# Paved guideway topology optimization for pedestrian traffic under Nash equilibrium

## 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 physics-based 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.

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## Notes

1. 1.

This function form is analogous to the well-known BPR function (U.S. Bureau of Public Roads), but other non-decreasing functions are also acceptable.

2. 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. 3.

Also called user equilibrium (UE) in the transportation literature, as originally described in Wardrop (1952).

4. 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. 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):

α 0 :

α :

$$\alpha _{\max \limits }$$ :

$$\bar {q}$$ :

traffic inflow or outflow at Ω

β :

α :

λ g :

adjoint vector in sensitivity analysis of constraint function g

λ J :

adjoint vector in sensitivity analysis of objective function J

Ω :

two-dimensional space

ρ :

discretized pedestrian density vector

$$\boldsymbol {\tilde {F}}$$ :

matrix storing the F of previous m AAR iterations

$$\boldsymbol {\tilde {R}}$$ :

matrix storing the RF of previous m AAR iterations

F :

global flux magnitude vector

G :

matrix mapping global degree of freedom vector Φi to ∇ϕ at the center of element

I :

identity matrix

$$\boldsymbol {k}^{0}_{\ell }$$ :

element stiffness matrix of element with κ = 1

K α :

global stiffness matrix

M :

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 :

κ value associated with element , assumed to be constant inside element

$$\kappa _{\min \limits }$$ :

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

$${\mathscr{U}}_{h}$$ :

finite-dimensional trial function space

ϕ :

total generalized cost

ϕ h :

finite-dimensional total generalized cost

Φ i :

value of ϕh at node i

ψ :

test function in weak form

ψ h :

finite-dimensional test function

Ψi :

values of ψh 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 :

area of element

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 p-norm of a n-dimensional vector

P r :

period of applying Anderson extrapolation in AAR

q :

traffic inflow or outflow in Ω

R :

r 0 :

T h :

finite element partition

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## 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 CMMI-1662825.

## Author information

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Correspondence to Xiaojia Shelly Zhang.

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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 fixed-point 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

$$\boldsymbol{R}_{F}(\boldsymbol{\alpha},\boldsymbol{F})={\sum}_{\ell}^{n}\kappa_{\ell}(\alpha_{\ell},|f|_{\ell})|\nabla_{E_{\ell}}\phi_{h}(\boldsymbol{F})|-\boldsymbol{F}=\boldsymbol{0},$$
(38)

where |f| denotes the flux in element , $$\boldsymbol {F}\in \mathbb {R}^{n\time 1}$$ is a vector collecting all the element fluxes, namely, F = [|f|1,...,|f|n]T; and κ(α,|f|) 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 RF(α,F) uses the flux vector F as the independent vector and Φ(F) is obtained from Φ = (Kα(κ))− 1Q, where we recall that κ = [κ1,...,κn]T.

Although defined in different forms and having different independent variables, we can show that RF(α,F) = 0 and R(α,Φ) = 0 are in fact equivalent in the sense that solution of RF(α,F) = 0 is also a solution of R(α,Φ) = 0 and vice versa. The advantage of using RF(α,F) in the AAR is that it allows us to use a non-zero initial guess for vector F which, according to Fig. 1, will can lead to well-conditioned matrix Kα. 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:

$$\boldsymbol{F}^{(k+1)} = \max\Big(\boldsymbol{F}^{(k)} + \mathbf{B}^{(k)} \boldsymbol{R}_{F}(\boldsymbol{\alpha},\boldsymbol{F}^{(k)}),\boldsymbol{0}\Big),$$
(39)

where F(k) is the flux vector at the k th AAR iteration, $$\max \limits (\cdot ,\cdot )$$ stands for the element-wise maximum operator between the two vectors, and the matrix B(k) is defined by:

$$\mathbf{B}^{(k)}{}={}\left\{{}\begin{array}{ll}{\theta \mathbf{I}} & {\text { if }(k{}+{}1) / Pr \notin \mathbb{N}} \\ {\theta \mathbf{I}-{}\left( {}\tilde{\mathbf{F}}^{(k)}{}+{}\theta \tilde{\mathbf{R}}^{(k)}{}\right){}\left( {}\tilde{\mathbf{R}}^{(k),T} \tilde{\mathbf{R}}^{(k)}{}\right)^{-1} \tilde{\mathbf{R}}^{(k),T}} & {\text { if }(k{}+{}1) / Pr \in \mathbb{N}}\end{array}\right.$$
(40)

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:

$$\tilde{\mathbf{F}}^{(k)}=\begin{bmatrix} {\varDelta}\boldsymbol{F}^{(k-m)} & {\varDelta} \boldsymbol{F}^{(k-m+1)} & \ldots& {\varDelta} \boldsymbol{F}^{(k-1)} \end{bmatrix}$$
(41)
$$\tilde{\mathbf{R}}^{(k)}=\begin{bmatrix} {\varDelta} \boldsymbol{R}_{F}(\boldsymbol{\alpha},\boldsymbol{F}^{(k-m)}) & {\varDelta} \boldsymbol{R}_{F}(\boldsymbol{\alpha},\boldsymbol{F}^{(k-m+1)}) & {\ldots} & {\varDelta} \boldsymbol{R}_{F}(\boldsymbol{\alpha},\boldsymbol{F}^{(k-1)}) \end{bmatrix},$$
(42)

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 quasi-Newton 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 Rr = 4. The AAR iteration is terminated when the 2-norm of the residual vector RF(α,F) is below 10− 3. The corresponding Φ = (Kα(κ))− 1Q 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

$$\boldsymbol{R}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})\approx\boldsymbol{R}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}}^{(k)})+\mathbf{K}^{\alpha}_{T}\Big(\boldsymbol{\kappa}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}}^{(k)})\Big){\varDelta}\boldsymbol{{\varPhi}}^{(k)}=\boldsymbol{0},$$
(43)

where $$\mathbf {K}^{\alpha }_{T}\doteq \partial \boldsymbol {R}/\partial \boldsymbol {{\varPhi }}$$ is the tangent stiffness matrix. Solving the linearized system gives

$${\varDelta}\boldsymbol{{\varPhi}}^{(k)}=\Big[\mathbf{K}^{\alpha}_{T}\Big(\boldsymbol{\kappa}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}}^{(k)})\Big)\Big]^{-1}\boldsymbol{R}(\boldsymbol{{\varPhi}}^{k}),$$
(44)

which leads to the recurrent update formula Φk+ 1 = Φk + ΔΦk until the 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

$$\begin{array}{@{}rcl@{}} \mathbf{K}^{\alpha}_{T}\Big(\boldsymbol{\kappa}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})\Big)&=&\frac{\partial \boldsymbol{R}}{\partial \boldsymbol{{\varPhi}}}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})\\ &=& \frac{\partial \mathbf{K}^{\alpha}}{\partial \boldsymbol{{\varPhi}}}\Big(\boldsymbol{\kappa}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})\Big) \boldsymbol{{\varPhi}} + \mathbf{K}^{\alpha}\Big(\boldsymbol{\kappa}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})\Big) \end{array}$$
(45)

To obtain an explicit expression of $$\mathbf {K}^{\alpha }_{T}$$, we first compute the following local matrix as

$$[\mathbf{k}^{{\varPhi}}_{\ell}]_{ij}=\frac{\partial \kappa_{\ell}}{\partial {\varPhi}_{j}} \Big({\sum}_{m} [\mathbf{k}^{0}_{\ell}]_{im}{\varPhi}_{m}\Big),$$
(46)

where the derivative κ/Φj can be computed as follows

$$\frac{\partial \kappa_{\ell}}{\partial {\varPhi}_{j}} = \frac{\partial \kappa_{\ell}}{\partial |\nabla_{E_{\ell}} \phi_{h}|} \frac{\partial |\nabla_{E_{\ell}} \phi_{h}|}{\partial {\varPhi}_{j}} = \frac{\partial \kappa_{\ell}}{\partial |\nabla_{E_{\ell}} \phi_{h}|}\frac{{\sum}_{i}[\mathbf{M}_{\ell}]_{ji}{\varPhi}_{i}}{2\sqrt{\boldsymbol{{\varPhi}}^{T}\mathbf{M}_{\ell}\boldsymbol{{\varPhi}}}}$$
(47)

Since κ 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

$$\begin{array}{@{}rcl@{}} \frac{d h}{d |\nabla_{E_{\ell}}\phi_{h}|}&=&\frac{\partial h(\kappa_{\ell},\alpha_{\ell},|\nabla_{E_{\ell}}\phi_{h}|)}{\partial \kappa_{\ell}}\frac{\partial \kappa_{\ell}}{\partial |\nabla_{E_{\ell}}\phi_{h}|}\\ &&+\frac{\partial h(\kappa_{\ell},\alpha_{\ell},|\nabla_{E_{\ell}}\phi_{h}|)}{\partial |\nabla_{E_{\ell}}\phi_{h}|}=0, \end{array}$$
(48)

which gives

$$\frac{\partial \kappa_{\ell}}{\partial |\nabla_{E_{\ell}}\phi_{h}|}=-\frac{\partial h(\kappa_{\ell},\alpha_{\ell},|\nabla_{E_{\ell}}\phi_{h}|)/\partial |\nabla_{E_{\ell}}\phi_{h}|}{\partial h(\kappa_{\ell},\alpha_{\ell},|\nabla_{E_{\ell}}\phi_{h}|)/\partial \kappa_{\ell}}$$
(49)

Finally, the global tangent stiffness matrix can be given by

$$\mathbf{K}^{\alpha}_{T}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})={\sum}_{\ell=1}^{M}\Big(\mathbf{k}^{{\varPhi}}_{\ell} + \kappa_{\ell}(\alpha_{\ell},|\nabla_{E_{\ell}}\phi_{h}|)\mathbf{k}^{0}_{\ell}\Big)$$
(50)

### Appendix B: Sensitivity analysis

The sensitivities of the objective function J and constraint function g with respect to the design variable z can be obtained from the adjoint method as

$$\begin{array}{@{}rcl@{}} &&{}\frac{\partial J}{\partial z_{\ell}}={\sum}_{m=1}^{n}\Big[\frac{\partial J(\boldsymbol{\alpha},\boldsymbol{\rho})}{\partial \alpha_{m}}+\frac{\partial J(\boldsymbol{\alpha},\boldsymbol{\rho})}{\partial \rho_{m}}\frac{\partial \rho_{m}}{\partial \alpha_{m}}+\boldsymbol{\lambda}_{J}^{T}\frac{\partial \boldsymbol{R}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})}{\partial \alpha_{m}}\Big]\frac{\partial \alpha_{m}}{\partial z_{\ell}}\quad \text{ and} \end{array}$$
(51)
$$\begin{array}{@{}rcl@{}} &&{}\frac{\partial g}{\partial z_{\ell}}={\sum}_{m=1}^{n}\Big[\frac{\partial g(\boldsymbol{\rho})}{\partial \rho_{m}}\frac{\partial \rho_{m}}{\partial \alpha_{m}} +\boldsymbol{\lambda}_{g}^{T}\frac{\partial \boldsymbol{R}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})}{\partial \alpha_{m}}\Big]\frac{\partial \alpha_{m}}{\partial z_{\ell}} \end{array}$$
(52)

respectively, where λJ and λg are the vectors of adjoint variables given by

$$\begin{array}{@{}rcl@{}} &&\boldsymbol{\lambda}_{J}=-\Big(\mathbf{K}^{\alpha}_{T}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})\Big)^{-T}\Big({\sum}_{m=1}^{n}\frac{\partial J(\boldsymbol{\alpha},\boldsymbol{\rho})}{\partial \rho_{m}}\frac{\partial \rho_{m}}{\partial \boldsymbol{{\varPhi}}}\Big)\quad\text{ and} \end{array}$$
(53)
$$\begin{array}{@{}rcl@{}} &&\boldsymbol{\lambda}_{g}=-\Big(\mathbf{K}^{\alpha}_{T}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})\Big)^{-T}\Big({\sum}_{m=1}^{n}\frac{\partial g(\boldsymbol{\rho})}{\partial \rho_{m}}\frac{\partial\rho_{m}}{\partial \boldsymbol{{\varPhi}}}\Big) \end{array}$$
(54)

respectively, with $$\mathbf {K}^{\alpha }_{T}$$ being evaluated at the converged solution of each optimization step and blackαm/z = [P]m.

In the above expressions for sensitivity analysis, the detailed expressions of J/αm, J/ρm, g/ρm and R/αm are given below:

$$\begin{array}{@{}rcl@{}} \frac{\partial J(\boldsymbol{\alpha},\boldsymbol{\rho})}{\partial \alpha_{m}}&=&\beta C_{B} v_{m}; \quad\quad\frac{\partial J(\boldsymbol{\alpha},\boldsymbol{\rho})}{\partial \rho_{m}}=(1-\beta) C_{T} v_{m}; \end{array}$$
(55)
$$\begin{array}{@{}rcl@{}} \frac{\partial g(\boldsymbol{\rho})}{\partial \rho_{m}}&=& \Big({\sum}_{j=1}^{n}\Big(\rho_{j}\Big)^{{p_{n}}}\Big)^{\frac{1}{{p_{n}}}-1}\Big(\rho_{m}\Big)^{{p_{n}}-1}; \end{array}$$
(56)
$$\begin{array}{@{}rcl@{}} \frac{\partial \boldsymbol{R}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})}{\partial \alpha_{m}}&=&\frac{\partial \mathbf{K}^{\alpha}(\boldsymbol{\alpha},\boldsymbol{{\varPhi}})}{\partial \alpha_{m}}\boldsymbol{{\varPhi}}=\frac{\partial \kappa_{m}}{\partial \alpha_{m}}\mathbf{k}^{0}_{m}\boldsymbol{{\varPhi}} \end{array}$$
(57)

Additionally, the detailed expressions of κm/αm 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

$$\frac{\partial \kappa_{m}}{\partial \alpha_{m}}=-\frac{\partial h(\kappa_{m},\alpha_{m},|\nabla_{E_{m}}\phi_{h}|)/\partial \alpha_{m}}{\partial h(\kappa_{m},\alpha_{m},|\nabla_{E_{m}}\phi_{h}|)/\partial \kappa_{m}},$$
(58)

with κm and $$|\nabla _{E_{m}}\phi _{h}|$$ being evaluated at the converged solution of each optimization step. Once κm/αm is obtained, we can further compute ρm/αm and ρm/Φ based on (34) as

$$\begin{array}{@{}rcl@{}} \frac{\partial \rho_{m}(\alpha_{m},\boldsymbol{{\varPhi}})}{\partial \alpha_{m}}&=&\frac{\partial \rho_{m}}{\partial \alpha_{m}}+\frac{\partial \rho_{m}}{\partial \kappa_{m}}\frac{\partial \kappa_{m}}{\partial \alpha_{m}}\\ &=&-|\nabla_{E_{m}}\phi_{h}|\frac{\kappa_{m}}{\alpha_{m}}\Bigg[\frac{b_{2}}{\alpha_{m}}+g\Big(\frac{\kappa_{m}|\nabla_{E_{m}}\phi_{h}|}{\alpha_{m}^{g+1}}\Big)^{g}\Bigg]\\ &&+ |\nabla_{E_{m}}\phi_{h}|\Bigg[\frac{b_{2}}{\alpha_{m}}+(g+1)\Big(\frac{\kappa_{m}|\nabla_{E_{m}}\phi_{h}|}{\alpha_{m}}\Big)^{g}\Bigg]\frac{\partial \kappa_{m}}{\partial \alpha_{m}} \end{array}$$
(59)

and

$$\begin{array}{@{}rcl@{}} \frac{\partial \rho_{m}(\alpha_{m},\boldsymbol{{\varPhi}})}{\partial \boldsymbol{{\varPhi}}}&=&\Bigg[\frac{\partial \rho_{m}}{\partial |\nabla_{E_{m}}\phi_{h}|}+\frac{\partial \rho_{m}}{\partial \kappa_{m}}\frac{\partial \kappa_{m}}{\partial |\nabla_{E_{m}}\phi_{h}|}\Bigg]\frac{\partial |\nabla_{E_{m}}\phi_{h}|}{\partial\boldsymbol{{\varPhi}}}\\ &=&\Big(\kappa_{m}+|\nabla_{E_{m}}\phi_{h}|\frac{\partial \kappa_{m}}{\partial |\nabla_{E_{m}}\phi_{h}|}\Big)\\ &&\times\Bigg[\frac{b_{2}}{\alpha_{m}}+(g+1)\Big(\frac{\kappa_{m}|\nabla_{E_{m}}\phi_{h}|}{\alpha_{m}}\Big)^{g}\Bigg]\frac{\mathbf{M}_{m}\boldsymbol{{\varPhi}}}{\sqrt{\boldsymbol{{\varPhi}}^{T}\mathbf{M}_{m}\boldsymbol{{\varPhi}}}}, \end{array}$$
(60)

where both κ/αm and $$\partial \kappa /\partial |\nabla _{E_{m}}\phi _{h}|$$ are obtained by evaluating (58) and (49) at the converged solution of each optimization step.

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Zhang, X.S., Li, W. & Ouyang, Y. Paved guideway topology optimization for pedestrian traffic under Nash equilibrium. Struct Multidisc Optim 63, 1405–1426 (2021). https://doi.org/10.1007/s00158-020-02767-1

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### Keywords

• Topology optimization
• Guideway network design
• Continuous traffic equilibrium
• Transportation engineering
• Nash equilibirum