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.

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|_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))^− 1Q, where we recall that κ = [κ₁,...,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 non-zero initial guess for vector F which, according to Fig. 1, will can lead to well-conditioned 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:

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

(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 R_r = 4. The AAR iteration is terminated when the ℓ₂-norm of the residual vector R_F(α,F) is below 10^− 3. The corresponding Φ = (K^α(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 Φ^k+ 1 = Φ^k + ΔΦ^k until the ℓ₂-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_ℓ/∂Φ_j 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_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

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

(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_ℓ 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 λ_J and λ_g 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 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:

$$ \begin{array}{@{}rcl@{}} \frac{\partial J(\boldsymbol{\alpha},\boldsymbol{\rho})}{\partial \alpha_{m}}&=&\beta C_{B} v_{m}; \quad\quad\frac{\partial J(\boldsymbol{\alpha},\boldsymbol{\rho})}{\partial \rho_{m}}=(1-\beta) C_{T} v_{m}; \end{array} $$

(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 ∂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

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

(58)

with 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

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

(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/∂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.