# American Institute of Mathematical Sciences

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September  2019, 8(3): 489-502. doi: 10.3934/eect.2019024

## A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments

 1 Departamento de Matemáticas, Universidade da Coruña, ETS de Ingenieros de Caminos, Canales y Puertos, Campus de Elviña, 15071 A Coruña, Spain 2 Departamento de Matemática Aplicada I, Universidade de Vigo, ETSI Telecomunicación, Campus As Lagoas Marcosende s/n, 36310 Vigo, Spain 3 Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy

* Corresponding author: José R. Fernández

Received  April 2018 Revised  November 2018 Published  May 2019

Fund Project: This work has been supported by Ministerio de Economía y Competitividad under the project MTM2015-66640-P (with the participation of FEDER).

In this paper we study, from the numerical point of view, a dynamic problem which models a suspension bridge system. This problem is written as a nonlinear system of hyperbolic partial differential equations in terms of the displacements of the bridge and of the cable. By using the respective velocities, its variational formulation leads to a coupled system of parabolic nonlinear variational equations. An existence and uniqueness result, and an exponential energy decay property, are recalled. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme. A discrete stability property is shown and a priori error estimates are proved, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some numerical results are shown to demonstrate the accuracy of the approximation and the behaviour of the solution.

Citation: Marco Campo, José R. Fernández, Maria Grazia Naso. A dynamic problem involving a coupled suspension bridge system: Numerical analysis and computational experiments. Evolution Equations & Control Theory, 2019, 8 (3) : 489-502. doi: 10.3934/eect.2019024
##### References:

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##### References:
Example 1: Asymptotic behaviour of the numerical scheme
Example 2: Oscillations of the bridge for different values of p
Example 3: Bridge and cable deformed configurations at final time for different values of $k_*^2.$
Example 1: Numerical errors for some discretization parameters
 $nd \downarrow k \to$ $10^{-1}$ $10^{-2}$ $10^{-3}$ $10^{-4}$ $10$ 0.1435116 0.0868544 0.0923330 0.0940042 $10^2$ 0.1639226 0.0174114 0.0070235 0.0069553 $10^3$ 0.1641941 0.0161108 0.0017435 0.0007232 $10^4$ 0.1646557 0.0163375 0.0015935 0.0001722
 $nd \downarrow k \to$ $10^{-1}$ $10^{-2}$ $10^{-3}$ $10^{-4}$ $10$ 0.1435116 0.0868544 0.0923330 0.0940042 $10^2$ 0.1639226 0.0174114 0.0070235 0.0069553 $10^3$ 0.1641941 0.0161108 0.0017435 0.0007232 $10^4$ 0.1646557 0.0163375 0.0015935 0.0001722
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