Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings

Ahmed El Kaimbillah; Oussama Bourihane; Bouazza Braikat; Mohammad Jamal; Foudil Mohri; Noureddine Damil

doi:10.3934/dcdss.2019113

Article Contents

2019, Volume 12, Issue 6: 1685-1708. Doi: 10.3934/dcdss.2019113

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Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings

1.
Laboratoire d'Ingénierie et Matériaux (LIMAT), Faculté des Sciences Ben M'Sik, Hassan Ⅱ University of Casablanca, Avenue Cdt Driss El Harti B.P. 7955, Sidi Othman, Casablanca, Morocco
2.
Laboratoire de Génie Mécanique (LGM), Faculté des Sciences et Techniques, Fès, Université Sidi Mohamed Ben Abdellah, Route d'Imouzzer B.P. 2202, Fès, Maroc
3.
Laboratoire d'Étude des Microstructures et de Mécanique des Matériaux (LEM3), Université de Lorraine, Metz, CNRS UMR 7239, Ile du Saulcy, 57057, France

* Corresponding author: Bouazza Braikat
* Corresponding author: Bouazza Braikat

Received: October 31, 2017

Revised: March 31, 2018

Published: April 2019

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Abstract

This paper aims to investigate, in large displacement and torsion context, the nonlinear dynamic behavior of thin-walled beams with open cross section subjected to various loadings by high-order implicit solvers. These homotopy transformations consist to modify the nonlinear discretized dynamic problem by introducing an arbitrary invertible pre-conditioner $ [K^\star] $ and an arbitrary path following parameter. The nonlinear strongly coupled equations of these structures are derived by using a $ 3D $ nonlinear dynamic model which accounts for large displacements and large torsion without any assumption on torsion angle amplitude. Coupling complex structural phenomena such that warping, bending-bending, and flexural-torsion are taken into account.

Two examples of great practical interest of nonlinear dynamic problems of various thin-walled beams with open section are presented to validate the efficiency and accuracy of high-order implicit solvers. The obtained results show that the proposed homotopy transformations reveal a few number of matrix triangulations. A comparison with Abaqus code is presented.

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Figure 1. Thin-walled beam with open cross section, co-ordinates of the point $M$ on the cross section contour

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Figure 2. Axial force $N$, bending moments $M_{y}$ and $M_{z}$, bimoment $B_{\omega}$ and St-Venant torsion moment $M_{sv}$

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Figure 3. Section beam under concentrated and distributed forces

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Figure 4. External dynamical loading and its time evolution applied on the U-mono-symmetrical thin-walled beam with open cross section

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Figure 5. Geometrical characteristics of sections $A$ and $B$

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Figure 6. Response curves obtained by the high-order implicit solver $Alg_3$ and by Abaqus code, Time evolution of displacement components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$

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Figure 7. Cantilever bi-symmetrical beam with steel I cross section under eccentric loading and its time evolution

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Figure 8. Thin-walled beam with steel I cross section under transverse eccentric force $F_{z}(t)$ and its point of application

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Figure 9. Response curves obtained by the high-order implicit solver $Alg_3$, by Abaqus code and by Sapountzakis: Time evolution of components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$

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Table 1. Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Influence of time step

Solvers	$Alg_1$		$Alg_2$		$Alg_3$
$\Delta t$	Optimal order	$Log\|Res\|$	Optimal order	$Log\|Res\|$	Optimal order	$Log\|Res\|$
$10^{-3}$	$10$	$-3.73$	$9$	$-3.71$	$8$	$-3.71$
$2\, 10^{-3}$	$12$	$-3.72$	$10$	$-3.71$	$9$	$-3.70$
$3\, 10^{-3}$	$13$	$-3.69$	$11$	$-3.67$	$10$	$-3.65$

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Table 2. Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Effect of truncation order

Solver	$Alg_1$			$Alg_2$			$Alg_3$
$p$	$IM$	$RHS$	$CPU(s)$	$IM$	$RHS$	$CPU(s)$	$IM$	$RHS$	$CPU(s)$
$7$	$a_{max}<1$			$a_{max}<1$			$a_{max}<1$
$8$	$a_{max}<1$			$a_{max}<1$			$2995$	$32000$	$3890$
$9$	$a_{max}<1$			$2810$	$36000$	$4252$	$2711$	$36000$	$4102$
$10$	2850	40000	4900	$2600$	$40000$	$4470$	$2480$	$40000$	$4262$
$15$	630	60000	12376	$612$	$60000$	$12023$	$520$	$60000$	$10210$
$20$	$320$	$80000$	$25896$	$309$	$80000$	$25000$	$280$	$80000$	$22640$

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Table 3. Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Influence of time step

Solver	$Alg_1$		$Alg_2$		$Alg_3$
$\Delta t$	Optimal order	$Log\|Res\|$	Optimal order	$Log\|Res\|$	Optimal order	$Log\|Res\|$
$10^{-3}$	$6$	$-5.23$	$4$	$-5.2$	$3$	$-5.13$
$2\, 10^{-3}$	$12$	$-5.10$	$9$	$-4.80$	$7$	$-4.62$
$3\, 10^{-3}$	$14$	$-4.91$	$11$	$-4.79$	$8$	$-4.60$
$4\, 10^{-3}$	$15$	$-4.88$	$12$	$-4.70$	$10$	$-4.55$

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Table 4. Comparison between three solvers $Alg_1$, $Alg_2$ and $Alg_3$: Effect of truncation order

Solver	$Alg_1$			$Alg_2$			$Alg_3$
$p$	$IM$	$RHS$	$CPU(s)$	$IM$	$RHS$	$CPU(s)$	$IM$	$RHS$	$CPU(s)$
$2$	$a_{max}<1$			$a_{max}<1$			$a_{max}<1$
$3$	$a_{max}<1$			$a_{max}<1$			$12$	$600$	$26$
$4$	$a_{max}<1$			$13$	$800$	$42$	$11$	$800$	$30$
$5$	$a_{max}<1$			$8$	$1000$	$46$	$6$	$1000$	$34$
$6$	$15$	$1200$	$102$	$7$	$1200$	$54$	$5$	$1200$	$36$
$7$	$4$	$1400$	$126$	$2$	$1400$	$65$	$1$	$1400$	$40$

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