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Numerical approximation of von Kármán viscoelastic plates

  • * Corresponding author: Martin Kružík

    * Corresponding author: Martin Kružík 

This paper is dedicated to Alexander Mielke in the occasion of his 60th birthday

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  • We consider metric gradient flows and their discretizations in time and space. We prove an abstract convergence result for time-space discretizations and identify their limits as curves of maximal slope. As an application, we consider a finite element approximation of a quasistatic evolution for viscoelastic von Kármán plates [44]. Computational experiments exploiting C1 finite elements are provided, too.

    Mathematics Subject Classification: Primary: 74D05, 74D10, 35A15, 35Q74, 49J45; Secondary: 49S05.


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  • Figure 1.  A rectangular mesh with Gauss integration points: midpoints are used for evaluation of Q1 elements (left) and four points for evaluation of Bogner-Fox-Schmit elements (right)

    Figure 2.  Time sequence of energy minimizers $ (u, v) $ in Benchmark Ⅰ. To emphasize the mesh deformation, the nodes displacement is magnified by a factor of 4

    Figure 3.  Time sequence of energy minimizers $ (u, v) $ in Benchmark Ⅱ. To emphasize the mesh deformation, the nodes displacement is magnified by a factor of 7

    Figure 4.  Example of $ C^1 $ approximation by the Bogner-Fox-Schmit rectangular elements: the function $ v_0 = (1-x_1^2)^2(1-x_2^2)^2 $ from Benchmark 1 is represented by its value (A), gradient components (C), (D) and the second mixed derivative (B) in all mesh nodes.

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