zbMATH — the first resource for mathematics

Nonlinear dynamics of slender structures: a new object-oriented framework. (English) Zbl 07037439
Summary: With this work, we present a new object-oriented framework to study the nonlinear dynamics of slender structures made of composite multilayer and hyperelastic materials, which combines finite element method and multibody system formalism with a robust integration scheme. Each mechanical system under consideration is represented as a collection of infinitely stiff components, such as rigid bodies, and flexible components like geometrically exact beams and solid-degenerate shells, which are spatially discretized into finite elements. The semi-discrete equations are temporally discretized for a fixed time increment with a momentum-preserving, energy-preserving/dissipative method, which allows the systematic annihilation of unresolved high-frequency content. As usual in multibody system dynamics, kinematic constraints are employed to render supports, joints and structural connections. The presented ideas are implemented following the object-oriented programming philosophy. The approach, which is perfectly suitable for wind energy or aeronautic applications, is finally tested and its potential is illustrated by means of numerical examples.

74 Mechanics of deformable solids
Full Text: DOI
[1] Reissner, E., On one-dimensional finite-strain beam theory: the plane problem, J Appl Math Phys, 23, 795-804, (1972) · Zbl 0248.73022
[2] Bathe, K-J; Bolourchi, S., Large displacement analysis of three-dimensional beam structures, Int J Numer Methods Eng, 14, 961-986, (1979) · Zbl 0404.73070
[3] Simo, JC, A finite strain beam formulation. The three-dimensional dynamic problem. Part I, Comput Methods Appl Mech Eng, 49, 55-70, (1985) · Zbl 0583.73037
[4] Cardona, A.; Géradin, M., A beam finite element non-linear theory with finite rotations, Int J Numer Methods Eng, 26, 2403-2438, (1988) · Zbl 0662.73049
[5] Romero, I.; Armero, F., An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy-momentum conserving scheme in dynamics, Int J Numer Methods Eng, 54, 1683-1716, (2002) · Zbl 1098.74713
[6] Armero, F.; Romero, I., On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part I: low-order methods for two model problems and nonlinear elastodynamics, Comput Methods Appl Mech Eng, 190, 2603-2649, (2001) · Zbl 1008.74035
[7] Armero, F.; Romero, I., On the formulation of high-frequency dissipative time-stepping algorithms for nonlinear dynamics. Part II: second-order methods, Comput Methods Appl Mech Eng, 190, 6783-6824, (2001) · Zbl 1068.74022
[8] Armero, F.; Romero, I., Energy-dissipative momentum-conserving time-stepping algorithms for the dynamics of nonlinear Cosserat rods, Comput Mech, 31, 3-26, (2003) · Zbl 1038.74670
[9] Betsch, P.; Steinmann, P., Constrained dynamics of geometrically exact beams, Comput Mech, 31, 49-59, (2003) · Zbl 1038.74580
[10] Yu, W.; Liao, L.; Hodges, DH; Volovoi, VV, Theory of initially twisted, composite, thin-walled beams, Thin-Walled Struct, 43, 1296-1311, (2005)
[11] Mäkinen, J., Total lagrangian Reissner’s geometrically exact beam element without singularities, Int J Numer Methods Eng, 70, 1009-1048, (2007) · Zbl 1194.74441
[12] Auricchio, F.; Carotenuto, P.; Reali, A., On the geometrically exact beam model: a consistent, effective and simple derivation from three-dimensional finite-elasticity, Int J Solids Struct, 45, 4766-4781, (2008) · Zbl 1169.74456
[13] Pimenta, PM; Campello, EM; Wriggers, P., An exact conserving algorithm for nonlinear dynamics with rotational dofs and general hyperelasticity. Part 1: rods, Comput Mech, 42, 715-732, (2008) · Zbl 1163.74561
[14] Romero, I., A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations, Multibody Syst Dyn, 20, 51-68, (2008) · Zbl 1142.74046
[15] Pai, PF, Problems in geometrically exact modeling of highly flexible beams, Thin-Walled Struct, 76, 65-76, (2014)
[16] Miranda, S.; Gutierrez, A.; Melchionda, D.; Patruno, L., Linearly elastic constitutive relations and consistency for GBT-based thin-walled beams, Thin-Walled Struct, 92, 55-64, (2015)
[17] Sprague MA, Jonkman JM, Jonkman B (2015) FAST modular framework for wind turbine simulation: new algorithms and numerical examples. In: 33rd Wind energy symposium, AIAA SciTech Forum, American Institute of Aeronautics and Astronautics
[18] Wang, Q.; Sprague, MA; Jonkman, J.; Johnson, N.; Jonkman, B., Beamdyn: a high-fidelity wind turbine blade solver in the fast modular framework, Wind Energy, 20, 1439-1462, (2017)
[19] Dvorkin, EN; Bathe, K-J, A continuum mechanics based four-node shell element for general non-linear analysis, Eng Comput, 1, 77-88, (1984)
[20] Bathe, K-J; Dvorkin, EN, A formulation of general shell elements—the use of mixed interpolation of tensorial components, Int J Numer Methods Eng, 22, 697-722, (1986) · Zbl 0585.73123
[21] Choi, CK; Paik, JG, An effective four node degenerated shell element for geometrically nonlinear analysis, Thin-Walled Struct, 24, 261-283, (1996)
[22] Betsch, P.; Stein, E., An assumed strain approach avoiding artificial thickness straining for a non-linear 4-node shell element, Commun Numer Methods Eng, 11, 899-909, (1995) · Zbl 0833.73051
[23] Betsch, P.; Stein, E., A nonlinear extensible 4-node shell element based on continuum theory and assumed strain interpolations, J Nonlinear Sci, 6, 169-199, (1996) · Zbl 0844.73075
[24] Simo, JC; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Int J Numer Methods Eng, 33, 1413-1449, (1992) · Zbl 0768.73082
[25] Bischoff, M.; Ramm, E., Shear deformable shell elements for large strains and rotations, Int J Numer Methods Eng, 40, 4427-4449, (1997) · Zbl 0892.73054
[26] Sansour, C.; Wriggers, P.; Sansour, J., Nonlinear dynamics of shells: theory, finite element formulation, and integration schemes, Nonlinear Dyn, 13, 279-305, (1997) · Zbl 0877.73038
[27] Betsch, P.; Menzel, A.; Stein, E., On the parametrization of finite rotations in computational mechanics: a classification of concepts with application to smooth shells, Comput Methods Appl Mech Eng, 155, 273-305, (1998) · Zbl 0947.74060
[28] Sansour, C.; Wagner, W.; Wriggers, P.; Sansour, J., An energy-momentum integration scheme and enhanced strain finite elements for the non-linear dynamics of shells, Int J Non-Linear Mech, 37, 951-966, (2002) · Zbl 1346.74175
[29] Romero, I.; Armero, F., Numerical integration of the stiff dynamics of geometrically exact shells: an energy-dissipative momentum-conserving scheme, Int J Numer Meth Eng, 54, 1043-1086, (2002) · Zbl 1047.74067
[30] Bauchau, OA; Choi, J-Y; Bottasso, CL, On the modeling of shells in multibody dynamics, Multibody Syst Dyn, 8, 459-489, (2002) · Zbl 1061.74033
[31] Aksu Ozkul, T., A finite element formulation for dynamic analysis of shells of general shape by using the Wilson-\(\theta \) method, Thin-Walled Struct, 42, 497-513, (2004)
[32] Betsch, P.; Sänger, N., On the use of geometrically exact shells in a conserving framework for flexible multibody dynamics, Comput Methods Appl Mech Eng, 198, 1609-1630, (2009) · Zbl 1227.74064
[33] Vaziri, A., Mechanics of highly deformed elastic shells, Thin-Walled Struct, 47, 692-700, (2009)
[34] Campello, EM; Pimenta, PM; Wriggers, P., An exact conserving algorithm for nonlinear dynamics with rotational dofs and general hyperelasticity. Part 2: shells, Comput Mech, 48, 195-211, (2011) · Zbl 1398.74311
[35] Wu, T-Y, Dynamic nonlinear analysis of shell structures using a vector form intrinsic finite element, Eng Struct, 56, 2028-2040, (2013)
[36] Ahmed, A.; Sluys, LJ, Implicit/explicit elastodynamics of isotropic and anisotropic plates and shells using a solid-like shell element, Eur J Mech A Solids, 43, 118-132, (2015) · Zbl 1406.74625
[37] Pietraszkiewicz, W.; Konopińska, V., Junctions in shell structures: a review, Thin-Walled Struct, 95, 310-334, (2015)
[38] Reinoso, J.; Blázquez, A., Application and finite element implementation of 7-parameter shell element for geometrically nonlinear analysis of layered CFRP composites, Compos Struct, 139, 263-276, (2016)
[39] Caliri, MF; Ferreira, AJM; Tita, V., A review on plate and shell theories for laminated and sandwich structures highlighting the finite element method, Compos Struct, 156, 63-77, (2016)
[40] Ota, NSN; Wilson, L.; Gay Neto, A.; Pellegrino, S.; Pimenta, PM, Nonlinear dynamic analysis of creased shells, Finite Elem Anal Des, 121, 64-74, (2016)
[41] Gebhardt, CG; Rolfes, R., On the nonlinear dynamics of shell structures: combining a mixed finite element formulation and a robust integration scheme, Thin-Walled Struct, 118, 56-72, (2017)
[42] Bucalem ML, Bathe K-J (2011) The mechanics of solids and structures—hierarchical modeling and the finite element solution. Springer, Berlin · Zbl 1216.74001
[43] Eisenberg, M.; Guy, R., A proof of the hairy ball theorem, Am Math Monthly, 86, 571-574, (1979) · Zbl 0433.57011
[44] Arnold VI (1989) Mathematical methods of classical mechanics. Springer, Berlin
[45] Heard WB (2006) Rigid body mechanics: mathematics, physics and applications. Wiley, Hoboken
[46] Romero I (2001) Formulation and analysis of dissipative algorithms for nonlinear elastodynamics. Ph.D. thesis, University of California, Berkeley
[47] Gebhardt CG (2012) Desarrollo de simulaciones numéricas del comportamiento aeroelástico de grandes turbinas eólicas de eje horizontal. Ph.D. thesis, Universidad Nacional de Córdoba
[48] Betsch, P.; Steinmann, P., Frame-indifferent beam finite elements based upon the geometrically exact beam theory, Int J Numer Methods Eng, 54, 1775-1788, (2001) · Zbl 1053.74041
[49] Gebhardt, CG; Matusevich, AE; Inaudi, JA, Coupled transverse and axial vibrations including warping effect in asymmetric short beams, J Eng Mech, 144, 04018043, (2018)
[50] Gay Neto, A., Simulation of mechanisms modeled by geometrically-exact beams using rodrigues rotation parameters, Comput Mech, 59, 459-481, (2017)
[51] Ghosh, S.; Roy, D., A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization, Comput Mech, 44, 103-118, (2009) · Zbl 1162.74386
[52] Romero, I., The interpolation of rotations and its application to finite element models of geometrically exact rods, Comput Mech, 34, 121-133, (2004) · Zbl 1138.74406
[53] Simo, JC; Rifai, MS, A class of mixed assumed strain methods and the method of incompatible modes, Int J Numer Methods Eng, 29, 1595-1638, (1990) · Zbl 0724.73222
[54] Ko, Y.; Lee, P-S; Bathe, K-J, A new mitc4+ shell element, Comput Struct, 182, 404-418, (2017)
[55] Kane, C.; Marsden, JE; Ortiz, M., Symplectic-energy-momentum preserving variational integrators, J Math Phys, 40, 3353-3371, (1999) · Zbl 0983.70014
[56] Simo, JC; Tarnow, N., A new energy and momentum conserving algorithm for the non-linear dynamics of shells, Int J Numer Methods Eng, 37, 2527-2549, (1994) · Zbl 0808.73072
[57] Harten, A.; Lax, B.; Leer, P., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev, 25, 35-61, (1983) · Zbl 0565.65051
[58] McLachlan, RI; Quispel, GRW; Robideux, N., Geometric integration using discrete gradients, Philos Trans Math Phys Eng Sci, 357, 1021-1045, (1999) · Zbl 0933.65143
[59] Gonzalez, O., Time integration and discrete Hamiltonian systems, J Nonlinear Sci, 6, 449-467, (1996) · Zbl 0866.58030
[60] Romero, I., An analysis of the stress formula for energy-momentum methods in nonlinear elastodynamics, Comput Mech, 50, 603-610, (2012) · Zbl 1312.74006
[61] Romero, I., Formulation and performance of variational integrators for rotating bodies, Comput Mech, 42, 825-836, (2008) · Zbl 1163.70302
[62] Leyendecker, S.; Marsden, J.; Ortiz, M., Variational integrators for constrained dynamical systems, Zeitschrift für Angewandte Mathematik und Mechanik, 88, 677-708, (2008) · Zbl 1153.70004
[63] Betsch, P., The discrete null space method for the energy consistent integration of constrained mechanical systems. Part i: holonomic constraints, Comput Methods Appl Mech Eng, 194, 5159-5190, (2005) · Zbl 1092.70002
[64] Betsch, P.; Leyendecker, S., The discrete null space method for the energy consistent integration of constrained mechanical systems. Part ii: multibody dynamics, Int J Numer Methods Eng, 67, 499-552, (2006) · Zbl 1110.70301
[65] Leyendecker, S.; Betsch, P.; Steinmann, P., The discrete null space method for the energy-consistent integration of constrained mechanical systems. Part iii: flexible multibody dynamics, Multibody Syst Dyn, 19, 45-72, (2008) · Zbl 1200.70003
[66] Betsch P (2016) Structure-preserving integrators in nonlinear structural dynamics and flexible multibody dynamics. Springer, Berlin · Zbl 1349.70008
[67] Klöppel, T.; Gee, MW; Wall, WA, A scaled thickness conditioning for solid- and solid-shell discretizations of thin-walled structures, Comput Methods Appl Mech Eng, 200, 1301-1310, (2011) · Zbl 1225.74093
[68] Simo, JC; Tarnow, N., The discrete energy-momentum method. Conserving algorithms for nonlinear elastodynamics, Zeitschrift für Angewandte Mathematik und Physik, 43, 757-792, (1992) · Zbl 0758.73001
[69] Romero I (2018) Coupling nonlinear beams and continua: Variational principles and finite element approximations. Int J Numer Methods Eng (in press)
[70] Wagner, W.; Gruttmann, F., Modeling of shell-beam transitions in the presence of finite rotations, Comput Assist Mech Eng Sci, 9, 4005-4018, (2002) · Zbl 1062.74054
[71] Lang, H.; Linn, J.; Arnold, M., Multi-body dynamics simulation of geometrically exact Cosserat rods, Multibody Syst Dyn, 25, 285-312, (2011) · Zbl 1271.74264
[72] Masud, A.; Tham, CL; Liu, WK, A stabilized 3-D co-rotational formulation for geometrically nonlinear analysis of multi-layered composite shells, Comput Mech, 26, 1-12, (2000) · Zbl 0980.74066
[73] Kuhl, D.; Crisfield, M., Energy-conserving and decaying algorithms in non-linear structural dynamics, Int J Numer Methods Eng, 45, 569-599, (1999) · Zbl 0946.74078
[74] Jonkman J, Butterfield S, Musial W, Scott G (2009) Definition of a 5-MW reference wind turbine for offshore system development. Technical report, National Renewable Energy Laboratory (NREL) Golden, CO
[75] Gebhardt, CG; Preidikman, S.; Massa, JC, Numerical simulations of the aerodynamic behavior of large horizontal-axis wind turbines, Int J Hydrog Energy, 35, 6005-6011, (2010)
[76] Gebhardt, CG; Preidikman, S.; Jørgensen, MH; Massa, JC, Non-linear aeroelastic behavior of large horizontal-axis wind turbines: a multibody system approach, Int J Hydrog Energy, 37, 14719-14724, (2012)
[77] Gebhardt, CG; Roccia, BA, Non-linear aeroelasticity: an approach to compute the response of three-blade large-scale horizontal-axis wind turbines, Renew Energy, 66, 495-514, (2014)
[78] Häfele, J.; Hübler, C.; Gebhardt, CG; Rolfes, R., An improved two-step soil-structure interaction modeling method for dynamical analyses of offshore wind turbines, Appl Ocean Res, 55, 141-150, (2016)
[79] Hübler, C.; Häfele, J.; Gebhardt, CG; Rolfes, R., Experimentally supported consideration of operating point dependent soil properties in coupled dynamics of offshore wind turbines, Mar Struct, 57, 18-37, (2018)
[80] Hübler, C.; Gebhardt, CG; Rolfes, R., Hierarchical four-step global sensitivity analysis of offshore wind turbines based on aeroelastic time domain simulations, Renew Energy, 111, 878-891, (2017)
[81] Hübler, C.; Gebhardt, CG; Rolfes, R., Development of a comprehensive data basis of scattering environmental conditions and simulation constraints for offshore wind turbines, Wind Energy Sci, 2, 491-505, (2017)
[82] Häfele, J.; Hübler, C.; Gebhardt, CG; Rolfes, R., A comprehensive fatigue load set reduction study for offshore wind turbines with jacket substructures, Renew Energy, 118, 99-112, (2018)
[83] Intel Corporation (2015) Intel\(^{\textregistered }\) Math Kernel Library 11.3 Developer Reference. https://software.intel.com/en-us/mkl. Accessed Oct 2017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.