The Effect of Nodal Deviation on the Reliability Performance of the Optimized Free-Form Single-Layer Reticulated Shell

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This study is helpful for the application of structural shape optimization in practical engineering.

Abstract

The free-form single-layer reticulated shell structure has the characteristics of complex shape, a high degree of static indeterminacy, and difficult node positioning in the construction process, and the nodal deviations that may occur in the construction stage have a significant impact on the reliability performance of the structure. In order to evaluate the influence of the nodal deviation on the reliability performance of the structure in the process of shape optimization, this paper takes the free-form surface of the rectangular plane as the initial structure. Shape optimization is carried out with the objective function of minimizing the strain energy under the uniform vertical load, and the influence of the nodal deviation on the reliability performance of the optimized structure is performed by analyzing changes in the structural response’s probability density function (PDF). The elastic modulus, yield strength, and nodal deviation of the material were selected as the basic random variables, and the PDF of the structural response was calculated using the probability density evolution method. In the case of considering and ignoring the nodal deviation, respectively, the PDF of the maximum displacement response of the structure under the same iteration step is calculated and compared. The results indicate that compared with the initial structure, the reliability performance of the optimized structure is significantly less sensitive to node deviations.

1. Introduction

Free-form surface structures have attracted much attention due to their excellent spatial expressiveness and outstanding mechanical properties. In recent years, with the development of computer-aided design, free-form surface structures have been widely used in large-span public buildings (e.g., Figure 1, Shanghai Sun Valley, and Figure 2, Centre Pompidou-Metz). The design of free-form surfaces must ensure both beautiful shapes and reasonable mechanical properties. Therefore, many scholars have conducted extensive research on the optimization of free-form surface structures. Cui et al. [1] took structural strain energy as the optimization target, applied vertical loads to the structure, used the height adjustment method to establish the mapping relationship between the coordinates of the surface control points and the mechanical response, and applied shape optimization to the design of large-span spatial structures. Li et al. [2] applied vertical loads to the structure and used genetic algorithms to improve the free-form surface creation method and proposed the “NURBS-GM” method. Wu et al. [3] established a numerical inverse hanging method based on the inverse hanging test method, applied vertical uniformly distributed loads to the structure, and provided an improvement idea for the shortcomings of the inverse hanging test method, such as low precision and similar shape after optimization. Ding et al. [4] proposed a node coordinate sensitivity analysis method, which established an explicit relationship between the control point displacement and the strain energy change through analytical derivation, and simultaneously monitored the volume change in the structure during the optimization process. In response to the optimization needs of complex shell structures, Falco et al. [5] proposed a structural sizing and shape optimization procedure to obtain optimum designs for plates and shells under dynamic loads. Shimoda et al. [6] proposed an algorithm that can increase the buckling strength substantially, especially under shape variation in the out-of-plane direction. Kairong et al. [7] introduced the bionic algorithm into the field of shape optimization. While ensuring the smoothness of the surface, the optimization efficiency was improved by more than 40%. Yang et al. [8] developed a method for dynamic reliability-based topology optimization of continuum structures with the optimization goal of minimizing the material volume. Wu et al. [9] proposed a unit topology encoding method for the topological optimization problem of lattice shell structures with a large scale of bars, which not only reduced the optimization variables but also reduced the probability of the occurrence of the mechanism system, thereby improving optimization efficiency.

In the optimization of free-form surfaces, strain energy is usually used as the optimization target, and the structural stiffness is improved by minimizing the structural strain energy [10,11]. As a defect-sensitive structure, the bearing capacity of a single-layer reticulated shell is greatly affected by the initial geometric defects. A small geometric deviation will cause a large change in the structural bearing capacity [12]. Therefore, in addition to strain energy, the influence of node deviations on structural reliability performance during structural optimization should also be examined. This study compared the influence of node deviation on the structure before and after performance-based optimization.

2. Formulation of the Shape Optimization Problem

This paper takes the free-form surface of the rectangular plane as the initial structure and the strain energy as the optimization target to perform shape optimization under vertical uniformly distributed loads. In order to improve the shape of the structure and make it more fluid, Bézier surfaces were employed to describe the structural geometry. The influence of node deviations on the reliability performance of the optimized structure is investigated through the PDF of the structural response. The PDF of the structural response is calculated via the probability density evolution method.

2.1. Shape Optimization Under Static Loads

The research object of this paper is performance-based shape optimization, and the shape optimization with the objective of strain energy optimization is in line with the goal of this paper, so it is selected as the optimization method. In this method, the applied distributed load is equivalently converted to concentrated nodal loads, with the total load remaining constant during the optimization process. Under the uniformly distributed load, the shape optimization of the structure is performed with the objective of minimizing the structural strain energy. This process can be mathematically formulated as follows:

$$\left\{\begin{array}{c}find\mathbf{X}\\ minC\left(\mathbf{X}\right)={\displaystyle \frac{1}{2}}{\mathbf{F}}^T\mathbf{U}\\ \mathbf{X}\in \left[{\mathbf{X}}^{\mathrm{L}},{\mathbf{X}}^{\mathrm{U}}\right]\end{array}\right.$$

(1)

where X = (x₁, x₂, x₃… x_w) represents the design variables, with w denoting the number of nodes. Each component of X corresponds to nodal coordinates; for example, x₁ includes the x-, y-, and z-coordinates of Node 1. Here, C is the structural strain energy, F is the load vector, U is the nodal displacement vector, and X^L and X^U are the lower and upper bounds of the design variables, respectively.

2.2. Shape Expression

In this study, Bézier surfaces are employed to describe the structural geometry, which can adapt to diverse complex design requirements. The computational method for Bézier surfaces is relatively simple and fast, offering high computational efficiency in practical applications.

The Bézier surface $\mathrm{S}\left(u,v\right)$ can be defined by the product of the control points ${\mathbf{P}}_{i,j}=(x_{\mathrm{i}\mathrm{j}},y_{\mathrm{i}\mathrm{j}},z_{\mathrm{i}\mathrm{j}})$ in two parametric directions and the Bernstein basis function ${\mathrm{B}}_{i,n}\left(u\right)$ [13], as expressed in Equations (2) and (3) as follows:

$$\mathrm{S}\left(u,v\right)=\left(\mathbf{x}\left(\mathit{u},\mathit{v}\right),\mathbf{y}\left(\mathit{u},\mathit{v}\right),\mathbf{z}\left(\mathit{u},\mathit{v}\right)\right)$$

(2)

where x, y, and z are

$$\left\{\begin{array}{c}\mathbf{x}\left(\mathit{u},\mathit{v}\right)={\displaystyle \sum _{i=0}^n} {\displaystyle \sum _{j=0}^m} {\mathbf{x}}_{\mathrm{i}\mathrm{j}}{\mathrm{B}}_{i,n}\left(u\right){\mathrm{B}}_{j,m}\left(v\right)\\ \mathbf{y}\left(\mathit{u},\mathit{v}\right)={\displaystyle \sum _{i=0}^n} {\displaystyle \sum _{j=0}^m} {\mathbf{y}}_{\mathrm{i}\mathrm{j}}{\mathrm{B}}_{i,n}\left(u\right){\mathrm{B}}_{j,m}\left(v\right)\\ \mathbf{z}\left(\mathit{u},\mathit{v}\right)={\displaystyle \sum _{i=0}^n} {\displaystyle \sum _{j=0}^m} {\mathbf{z}}_{\mathrm{i}\mathrm{j}}{\mathrm{B}}_{i,n}\left(u\right){\mathrm{B}}_{j,m}\left(v\right)\end{array}\right.$$

(3)

where ${\mathbf{x}}_{\mathrm{i}\mathrm{j}}$, ${\mathbf{y}}_{\mathrm{i}\mathrm{j}}$, and ${\mathbf{z}}_{\mathrm{i}\mathrm{j}}$ represent the x, y, and z coordinates of the control point n in the ii-th row and j-th column, respectively. The Bernstein basis function ${\mathrm{B}}_{i,n}\left(u\right)$ is defined as follows:

$${\mathrm{B}}_{i,n}\left(u\right)={\displaystyle \frac{n!}{i!\left(n-i\right)!}}{u}^i{\left(1-u\right)}^{n-i},\mathrm{u}\in [0,1]$$

(4)

Here, n and m denote the degrees of the Bernstein basis functions in the u and v directions, respectively. In this study, both n and m are 5. The parameters u and v correspond to a specific position on the surface. Substituting them into the surface equation S will give the x, y, and z coordinates of the point. For any n and m, $\mathrm{S}\left(0,0\right)$ = ${\mathbf{P}}_{0,0}$, $\mathrm{S}\left(0,1\right)$ = ${\mathbf{P}}_{0,m}$, $\mathrm{S}\left(1,0\right)$ = ${\mathbf{P}}_{n,0}$, and $\mathrm{S}\left(1,1\right)$ = ${\mathbf{P}}_{n,m}$; that is, the surface is interpolated at the four corner points of the control grid. To determine the u and v values corresponding to a known (x, y) coordinate on the surface, substitute x and y into the surface equations $\mathrm{S}\left(u,v\right)$, forming a system of bivariate equations. Solving these equations numerically yields the (u,v) parameters.

2.3. Probability Density Evolution Method

In conservative stochastic systems, fundamental principles analogous to mass, momentum, and energy conservation exist—termed the probability conservation principle [14]. It states that for a stochastic system with time-invariant random factors, probability is preserved during the evolution of system states. Based on this principle, the generalized probability density evolution equation is derived [15] as follows:

$${\displaystyle \frac{\partial P_{Y\mathit{\theta }}\left(\mathit{y},\mathit{\theta },\tau \right)}{\partial \tau }}+\sum _{i=1}^r {\dot{Y}}_i\left(\mathit{\theta },\tau \right){\displaystyle \frac{\partial P_{Y\mathit{\theta }}\left(\mathit{y},\mathit{\theta },\tau \right)}{\partial y_i}}=0$$

(5)

In the formula, τ is the system state evolution parameter (which can be regarded as a generalized time parameter), Y represents the system state (such as the displacement of a key point), θ is a basic random variable (such as material yield strength, material Young’s modulus, etc.), which is the cause of randomness, $P_{\mathrm{Y}\theta }\left(y,\theta ,\tau \right)$ is the joint probability distribution density of $\left(y,\theta \right)$ at time τ, and r is the dimension of the state variable Y. When r = 1, only a specific physical quantity is examined (the physical quantity examined in this article is the displacement of the structure), and the above formula degenerates into the following:

$${\displaystyle \frac{\partial P_{Y\mathit{\theta }}\left(\mathit{y},\mathit{\theta },\tau \right)}{\partial \tau }}+{\dot{Y}}_i\left(\mathit{\theta },\tau \right){\displaystyle \frac{\partial P_{Y\mathit{\theta }}\left(\mathit{y},\mathit{\theta },\tau \right)}{\partial \mathrm{y}}}=0$$

(6)

For static problems, a proportional loading mechanism is introduced, where the loading parameter (e.g., load magnitude or displacement) serves as τ.

Since analytical solutions for most engineering problems are intractable, the generalized probability density evolution equation is typically solved numerically. The procedure involves [16] the following:

• The value range of the basic random parameter ${\Omega }_{\theta }$ is divided, and the qth region is set as ${\Omega }_q$. The probability of assigning each part is calculated as $P_q$. In each divided unit, it is defined as follows:

  $$P_y^{\left(q\right)}\left(y,\tau \right)={\int }_{{\Omega }_q}{p}_{\mathrm{Y}\mathit{\theta }}\left(y,\mathit{\theta },\tau \right)d\mathit{\theta }$$

  (7)
  Then, Equation (7) will become a series of new equations as follows:

  $${\displaystyle \frac{\partial P_y^{\left(q\right)}\left(\mathit{y},\tau \right)}{\partial \tau }}+{\dot{Y}}_i\left(\tau \right){\displaystyle \frac{\partial P_y^{\left(q\right)}\left(\mathit{y},\tau \right)}{\partial \mathrm{y}}}=0,q=1,2,3\dots s$$

  (8)
• Select representative points ${\mathit{\theta }}_q$ within each ${\Omega }_q$. Solve the physical equations deterministically at these points to obtain $\dot{Y}\left({\mathit{\theta }}_q,{\tau }_m\right)$, where ${\tau }_m=m·\Delta \tau$ and $\Delta \tau$ is the time step for solving physical equations.
• Substitute the physical solutions into Equation (8) with initial conditions and solve numerically. The joint PDF $p_{\mathrm{Y}\mathit{\theta }}\left(y_j,\mathit{\theta },{\tau }_k\right)$ is computed at discrete points $y_j=y_0+j·\Delta y$ ($\Delta y$ is the discrete step of y) and ${\tau }_k=k·\Delta \overline{\tau }$ ($\Delta \overline{\tau }$ is the time step for the probability density evolution equation), and ∆τ is usually smaller than $\Delta \overline{\tau }$.
• Integrate the probability density solution $p_{\mathrm{Y}\mathit{\theta }}\left(y_j,\mathit{\theta },{\tau }_k\right)$ obtained in the previous step with respect to $\mathit{\theta }$ to obtain the probability density solution $p_{\mathrm{Y}}\left(y_j,\tau \right)$ of the system response y.

An open-source code [17] is employed to solve the generalized probability density evolution equation, which implements the theoretical framework developed by Li Jie, Chen Jianbing, and colleagues at Tongji University [14,18,19,20,21].

3. Numerical Examples

The shape optimization of the free-form surface of the rectangular plane is carried out with the objective function of minimizing strain energy, and the shapes of several typical iterative steps in the optimization process are obtained. Then, the PDF of the response under vertical load is calculated for these typical iterative steps.

3.1. Shape Optimization

In the example, the span of the structure is 30 m, the height is 5 m, and the shape is shown in Figure 3. In order to make the optimized structure have better symmetry, 1/4 of the structure is taken as the optimization area, and the initial control points are triangular meshes, as shown in Figure 3b. After the initial control points are symmetrical along the hypotenuse, a 6 × 6 rectangular control point mesh is obtained. The coordinates of the initial control points are shown in

Table 1. Coordinates of the initial control point.

Point Number	x	y	z	Point Number	x	y	z
1	0.00	0.00	5.00	12	6.00	3.00	3.99
2	3.00	0.00	4.82	13	9.00	3.00	3.10
3	6.00	0.00	4.27	14	12.00	3.00	1.80
4	9.00	0.00	3.32	15	15.00	3.00	0.00
5	12.00	0.00	1.93	16	9.00	6.00	2.47
6	15.00	0.00	0.00	17	12.00	6.00	1.42
7	3.00	3.00	4.51	18	15.00	6.00	0.00
8	6.00	6.00	3.19	19	12.00	9.00	0.87
9	9.00	9.00	1.52	20	15.00	9.00	0.00
10	12.00	12.00	0.30	21	15.00	12.00	0.00
11	15.00	15.00	0.00

. After generating the Bézier surface according to the rectangular control points, the points with the same x and y coordinates as each control point are found on the surface, and their coordinates are used as the node coordinates of the structure. Currently, the node coordinates of 1/4 of the structure are obtained (the green area in Figure 3b). The remaining node coordinates are obtained by symmetry along the x-axis (the blue area in Figure 3b) and the y-axis (the red area in Figure 3b) of these points. The design variables are the control points other than the boundary constraint points shown in Figure 3b, among which the 6th, 11th, 15th, 18th, 20th, and 21st points are constraint points. When the structure is iterated, the coordinates of the control points will change, resulting in changes in the shape of the surface.

The structural members were simulated using the beam188 unit in Ansys (Pittsburgh, PA, USA), with an elastic modulus of E = 2.1 × 10¹¹ Pa, a Poisson’s ratio of v = 0.3, a density of ρ = 7850 kg/m³, and a member cross-section of a circular ring with inner and outer radii of 0.1 m and 0.11 m, respectively. The mass21 unit was used to apply concentrated mass to the structure; the support was a fixed hinge, and displacement constraints were applied in three directions. According to the provisions of the Code for Seismic Design of Buildings [22], in the earthquake dynamic time history analysis, the representative value of the gravity load on the structural roof is recommended to be the standard value of the constant load plus 50% of the standard value of the live load. The dead load is the dead weight of the material. The dead weight of the load-bearing components is considered in Ansys through density and does not need to be added separately. Here, only the dead weight of the additional components (such as roof trusses, etc.) needs to be considered, which is taken as 0.43 kN/m² (including 0.3 kN/m² of glass roof and 0.13 kN/m² of steel roof trusses); the live load is mainly the wind load and roof load, which is taken as 2.64 kN/m² (including 2 kN/m² of accessible roof and 0.64 kN/m² of wind load), which is equivalent to a roof mass of 180 kg/m² and converted to the node. After conversion, the mass of each node is 1338.84 kg.

A vertical static load of 10 kN/m² is applied to the structure, and it is equivalent to the node load. The total size remains unchanged, and the shape of the structure is optimized with vertical strain energy as the optimization target. The upper and lower limits of the optimization variables in the horizontal direction are set to ±1 m, and the height range of the structure is 0–15 m. The convergence condition is that the difference between the two objective function values is less than the default value (1 × 10⁻⁶), and the structure converges after 197 iterations. Figure 4 shows the structural shape of a typical iteration step in the optimization process.

Figure 5 and Figure 6 are, respectively, the von Mises stress cloud diagrams of the initial structure and the optimized structure at the same optimized load.

It can be seen in Figure 5 and Figure 6 that the maximum von Mises stress of the optimized structure decreases from 1.53 × 10⁸ Pa to 3.18 × 10⁷ Pa, which is 20.78% of the initial structure. In addition, the stress distribution is more even, which means that the optimized structure is more reasonably stressed.

The strain energy of the optimized structure decreases from the initial 44.9 kJ to 3.12 kJ, which is 6.94% of the initial structure. The information on the typical iterative step structure during the optimization process is listed in

Table 2. Changes in structural response during the optimization process.

Iterations	0	3	10	50	197
Relative Strain Energy 1	100.00%	37.09%	23.19%	7.66%	6.94%
Relative Volume 2	100.00%	99.56%	99.24%	119.07%	126.24%
Structural Height (m)	5	4.336	4.994	12.055	14.401

¹ Relative strain energy: the ratio of the strain energy of the structure at the current iteration to the initial structure’s strain energy. ² Relative volume: the ratio of the volume of the structure at the current iteration to the initial structure’s volume.

.

Combining Figure 4 and Table 2, it can be seen that as the optimization progresses, the structural strain energy decreases, while the height increases and the cross-section approaches the catenary shape, which is mechanically reasonable. The volume of the optimized structure increases by about 26%, which is acceptable compared to the performance improvement, indicating that the material utilization rate of the optimized structure is improved. It is worth noting that although the increase in structural height resulting from optimization reduced the strain energy of the structure, it may worsen its other indicators, such as weight.

3.2. Influence of Node Deviations on the Reliability Performance of Optimized Structures

In order to study the influence of node deviations on the structure, the optimization results in Section 3.1 are taken as the research object, and the probability density evolution theory is used to calculate the PDF of the maximum displacement response of the structure when the structure is subjected to a vertical uniformly distributed load at each typical iteration step. Two groups of basic random variables are used in calculating the PDF of the structural response, one of which only contains the material yield strength and elastic modulus (

Table 3. Parameters and sources of random variables.

Random Variable	Probability Distribution Model	Source	Mean	Coefficient of Variation
Yield Strength of Material (MPa)	Lognormal	Article [23]	210	0.081
Elastic Modulus (×105 Mpa)	Normal	JCSS [24]	2.1	0.03

), and the other group adds node deviations on the basis of Table 3 (

Table 4. Parameters and sources of random variables (considering node deviations).

Random Variable	Probability Distribution Model	Source	Mean	Coefficient of Variation
Yield Strength of Material (MPa)	Lognormal	Article [23]	210	0.081
Elastic Modulus (×105 Mpa)	Normal	JCSS [24]	2.1	0.03
Node deviations (m)	Normal	JCSS [24]	0	1/1000 L 1

¹ L is the length of the member.

). The other parameters of the two groups are the same. When the range is divided, the two groups of random variables are divided into 365 regions based on the GF discrepancy-based technique [18].

After trial calculation, the bearing capacity of the initial structure is about $30\mathrm{k}\mathrm{N}/{\mathrm{m}}^2$. The PDF of the maximum node displacement response of the structure is calculated using this load. The calculation results are shown in Figure 7.

As shown in Figure 7, during the initial iteration stages—particularly for the initial structure—the PDF of the structural response calculated with nodal deviations exhibits significant discrepancies compared to those without nodal deviations. As optimization progresses, the gap between the two curves narrows, indicating that the influence of nodal deviations on the PDF of the structural response diminishes.

In order to better compare the effect of nodal deviations on the structure, we calculated the expected value and variance of the displacement response of the initial structure and the optimized structure based on the data in Figure 7 as follows:

• Initial structure (without nodal deviations):
  Expected displacement: −2.73 × 10⁻² m (upward direction defined as positive);
  Variance: 6.68 × 10⁻³ m².
  Initial structure (with nodal deviations):
  Expected displacement: −2.84 × 10⁻² m;
  Variance: 1.42 × 10⁻³ m²;
  Relative differences: 4.05% in expectation and 78.69% in variance.
• Optimized structure (without nodal deviations):
  Expected displacement: −9.90 × 10⁻³ m;
  Variance: 1.09 × 10⁻⁷ m².
  Initial structure (with nodal deviations):
  Expected displacement: −9.89 × 10⁻³ m;
  Variance: 1.22 × 10⁻⁷ m²;
  Relative differences: 0.11% in expectation and 11.76% in variance.

The above data show that compared with the initial structure, the difference in the expected value of the response of the optimized structure in the two cases is smaller (from 4.05% to 0.11%), and the difference in variance is also reduced (from 78.69% to 11.76%). This shows that the nodal deviations have less impact on the optimized structure.

4. Conclusions

To assess the effect of nodal deviations on structural reliability performance during shape optimization, this study performs shape optimization of a free-form surface of the rectangular plane with the objective of minimizing strain energy. The influence of nodal deviations was analyzed through the PDF of structural responses. The PDF of the maximum nodal displacement responses under a vertical uniformly distributed load was calculated for representative iteration steps during the optimization process. Through comparative analysis of the PDF results derived from two sets of random variables (with and without nodal deviations), the convergence of two PDF curves and reduction in discrepancies of displacement expectations were observed. Compared with the initial structure, the PDF of the maximum nodal response of the optimized structure shows a higher degree of coincidence, including smaller displacement expectations (from 4.05% to 0.11%) and smaller variances (from 78.69% to 11.76%), demonstrating that adjusting control point coordinates during strain energy minimization inherently suppresses the influence of nodal deviations.

Following shape optimization targeting strain energy minimization, the structure not only achieved performance enhancement but also showed significantly reduced sensitivity to nodal deviations. This finding holds significant implications for the practical application of shape optimization in engineering, and other types of loads may be studied in the future.