Numerical approximation of von Kármán viscoelastic plates

Manuel Friedrich; Martin Kružík; Jan Valdman

doi:10.3934/dcdss.2020322

Article Contents

2021, Volume 14, Issue 1: 299-319. Doi: 10.3934/dcdss.2020322

This issue Previous Article Contraction and regularizing properties of heat flows in metric measure spaces Next Article Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system

Numerical approximation of von Kármán viscoelastic plates

1.
Institute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, D-48149 Münster, Germany
2.
Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou vĕží 4, CZ-182 00 Praha 8, Czechia, Faculty of Civil Engineering, Czech Technical University, Thákurova 7, CZ-166 29 Praha 6, Czechia
3.
Czech Academy of Sciences, Institute of Information Theory and Automation, Pod vodárenskou vĕží 4, CZ-182 00 Praha 8, Czechia, Institute of Mathematics, University of South Bohemia, Branišovská 1760, CZ-370 05 České Budĕjovice, Czechia

^* Corresponding author: Martin Kružík
^* Corresponding author: Martin Kružík

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

Received: February 28, 2019

Revised: September 30, 2019

Early access: April 2020

Published: January 2021

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Abstract

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.

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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)

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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

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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

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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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