城市地下预制管廊拓扑设计综述
(1. 宁波明森建筑设计院有限公司, 宁波 315122
2. 哈尔滨工业大学土木工程学院, 哈尔滨 150090
3. 宁波建工工程集团有限公司, 宁波 315199)
摘要: 与传统城市地下综合管廊相比,预制管廊在工厂制作和现场拼装的基础上,提高了现场作业效率,减少了施工材料运输困难。预制管廊的整体抗震性能及其管片上的应力分布已被广泛研究。考虑到成本控制,且预制管廊不同连接节点角部对应多种不同应变分布现象,可以通过拓扑优化的方式实现管廊管片材料最佳分布形式,从而优化管片应力分布,降低材料成本。关于地下管廊拓扑设计并对相应优化结果进行非线性分析的研究较为缺乏,因此,本文回顾了部分拓扑算法,并重点讨论了固体各向同性材料惩罚法(SIMP)以及双向渐进结构优化法(BESO)分别在箱式管廊与圆形管廊优化应用中的可行性。针对不同截面形状的管廊管片,需要开发相应的拓扑优化体系,并且需要注意断面更加复杂的多舱型管廊的优化迭代计算效率。针对初步优化结果进行后续详细设计和抗震性能分析,能够使优化后的城市地下预制管廊具有更好的安全性与可靠性,从而可以将其投入到实际工程项目中。
关键词: 城市地下预制管廊, 拓扑优化, SIMP法, BESO法
引用格式: 江朝宏, 李志豪, 黄金桥, 等.城市地下预制管廊拓扑设计综述[J]. 工程材料与结构, 2025, 4(4): 120-131.
文章类型: 综 述
收稿日期: 2025-11-27
接收日期: 2025-12-10
出版日期: 2025-12-28
1 Introduction
The urban underground comprehensive pipe gallery refers to a gallery style structure used to arrange various municipal public pipelines in the shallow space below the urban ground surface，which can solve the contradiction between urban expansion and land scarcity.The construction of comprehensive pipe galleries originated in Europe in the 19th century and it is one of the most advanced regions in the world for underground space development and utilization[1].France began to plan its drainage network in Paris in 1833 systematically，while constructing underground pipe galleries.In the 20th century，the United States，Canada，China，Spain，Russia，Japan，and Hungary also began to construct underground pipe galleries.Although the United States and Canada have vast territories，their cities are highly concentrated，and the conflict between urban public space and land use is still very acute[2].This type of contradiction generally occurs in subsequent urban construction in other countries，and the research on different aspects of underground pipe galleries is continuously progressing.
The study on the design of pipe galleries and the excavation of soil required for different environments has received widespread attention.Marshall and Haji[3] theoretically derived and analyzed the interaction between underground pipe galleries and soil from factors such as pipe gallery burial depth，soil stiffness，and force transmission methods.Sharma et al.[4] conducted research on the impact of large excavation on adjacent structures，and pointed out that the settlement deformation caused by large excavation was an important factor affecting the stiffness of the structure.
The impact of the construction of underground comprehensive pipe galleries on urban environment has also been taken seriously.Hunt et al.[5] provided a detailed explanation of the layout plan for the pipe gallery，including influencing factors such as burial depth，location，and construction method on the environment.The advantages and disadvantages of various forms of pipe galleries were analyzed.Robati and Atabi[6] elaborated on the impact of comprehensive pipe galleries on the urban environment during the construction and operation stages through extensive data statistics，and clarified that establishing urban underground comprehensive pipe galleries was in line with urban design and sustainable development policies and concepts.Namin et al.[7] also elaborated on the important role of pipe gallery construction in environmental improvement from aspects such as ecology，surface water，and air quality.
On the basis of sustainable urban development，the seismic performance and safety of underground pipe galleries need to be guaranteed.Li et al.[8,9] conducted shaking table tests on a soil structure integrated system with an underground comprehensive pipe gallery structure under the condition of soil free-field.The proposed layered shear sand box modeling method model can basically reflect the response characteristics of the structure and approximate the simulation test situation.Tan et al.[10] evaluated the safety of the comprehensive pipe gallery using fuzzy theory，taking into account existing conditions such as steel corrosion，concrete cracking，structural water seepage，and soil liquefaction.
In addition to the seismic performance of the underground pipe gallery itself and its impact on the external environment，its construction cost cannot be ignored.Petrukhin et al.[11] introduced the use of shield tunneling technology to construct pipe galleries and pointed out that the cost of constructing comprehensive pipe galleries varies in different regions.Compared to traditional cast-in-place underground pipe galleries，prefabricated comprehensive pipe galleries can be constructed by combining factory prefabrication and assembly，greatly reducing the difficulty of transportation and on-site operations，ensuring safety，economy，and significant social benefits.The seismic performance of various prefabricated pipe galleries has been studied through finite element simulation and scaling tests[12,13].Moreover，prefabricated pipe galleries in different geographical regions have also been studied and there are some noteworthy findings.Duan et al.[14] found in their study on the seismic characteristics of prefabricated pipe galleries in loess areas that the acceleration response of built-in pipelines was higher than that of pipe galleries，and strain concentration was prone to occur at the top and bottom corners.
Combining cost control and the phenomenon of various strain distributions at different corners of prefabricated pipe gallery segments，topology optimization of prefabricated underground pipe gallery segments can be carried out to achieve the optimal distribution of segment materials，which can optimize stress distribution and reduce material costs.The solid isotropic material with penalization(SIMP)method[15]，which introduces the volumetric density of an element representing the material properties and its constitutive relations at the micro level，can increase the stiffness of the pipe segment while relatively reducing its volume.The evolutionary structure optimization(ESO)[16] optimizes the stress distribution of the structure by limiting the local maximum principal stress，which also meets the optimization objectives of the pipe segments.The bi-directional evolutionary topology optimization(BESO)[17] is based on ESO method and uses a more efficient heuristic algorithm to remove inefficient elements from the structure.At present，there are limited researches on optimizing prefabricated underground pipe galleries and simulating the seismic performance of optimized pipe segments based on the above topology algorithms.
Considering the above factors，the present paper reviews research status of prefabricated pipe galleries and discusses the feasibility of different topology methods for optimizing pipe gallery segments with different cross-sections and soil conditions.In the future practical applications，the prefabricated pipe galleries can achieve light weighting while meeting design requirements by adding structural details to the optimized results.
2 Prefabricated pipe gallery
The characteristic of prefabricated comprehensive pipe gallery is that each section of it is divided into multiple components and installed on site.This construction method appropriately increases the length of individual segments and the diversity of cross-sectional forms for design.It also accelerates on-site construction speed and has significant characteristics such as multifunctionality，multiple compartments，and complex systems.In China，the project and construction of underground comprehensive prefabricated pipe galleries is an important component of achieving coordinated planning and scientific management of urban infrastructure.In the study of the performance of underground pipe galleries with different structural types，full-scale tests were conducted based on the comprehensive pipe gallery project of the Shanghai World Expo Park[18,19].The segmental assembly technology of post tensioning and prestressing was adopted，which effectively solved the functional requirements of waterproofing and joint deformation.
Earthquakes are the main factors causing damage to underground pipe galleries.To explore the seismic response of prefabricated single cabin underground pipe galleries，Lei[20] established a finite element model of pipe gallery and soil based on finite element software.Three sets of natural seismic wave inputs with different peak accelerations were selected，and the nonlinear factors of materials and contact between structures and soil were comprehensively considered.Simulation analysis was conducted from three aspects:interface displacement，stress distribution，and material damage.The results indicated that the junction between the side wall of the pipe gallery and the top plate，as well as the position of the socket，were vulnerable areas.In addition，in the seismic resistant part of the prefabricated assembly comprehensive pipe gallery structure，the nodes are the weak links of the structure，and their mechanical properties directly affect the deformation and bearing capacity of the overall structure.Kuang et al.[21] proposed a new type of joint connection called “U-shaped ferrule joint bars connection” for prefabricated double compartment comprehensive pipe gallery structure，as shown in Fig.1，and verified the effectiveness of this connection technology through mechanical performance tests on four full-scale specimens of the nodes.
Fig.1 Prefabricated double cabin pipe gallery[21]
In summary，it can be found that the seismic performance of the pipe gallery segment sidewalls and nodes needs to be further improved，and optimizing their own material distribution may achieve this goal.These optimization approaches need to be discussed in detail.
3 Topology optimization methods
Structural optimization design refers to determining the optimal structural design scheme based on certain objectives under given constraints.It can be divided into three levels:cross-section，shape，and topology.Topology optimization is used to optimize the component layout and node connection relationships of a structure.Since the 1980s，topology optimization methods have gradually evolved into objective functions，constraints，design domains，and mathematical algorithms.The homogenizatio
，SIMP[15]，ESO[16]，BESO[17]，and adaptive design domain[23] methods have been proposed successively，as listed in Table 1[24].The objective function，design variables，and constraints describe the optimization model throughout the entire optimization process.Many classic optimization algorithms that link these parameters，such as optimality criteria[25]，multiplier[26]，and sequential linear programming methods，have been used to solve some representative examples in structural optimization[27].
Table 1 Categorization of topology optimization methods[24]
|
Optimization Method |
Author |
Year |
Reference |
Objective Function |
Constraint Condition |
Design Domain |
|
Homogenization Method |
Bendsøe M P Kikuchi N |
1988 |
Generating optimal topologies in structural design using a homogenization method |
Strain energy |
Volume limit |
Fixed |
|
SIMP:Solid Isotropic Material with Penalization |
Bendsøe M P |
1989 |
Optimal shape design as a material distribution problem |
Strain energy |
Volume limit |
Fixed |
|
ESO:Evolutionary Structure Optimization |
Xie Y M Steven G P |
1993 |
A simple evolutionary procedure for structural optimization |
Principal stress |
Maximum principal stress limit |
Fixed |
|
Bi-directional Evolutionary Topology Optimization |
Huang X Xie Y M |
2009 |
Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials |
Strain energy |
Volume limit |
Fixed |
|
Adaptive Design Domain |
Yi Rong |
2022 |
Structural topology optimization with an adaptive design domain |
Strain energy |
Volume limit |
Self-adaption |
For the SIMP method，the topology problem is described by Eq.(1).The design variable xe is the relative density of single element，and N is the number of elements discretizing the design domain.U，F and K are the global displacement vector，force vector，and stiffness matrix.Correspondingly，ue and k0 are the element displacement vector and stiffness matrix.The factor p is the penalization power.Generally，the design responses in the SIMP method are the strain energy c and volume V，which are functions of the design variable vector x.The ratio f of the demand volume V(x)to the original volume 
is the volume fraction，which plays a constraint role in the optimization process.Moreover，the minimum density of each element is greater than 0 to prevent the singularity.
(1)
The disadvantage of SIMP method is that it cannot optimize structures with mixed loading of force control and displacement control.In a feasibility study[24]，the force-controlled loading was selected so that the goal of the objective function was to minimize the strain energy of the structure to maximize its stiffness under a specific volume constraint.The reason why displacement-controlled loading is not used is that the influence of gravity and axial compression ratio is considered during the modeling.For topology optimization using strain energy as the objective function，the stiffness of the structure is maximized by minimizing the strain energy when the structure is only under force loading.This is achieved by maximizing the strain energy when the structure is only under displacement loading.In the case of gravity and vertical load，the application of displacement loading results in the simultaneously existing force and displacement.The maximization or minimization of the overall strain energy cannot maximize the structural stiffness，which leads to logic errors in the topology program.Although the SIMP method has certain drawbacks，it is suitable for static problems executed by the ABAQUS topology optimization module(ATOM).
Compared with the SIMP method，the ESO method uses the maximum stress standard for all variables to retain elements in the structure that are close to the maximum stress limit.In addition，the implementation of the BESO method mainly includes two basic steps:first deleting inefficient material elements，and then adding efficient material elements.The difference between it and the SIMP method is that the recognition of elements has only absolute densities of 0 or 1，without intermediate relative densities in the iterative process of optimization.In the next section，these topology optimization methods will be discussed in the application of some typical pipe gallery structures，and the problems they may encounter in the application process of prefabricated underground pipe galleries will be analyzed.
4 Application of topological design in pipe galleries
4.1 SIMP method for box-type pipe galleries
The SIMP method can combine the finite element method with the relative volume density of elements，making it applicable to many structural optimization scenarios，such as reinforcing layers[28]，shear walls[29] and beams[30].For prefabricated box-type pipe galleries，the pipe segment is a conventional load-bearing plate and shell structure.The roof plate bears the water and soil pressure above，as well as the load of vehicles，while the wall plates on both sides bear the lateral water and soil pressure.According to regulations，in practical engineering such as excavation and support design of foundation pits，tunnel side walls，and dock design，it is necessary to calculate the stress on the pipe segment based on active soil pressure.The bottom plate is subjected to the foundation reaction force exerted on it by the foundation.In design practice，the simulation of the foundation generally adopts the Winkler elastic foundation model[31]，as shown in Fig.2.
Fig.2 Winkler elastic foundation model[31]
In the Winkler foundation model，there is a corresponding relationship between the displacement and the reaction force of the foundation，which has an additional stiffness.Therefore，the stiffness matrix Ks of the foundation bed should be considered and added to the Eq.(1).Based on the load balance condition，the updated relationship between the global displacement vector U and force vector F is shown in Eq.(2).The F in the equation is still an external load，which maintains its essence.The reaction effect of the foundation is reflected by increasing the overall stiffness of the system，rather than acting as an additional downward force on F.
(2)
In addition to satisfying the constraint of mechanical equilibrium conditions，according to the relevant provisions of GB50007 standard in China[32]，the design of pipe galleries should also meet the requirements of base deformation.For the Winkler elastic foundation model，the base settlement displacement is proportional to the base pressure.The maximum base reaction can be limited by restricting the maximum settlement displacement of the foundation.Therefore，the constraint function needs to include settlement displacement constraints to meet the requirements of foundation bearing capacity，as shown in Eq.(3).In the formula，uz is the foundation settlement displacement，and Umax is the settlement displacement limit，which is determined by the reaction coefficient of the foundation bed and the specific project.
(3)
In summary，the optimization steps of the SIMP method based on the Winkler elastic foundation model are as follows:(1)Divide the initial design domain with finite element mesh;(2)Define the global stiffness matrix，which includes the stiffness matrix of the pipe gallery segment structure and the stiffness matrix of the foundation.And set boundary conditions，loads，and element characteristics;(3)Set the optimization objective function as the minimum strain energy;(4)Set constraint conditions:volume constraint and foundation bearing capacity constraint;(5)Perform program iterative optimization through optimization criteria.Finally，the overall mathematical description of the topology optimization problem for box-type pipe galleries based on SIMP method is shown in Eq.(4).
(4)
4.2 BESO method for circular pipe galleries
Regarding the BESO method，it has been adopted by many research institutes and improved in some structural optimizations due to its high efficiency.Amaral et al.[33] proposed a novel multi-material topology method based on BESO to optimize two-pile cap structures.The optimized two-pile cap structure was evaluated through weight analysis and nonlinear finite element analysis，which was to determine potential failure modes.The resulting design scheme achieved the safety level required by the design load while considering manufacturing constraints and reducing concrete volume.
As for circular pipe galleries，their cross-section is relatively simple.In the majority of cases，there is no issue of haunching between roof plates and wall plates in the circular pipe segments，which needs to be considered in box type pipe galleries，enabling them to be optimized by more effective approach such as the BESO method.In the traditional BESO method，some deleted elements may be added back to the structure during the process of deleting and adding elements.This causes oscillations in the iterations of structural optimization，resulting in low computational efficiency.The rational approximation of material properties[34](RAMP)interpolation model allows all solid and empty elements to be grouped together for element operations，and its element elastic modulus can be expressed by Eq.(5).In this formula，v is the ratio of the stiffness change value ΔE to the original stiffness E0，and ρi is the density value of a single element.Assuming that the Poissons ratio of the material is independent of the element density，the structural stiffness matrix K can be represented by the initial element stiffness matrix 
and element density ρi，as shown in Eq.(6).
(5)
(6)
For circular pipe galleries that exhibit circular geometric features as a whole，it is necessary to discretize them in Cartesian coordinates to facilitate numerical analysis using finite element methods.It should be clarified that the Cartesian coordinate is not directly used to describe the overall spatial form of the circular structure，but to construct a rectangular mesh model by unfolding and mapping it to a planar region.After modeling and calculating the original volume of the elements by the Cartesian coordinate，the structural elements are divided into two types:the first is designable elements，and the second is non-designable elements.The number of non-designable elements is represented by P，and their density is represented by ρip.During the optimization iteration process，the topological variables of these elements remain unchanged.The number of designable elements is represented by Q，and their density is represented by ρnq.During the optimization iteration process，ρnq varies between[0，1].Therefore，the optimization problem of minimizing the flexibility of the volume constrained structure can be represented by Eq.(7).
(7)
In Eq.(7)，Csum represents the flexibility of the structure under all load conditions to fully consider the importance of the various conditions that affect the pipe gallery.And Cr represents the flexibility of the structure under the action of single condition，while wr represents the weight factor.Within the structural volume constraint V*，Vnq represents the original volume of designable elements，while Vip represents the original volume of non-designable elements.Non-designable elements are not allowed to be changed during optimization.For the pipe gallery，in order to meet the design specifications of the structure，the area where the pipe segments contact the external soil is designated as a frozen zone.And the addition and deletion of elements are carried out from the inside of the structure.The lower limit of topological variables ρi is not zero to ensure the non-singularity of the structure.
In summary，in the BESO method，the initial design domain is generally discretized by finite element mesh first.In terms of circular pipe galleries，it is necessary to recalculate the volume of each element using Cartesian coordinate.And then，the stiffness sensitivity of each element is calculated based on the RAMP interpolation model.Finally，whether to retain the elements through deletion and addition criteria is determined.
4.3 Refinement，validation，and computational efficiency
Both the SIMP method highly compatible with ATOM and the BESO method based on the RAMP interpolation model only have preliminary optimization results that have not yet been designed in detail.Bruggi[35] pointed out that the strain energy calculated by the optimization program may only be a rough elastic approximation of the actual mechanical behavior of the structure.After determining the optimized shape and material distribution of the pipe gallery segments，the detailed design and fabrication of them are important for the practical application.In addition，a subsequent appropriate nonlinear finite element analysis can more reliably study their responses to various dynamic loads or seismic effects.
The rationality of optimization results for pipe galleries also needs to be considered.However，there is little research on the rationality verification standards or indicators for the topological design results of underground pipe galleries.So，it is necessary to refer to other optimized structural types to find suitable validation standards.The strut-and-tie models were popularized in[36] for various concrete structures.A method was proposed in[37] to automatically generate optimization-based strut-and-tie models for extractions of results.Xia et al.[38] proposed three methods to evaluate the results of strut-and-tie models extracted from different original structures，which are shown in Fig.3，including applicability and rationality indicators.Due to the significant difference between axial force and shear force values in the cross-section of the rods in this model，it is very appropriate to use the proportion of axial force in the total stress of the cross-section as a criterion for applicability evaluation.Taking this as a reference，the proportion of radial principal stress in the wall plate of the underground pipe gallery to the total stress distribution may become an indicator for evaluating the rationality of optimized result.
Fig.3 Principal stress plots of strut-and-tie models[38]
In addition，optimizing the nodes of the pipe gallery is also a difficult problem.At present，most optimization processes start from the overall component rather than local nodes.This means that the L-shaped nodes split from box-type pipe gallery and the arc-shaped nodes split from circular pipe gallery need to be equivalently verified with the overall optimization results.This local optimization method can improve computational efficiency for symmetric structures，but the equivalent boundary conditions connecting nodes need to be further refined.Moreover，the local optimization is more important in reducing computational time costs for multi-cabin pipe galleries，which needs further research in the future.
5 Conclusions
On the basis of factory prefabrication and on-site assembly，the prefabricated underground comprehensive pipe gallery has improved the efficiency of on-site operations and reduced transportation difficulties.In terms of topological design for pipe galleries，the significance of optimization lies not only in reducing material costs，but also in optimizing stress distribution，which can directly improve overall load-bearing capacity.Moreover，topological design has certain potential in reducing carbon emissions throughout the entire lifecycle of underground pipe galleries due to the reduction of material consumption.
In this review，some topological algorithms have been mentioned，and the potential applications of SIMP method in box-type pipe galleries and BESO method in circular pipe galleries are mainly discussed.After considering the characteristics of different cross-sectional types，the corresponding optimized segments need to be refined for actual projects.The rationality of the optimization results also needs to be verified.When the appropriate optimization method is determined，improving its computational efficiency is more useful for larger multi-cabin pipe galleries.It is worth further research in the future to form a complete and reliable optimization system for pipe galleries.
[③] *通讯作者 Corresponding author:李志豪，709378130@qq.com
收稿日期:2025-11-27; 录用日期:2025-12-10; 发表日期:2025-12-28
基金项目:This review was sponsored by the Key Research and Development Program of Ningbo City(No.2023Z101).
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A Review on Topological Design for Urban Underground Prefabricated Pipe Galleries
(1. Ningbo Mingsen Architectural Design Institute Co. , Ltd, Ningbo 315122, China
2. School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China
3. Ningbo Construction Engineering Group Co. , Ltd, Ningbo 315199, China)
Abstract: Compared with traditional urban underground comprehensive pipe galleries, prefabricated galleries enhance on-site construction efficiency and alleviate transportation difficulties through factory prefabrication and on-site assembly. The overall seismic performance of such galleries and the stress distribution within their segments has been extensively investigated. Considering cost constraints and the distinct strain distributions observed at segment joints, topology optimization can be employed to determine the optimal material layout, thereby improving stress performance and reducing material usage. However, research on the topological design of pipe-gallery segments and the nonlinear evaluation of optimization results remains limited. This paper reviews representative topology-optimization algorithms and evaluates the applicability of the SIMP method for box-type galleries and the BESO method for circular galleries. Dedicated topology-optimization frameworks must be developed to accommodate different cross-sectional geometries, and computational efficiency becomes critical for complex multi-chamber configurations. Conducting detailed design and seismic-performance assessments based on preliminary optimization results enhances the safety and reliability of prefabricated pipe galleries, thereby supporting their application in real engineering projects.
Keywords: Urban underground prefabricated pipe galleries, topology optimization, SIMP method, BESO method
Citation: JIANG Chaohong, LI Zhihao, HUANG Jinqiao, et al. A review on topological design for urban underground prefabricated pipe galleries[J]. Engineering Materials and Structures 2025, 4(4): 120-131.