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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 |
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.
References:
[1] |
Abaqus,
Version 6.11 Documentation, Dassautt Systemes Simulia Corp, Providence, RI, USA, 2011. |
[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. |
[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.
|
[4] |
K. J. Bathe, Finite Elements Procedures, Prentice-Hall, New Jersey, 1996. |
[5] |
J. L. Batoz and G. Dhatt, Modélisation des structures par éléments finis, Hermès, Paris, 1990. |
[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.
|
[7] |
P. Betsch and P. Steinmann,
Constrained dynamics of geometrically exact beams, Computational Mechanics, 31 (2003), 49-59.
|
[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.
|
[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.
|
[10] |
M. A. Crisfield,
Nonlinear Finite Elements Analysis of Solids and Structures, John Willey and Sons, 1991. |
[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.
|
[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.
|
[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. |
[14] |
D. Haijuan,
Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open section, Thin-Walled Structures, 46 (2008), 107-112.
|
[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. |
[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. |
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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. |
[25] |
H. Mottaqui, B. Braikat and N. Damil,
Local parameterization and the asymptotic numerical method, Mathematical Modelling of Natural Phenomena, 5 (2010), 16-22.
|
[26] |
N. Newmark,
A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceeding of ASCE, (1959), 67-94.
|
[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. |
[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.
|
[29] |
V. Z. Vlasov, Thin walled elastic beams, Eyrolles, French translation: Pièces longues en voiles minces, Paris, 1965. |
[30] |
O. C. Zienkiewicz and R. Taylor, The Finite Element Method, Solid and Fluid Mechanics and Non-linearity, Book Company, 1987. |
show all references
References:
[1] |
Abaqus,
Version 6.11 Documentation, Dassautt Systemes Simulia Corp, Providence, RI, USA, 2011. |
[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. |
[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.
|
[4] |
K. J. Bathe, Finite Elements Procedures, Prentice-Hall, New Jersey, 1996. |
[5] |
J. L. Batoz and G. Dhatt, Modélisation des structures par éléments finis, Hermès, Paris, 1990. |
[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.
|
[7] |
P. Betsch and P. Steinmann,
Constrained dynamics of geometrically exact beams, Computational Mechanics, 31 (2003), 49-59.
|
[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.
|
[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.
|
[10] |
M. A. Crisfield,
Nonlinear Finite Elements Analysis of Solids and Structures, John Willey and Sons, 1991. |
[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.
|
[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.
|
[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. |
[14] |
D. Haijuan,
Nonlinear free vibration analysis of asymmetric thin-walled circularly curved beams with open section, Thin-Walled Structures, 46 (2008), 107-112.
|
[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. |
[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. |
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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.
|
[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. |
[25] |
H. Mottaqui, B. Braikat and N. Damil,
Local parameterization and the asymptotic numerical method, Mathematical Modelling of Natural Phenomena, 5 (2010), 16-22.
|
[26] |
N. Newmark,
A method of computation for structural dynamics, Journal of the Engineering Mechanics Division, Proceeding of ASCE, (1959), 67-94.
|
[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. |
[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.
|
[29] |
V. Z. Vlasov, Thin walled elastic beams, Eyrolles, French translation: Pièces longues en voiles minces, Paris, 1965. |
[30] |
O. C. Zienkiewicz and R. Taylor, The Finite Element Method, Solid and Fluid Mechanics and Non-linearity, Book Company, 1987. |








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