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October  2019, 12(6): 1685-1708. doi: 10.3934/dcdss.2019113

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

Received  November 2017 Revised  April 2018 Published  November 2018

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.

Citation: Ahmed El Kaimbillah, Oussama Bourihane, Bouazza Braikat, Mohammad Jamal, Foudil Mohri, Noureddine Damil. Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1685-1708. doi: 10.3934/dcdss.2019113
References:
[1]

Abaqus, Version 6.11 Documentation, Dassautt Systemes Simulia Corp, Providence, RI, USA, 2011. Google Scholar

[2]

E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer series in Computational Mathematics, 1990. doi: 10.1007/978-3-642-61257-2.  Google Scholar

[3]

R. D. AmbrosiniJ. D. Riera and R. F. Danesi, Dynamic analysis of thin-walled and variable open section beams with shear flexibility, International Journal for Numerical Methods in Engineering, 38 (1995), 2867-2885.   Google Scholar

[4]

K. J. Bathe, Finite Elements Procedures, Prentice-Hall, New Jersey, 1996. Google Scholar

[5]

J. L. Batoz and G. Dhatt, Modélisation des structures par éléments finis, Hermès, Paris, 1990. Google Scholar

[6]

K. BehdinanM. C. Stylianou and B. Tabarrok, Co-rotational dynamic analysis of flexible beams, Computer Methods in Applied Mechanics and Engineering, 154 (1998), 151-161.   Google Scholar

[7]

P. Betsch and P. Steinmann, Constrained dynamics of geometrically exact beams, Computational Mechanics, 31 (2003), 49-59.   Google Scholar

[8]

O. BourihaneB. BraikatM. JamalF. Mohri and N. Damil, Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm, Engineering Structures, 120 (2016), 133-146.   Google Scholar

[9]

S. BoutmirB. BraikatM. JamalN. DamilB. Cochelin and M. Potier-Ferry, Des solveurs implicites d'ordre supérieurs pour les problèmes de dynamique non linéaire des structures, Revue Européenne des Eléments Finis, 13 (2004), 449-460.   Google Scholar

[10]

M. A. Crisfield, Nonlinear Finite Elements Analysis of Solids and Structures, John Willey and Sons, 1991. Google Scholar

[11]

E. Dale Martin, A technique for accelerating iterative convergence in numerical integration with application in transonic aerodynamics, Lectures notes in Physics, 47 (1976), 123-139.   Google Scholar

[12]

A. Ed-dinariH. MottaquiB. BraikatM. JamalF. Mohri and N. Damil, Large torsion analysis of thin-walled open sections beams by the asymptotic numerical method, Engineering Structures, 81 (2014), 240-255.   Google Scholar

[13]

Y. GuevelG. Girault and J. M. Cadou, Numerical comparisons of high-order nonlinear solvers for the transient $ \textbf{N} $avier-$ \textbf{S} $tokes equations based on homotopy and perturbation techniques, Journal of Computational and Applied Mathematics, 289 (2015), 356-370.  doi: 10.1016/j.cam.2014.12.008.  Google Scholar

[14]

D. Haijuan, Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open section, Thin-Walled Structures, 46 (2008), 107-112.   Google Scholar

[15]

M. JamalB. BraikatS. BoutmirN. Damil and M. Potier-Ferry, A high order implicit algorithm for solving instationary nonlinear problems, Computational Mechanics, 28 (2002), 375-380.  doi: 10.1007/s00466-002-0301-7.  Google Scholar

[16]

T. N. LeJ. M. Battini and M. Hjiaj, Efficient formulation for dynamics of corotational 2D beams, Computational Mechanics, 48 (2011), 153-161.  doi: 10.1007/s00466-011-0585-6.  Google Scholar

[17]

T. N. LeJ. M. Battini and M. Hjiaj, Corotational formulation for nonlinear dynamics of beams with arbitrary thin-walled open cross-sections, Computer and Structures, 134 (2014), 112-127.   Google Scholar

[18]

S. MesmoudiA. TimesliB. BraikatH. Lahmam and H. Zahrouni, A 2D mechanical--thermal coupled model to simulate material mixing observed in friction stir welding process, Engineering with Computers, (2017), 1-11.   Google Scholar

[19]

F. MohriN. Damil and M. Potier Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683.   Google Scholar

[20]

F. MohriL. Azrar and M. Potier-Ferry, Vibration analysis of buckled thin-walled beams with open sections, Journal of Sound and Vibration, 275 (2004), 434-446.   Google Scholar

[21]

F. MohriN. Damil and M. Potier-Ferry, Linear and nonlinear stability analyses of thin-walled beams with monsymmetric sections, Thin-Walled Structures, 48 (2010), 299-315.   Google Scholar

[22]

F. MohriN. Damil and M. Potier-Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683.   Google Scholar

[23]

F. MohriA. Ed-dinari and N. Damil, A beam finite element for nonlinear analysis of thin-walled elements, Thin Walled Structures, 46 (2008), 981-990.   Google Scholar

[24]

H. MottaquiB. Braikat and N. Damil, Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1701-1709.  doi: 10.1016/j.cma.2010.01.020.  Google Scholar

[25]

H. MottaquiB. Braikat and N. Damil, Local parameterization and the asymptotic numerical method, Mathematical Modelling of Natural Phenomena, 5 (2010), 16-22.   Google Scholar

[26]

N. Newmark, A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceeding of ASCE, (1959), 67-94.   Google Scholar

[27]

E. J. Sapountzakis and I. C. Dikaros, Nonlinear flexural-torsional dynamic analysis of beams of variable doubly symmetric cross section-application to wind turbine towers, Nonlinear Dynamics, 73 (2013), 199-227.  doi: 10.1007/s11071-013-0779-x.  Google Scholar

[28]

A. TimesliB. BraikatH. Lahmam and H. Zahrouni, A new algorithm based on moving least square method to simulate material mixing in friction stir welding, Engineering Analysis with Boundary Elements, 50 (2015), 372-380.   Google Scholar

[29]

V. Z. Vlasov, Thin walled elastic beams, Eyrolles, French translation: Pièces longues en voiles minces, Paris, 1965. Google Scholar

[30]

O. C. Zienkiewicz and R. Taylor, The Finite Element Method, Solid and Fluid Mechanics and Non-linearity, Book Company, 1987. Google Scholar

show all references

References:
[1]

Abaqus, Version 6.11 Documentation, Dassautt Systemes Simulia Corp, Providence, RI, USA, 2011. Google Scholar

[2]

E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer series in Computational Mathematics, 1990. doi: 10.1007/978-3-642-61257-2.  Google Scholar

[3]

R. D. AmbrosiniJ. D. Riera and R. F. Danesi, Dynamic analysis of thin-walled and variable open section beams with shear flexibility, International Journal for Numerical Methods in Engineering, 38 (1995), 2867-2885.   Google Scholar

[4]

K. J. Bathe, Finite Elements Procedures, Prentice-Hall, New Jersey, 1996. Google Scholar

[5]

J. L. Batoz and G. Dhatt, Modélisation des structures par éléments finis, Hermès, Paris, 1990. Google Scholar

[6]

K. BehdinanM. C. Stylianou and B. Tabarrok, Co-rotational dynamic analysis of flexible beams, Computer Methods in Applied Mechanics and Engineering, 154 (1998), 151-161.   Google Scholar

[7]

P. Betsch and P. Steinmann, Constrained dynamics of geometrically exact beams, Computational Mechanics, 31 (2003), 49-59.   Google Scholar

[8]

O. BourihaneB. BraikatM. JamalF. Mohri and N. Damil, Dynamic analysis of a thin-walled beam with open cross section subjected to dynamic loads using a high-order implicit algorithm, Engineering Structures, 120 (2016), 133-146.   Google Scholar

[9]

S. BoutmirB. BraikatM. JamalN. DamilB. Cochelin and M. Potier-Ferry, Des solveurs implicites d'ordre supérieurs pour les problèmes de dynamique non linéaire des structures, Revue Européenne des Eléments Finis, 13 (2004), 449-460.   Google Scholar

[10]

M. A. Crisfield, Nonlinear Finite Elements Analysis of Solids and Structures, John Willey and Sons, 1991. Google Scholar

[11]

E. Dale Martin, A technique for accelerating iterative convergence in numerical integration with application in transonic aerodynamics, Lectures notes in Physics, 47 (1976), 123-139.   Google Scholar

[12]

A. Ed-dinariH. MottaquiB. BraikatM. JamalF. Mohri and N. Damil, Large torsion analysis of thin-walled open sections beams by the asymptotic numerical method, Engineering Structures, 81 (2014), 240-255.   Google Scholar

[13]

Y. GuevelG. Girault and J. M. Cadou, Numerical comparisons of high-order nonlinear solvers for the transient $ \textbf{N} $avier-$ \textbf{S} $tokes equations based on homotopy and perturbation techniques, Journal of Computational and Applied Mathematics, 289 (2015), 356-370.  doi: 10.1016/j.cam.2014.12.008.  Google Scholar

[14]

D. Haijuan, Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open section, Thin-Walled Structures, 46 (2008), 107-112.   Google Scholar

[15]

M. JamalB. BraikatS. BoutmirN. Damil and M. Potier-Ferry, A high order implicit algorithm for solving instationary nonlinear problems, Computational Mechanics, 28 (2002), 375-380.  doi: 10.1007/s00466-002-0301-7.  Google Scholar

[16]

T. N. LeJ. M. Battini and M. Hjiaj, Efficient formulation for dynamics of corotational 2D beams, Computational Mechanics, 48 (2011), 153-161.  doi: 10.1007/s00466-011-0585-6.  Google Scholar

[17]

T. N. LeJ. M. Battini and M. Hjiaj, Corotational formulation for nonlinear dynamics of beams with arbitrary thin-walled open cross-sections, Computer and Structures, 134 (2014), 112-127.   Google Scholar

[18]

S. MesmoudiA. TimesliB. BraikatH. Lahmam and H. Zahrouni, A 2D mechanical--thermal coupled model to simulate material mixing observed in friction stir welding process, Engineering with Computers, (2017), 1-11.   Google Scholar

[19]

F. MohriN. Damil and M. Potier Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683.   Google Scholar

[20]

F. MohriL. Azrar and M. Potier-Ferry, Vibration analysis of buckled thin-walled beams with open sections, Journal of Sound and Vibration, 275 (2004), 434-446.   Google Scholar

[21]

F. MohriN. Damil and M. Potier-Ferry, Linear and nonlinear stability analyses of thin-walled beams with monsymmetric sections, Thin-Walled Structures, 48 (2010), 299-315.   Google Scholar

[22]

F. MohriN. Damil and M. Potier-Ferry, Large torsion finite element model for thin-walled beams, Computers and Structures, 86 (2008), 671-683.   Google Scholar

[23]

F. MohriA. Ed-dinari and N. Damil, A beam finite element for nonlinear analysis of thin-walled elements, Thin Walled Structures, 46 (2008), 981-990.   Google Scholar

[24]

H. MottaquiB. Braikat and N. Damil, Discussion about parameterization in the asymptotic numerical method: Application to nonlinear elastic shells, Computer Methods in Applied Mechanics and Engineering, 199 (2010), 1701-1709.  doi: 10.1016/j.cma.2010.01.020.  Google Scholar

[25]

H. MottaquiB. Braikat and N. Damil, Local parameterization and the asymptotic numerical method, Mathematical Modelling of Natural Phenomena, 5 (2010), 16-22.   Google Scholar

[26]

N. Newmark, A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceeding of ASCE, (1959), 67-94.   Google Scholar

[27]

E. J. Sapountzakis and I. C. Dikaros, Nonlinear flexural-torsional dynamic analysis of beams of variable doubly symmetric cross section-application to wind turbine towers, Nonlinear Dynamics, 73 (2013), 199-227.  doi: 10.1007/s11071-013-0779-x.  Google Scholar

[28]

A. TimesliB. BraikatH. Lahmam and H. Zahrouni, A new algorithm based on moving least square method to simulate material mixing in friction stir welding, Engineering Analysis with Boundary Elements, 50 (2015), 372-380.   Google Scholar

[29]

V. Z. Vlasov, Thin walled elastic beams, Eyrolles, French translation: Pièces longues en voiles minces, Paris, 1965. Google Scholar

[30]

O. C. Zienkiewicz and R. Taylor, The Finite Element Method, Solid and Fluid Mechanics and Non-linearity, Book Company, 1987. Google Scholar

Figure 1.  Thin-walled beam with open cross section, co-ordinates of the point $M$ on the cross section contour
Figure 2.  Axial force $N$, bending moments $M_{y}$ and $M_{z}$, bimoment $B_{\omega}$ and St-Venant torsion moment $M_{sv}$
Figure 3.  Section beam under concentrated and distributed forces
Figure 4.  External dynamical loading and its time evolution applied on the U-mono-symmetrical thin-walled beam with open cross section
Figure 5.  Geometrical characteristics of sections $A$ and $B$
Figure 6.  Response curves obtained by the high-order implicit solver $Alg_3$ and by Abaqus code, Time evolution of displacement components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$
Figure 7.  Cantilever bi-symmetrical beam with steel I cross section under eccentric loading and its time evolution
Figure 8.  Thin-walled beam with steel I cross section under transverse eccentric force $F_{z}(t)$ and its point of application
Figure 9.  Response curves obtained by the high-order implicit solver $Alg_3$, by Abaqus code and by Sapountzakis: Time evolution of components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$
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$$-3.65$
Solvers $Alg_1$ $Alg_2$ $Alg_3$
$\Delta t$Optimal order$Log|Res|$Optimal order$Log|Res|$Optimal order$Log|Res|$
$10^{-3}$$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$
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$$4102$
$10$2850400004900$2600$$40000$$4470$$2480$$40000$$4262$
$15$6306000012376$612$$60000$$12023$$520$$60000$$10210$
$20$$320$$80000$$25896$$309$$80000$$25000$$280$$80000$$22640$
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$$4102$
$10$2850400004900$2600$$40000$$4470$$2480$$40000$$4262$
$15$6306000012376$612$$60000$$12023$$520$$60000$$10210$
$20$$320$$80000$$25896$$309$$80000$$25000$$280$$80000$$22640$
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$
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$
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$
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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