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

  • * Corresponding author: José R. Fernández

    * Corresponding author: José R. Fernández 
This work has been supported by Ministerio de Economía y Competitividad under the project MTM2015-66640-P (with the participation of FEDER).
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  • 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.

    Mathematics Subject Classification: Primary: 74B20, 65M60, 65M15; Secondary: 74K10, 74H15.


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  • Figure 1.  Example 1: Asymptotic behaviour of the numerical scheme

    Figure 2.  Example 2: Oscillations of the bridge for different values of p

    Figure 3.  Example 3: Bridge and cable deformed configurations at final time for different values of $ k_*^2. $

    Table 1.  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
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