Abstract
Without proper flow channelization, congestion and overcrowding in pedestrian traffic may lead to significant inefficiency and safety hazards. Thus, the design of guideway networks that provide a fine balance between traffic congestion and infrastructure construction investment is vital. This paper presents a mathematical formulation and topology optimization framework for paved pedestrian guideway design under physicsbased traffic equilibrium in a continuous space. Pedestrians are homogeneous, and their destination and path choices under the Nash equilibrium condition are described by a set of nonlinear partial differential equations. The design framework optimizes the deployment of pavement, which alters the road capacity and directly affects pedestrians’ free flow travel speed. A maximum crowd density constraint is included in the design model to address public safety concerns (e.g., over stampede risks). A series of numerical experiments are conducted to illustrate the effectiveness of the proposed model as well as solution techniques. The proposed framework, which builds on the traffic equilibrium theory, produces optimized guideway designs with controllable maximum pedestrian density, accounts for budget constraints (through an adjustable multiplier that balances pavement construction and travel costs), and allows for control of the spatial configuration of road branches. Comparison with lamellar structures and more conventional guideway designs demonstrates better performance of the outcomes from the proposed modeling and optimization framework.
This is a preview of subscription content, access via your institution.
Notes
 1.
This function form is analogous to the wellknown BPR function (U.S. Bureau of Public Roads), but other nondecreasing functions are also acceptable.
 2.
The choice of facility i depends on the pedestrian’s location x, i.e., i(x). However, we note that \(\boldsymbol {x} \in {\mathscr{A}}_{i}\) equivalently indicates the facility choice of the pedestrians at x. Hence, we simply use facility index i without the argument, for notation convenience.
 3.
Also called user equilibrium (UE) in the transportation literature, as originally described in Wardrop (1952).
 4.
Wadbro and Noreland (2019) used an interesting linear heat conduction model analogy to approach a similar problem; i.e. it assumes a linear relation between the local flux f(x) and the potential gradient ∇ϕ(x) with a constant conductivity factor κ. The model we introduced in (16)–(19), based on the Nash equilibrium condition of travelers in the continuous domain, shows that the κ in (17) should be a nonlinear function of ∇ϕ, see (15).
 5.
The traffic equilibrium model is based on “macroscopic” fluid approximation, where pedestrians are described not as discrete particles but by continuous flux. Hence, the PDE and optimization model do not impose any requirement on the minimum thickness of guideway paths. The congestion delay is dictated by the ratio of flux intensity f to the capacity α, and hence the optimization model tends to provide capacity to the area where flux concentrates. When traffic is dispersed and relatively light (e.g., near the entrances and exits), it is reasonable for the optimization model to yield very thin roads. The density filter is hence needed as a regularization approach to control the length scale to avoid overly thin roads—for practical construction convenience and aesthetic purposes.
Abbreviations
 α(x):

road capacity at location x
 α _{0} :

minimum road capacity
 α _{ ℓ } :

road capacity of element ℓ
 \(\alpha _{\max \limits }\) :

maximum road capacity
 \(\bar {q}\) :

traffic inflow or outflow at ∂Ω
 β :

shadow price of construction
 α :

discretized road capacity vector
 λ _{ g } :

adjoint vector in sensitivity analysis of constraint function g
 λ _{ J } :

adjoint vector in sensitivity analysis of objective function J
 Ω :

twodimensional space
 ρ :

discretized pedestrian density vector
 \(\boldsymbol {\tilde {F}}\) :

matrix storing the F of previous m AAR iterations
 \(\boldsymbol {\tilde {R}}\) :

matrix storing the R_{F} of previous m AAR iterations
 F :

global flux magnitude vector
 G _{ ℓ } :

matrix mapping global degree of freedom vector Φ_{i} to ∇ϕ at the center of element ℓ
 I :

identity matrix
 \(\boldsymbol {k}^{0}_{\ell }\) :

element stiffness matrix of element ℓ with κ_{ℓ} = 1
 K ^{α} :

global stiffness matrix
 M _{ ℓ } :

matrix defined by \(\boldsymbol {M_{\ell }} := \boldsymbol {G_{\ell }}^{T} \boldsymbol {G_{\ell }}\)
 n(x):

unit normal vector at of ∂Ω
 P :

density filter matrix
 Q :

global traffic inflow/outflow vector
 R :

residual vector in Newton’s method
 R _{ F } :

global residual vector in AAR method
 v :

vector of element area
 x _{ i } :

Location of facility i
 z :

design variable vector
 z _{ i n i } :

initial design variable vector
 \(\boldsymbol {z}_{\max \limits }\) :

upper bounds of design variable
 η :

Free flow travel time per unit distance
 κ :

variable defined as \(\kappa :=\frac {\mid \mathbf {f}\mid }{c}\)
 κ _{ ℓ } :

κ value associated with element ℓ, assumed to be constant inside element ℓ
 \(\kappa _{\min \limits }\) :

a small value to void numerical singularities
 f :

flux vector of pedestrian flows
 \({\mathscr{A}}_{i}\) :

area from where pedestrian travels to facility i
 \({\mathscr{H}}_{1}\) :

Sobolev space
 \({\mathscr{U}}\) :

space of trial functions
 \({\mathscr{U}}^{0}\) :

space of test functions
 \({\mathscr{U}}_{h}\) :

finitedimensional trial function space
 ϕ :

total generalized cost
 ϕ _{ h } :

finitedimensional total generalized cost
 Φ _{ i } :

value of ϕ_{h} at node i
 ψ :

test function in weak form
 ψ _{ h } :

finitedimensional test function
 Ψ_{i} :

values of ψ_{h} at node i
 ρ :

pedestrian density
 \(\rho _{\max \limits }\) :

upper bound for pedestrian density
 τ _{ o p t } :

tolerance of change of design variable for terminating optimization
 x :

position vector
 𝜃 :

step size in AAR
 b _{1} :

monetary cost to travel a unit distance
 b _{2} :

ratio of capacity to free flow travel speed
 c :

generalized cost to travel a unit distance
 C _{ R } :

prorated unit cost for road construction
 C _{ T } :

monetary value of one unit pedestrian time
 E _{ ℓ } :

area of element ℓ
 f(α(x)):

prorated cost per unit time for setting a capacity of α(x) for a unit area near x
 g :

parameter characterizing travel cost sensitivity to congestion
 h _{ e } :

width of a square quadrilateral element
 J :

objective function
 J _{ R } :

total construction cost
 J _{ T } :

total transportation cost
 M :

number of nodes in an finite element mesh
 N _{ i } :

global Lagrange basis function associated with node i
 \(N_{\max \limits }\) :

maximum number of optimization steps
 p :

travel path
 p ^{n} :

power in pnorm of a ndimensional vector
 P r :

period of applying Anderson extrapolation in AAR
 q :

traffic inflow or outflow in Ω
 R :

filter radius
 r _{0} :

radius of facilities
 T _{ h } :

finite element partition
References
Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Gigavoxel computational morphogenesis for structural design. Nature 550:84
Alexandersen J, Andreasen C (2020) A review of topology optimisation for fluidbased problems. Fluids 5(29)
Alexandersen J, Sigmund O, Aage N (2016) Large scale threedimensional topology optimisation of heat sinks cooled by natural convection. Int J Heat Mass Transfer 100:876–891
An S, Cui N, Bai Y, Xie W, Chen M, Ouyang Y (2015) Reliable emergency service facility location under facility disruption, enroute congestion and infacility queuing. Transp Res E 82:199–216
Anderson DG (1965) Iterative procedures for nonlinear integral equations. J ACM (JACM) 12 (4):547–560
Bai Y, Hwang T, Kang S, Ouyang Y (2011) Biofuel refinery location and supply chain planning under traffic congestion. Trans Res Part B Meth 45(1):162–175
Bai Y, Ouyang Y, Pang J (2016) Enhanced models and improved solution for competitive biofuel supply chain design under land use constraints. Eur J Oper Res 249(1):281–297
Banerjee AS, Suryanarayana P, Pask JE (2016) Periodic pulay method for robust and efficient convergence acceleration of selfconsistent field iterations. Chem Phys Lett 647:31–35
BBC News (2015) Shanghai new year crush kills 36. Web link: https://www.bbc.com/news/worldasiachina30646918
Beghini LL, Beghini A, Katz N, Baker WF, Paulino GH (2014) Connecting architecture and engineering through structural topology optimization. Eng Struct 59:716–726
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Meth Appl Mechan Eng 71(2):197–224. https://doi.org/10.1016/00457825(88)900862. http://www.sciencedirect.com/science/article/pii/0045782588900862
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, Berlin
Borrvall T, Petersson J (2003) Topology optimization of fluids in stokes flow. Int J Numer Methods Fluids 41(1):77–107
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158
Christiansen RE, Wang F, Sigmund O (2019) Designing photonic topological insulators with quantumspinhall edge states using topology optimization. Nanophotonics 8:1363–1369
Clausen A, Wang F, Jensen JS, Sigmund O, Lewis JA (2015) Topology optimized architectures with programmable poisson’s ratio over large deformations. Adv Mater 27(37):5523–5527
Da D, Yvonnet J, Xia L, Li G (2018) Topology optimization of particlematrix composites for optimal fracture resistance taking into account interfacial damage. Int J Numer Methods Eng 115(5):604–626
Deaton JD, Grandhi RV (2014) A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct Multidiscip Optim 49(1):1–38. https://doi.org/10.1007/s001580130956z
Duysinx P, Sigmund O (1998) New developments in handling stress constraints in optimal material distribution, 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, American Institute of Aeronautics and Astronautics, 0. https://doi.org/10.2514/6.19984906
GersborgHansen A, Bendsøe MP, Sigmund O (2006) Topology optimization of heat conduction problems using the finite volume method. Struct Multidiscip Optim 31(4):251–259. https://doi.org/10.1007/s0015800505843
Gladstone R (2015) Death toll from hajj stampede reaches 2,411 in new estimate
Guenther R, Lee J (1996) Partial differential equations of mathematical physics and integral equations. Dover books on mathematics, Dover Publications
Hajibabai L, Ouyang Y (2013) Integrated planning of supply chain networks and multimodal transportation infrastructure expansion: model development and application to the biofuel industry. ComputAided Civ Inf Eng 28(4):247–259
Hajibabai L, Bai Y, Ouyang Y (2014) Joint optimization of freight facility location and pavement infrastructure rehabilitation under network traffic equilibrium. Trans Res Part B Meth 63:38–52
Helbing D (1991) A mathematical model for the behavior of pedestrians. Behav Sci 36(4):298–310
Helbing D, Molnar P (1995) Social force model for pedestrian dynamics. Phys Rev E 51 (5):4282
Helbing D, Buzna L, Johansson A, Werner T (2005) Selforganized pedestrian crowd dynamics: experiments, simulations, and design solutions. Trans Sci 39(1):1–24
Helbing Dirk JK, Molnar P (1997) Modelling the evolution of human trail systems. Nature 388:47–50
Hoogendoorn S, Campanella M, Daamen W (2011) Fundamental Diagrams for Pedestrian Networks. In: Peacock R, Kuligowski E, Averill J (eds) Pedestrian and Evacuation Dynamics. Springer, Boston, MA, pp 255–264
Jiang L, Li J, Shen C, Yang S, Han Z (2014) Obstacle optimization for panic flowreducing the tangential momentum increases the escape speed. PloS one 9(12) e115:463
Johansson A, Helbing D (2007) Pedestrian flow optimization with a genetic algorithm based on boolean grids. Springer, pp 267–272
Konur D, Geunes J (2011) Analysis of traffic congestion costs in a competitive supply chain. Trans Res Part E Logist Trans Rev 47(1):1–17
Konur D, Geunes J (2012) Competitive multifacility location games with nonidentical firms and convex traffic congestion costs. Trans Res Part E Logist Trans Rev 48(1):373–385
Li AC, Nozick L, Xu N, Davidson R (2012) Shelter location and transportation planning under hurricane conditions. Trans Res Part E Logist Trans Rev 48(4):715–729
Little JDC (1961) A proof for the queuing formula: L = w. Oper Res 9:296–435
Maute K, Allen M (2004) Conceptual design of aeroelastic structures by topology optimization. Struct Multidiscip Optim 27(1):27–42. https://doi.org/10.1007/s001580030362z
Ouyang Y, Wang Z , Yang H (2015) Facility location design under continuous traffic equilibrium. Transp Res Part B 81(1):18–33
Russ JB, Waisman H (2020) A novel topology optimization formulation for enhancing fracture resistance with a single quasibrittle material. Int J Numer Methods Eng 121(13):2827–2856
Ryu JC, Park FC, Kim YY (2012) Mobile robot path planning algorithm by equivalent conduction heat flow topology optimization. Struct Multidiscip Optim 45(5):703–715
Sherali HD, Carter TB, Hobeika AG (1991) A locationallocation model and algorithm for evacuation planning under hurricane/ flood conditions. Trans Res Part B Meth 25(6):439–452
Sigmund O (1994) Materials with prescribed constitutive parameters: an inverse homogenization problem. Int J Solids Struct 31(17):2313–2329
Sigmund O (1995) Tailoring materials with prescribed elastic properties. Mech Mater 20(4):351–368
Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48 (6):1031–1055. https://doi.org/10.1007/s0015801309786
Suryanarayana P, Pratapa PP, Pask JE (2019) Alternating andersonrichardson method: an efficient alternative to preconditioned krylov methods for large, sparse linear systems. Comput Phys Commun 234:278–285
Svanberg K (1987) The method of moving asymptotes – a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373
Wadbro E, Noreland D (2019) Continuous transportation as a material distribution topology optimization problem. Struct Multidiscip Optim 59(5):1471–1482
Wang F, Sigmund O, Jensen JS (2014) Design of materials with prescribed nonlinear properties. Journal of the Mechanics and Physics of Solids 69:156–174
Wang F, Christiansen RE, Yu Y, Mørk J, Sigmund O (2018) Maximizing the quality factor to mode volume ratio for ultrasmall photonic crystal cavities. Appl Phys Lett 113(24):241, 101
Wang Z (2017) Planning Service Facilities and Infrastructures Under Continuous Traffic Equilibrium. PhD Dissertation. University of Illinois at UrbanaChampaign
Wang Z, Ouyang Y (2016) On solving a class of continuous traffic equilibrium problems and planning facility location under congestion. Revision Under Review
Wang Z, Xie S, Ouyang Y (2019) Planning facility location in a continuous space under congestion and disruption risks. Under Revision
Wardrop J. (1952) Some Theoretical Aspects of Road Traffic Research. ICE Proceedings: Engineering Divisions. pp. 325–362.
Xia L, Breitkopf P (2015) Design of materials using topology optimization and energybased homogenization approach in matlab. Struct Multidiscip Optim 52(6):1229–1241. https://doi.org/10.1007/s0015801512940
Yan S, Wang F, Sigmund O (2018) On the nonoptimality of tree structures for heat conduction. Int J Heat Mass Trans 122:660–680 . https://doi.org/10.1016/j.ijheatmasstransfer.2018.01.114. http://www.sciencedirect.com/science/article/pii/S0017931017351566http://www.sciencedirect.com/science/article/pii/S0017931017351566
Yang H (1996) A spatial price equilibrium model with congestion effects. Ann Reg Sci 30 (4):359–371
Yang H, Wong S (2000) A continuous equilibrium model for estimating market areas of competitive facilities with elastic demand and market externality. Transp Sci 34(2):216–227
Yang H, Yagar S, Iida Y (1994) Traffic assignment in a congested discrete/continuous transportation system. Trans Res Part B Meth 28(2):161–174
Zhang X, Ramos AS Jr, Paulino GH (2017) Material nonlinear topology design using the ground structure method with a discrete filter scheme. Struct Multidiscip Optim 55(6):2045–2072
Zhang XS, de Sturler E, Paulino GH (2017) Stochastic sampling for deterministic structural topology optimization with many load cases: Densitybased and ground structure approaches. Comput Methods Appl Mech Eng 325:463–487
Zhang XS, Paulino GH, Ramos AS Jr (2018) Multimaterial topology optimization with multiple volume constraints: a ground structure approach involving material nonlinearity. Struct Multidiscip Optim 57:161–182
Zhu JH, Zhang WH, Xia L (2016) Topology optimization in aircraft and aerospace structures design. Arch Comput Methods Eng 23(4):595–622. https://doi.org/10.1007/s1183101591512
Acknowledgments
The authors would like to thank Dr. Ole Sigmund for suggesting Example 1. We would also like to acknowledge Dr. K. Svanberg for providing the MMA code.
Funding
The work by X. S. Zhang and W. Li were supported in part by the University of Illinois. The work by Y. Ouyang was supported in part by the U.S. National Science Foundation via Grant CMMI1662825.
Author information
Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Replication of results
Data are available from authors upon request.
Responsible Editor: Mehmet Polat Saka
Appendices
Appendix A: A hybrid solution scheme for the nonlinear state equations
To address the challenges described in Section 3.1 associated with solving the nonlinear state equation (29), we propose a robust hybrid solution strategy which combines AAR, a fixedpoint iteration method proposed by Suryanarayana et al. (2019) and Banerjee et al. (2016), and the standard Newton’s method. In the proposed solution strategy, we first use the AAR to bring the iterate close to the solution and then apply the Newtwon’s method to achieve fast convergence.
In the AAR, in order to avoid the difficulty related to the initial guess Φ = 0, we introduce a new global residual vector as
where f_{ℓ} denotes the flux in element ℓ, \(\boldsymbol {F}\in \mathbb {R}^{n\time 1}\) is a vector collecting all the element fluxes, namely, F = [f_{1},...,f_{n}]^{T}; and κ(α_{ℓ},f_{ℓ}) is defined by relation (13). We note that, unlike the original residual vector R(α,Φ) defined in (30), where Φ is the independent vector, the new residual vector R_{F}(α,F) uses the flux vector F as the independent vector and Φ(F) is obtained from Φ = (K^{α}(κ))^{− 1}Q, where we recall that κ = [κ_{1},...,κ_{n}]^{T}.
Although defined in different forms and having different independent variables, we can show that R_{F}(α,F) = 0 and R(α,Φ) = 0 are in fact equivalent in the sense that solution of R_{F}(α,F) = 0 is also a solution of R(α,Φ) = 0 and vice versa. The advantage of using R_{F}(α,F) in the AAR is that it allows us to use a nonzero initial guess for vector F which, according to Fig. 1, will can lead to wellconditioned matrix K^{α}. In our implementation, we use F = 1 as the initial guess for AAR in the first optimization step and, in the subsequent optimization steps, F is initialized using the converged F from the previous optimization step. This choice has been shown to be effective and robust for all the numerical examples in this work.
The AAR iteration to update the flux vector F as:
where F^{(k)} is the flux vector at the k th AAR iteration, \(\max \limits (\cdot ,\cdot )\) stands for the elementwise maximum operator between the two vectors, and the matrix B^{(k)} is defined by:
where 𝜃 is the step size, Pr is the period of applying Anderson mixing (Anderson 1965), and \(\tilde {\mathbf {F}}^{(k)} \in \mathbb {R}^{n\times m}\) and \(\tilde {\mathbf {R}}^{(k)} \in \mathbb {R}^{n\times m}\) are matrices collecting history information of flux and residual vectors:
and ΔF^{(j)} = F^{(j+ 1)} −F^{(j)}, \({\varDelta } \boldsymbol {R}_{F}(\boldsymbol {\alpha },\boldsymbol {F}^{(j)})=\boldsymbol {R}_{F}(\boldsymbol {\alpha },\boldsymbol {F}^{(j+1)})\boldsymbol {R}_{F}(\boldsymbol {\alpha },\boldsymbol {F}^{(j)})\). We note that, by defining matrix B using (40), we essentially apply a quasiNewton Anderson mixing (Anderson 1965) every Pr iterations. In other iterations, the simple Richardson iteration is used. For all the examples in this work, we use 𝜃 = 0.5, m = 5, and R_{r} = 4. The AAR iteration is terminated when the ℓ_{2}norm of the residual vector R_{F}(α,F) is below 10^{− 3}. The corresponding Φ = (K^{α}(κ))^{− 1}Q is then taken as the initial guess of the Newton’s method described below.
In the Newton’s method, we will switch back to the original residual vector R in (30). At iteration k of the Newton’s method, we linearize the above nonlinear system of equations as
where \(\mathbf {K}^{\alpha }_{T}\doteq \partial \boldsymbol {R}/\partial \boldsymbol {{\varPhi }}\) is the tangent stiffness matrix. Solving the linearized system gives
which leads to the recurrent update formula Φ^{k+ 1} = Φ^{k} + ΔΦ^{k} until the ℓ_{2}norm of the residual vector is below tolerance 10^{− 5}.
A consistently linearized tangent stiffness matrix is essential to ensure the convergence of the Newton’s method. Thus, in the remainder of this appendix, a detailed derivation of the consistent tangent stiffness matrix \(\mathbf {K}^{\alpha }_{T}\) is provided.
By definition and using the chain rule, we have
To obtain an explicit expression of \(\mathbf {K}^{\alpha }_{T}\), we first compute the following local matrix as
where the derivative ∂κ_{ℓ}/∂Φ_{j} can be computed as follows
Since κ_{ℓ} is defined implicitly through \(h(\kappa _{\ell },\alpha _{\ell },\nabla _{E_{\ell }}\phi _{h})=0\), see (15) and Footnote 4, we can compute the derivative \(\partial \kappa _{\ell }/\partial \nabla _{E_{\ell }}\phi _{h}\) as
which gives
Finally, the global tangent stiffness matrix can be given by
Appendix B: Sensitivity analysis
The sensitivities of the objective function J and constraint function g with respect to the design variable z_{ℓ} can be obtained from the adjoint method as
respectively, where λ_{J} and λ_{g} are the vectors of adjoint variables given by
respectively, with \(\mathbf {K}^{\alpha }_{T}\) being evaluated at the converged solution of each optimization step and black∂α_{m}/∂z_{ℓ} = [P]_{mℓ}.
In the above expressions for sensitivity analysis, the detailed expressions of ∂J/∂α_{m}, ∂J/∂ρ_{m}, ∂g/∂ρ_{m} and ∂R/∂α_{m} are given below:
Additionally, the detailed expressions of ∂κ_{m}/∂α_{m} in the above expressions can be obtained in the similar manner as \(\partial \kappa _{m}/\partial \nabla _{E_{m}}\phi _{h}\) (i.e., (48)–(49)) as
with κ_{m} and \(\nabla _{E_{m}}\phi _{h}\) being evaluated at the converged solution of each optimization step. Once ∂κ_{m}/∂α_{m} is obtained, we can further compute ∂ρ_{m}/∂α_{m} and ∂ρ_{m}/∂Φ based on (34) as
and
where both ∂κ/∂α_{m} and \(\partial \kappa /\partial \nabla _{E_{m}}\phi _{h}\) are obtained by evaluating (58) and (49) at the converged solution of each optimization step.
Rights and permissions
About this article
Cite this article
Zhang, X.S., Li, W. & Ouyang, Y. Paved guideway topology optimization for pedestrian traffic under Nash equilibrium. Struct Multidisc Optim (2021). https://doi.org/10.1007/s00158020027671
Received:
Revised:
Accepted:
Published:
Keywords
 Topology optimization
 Guideway network design
 Continuous traffic equilibrium
 Transportation engineering
 Nash equilibirum