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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. Ambrosini, J. 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. Behdinan, M. 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. Bourihane, B. Braikat, M. Jamal, F. 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. Boutmir, B. Braikat, M. Jamal, N. Damil, B. 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-dinari, H. Mottaqui, B. Braikat, M. Jamal, F. 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. Guevel, G. 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. Jamal, B. Braikat, S. Boutmir, N. 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. Le, J. 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. Le, J. 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. Mesmoudi, A. Timesli, B. Braikat, H. 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. Mohri, N. 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. Mohri, L. 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. Mohri, N. 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. Mohri, N. 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. Mohri, A. 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. Mottaqui, B. 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. Mottaqui, B. 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. Timesli, B. Braikat, H. 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. Ambrosini, J. 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. Behdinan, M. 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. Bourihane, B. Braikat, M. Jamal, F. 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. Boutmir, B. Braikat, M. Jamal, N. Damil, B. 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-dinari, H. Mottaqui, B. Braikat, M. Jamal, F. 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. Guevel, G. 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. Jamal, B. Braikat, S. Boutmir, N. 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. Le, J. 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. Le, J. 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. Mesmoudi, A. Timesli, B. Braikat, H. 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. Mohri, N. 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. Mohri, L. 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. Mohri, N. 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. Mohri, N. 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. Mohri, A. 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. Mottaqui, B. 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. Mottaqui, B. 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. Timesli, B. Braikat, H. 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
Thin-walled beam with open cross section, co-ordinates of the point $M$ on the cross section contour
Axial force $N$, bending moments $M_{y}$ and $M_{z}$, bimoment $B_{\omega}$ and St-Venant torsion moment $M_{sv}$
Section beam under concentrated and distributed forces
External dynamical loading and its time evolution applied on the U-mono-symmetrical thin-walled beam with open cross section
Geometrical characteristics of sections $A$ and $B$
Response curves obtained by the high-order implicit solver $Alg_3$ and by Abaqus code, Time evolution of displacement components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$
Cantilever bi-symmetrical beam with steel I cross section under eccentric loading and its time evolution
Thin-walled beam with steel I cross section under transverse eccentric force $F_{z}(t)$ and its point of application
Response curves obtained by the high-order implicit solver $Alg_3$, by Abaqus code and by Sapountzakis: Time evolution of components $(u(L, t), v(L, t), w(L, t), \theta_x(L, t))$
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 $Log|Res|$ $10^{-3}$ $10$ $-3.73$ $9$ $-3.71$ $8$ $-3.71$ $2\, 10^{-3}$ $12$ $-3.72$ $10$ $-3.71$ $9$ $-3.70$ $3\, 10^{-3}$ $13$ $-3.69$ $11$ $-3.67$ $10$ $-3.65$
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 $612$ $60000$ $12023$ $520$ $60000$ $10210$ $20$ $320$ $80000$ $25896$ $309$ $80000$ $25000$ $280$ $80000$ $22640$
 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 $612$ $60000$ $12023$ $520$ $60000$ $10210$ $20$ $320$ $80000$ $25896$ $309$ $80000$ $25000$ $280$ $80000$ $22640$
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 $Log|Res|$ $10^{-3}$ $6$ $-5.23$ $4$ $-5.2$ $3$ $-5.13$ $2\, 10^{-3}$ $12$ $-5.10$ $9$ $-4.80$ $7$ $-4.62$ $3\, 10^{-3}$ $14$ $-4.91$ $11$ $-4.79$ $8$ $-4.60$ $4\, 10^{-3}$ $15$ $-4.88$ $12$ $-4.70$ $10$ $-4.55$
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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