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Topology optimization of high-speed rail bridges considering passenger comfort

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Worldwide growth in high-speed rail (HSR) networks has brought a demand for improved structural performance of the bridges that make up a large portion of many of these HSR systems. While some improvement has been made via optimization either of the bridge or the passenger cars, the design criteria of passenger comfort has yet to be addressed in bridge optimization. The transient dynamics of vehicle–bridge interaction between a high-speed train and bridge make such structural optimization of these systems challenging. In this paper, we derive an approach for topology optimization of high-speed rail bridges including vehicle–bridge interaction that enables direct consideration of the passenger comfort in the objective function. Assuming constant contact between the vehicle’s wheels and the bridge, the two systems are combined into a single-state space system. The resulting system matrices are time dependent, as they are a function of the wheel contact locations which change as the vehicles move over the bridge. The equations of motion and the adjoint sensitivities are derived and solved numerically in the time domain. Several numerical examples are provided based on high-speed rail applications that minimize a multi-objective function comprised of bridge and vehicle responses, including passenger comfort. These examples generate topologies that improve passenger comfort at only a small cost to the bridge response and demonstrate the dependence of optimal topology on train speed and length. The proposed method offers the potential for improving high-speed rail passenger comfort through optimization of bridge topology by accounting for the vehicle–bridge interaction effects.

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This work made use of the Illinois Campus Cluster, a computing resource that is operated by the Illinois Campus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA) and which is supported by funds from the University of Illinois at Urbana-Champaign.

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Correspondence to Thomas Golecki.

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Appendix A: Nomenclature

Appendix A: Nomenclature

State space system variables


\({\mathbf{A}}_{\bullet }\)

System matrix

\({\mathbf{B}}_{\bullet }\)

Input matrix

\({\mathbf{C}}_{\bullet }\)

Output matrix

\({\mathbf{D}}_{\bullet }\)

Feedthrough matrix

\({\mathbf{x}}_{\bullet }\)

State vector

\({\mathbf{y}}_{\bullet }\)

Output vector

\({\mathbf{A}}_{\mathrm{a}}, {\mathbf{B}}_{\mathrm{a}}, {\mathbf{C}}_{\mathrm{a}}, {\mathbf{D}}_{\mathrm{a}},{{\varvec{x}}}_{\mathrm{a}},{{\varvec{y}}}_{\mathrm{a}}\)

State space system variables of the augmented system

\({\mathbf{A}}_{\mathrm{f}}, {\mathbf{B}}_{\mathrm{f}}, {\mathbf{C}}_{\mathrm{f}}, {\mathbf{D}}_{\mathrm{f}},{{\varvec{x}}}_{\mathrm{f}},{{\varvec{y}}}_{\mathrm{f}}\)

State space system variables of the filter system

\({\mathbf{A}}_{\mathrm{vb}}, {\mathbf{B}}_{\mathrm{vb}}, {\mathbf{C}}_{\mathrm{vb}}, {\mathbf{D}}_{\mathrm{vb}},{{\varvec{x}}}_{\mathrm{vb}},{{\varvec{y}}}_{\mathrm{vb}}\)

State space system variables of the vehicle–bridge system

Structural system variables


\({\mathbf{M}}_{\mathrm{b}},{\mathbf{C}}_{\mathrm{b}}, {\mathbf{K}}_{\mathrm{b}}\)

Mass, damping, and stiffness matrices of the bridge

\({ \widetilde{\mathbf{M}}}_{\mathrm{b}}, {\widetilde{\mathbf{C}}}_{\mathrm{b}}, {\widetilde{\mathbf{K}}}_{\mathrm{b}}\)

Guyan reduced mass, damping, and stiffness matrices of the bridge

\({{\mathbf{M}}_{\mathrm{vb}},\mathbf{C}}_{\mathrm{vb}},\boldsymbol{ }{\mathbf{K}}_{\mathrm{vb}},\boldsymbol{ }{\mathbf{G}}_{\mathrm{vb}}\)

Mass, damping, stiffness, and load effect matrices of the vehicle–bridge system

\(\left[\begin{array}{cc}{{\varvec{m}}}_{\mathrm{uu}}& {{\varvec{m}}}_{\mathrm{uw}}\\ {{\varvec{m}}}_{\mathrm{wu}}& {{\varvec{m}}}_{\mathrm{ww}}\end{array}\right]\)

Partitioned mass matrix of the vehicle(s)

\(\left[\begin{array}{cc}{{\varvec{c}}}_{\mathrm{uu}}& {{\varvec{c}}}_{\mathrm{uw}}\\ {{\varvec{c}}}_{\mathrm{wu}}& {{\varvec{c}}}_{\mathrm{ww}}\end{array}\right]\)

Partitioned damping matrix of the vehicle(s)

\(\left[\begin{array}{cc}{{\varvec{k}}}_{\mathrm{uu}}& {{\varvec{k}}}_{\mathrm{uw}}\\ {{\varvec{k}}}_{\mathrm{wu}}& {{\varvec{k}}}_{\mathrm{ww}}\end{array}\right]\)

Partitioned stiffness matrix of the vehicle(s)

Other variables



Penalized elastic modulus


Elastic modulus of full density material


Gravitational constant


Identity matrix


Objective function

\({ J}_{\mathrm{c}}\)

Objective function combining bridge displacement and passenger comfort

\({ J}_{\mathrm{pc}}\)

Passenger comfort objective function

\({ K}_{\mathrm{p}}, {C}_{\mathrm{p}}\)

Stiffness and damping of vehicle primary suspension system defined in Table 2

\({ K}_{\mathrm{s}}, {C}_{\mathrm{s}}\)

Stiffness and damping of vehicle secondary suspension system defined in Table 2


Lagrangian form of objective function

\({ L}_{\mathrm{c}}\)

Distance from vehicle body c.g. to bogie c.g. defined in Table 2

\({ L}_{\mathrm{t}}\)

Distance from bogie c.g. to wheel defined in Table 2


Indicator of whether the \(n\) th car is on the span

\({ n}_{\mathrm{r}}\)

Number of retained DOF in Guyan reduction

\({ M}_{\mathrm{c}}, {I}_{\mathrm{c}}\)

Mass and inertia of the car body defined in Table 2

\({ M}_{\mathrm{t}}, {I}_{\mathrm{t}}\)

Mass and inertia of the bogie defined in Table 2

\({ M}_{\mathrm{w}}\)

Mass of a car wheel defined in Table 2

\({ N}_{\mathrm{el}}\)

Number of elements in the bridge model

\({\boldsymbol{ }{\varvec{N}}}_{\mathrm{vi}}\)

Function to interpolate contact location displacements from the bridge displacements


Stiffness penalty


Mass penalty

\(\mathrm{ SCI}\)

Sperling’s Comfort Index


Total simulation time


Transformation matrix used in Guyan reduction



\({\boldsymbol{ }{\varvec{u}}}_{\mathrm{b}}\)

Bridge DOF

\({ \widetilde{{\varvec{u}}}}_{\mathrm{b}}\)

DOF in the Guyan reduced bridge model

\({\boldsymbol{ }{\varvec{u}}}_{\mathrm{u}}\)

DOF of vehicle upper (non-contact) portion, shown in Fig. 1

\({\boldsymbol{ }{\varvec{u}}}_{\mathrm{w}}\)

DOF of vehicle wheels, shown in Fig. 1


Material volume used in the design

\({ V}_{\mathrm{max}}\)

Material volume constraint


Vehicle velocity


Contact location of the \(k\) th wheel

\({\boldsymbol{ }{\varvec{y}}}_{\mathrm{bd}}\)

Bridge displacement outputs

\({\boldsymbol{ }{\varvec{y}}}_{\mathrm{c}}\)

Combined bridge displacement and passenger comfort outputs


Vector of passenger comfort outputs for multiple train cars

\({ y}_{\mathrm{pc}}\)

Passenger comfort output for a single car

\(\boldsymbol{ }{\varvec{z}}\)

Vector of element densities

 \({z}_{\mathrm{min}}, {z}_{\mathrm{max}}\)

Bounds of element densities

\({ \alpha }_{1}, {\alpha }_{2}\)

Weight factors for passenger comfort and bridge displacement portions of the combined objective


Adjoint variable to compute sensitivities

\(\boldsymbol{ }{\varvec{\lambda}}\)

Adjoint variable to compute sensitivities

 \(\rho (z)\)

Penalized mass density

\({ \rho }_{0}\)

Mass density of solid material

 \(\psi \left({\varvec{y}}(t,{\varvec{z}})\right)\)

Time domain function of system responses

\({ \psi }_{\mathrm{bd}}\)

Bridge displacement responses

\({ \psi }_{\mathrm{c}}\)

Combined bridge displacement and passenger comfort responses

\({\psi }_{\mathrm{m}}\)

Combined passenger comfort responses from multiple train cars

\({\psi }_{\mathrm{pc}}\)

Passenger comfort response from a single train car

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Golecki, T., Gomez, F., Carrion, J. et al. Topology optimization of high-speed rail bridges considering passenger comfort. Struct Multidisc Optim 66, 215 (2023).

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