Research on Tunnel Segment Layout Technology Based on Intelligent Optimization Methods

Zhuoyu CUI¹,², Wenqi DING¹,²,*, Yude SHEN¹,², Jixiang TANG¹,², Chang MA¹,²

¹Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China
²Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China

*Corresponding author: dingwq@tongji.edu.cn

Abstract

Segment assembly is a critical process in shield tunneling, where the accuracy and efficiency of segment layout design significantly impact construction quality. Traditional methods relying on manual experience and mathematical formulas suffer from low precision and inefficiency. To address these limitations, this study proposes an optimized segment layout model based on assembly posture, incorporating modern optimization techniques. Taking a metro shield tunneling project as a case study, we investigate the performance and stability of the Simulated Annealing (SA) method and the Greedy-Simulated Annealing (GSA) hybrid method under different hyperparameters using a grid search approach. The results demonstrate that: (1) The SA method achieves effective optimization under appropriate conditions, with the optimal solution improving the objective function value by 23% compared to traditional methods. (2) The SA method exhibits sensitivity to initial solutions, which can be categorized into three types based on convergence behavior: fluctuating-decreasing, fluctuating-stable, and stable. (3) In terms of layout optimization performance: SA > GSA > Greedy Algorithm (traditional method). However, the GSA method demonstrates superior stability over SA in terms of solution distribution. The findings provide a reference for segment layout design, enabling the selection of appropriate algorithms under varying conditions. This research contributes to enhancing the precision and efficiency of shield tunneling construction.

Keywords: Shield Tunnel; Segment Lining Layout; Mathematical Model; Simulated Annealing; Hyperparameter Search

Introduction

Segment-by-segment assembly in tunnel construction demands high precision control while involving substantial workload. Discussions on tunnel segment layout methods have long attracted attention, and researchers have achieved fruitful results in recent years. Segment layout technology refers to the process of determining the assembly posture of each ring of segments during tunnel design, based on the tunnel design axis and proposed segment types within certain tolerance limits, using mathematical theory or design experience [1,2]. Current layout techniques primarily focus on algorithmic improvements and real-time correction technologies.

Li Weiping [3] established a rotating cone model by exploring the core algorithm for universal tapered segment layout in shield tunnels, completely determining the basic steps and fixed-point algorithms for horizontal and vertical curves from a pure mathematical theory perspective, yielding corresponding mathematical formulas with strong theoretical practicality. Ruan Chengzhi [4] simplified the problem by ignoring vertical curve factors according to specification requirements, categorizing layout issues for circular curve sections and transition curve sections, and proposing corresponding calculation formulas to facilitate practical layout operations. Wang Zhengbin et al. [5] abstracted segment layout as three-dimensional spatial computation through mathematical modeling. Wu Haibin et al. [6] proposed complete mathematical derivation formulas through complex mathematical derivations, matrixizing and parameterizing shield tunnel segment layout. Due to the numerous connection combinations of segments, the computational difficulty increases exponentially in mathematical terms. Many scholars have attempted to adopt various simplified models in engineering practice to reduce computational load [7,8]: using a small number of fixed combination forms to meet shield tunnel segment assembly requirements within certain accuracy limits. Currently, different scholars have established segment layout algorithms through various curve fitting methods [9-11].

Traditional layout algorithms are limited by computational capacity and time, making them suitable only for local layout of small numbers of segment rings. Building upon existing research, this study combines segment layout technology with modern intelligent optimization algorithms to improve layout accuracy while reducing time complexity, making it applicable to global layout of multiple segment rings. Furthermore, we conduct parameter search analysis based on algorithm characteristics and compare the layout effects of different algorithms.

1. Segment Layout Model Based on Assembly Posture

When using universal tapered segments to fit the tunnel axis, the conventional approach employs piecewise polylines to approximate smooth curved alignments. Tunnel design axes are typically three-dimensional spatial curves, with straight sections being easy to design, while planar curve sections and vertical curve sections rely on the taper of universal tapered segment rings for fitting. Based on this concept, segment layout technology can be abstracted as shown in [FIGURE:1]: starting from an initial plane, the assembly condition of the next segment ring is determined based on the current reference plane αi (initially the initial plane, subsequently the forward face αi+1 of the previous ring), the coordinates of reference plane center Vi (xi, yi, zi), the direction vector ni = (nxi, nyi, nzi) corresponding to the reference plane center, and the tunnel design axis.

Key parameters include:
- Vi: Center point of the current segment ring's forward face
- ni: Normal vector of the current segment ring's forward face, pointing toward the tunnel excavation direction
- φi: Assembly posture of the current segment ring, expressed as a sequence number ∈ [0,1,2...,m-1], where m is the number of axial connection types

In actual engineering, universal tapered segment rings feature multiple connection methods at ring joints (denoted as m). The assembly posture multiplied by m/2π represents the angle between the reference line corresponding to the key segment of the previous ring's forward face and the reference line 0 corresponding to the key segment of the current ring's reference plane, with counterclockwise defined as positive. As shown in FIGURE:2 and (b), the outer ring represents the current segment ring's reference plane, while the inner ring represents the previous ring's forward face. The two planes are in close contact but exhibit a corresponding angle due to different assembly positions, which is the angle between lines li and li-1.

Based on the current ring's assembly information (Vi, ni, ni+1, φi) and the tunnel design axis, the next ring's assembly information Si+1 = (Vi+1, ni+1, ni+2, φi+1) can be determined through fundamental mathematical calculations and optimization algorithms. In practical applications, Vi+1 and ni+1 can be directly obtained from current segment information; φi+1 can be selected as the optimal assembly posture through enumeration of all possible postures; and ni+2 can be determined through geometric relationships after φi+1 is fixed. This enables ring-by-ring calculation to determine the assembly information for all segment rings.

2. Segment Layout Optimization Algorithms

2.1 Greedy Algorithm

The greedy algorithm, also known as the greedy strategy, is a common approach for solving optimization problems. Its principle is to select the decision that optimizes the target variable at each step. The entire decision-making process only considers whether the current decision is locally optimal, without regard for global optimality or the impact on subsequent decisions.

Current shield tunnel segment layout methods employ the greedy algorithm concept: determining the layout position and method of the next ring (or next two rings) through traversal based on the current ring's position. The greedy algorithm is suitable for scenarios requiring rapid results but suffers from insufficient accuracy. When exploring multi-ring or full-line layout methods, local optimality becomes inadequate in terms of precision. However, global traversal would lead to exponential computational disaster, necessitating the introduction of intelligent algorithms to improve computational efficiency.

2.2 Simulated Annealing Method

Simulated annealing is a stochastic optimization algorithm based on the physical process of metal solid annealing. Its essence is that solid particles have high internal energy, strong activity, and greater instability at high temperatures. As temperature decreases, internal particle energy reduces, activity weakens, and stability increases. The algorithm conducts a certain number of Metropolis trials at each temperature, accepting new states with lower energy, and otherwise accepting states with increased energy with a certain probability. This probability is typically related to the energy difference between new and current states—larger energy differences yield higher acceptance probabilities. This optimization approach avoids falling into local minima and attempts to find global optimal solutions.

2.3 Improved Mathematical Model Based on Simulated Annealing

Simulated annealing is a powerful optimization algorithm that can escape local optima even when the objective function deviates from minima. However, existing theory suggests that simulated annealing performance excessively depends on initial solution selection, posing significant challenges for hyperparameter optimization and initial solution choice. For shield tunnel segment layout, segment assembly postures are restricted to specific discrete values due to axial connection constraints, differing from simulated annealing's conventional approach of applying micro-perturbations to continuous variables. This requires targeted modifications to adapt the algorithm to practical engineering problems.

In this engineering problem, segment assembly information can be calculated through stepwise transmission theory. That is, with the initial plane and center point determined, only the assembly posture of each ring needs to be determined to ascertain the complete segment layout.

The optimization problem uses the sequence of assembly postures {φ₁, φ₂, φ₃, ...} as independent variables, and the sum of squared perpendicular distances from segment ring centers to the tunnel design axis as the objective variable to measure layout quality.

The objective function is:
$$ \min \sum_{i=1}^{n} (V_iH_i^2 + V_{i+1}H_{i+1}^2) $$

Where:
- ViHi: Distance from the current segment ring's reference plane center Vi to the tunnel design axis
- Vi+1Hi+1: Distance from the current segment ring's forward face center Vi+1 to the tunnel design axis
- Hi, Hi+1: Points on the tunnel design axis nearest to Vi and Vi+1, respectively

Based on requirements for staggered segment joints during actual assembly, control conditions are added. Different numbers of longitudinal joint connection bolts and different segment ring divisions create varying conditions; this study considers only the simple case where φi ≠ 0.

The simulated annealing process is illustrated in [FIGURE:3], where:
- X: Number of solution updates at current temperature
- x: Maximum number of solution updates at current temperature
- Y: Number of consecutive non-updated solutions at current temperature
- y: Maximum number of consecutive non-updated solutions at current temperature
- T: Current temperature in simulated annealing
- t: Minimum acceptable temperature
- k: Probability of accepting non-optimal solution replacement

The stopping condition for simulated annealing is T < t.

2.4 Greedy-Simulated Annealing Algorithm

During new solution generation from the initial solution in the simulated annealing model, a random independent variable φi is perturbed. However, according to the definition, φi represents the position change of the current ring relative to the previous ring. Therefore, when φi is perturbed to φi+1, the subsequent φi+1, φi+2, φi+3... in the original solution must change accordingly due to three-dimensional spatial connection transmission effects, rather than being directly transferable. This is schematically shown in [FIGURE:4].

Considering the poor interpretability of the simulated annealing method, we propose the following improvement: the initial perturbation remains unchanged, while φi+1, φi+2, φi+3... are determined according to the greedy algorithm to form a new solution. From the perspective of solution variation, this rapidly perturbs the original solution to generate new solutions. Meanwhile, since all nodes after the perturbation point are generated by the greedy algorithm, the new solution maintains substantial consistency with the original solution in format, conforming to the micro-perturbation characteristic of simulated annealing. This resolves the issue of meaningless nodes after the perturbation point in the original simulated annealing method, as shown in [FIGURE:5].

3. Hyperparameter Analysis

3.1 Engineering Background

Based on a metro shield tunnel construction project, this study investigates the influence of different model hyperparameters on fitting effectiveness. The shield interval is a spatial curve with a horizontal curve radius of 500R and a vertical curve gradient of approximately 30‰. Segment parameters include: inner diameter 5400mm, thickness 300mm, ring width 1500mm, 6 blocks per ring, and 10 longitudinal assembly positions. Fifty segment rings from the curve section were selected as the research object, using the objective function value of these 50 rings as the evaluation metric.

Hyperparameter selection for x, y, and k in the simulated annealing model employs grid search. The model considers 10 randomly generated initial solutions for subsequent calculations. Given that x and y are both related to iteration count, and x and k are both related to annealing acceptance probability, preliminary experiments were conducted to explore inter-parameter relationships before investigating the relationship between initial solutions and hyperparameters.

The hyperparameter iteration ranges are:
- x: 10 to 100, step size 10
- y: 10 to 100, step size 10
- k: 10 to 300, step size 10

3.2 Iteration Count Hyperparameter Analysis

As iteration-related hyperparameters in simulated annealing, x and y directly affect algorithm effectiveness. Using x and y values as axes and objective function values as the z-axis, a three-dimensional coordinate system is established as shown in [FIGURE:6]. The results indicate that the objective function value generally decreases with increasing x and y, rapidly at first, but exhibits obvious fluctuations when x and y exceed 50. At this point, the iteration steps have reached the marginal range of the simulated annealing algorithm, and further increasing iteration steps yields limited improvement. Moreover, increasing x is more beneficial for reducing the objective function value, while increasing y has some effect but is less significant.

In [FIGURE:7], the red plane represents the 95th percentile of all objective function solutions, showing that nearly all 95th percentile points are distributed in regions where x and/or y exceed 50. For subsequent analysis, we define the larger of x and y as the iteration count.

3.3 Iteration Probability Hyperparameter Analysis

Parameters x and k relate to solution replacement probability. Their relationship can be established directly through the relevant formula and expected simulated annealing iteration behavior.

The acceptance probability is:
$$ p = \exp(-\Delta/T/k) $$

When k = 300 and iterating to 100 generations (x = 100):
$$ T = 100 \times 0.95^{1.77} \approx 1.77 $$

With Δ = 10:
$$ p = \exp(-10/1.77) \approx 0.003 $$

This result indicates that when x = 100 and Δ = 10, setting k to 300 maintains sufficient stability during the final iteration stage.

3.4 Initial Solution Hyperparameter Analysis

The initial solution is the most critical parameter in simulated annealing. As an intelligent optimization search method, simulated annealing is highly sensitive to initial solutions, where a good initial solution may yield better final results. This study selected 10 initial solutions for exploration.

The distribution of objective function values for these solutions under different hyperparameters is shown in [FIGURE:7]. The 10th solution is the greedy algorithm solution, represented by a red dashed line due to its invariance across hyperparameters. The greedy algorithm yields an objective function value of 727.9. Except for Initial Solution 1, all other initial solutions produce objective function values below 727.9, with the minimum value being 556.8. Eighty percent of initial solutions outperform the greedy algorithm, optimizing the objective function value by 23%.

The objective function distributions for different initial solutions exhibit long-tailed, spindle-shaped patterns. The long tail indicates large initial objective function values that rapidly decrease during early iterations. The spindle shape suggests that simulated annealing gradually reduces the objective function value, with fluctuations and residence at local optima creating high solution density in certain ranges. The pointed tip indicates that simulated annealing explores some relatively optimal solutions with low probability, appearing as a pointed tip in the distribution. Initial Solution 1 falls into a local optimum that cannot be escaped within the current hyperparameter search range.

Further analysis of the relationship between initial solutions and iteration count is shown in [FIGURE:8]. Based on objective function variation trends with iteration count, three categories emerge:
1. Fluctuating-Decreasing: Represented by Initial Solutions 2, 4, and 5, showing gradual decrease with fluctuations as iteration count increases, reaching lower objective function values.
2. Fluctuating-Stable: Represented by Initial Solutions 3, 6, 7, 8, and 9, showing fluctuations around the initial solution's objective function value. These solutions start with relatively low objective function values but haven't fully converged due to insufficient total iteration steps.
3. Stable: Represented by Initial Solutions 1 and 10 (greedy solution), showing no fluctuation with increasing iteration steps.

After identifying initial solution characteristics, we examine the objective function fluctuation of the optimal solution across all iterations, as shown in [FIGURE:9]. The objective function value alternates between peaks and valleys, rapidly decreasing at peaks to reach valleys, then jumping back to peaks after stabilizing for some time at valleys. This demonstrates a jump-descent pattern. Overall, the valley-corresponding objective function values gradually decrease, indicating that simulated annealing effectively escapes local optima to find new local optima until finally stabilizing at a certain objective function value. The traditional greedy algorithm's optimal objective function value is 727.9, while simulated annealing achieves 556.8—a 23% improvement.

3.5 Model Comparison Analysis

Comparing the objective function distributions of three methods—greedy algorithm, simulated annealing, and greedy-simulated annealing—FIGURE:10 and (b) show cumulative distributions where the x-axis represents objective function values and the y-axis represents the cumulative proportion of solutions exceeding the x-axis value.

As shown in FIGURE:10, the GSA method's solutions are distributed in the 600-1200 range, while only 80% of SA solutions fall in this range, demonstrating GSA's superior stability.

As shown in FIGURE:10, the traditional greedy algorithm's objective function value is 727.9. Approximately 12% of SA solutions and 5% of GSA solutions fall below 727.9. The SA minimum is 556.8 (23% improvement over greedy), while the GSA minimum is 658.2 (10% improvement). SA outperforms GSA in solution quality.

Based on this comparative analysis, we conclude:
1. Simulated annealing demonstrates strong optimization capability under appropriate conditions, with 80% of initial solutions producing better layout effects than traditional methods. The optimal solution improves the objective function value by 23%.
2. Simulated annealing exhibits initial solution dependency. Different initial solutions can be categorized into three types based on iteration characteristics: fluctuating-decreasing, fluctuating-stable, and stable. Fluctuating-decreasing solutions (approximately 30% of all solutions) are most effective for substantially reducing objective function values.
3. We propose the GSA method with high interpretability and low initial solution dependency. GSA improves the objective function value by 10% over traditional methods. The performance ranking is: SA > GSA > traditional method. GSA produces more concentrated solution distributions and stronger algorithmic stability. Traditional methods are limited to local layout due to computational constraints, while the proposed algorithms enable higher-precision, higher-efficiency global layout with stable, high-quality outputs.

References

[1] Zhang Zhihua. Research on Optimized Segment Layout Method for Shield Tunnels [D]. Hubei: Huazhong University of Science and Technology, 2012.

[2] Liu Xuezeng, Liu Xingen, Zhou Decheng, et al. Integrated Research on 3D Segment Layout Correction and Structural Analysis for Shield Tunnels [C]//Proceedings of the 7th China-Japan Shield Tunnel Technology Exchange Conference. 2013: 78-83.

[3] Li Weiping, Zheng Guoping. Research on Core Algorithms of Universal Tapered Segment Layout System for Shield Tunnels [J]. Modern Tunnelling Technology, 2008, 45(5): 34-37, 43.

[4] Ruan Chengzhi. Research on Segment Layout Analysis Method for Shield Tunnels [J]. Modern Tunnelling Technology, 2021, 58(4): 224-228.

[5] Wang Zhengbin, Liu Guoshan, Zhu Lianchen, et al. Research on Universal Tapered Segment Layout Technology for Complex Alignment Shield Tunnels Based on Mathematical Models [J]. Sichuan Architecture, 2020, 40(2): 70-71, 75.

[6] Wu Haibin, He Chuan, Yan Qixiang, et al. Universal Tapered Segment Shield Tunnel Curve Alignment Fitting Algorithm and Application [J]. Journal of the China Railway Society, 2016, 38(10): 90-97.

[7] Du Zhanjun, Wu Jifeng, Xu Chen. Research on Universal Segment Position Optimization and Layout Application for Shield Tunnels [J]. Construction Technology, 2021, 50(19): 6-11.

[8] Bao Heli, Jiang Hong, Pan Weiqiang. Research on Quasi-Universal Ring Segment Layout Technology for Rectangular Shield Tunnels [J]. Tunnel Construction, 2022, 42(z1): 232-237.

[9] Liu Fenghua. Research on Universal Segment Fitting Layout and Selection Technology for Shield Tunnels [D]. Shanghai: Tongji University, 2007.

[10] Song Ruiheng. Development of Management Software for Universal Segment Layout and Dynamic Correction in Shield Tunnels [D]. Shanghai: Shanghai Jiao Tong University, 2008.

[11] Zhao Yunhui, Si Dianhao, Zhang Yang, et al. Research on Shield Tunnel Design Axis and Segment Layout Algorithm Based on Revit [J]. Information Technology of Civil Engineering and Architecture, 2022, 14(5): 1-6.