An Improved PSO-Based Approach for Automated Form-Finding of Cable-Truss Structures

1. Introduction

Cable-truss structures, known for their structural efficiency, rapid construction, and lightweight properties, have been widely adopted as roof systems for large stadiums worldwide [1,2,3,4,5,6,7,8,9]. As shown in Figure 1, two common configurations are used in engineering practice: one with two inner hoop cables and the other with a single inner hoop cable. Cable-truss structures, as an application of tensegrity systems, rely essentially on a self-equilibrated stress state between discontinuous compressive struts and continuous tensile cables to achieve structural rigidity. In fact, the initial prestress distribution and the geometry are two interdependent factors that govern the stiffness of such structures. In practical design, determining these two parameters is crucial, yet complicated by their mutual coupling—especially when structural self-weight is taken into account.

The determination of initial prestress distribution and geometry, collectively known as form-finding analysis, can be accomplished through several established approaches. Classical methods include the force density method [10,11,12], singular value decomposition (SVD) [13,14,15,16,17], the dynamic relaxation method [18,19], and the finite element method [20,21,22]. In addition to classical approaches, intelligent algorithms have been increasingly applied to the form-finding and optimization of cable-strut structures, with developments primarily evolving along three interconnected directions. The first direction involves the application and enhancement of metaheuristic algorithms. Early applications include the use of Particle Swarm Optimization (PSO) by Chen et al. to efficiently determine optimal feasible prestress modes for structures with predefined geometry [23]. Subsequent research has focused on developing enhanced variants to improve global search performance, such as a tabu search-enhanced fruit fly optimization algorithm (TS-FOA) for cable dome force-finding [24] and a hybrid Grey Wolf-Fruit Fly Optimization Algorithm (GWFOA) for stable prestress optimization [25]. The second direction advances machine learning-aided computational frameworks for higher efficiency and data-driven insights, exemplified by an integrated artificial neural network and finite element analysis framework for force identification [26], and a boosting tree with bootstrap technique (BTWBT) for precise displacement control with minimal samples [27]. The third direction focuses on interactive design and performance optimization for specific structural configurations, including methods developed for novel conical cable domes [28] and standardized geometric-force relationships for general cable domes [29], as well as multi-objective actuator placement design using combined numerical-optimization techniques [30]. Compared to traditional methods, these intelligent approaches offer significant advantages: they enhance global search capabilities to effectively avoid local optima and substantially improve computational efficiency. Furthermore, these frameworks can flexibly incorporate practical engineering conditions like external loads and simultaneously optimize multiple performance criteria, providing powerful tools for the automated and high-performance design of complex cable-strut systems. However, traditional algorithms often rely on complex matrix computations, while the intelligent algorithms built upon them typically demand the development of sophisticated programs and substantial computational power, which poses challenges and practical application barriers for engineering designers.

For relatively regular systems such as cable domes and cable-truss structures, simpler design methodologies have also been developed to efficiently determine prestress distribution and geometry, benefiting from their inherent structural simplicity [31,32,33,34,35,36]. These methods typically do not require complex matrix computations or specialized software. In particular, a form-finding method based on a partial balance strategy has been developed for cable-truss structures [34,35,36]. This approach achieves structural equilibrium by adjusting the prestress and geometry of the lower chord cables while keeping the shape of the upper chord cables unchanged. However, the adjustment of the lower chord cables in the method relies heavily on experience and trial calculations. Furthermore, its computational efficiency is often low when optimizing a large number of parameters simultaneously.

In this paper, considering the need to optimize a large number of parameters simultaneously, an improved Particle Swarm Optimization (PSO) algorithm is introduced into this strategy. The improved PSO incorporates logistic chaotic mapping during initialization to ensure a uniform distribution of search agents, and a mutation operation within the iteration loop to maintain population diversity and avoid premature convergence. This hybrid approach integrates the simplicity of the analytical strategy with the adaptive, global search capability of the intelligent algorithm, enabling the geometry of the lower chord cables to be determined rapidly, automatically, and reliably within a specified feasible range of horizontal inclination angles. Consequently, the form-finding analysis is completed with enhanced computational efficiency, robustness, and broader applicability.

This paper is structured as follows. Section 2 proposes a simplified method for determining the initial prestress and geometry of two types of cable-truss structures. Section 3 then details the form-finding process using the improved PSO algorithm. In Section 4, two illustrative examples are presented to validate the proposed method and demonstrate its application in initial prestress and geometry design. Finally, Section 5 summarizes the key findings and conclusions of the study.

2. Initial Prestress and Geometry of Cable-Truss Structure

2.1. Initial Prestress of Type I Cable-Truss

Based on geometric symmetry, the computational model for the type I cable-truss is illustrated in Figure 2. Here, H₁, and $H_1^{\text{'}}$ denote the initial prestress of the upper hoop cable and lower hoop cable, respectively. The symbols T_i, B_i, and V_i represent the initial prestress in the upper chord cables, lower chord cables, and vertical members, respectively. Furthermore, F_i and $F_i^{\text{'}}$ correspond to the equivalent nodal loads applied at the upper and lower chord nodes, which account for the self-weight of the structure. The angle α_i is defined as the inclination of the upper chord cable relative to the horizontal plane, while β_i denotes the corresponding angle for the lower chord cable.

The configuration of the upper chord cables is predetermined, with α_i serving as a known parameter. Given the initial prestress in the upper hoop cable (H₁), the initial prestress for all remaining members, as well as the geometry of the lower chord cables, can be correspondingly determined.

For the upper chord cable and vertical strut, when i=1:

$$\left.\begin{array}{l}{T}_1=2H_1\sin (\pi /n)/\cos {\alpha }_1\\ V_1=-2H_1\sin (\pi /n)\tan {\alpha }_1-F_1\end{array}\right\}$$

(1)

where n denotes the number of divisions of the cable-truss structure in the circular direction.

When i≥2, the initial prestress distribution in the upper chord cable and the vertical strut can be expressed as follows:

$$\left.\begin{array}{l}{T}_i=T_{i-1}\cos {\alpha }_{i-1}/\cos {\alpha }_i\\ V_i=T_{i-1}\sin {\alpha }_{i-1}-T_i\sin {\alpha }_i-F_i\end{array}\right\}$$

(2)

From Eq. (2), the expressions for T_i and V_i can be derived as:

$$\left.\begin{array}{l}{T}_i=2H_1\sin (\pi /n)/\cos {\alpha }_i\\ V_i=2H_1\sin (\pi /n)(\tan {\alpha }_{i-1}-\tan {\alpha }_i)-F_i\end{array}\right\}$$

(3)

Regarding the lower chord cable, when i=1:

$$\left.\begin{array}{l}{B}_1=(-V_1+F_1^{\text{'}})/\sin {\beta }_1\\ H_1^{\text{'}}=B_1\cos {\beta }_1/2/\sin (\pi /n)\end{array}\right\}$$

(4)

When i≥2, the initial prestress distribution and the geometry of the lower chord cable can be expressed as follows:

$$\left.\begin{array}{l}{B}_i=B_{i-1}\cos {\beta }_{i-1}/\cos {\beta }_i\\ {\beta }_i=\arctan ({\displaystyle \frac{B_{i-1}\sin {\beta }_{i-1}-V_i+F_i^{\text{'}}}{B_{i-1}\cos {\beta }_{i-1}}})\end{array}\right\}$$

(5)

Using Eq. (5), the expressions for B_i and β_i are obtained as:

$$\left.\begin{array}{l}{B}_i=B_1\cos {\beta }_1/\cos {\beta }_i\\ {\beta }_i=\arctan (\tan {\beta }_{i-1}+{\displaystyle \frac{-V_i+F_i^{\text{'}}}{B_1\cos {\beta }_1}})\end{array}\right\}$$

(6)

2.2. Initial Prestress of Type II Cable-Truss

Figure 3 presents the computational model for the type II cable-truss. Similar to the initial prestress design for the type I cable-truss, the determination of the initial prestress distribution for the type II cable-truss also relies on maintaining the predefined shape of the upper chord cables. Under the assumption that the initial prestress in the inner hoop cable (H₁) is specified, the initial prestress for all other structural members and the geometry of the lower chord can be subsequently derived.

For node 1:

$$\left.\begin{array}{l}{T}_1={\displaystyle \frac{2H_1\sin (\pi /n)\sin {\beta }_1+F_1\cos {\beta }_1}{\sin ({\alpha }_1+{\beta }_1)}}\\ B_1={\displaystyle \frac{2H_1\sin (\pi /n)\sin {\alpha }_1-F_1\cos {\alpha }_1}{\sin ({\alpha }_1+{\beta }_1)}}\end{array}\right\}$$

(7)

When i≥2, the prestress distributions in the upper chord cable and the inhaul cable can be expressed as follows:

$$\left.\begin{array}{l}{T}_i=T_1\cos {\alpha }_1/\cos {\alpha }_i\\ V_i=T_1\cos {\alpha }_1(\tan {\alpha }_i-\tan {\alpha }_{i-1})-F_i\end{array}\right\}$$

(8)

When i≥2, the prestress distributions and the geometry of the lower chord can be expressed as follows:

$$\left.\begin{array}{l}{B}_i=B_1\cos {\beta }_1/\cos {\beta }_i\\ {\beta }_i=\arctan (\tan {\beta }_{i-1}-{\displaystyle \frac{-V_i+F_i^{\text{'}}}{B_1\cos {\beta }_1}})\end{array}\right\}$$

(9)

2.3. Geometry of the Cable-Truss

The geometry of the cable-truss structure is determined through a strategy that adjusts the lower chord to meet the target geometry of the upper chord, while accounting for self-weight. The adjustment is governed by the parameter β₁, which determines the new geometry of the lower chord through Eqs. (6) and (9). Setting the vertical coordinate of node 4 (type I cable-truss) and node 1 (type II cable-truss) to zero, the vertical coordinates of the lower chord nodes are determined relative to these references and can be expressed as:

$$Z_{i\text{'}}=-{\displaystyle \sum \Delta l_i\tan {\beta }_i}$$

(10)

where Δl_i represents the horizontal distance between node (i+1)’ and node i’.

However, the equivalent nodal loads applied at both the upper and lower chord nodes are altered by modifications to the lower chord geometry. Consequently, the prestress and the geometry of the cable-truss structure must be recalculated iteratively. For each iteration, the vertical coordinates of the lower chord nodes are updated. Their vertical coordinates are calculated based on Eq. (10). To meet engineering accuracy requirements, the following constraints are imposed on the convergence values at the inner vertical strut and the outer ring support locations, specifically at nodes 1’ and node 4’:

$${\Delta }_z({\beta }_1)=Z_{i\text{'},n}-Z_{i\text{'}}\le 1\times {10}^{-3}m$$

(11)

where Z_i’,n and Z_i’ represent the vertical coordinate of the lower chord node i’ after the nth iteration and at its initial position, respectively.

3. Form-Finding of the Cable-Truss Using PSO Algorithm

First introduced by Eberhart and Kennedy, the particle swarm optimization (PSO) algorithm is a widely adopted evolutionary technique inspired by the social foraging behavior of bird flocks [37,38]. Through collective cooperation among individuals, PSO efficiently locates optimal solutions, exhibiting notable robustness and performance in various applications. As a result, it has become a valuable tool across scientific and engineering disciplines, as evidenced by its extensive adoption in numerous studies.

The algorithm begins by initializing a population of particles, with each particle representing a random candidate solution in the search space. A fitness function evaluates the quality of the position of every particle. During the search process, each particle moves with an adaptive velocity, dynamically adjusting its trajectory based on two key factors: its own best historical position and the best position found by the entire swarm. This iterative updating mechanism effectively balances individual experience and collective intelligence, guiding the swarm toward the optimal region over successive generations.

The velocity update is mathematically formulated to incorporate both cognitive and social influences. Specifically, the velocity of each particle is adjusted according to the following equation:

$$v_i(t+1)=w{v}_i(t)+c_1{r}_1[pbes{t}_i(t)-x_i(t)]+c_2{r}_2[gbest(t)-x_i(t)]$$

(12)

$$w=0.7-0.5t/N$$

(13)

where v_i(t) and v_i(t+1) denote the velocity vectors of particle i at iterations t and t+1, respectively; w is the inertia weight, which controls the influence of the previous velocity; t is the current iteration number; N is the maximum iteration number; c₁ and c₂ are cognitive and social parameters, which are taken as 1.5 in this paper; r₁ and r₂ are random numbers uniformly distributed between [0,1]; pbest_i and gbest represent the personal best position ever found by particle i and the global best position ever found by the entire swarm, respectively; x_i(t) represents position vector of particle i at iteration t. In the context of our specific optimization problem, the components of this vector x_i(t) are the angles β₁ to be optimized.

The form-finding analysis, which incorporates structural self-weight, is formulated as a geometry optimization problem for the lower chord cables. This optimization problem is mathematically defined as determining the optimal set of angles β₁. The fitness function is:

$$F({\beta }_1)={\Delta }_z\left({\beta }_1\right)$$

(12)

For cable-truss structures with a circular plan, the form-finding analysis, leveraging structural symmetry, can be conducted on a single radial cable-truss frame, requiring the optimization of only the β₁ variable. However, for elliptical or other asymmetric plans, the analysis must encompass all radial cable-truss frames, necessitating the simultaneous optimization of numerous parameters. When applying the traditional PSO algorithm to such multi-parameter problems, a large population size and high iteration count are typically required, and the process often converges prematurely to local optima. To overcome these limitations of the traditional PSO in multi-parameter optimization, an improved PSO algorithm is proposed.

The improvements of the PSO algorithm primarily focus on two aspects: first, during the initialization phase, the initial diversity is enhanced by using logistic chaotic mapping to generate an ergodic sequence, which ensures a more uniform distribution of particles across the search space and avoids clustering in local regions; second, within the iterative loop, the global search capability is strengthened through the introduction of a mutation operation, helping to prevent premature convergence to local optima and increasing the likelihood of locating the global optimal solution. The detailed program code is provided in the Appendix. A flowchart of the form-finding method using the improved PSO algorithm to determine the initial prestress and the geometry of the cable-truss is depicted in Figure 4.

4. Illustrative Examples

To illustrate the application and validate the efficacy of the improved PSO algorithm in determining the initial prestress and geometry of cable-truss structures, the two aforementioned structural types are employed as case studies. Their design parameters are specified as follows. The elastic modulus of the cables and the struts are 1.65×10⁸ kN/m² and 2.06×10⁸ kN/m² [39,40], respectively. Both the cables and struts have a density of 7.85×10³ kg/m³.

4.1. Type I Cable-Truss

Figure 5(a) shows a type I cable-truss structure described from Ref. [34], with 36 sections in the circular direction. It has an outer diameter of 200 m and an inner diameter of 100 m. The cable-truss structure has a cantilever span of 50 m, with its upper and lower chords divided radially into five equal segments. Its initial configuration, geometric sizes and member group are shown in Figure 5(b). The initial prestress H₁ of the upper hoop cable is set to 19,780.42 kN. Two types of cables with cross-sections of 73ϕ7 and 109ϕ7 are employed in the case study, with cross-sectional areas of 2.811 × 10⁻³ m² and 4.197 × 10⁻³ m², respectively. The vertical struts are steel tubes with a diameter of 450 mm and a wall thickness of 10 mm.

The computational parameters are set as follows: a swarm size of 100 particles, a search range for the design variable (angle β₁) from 0 to π/4, and a maximum of 50 iterations. Form-finding analyses of the cable-truss structure were conducted under two conditions: ignoring and considering self-weight. The corresponding iteration histories are presented in Figure 6. It can be observed that the traditional PSO algorithm exhibits a stable convergence process, as the fitness value converges to the optimum after only 27 and 26 iterations for the two cases, respectively.

To intuitively illustrate the variation of the fitness function with the design variable β₁, Figure 7 plots the function values evaluated using 100 particles uniformly distributed in the search range from 0 to π/4. The optimal results of the form-finding analysis for both the self-weight ignored and self-weight considered cases are also marked in Figure 7. The corresponding optimal fitness values, Δ_z(β₁), are 0.0003 and 0.0006, with the globally optimal solutions β_1,gbest found at 0.14401 and 0.14049, respectively.

The initial prestress distributions of the cable-truss structure obtained using the PSO algorithm are presented in Table 1. The table shows that when the structural self-weight is ignored, the results are in close agreement with those reported in Ref. [34]. Specifically, the prestress in the upper hoop cables, the upper chord cables, and the vertical struts are identical, while minor discrepancies exist in the lower hoop cables and the lower chord cables. When the self-weight is considered, the prestress in the upper hoop cables and the upper chord cables remain identical. However, the prestress in the lower hoop cables and the lower chord cables increase by approximately 15%, and those in the vertical struts increase by 5% to 13%. This difference arises because the form-finding analysis in the present work is based on keeping the shape of the upper chord unchanged and adjusting the geometry of the lower chord to determine the prestress distribution and final geometry of the cable-truss.

Figure 8 illustrates the final geometry of the cable-truss structure after form-finding. When the self-weight is ignored, the geometry of the lower chord cable remains largely consistent with its initial configuration. In contrast, when the self-weight is considered, while keeping the positions of the two end nodes of the lower chord cable unchanged, the intermediate nodes of the lower chord cable exhibit downward displacement relative to the initial shape.

To verify the correctness of the form-finding results, a finite element analysis was carried out using ANSYS. The prestress distribution corresponding to the final geometry of the cable-truss structure was applied, and the resulting vertical displacements are shown in Figure 9. It can be observed that the vertical displacements of the structure are negligible, indicating that the final geometry matches the prestress distribution well and, in turn, validating the accuracy of the traditional PSO algorithm.

4.2. Type II Cable-Truss

Figure 10(a) shows a type II cable-truss structure with an elliptical plane, adapted from Ref. [36]. The inner elliptical hoop measures between 130.503 m and 159.497 m, and the outer elliptical hoop ranges from 230.503 m to 259.497 m. The structure has 40 identical cable-truss frames along the elliptical layout. Each cable-truss frame features a 50 m cantilever span, and its upper and lower chords are radially subdivided into five equal segments. The initial configuration, geometric sizes and member group are shown in Figure 10(b).

Its 3D model is presented in Figure 10I, and the inner hoop cable features a hyperbolic paraboloid shape. During the form-finding process, the geometries of the inner hoop cable and the upper chord cable remain unchanged. The initial prestress in the hoop cables are denoted sequentially as H₁, H₂, …, H₄₀, where H_i represents the initial prestress in the hoop cable located between the ith and (i+1)th cable-truss frame. In total, 40 segments of inner hoop cables are defined in this manner. By decomposing the prestress in each adjacent hoop cable segment into radial and vertical components, equivalent nodal forces F_r and F_z in the radial and vertical directions at the inner hoop cable nodes can be obtained. The equivalent nodal force F_r is equivalent to H₁, and the equivalent nodal force is then combined with the force F₁ shown in Figure 3, allowing the initial prestress of each cable-truss frame to be calculated according to Eqs. (7)–(9). In this case, the initial prestress H₁ is set to 25,000 kN. The inner hoop cable adopts a cross-section of 10 × 73ϕ7. Two cable cross-sections were used: 31ϕ7 for the upper and lower chord cables, and 7ϕ7 for the inhaul cables. Their corresponding cross-sectional areas are 1.193 × 10⁻³ m² and 2.69 × 10⁻⁴ m², respectively.

As illustrated in Figure 11(a), it requires simultaneous calculation for 40 cable-truss frames, which involves optimizing 40 parameters (β_1,1 to β_40,1). To ensure the convergence of the form-finding results for all 40 cable-truss frames, an effective convergence criterion is adopted: it is only necessary to ensure that the sum of Δ_z(β_i,1) for the 40 frames satisfies the requirement of Eq. (11). Once this condition is met, the form-finding result of each individual cable-truss frame is guaranteed to fulfill the requirement. Then the fitness function is modified as follow.

$$F({\beta }_1)={\displaystyle \sum _{i=1}^{40}{\Delta }_z\left({\beta }_{i,1}\right)}$$

(13)

When the traditional PSO algorithm is employed, a larger swarm size of 5000 particles is adopted for the type II structure, with the maximum iteration number increased to 1000, while the search range remains 0 to π/4. In contrast, when the improved PSO algorithm is applied, a swarm size of 1000 particles and a maximum of 500 iterations are used. The corresponding iteration histories are shown in Figure 11(a) and (b). It can be observed that the traditional PSO algorithm exhibits a stable convergence process for the cable-truss considering self-weight, as the fitness value converges to the optimum after 398 iterations. Although the calculation results can converge, a large number of particles and iterations are required, indicating that the traditional PSO algorithm suffers from low computational efficiency. An improved PSO algorithm is adopted, which requires only 1000 particles to complete the computation within 228 iterations. These results demonstrate that for simultaneously optimizing multiple parameters, the improved PSO algorithm exhibits significantly higher computational efficiency. To separately verify the effectiveness of the two key improvements—namely, the use of logistic chaotic mapping in the initialization phase and the introduction of the mutation operation during iteration—case studies were conducted employing only one of these strategies at a time. The corresponding results are presented in Figure 11(c) and (d). When logistic chaotic mapping was not applied, the computation converged after 302 iterations. In contrast, when the mutation operation was omitted, the algorithm failed to converge. These findings indicate that, within the improved PSO framework, the incorporation of the mutation operation during the iterative phase is the most critical factor for enhancing computational efficiency.

To visually illustrate the relationship between the fitness function and the design variable β₁ for cable-truss frames 1 to 11, Figure 12 plots the function values computed using 100 particles uniformly distributed over the search range of 0 to π/4. The optimal results for the two cases, namely ignoring self-weight and considering self-weight, are also marked in Figure 12. The globally optimal solutions β_1,gbest for the two cases are obtained as follows:

Case 1 (self-weight ignored):

β_1,gbest = [0.16471, 0.16372, 0.16062, 0.15511, 0.09237, 0.14248, 0.13793, 0.13349, 0.12969, 0.12710, 0.12617]

Case 2 (self-weight considered):

β_1,gbest = [0.16685, 0.16587, 0.16281, 0.15734, 0.09085, 0.14544, 0.14068, 0.13600, 0.13194, 0.12916, 0.12816]

The prestress distributions for the two cases of ignoring and considering self-weight, calculated using the improved PSO algorithm, are listed in Table 2 and Table 3. The results are in close agreement with those reported in Ref. [36]. Compared to the case where self-weight is ignored, the prestress in the upper chord cables increases when self-weight is considered, while that in the lower chord cables decreases, and a slight reduction is observed in the inhaul cables.

The vertical displacements at the nodes of the lower chord cables, obtained for the cases ignoring and considering structural self-weight, are provided in Table 4 and Table 5. A comparison with the data in Ref. [36] demonstrates very close agreement. Figure 13 presents the vertical displacement contours of the cable-truss structure for the two cases. It can be observed that the displacements in the equilibrium state are negligible. This confirms that the initial prestress distribution matches the structural geometry, and also verifies the correctness of the form-finding results obtained using the improved PSO algorithm.

5. Conclusions

This study developed and validated an integrated form-finding methodology for cable-truss structures by combining a simplified mechanical model with an improved PSO algorithm. The main conclusions are as follows:

1) The complex form-finding problem was successfully transformed into an optimization problem based on key design parameters, specifically the inclination angles of the lower chords. The automation of this process via the improved PSO algorithm, which employs chaotic initialization and mutation strategies, effectively eliminates the dependency on empirical trial adjustments found in traditional methods, thereby enhancing design efficiency, reliability, and solution quality.

2) The improved PSO algorithm converged rapidly and stably in high-dimensional parameter optimization. Compared with the traditional PSO, it achieved equally accurate solutions using a significantly smaller population size and fewer iterations, demonstrating superior computational efficiency and robustness. This performance makes the approach particularly suitable for practical engineering applications where both accuracy and speed are essential.

3) Ablation studies confirmed the individual and synergistic effectiveness of the two core algorithmic enhancements. The logistic chaotic mapping ensured a more uniform initial particle distribution, providing a better starting point for the search. More critically, the introduced mutation operation proved indispensable for maintaining population diversity during iteration, effectively preventing premature convergence and enabling the algorithm to robustly locate the global optimum in complex search spaces.

4) Numerical examples of two typical cable-truss structures—a circular Type I and an elliptical Type II—demonstrated that the proposed method yields precise prestress distributions and geometry under two distinct conditions: ignoring and considering self-weight. The results show excellent agreement with established reference solutions, confirming the correctness and effectiveness of the method.