Numerical schemes

Objective functions

Six objective functions were used to investigate the interrelationships among the optimized domes and their mechanical properties. Details of the objective functions have been described in a previous study.²⁵ The standard deviation of the member length L is defined as

δ = \sqrt{\frac{1}{N_{m}} \sum_{i = 1}^{N_{m}} {(L_{ave} - L_{i})}^{2}},

(8)

where $N_{m}$ , $L_{i}$ , and $L_{ave}$ are the number of members, length of the i-th member, and average member length, respectively. The total volume of the members is defined as

V = A \sum_{i = 1}^{N_{m}} L_{i},

(9)

where A denotes the cross-sectional area of the member. The strain energy was calculated using the nodal displacement vector $D$ and global stiffness matrix $K_{E}$ .

U = \frac{1}{2} D^{T} K_{E} D

(10)

The buckling safety factor $μ_{i}$ is defined as the ratio of the member buckling load $P_{i}^{c r}$ to the axial force of the i-th member $P_{i}$ . Its minimum value is defined as

μ_{i} = \frac{P_{i}^{c r}}{P_{i}} = \frac{π^{2} E I}{P_{i} {L_{i}}^{2}},

(11)

μ_{\min} = \min {μ_{1}, μ_{2}, . . ., μ_{N_{m}}} .

(12)

The linear buckling load factor $λ_{i}$ is calculated by solving an eigenvalue problem. The minimum value of $λ_{i}$ (i.e., first buckling mode) is defined as

λ_{\min} = \min {λ_{1}, λ_{2}, . . ., λ_{N_{f}}},

(13)

where N_f denotes the number of degrees of freedom. $σ_{i}$ is the maximum normal stress on the cross-section of the i-th member. The maximum value of $σ_{i}$ is defined as

σ_{\max} = \max {σ_{1}, σ_{2}, . . ., σ_{N_{m}}} .

(14)

Geometrical and material properties

Sixty-four lattice domes with different geometries were created, where m and n ranged from 4 to 7, the half-open angle θ was 20°, 30°, 45°, or 60°, and S was fixed at 45.0 m. The cross-section of the members were circular tubes with an external radius of 69.9 mm and a thickness of 4.5 mm. The members were composed of SN400 (structural steel manufactured in Japan), where the Young’s modulus, Poisson’s ratio, and specific gravity were 205000 MPa, 0.3, and 77 kN/m³, respectively.²⁶ The initial shape of the geodesic dome was created using a Formian script, which is based on the Formex algebra.^27,28 The external shape and nodal position of the geodesic domes were uniquely determined on the Formian script using m, n, θ, and S. Rigid connections were used for the member joints. A pinned support was imposed on the periphery of the domes. The self-weight of the members was considered as the vertical nodal force.

Optimization problems

For the grid pattern optimization of lattice domes in a previous study,²⁵ nodal coordinates were used as design variables under a geometrical constraint, where the nodes only moved on a spherical surface to maintain the external shape of the initial domes. This optimization scheme has a high computational complexity because many iterative calculations are required to reach the optimal solution. This study considered the distance between the ring radius r_i as a design variable.

d_{i} = r_{i} - r_{i - 1}

(15)

From the geometrical relationship, $d_{m}$ , which is the $d_{i}$ at the end, can be calculated as

d_{m} = \frac{S}{2} - \sum_{i = 1}^{m - 1} d_{i} .

(16)

Therefore, the number of design variables is m-1, significantly less than that of the authors’ previous optimization problem. Design variable d_i can maintain the external shape of the initial domes after optimization; however, at least one axis of symmetry for the grid patterns is also preserved. Thus, the design variable d_i limits the variation in the grid pattern of the optimal domes. The following section validates this configuration.

The six optimization problems were defined with the objective function under a constraint.

Case 1: Minimize $δ$ ,

Case 2: Minimize $V$ ,

Case 3: Minimize $U$ ,

Case 4: Minimize $1 / μ_{\min}$ ,

Case 5: Minimize $1 / λ_{\min}$ ,

Case 6: Minimize $σ_{\max}$ ,

under the constraint

d_{i 0} - \frac{L_{a v e}}{2} \leq d_{i 0} \leq d_{i 0} + \frac{L_{a v e}}{2},

where d_i0 is the initial value of d_i. The constraint was imposed to prevent the rings from overlapping and intersecting. Sequential quadratic programming in MATLAB was used as the optimization algorithm.

Optimization results

The present optimal domes were compared with those in a previous study.²⁵ The grid patterns of both optimal domes with m = 4, n = 4, and θ = 20° were compared to investigate the geometric characteristics of the optimal grid patterns for the previous and present optimizations, as shown in Figure 3. The optimal grid patterns in Cases 1 and 4 were similar, whereas those in the other cases differed. In particular, asymmetric patterns emerged in Cases 5 and 6 of the previous optimization. The convergence rates to the previous and present optimal solutions were compared to investigate the computational complexity. The ratio of the optimized to the initial values of the objective functions is defined as

p = \frac{g_{o p t}}{g_{I}},

(17)

where $g_{o p t}$ and $g_{I}$ are the values of the optimized and initial objective functions, respectively. Figure 4 shows the relationship between p and the number of iterations for a dome with m = 4, n = 4, and θ = 20°. Table 1 lists the number of iterations required for convergence. These results indicate that the previous optimization required approximately 19–92 times more iterations than the present optimization for convergence. The present optimal solution was superior to the previous solution, particularly in Case 6.

Figure 3.

Plan views of optimized domes using the previous and present methods. The red dotted and blue solid lines indicate members of the initial and optimized domes, respectively.

Figure 4.

Convergence curves of the objective functions of the dome with m = 4, n = 4, and θ = 20°.

Table 1.

Number of iterations to convergency.

Case number	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6
Present	40	50	116	431	584	203
Previous	2736	2646	2578	11,072	10,989	18,663
Previous/present	68	53	22	26	19	92

The relative differences in the functions before and after optimization are defined as ( )^r. Table 2 lists the average values of ( )^r for the 64 domes, where all functions were evaluated to investigate their interrelationships. The gray area indicates the objective function value for each case. The objective function values were reduced in all cases of the present and previous optimizations; however, the reduction rate of the previous optimization was bigger than that of present optimization owing to the stronger geometrical constraint of the present optimization. Functions other than the objective functions in Cases 2, 3, 5, and 6 deteriorated significantly, whereas the functions in Cases 1 and 4 deteriorated slightly. This trend is common in the previous and present optimizations.

Table 2.

Average of relative values of the objective functions.

Note: The gray area indicates the objective function value for each case.”in the above part.

Figure 5 compares the objective functions for the previous and present optimal domes. Table 3 lists the coefficients of determination R² calculated from Figure 5. The R² values of V and U were nearly 1.0 in all cases. This indicates that V and U are insensitive to the choice of objective functions and design variables. The R² values of δ, 1/µ_min, 1/λ_min, and σ_max fluctuated in Cases 2, 3, 5, and 6, whereas these values were nearly 1.0 in Cases 1 and 4. Therefore, the previous and present optimal domes in Cases 1 and 4 were geometrically and mechanically similar. This result is consistent with the geometrical comparison of the grid patterns shown in Figure 3.

Figure 5.

(a) δ, (b) V, (c) U, (d) 1/µ_min, (e) 1/λ_min, and (f) σ_max values on the previous and present models after optimization.

Table 3.

R² value of regression analysis.

Functions	Geometric		Mechanical
Functions	δ (%)	V (%)	U	1/µ_min	1/λ_min	σ _max
Case 1	0.9852	0.9999	0.9977	0.9949	0.9979	0.9781
Case 2	0.8311	0.9994	0.9985	0.8772	0.9311	0.8008
Case 3	0.3928	0.9964	0.9985	0.6857	0.8833	0.5361
Case 4	0.9748	0.9999	0.9991	0.9952	0.9972	0.9834
Case 5	0.3272	0.9961	0.9899	0.6639	0.9963	0.6468
Case 6	0.3743	0.9985	0.9952	0.5899	0.9084	0.9079

The comparisons between the previous and present optimal domes are summarized as follows:

(1) In Cases 1 and 4, the grid patterns of both domes were geometrically similar, and the previous grid patterns maintained symmetry. However, the grid patterns of both domes differed in Cases 2, 3, 5, and 6 because some previous grid patterns were asymmetric.

(2) The computational complexity of reaching the optimal solution using the previous optimization scheme was 10–100 times greater than that of the present optimization scheme.

(3) For the present and previous optimal domes, the function values other than the objective functions deteriorated slightly in Cases 1 and 4 but deteriorated significantly in Cases 2, 3, 5, and 6.

(4) The coefficient of determination R² of the objective functions for the previous and present optimal domes indicates that both models exhibited similar mechanical and geometrical properties in Cases 1 and 4.

These results indicate that the mechanical performance of the present optimal domes is comparable to that of the previous domes in Cases 1 and 4. However, the present domes are less computationally complex than the previous ones.

Conclusions

This study presented a geometric optimization technique for geodesic domes with low computational complexity using a precise mathematical expression proposed by Saka,¹⁶ in which the grid pattern symmetry was maintained during optimization. Sixty-four geodesic domes were optimized using six objective functions. The results were compared with those of a previous study on optimal domes.²⁵ The previous and present optimal domes exhibited significant mechanical and geometric similarities when the objective function was considered as the standard deviation of the member length and the minimum buckling safety factor. In addition, the computational complexity of reaching the optimal solution using the previous optimization scheme was approximately 10–100 times higher than that of the present optimization scheme. The proposed scheme can be used to optimize lattice domes with less computational complexity while ensuring sufficient mechanical performance and retaining their intended functionality and aesthetics.

Footnotes

Acknowledgements

I express my gratitude to Mr. Shuto Mori, a former graduate student at Niigata University, for assisting me in arranging the numerical results.

Declaration of conflicting interests

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

Funding

The author received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Masaki Teranishi

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