Structural optimization; Equivalent static load (ESL) method; Flexible multibody systems; Nonlinear finite element approach; Lie group formalism
Abstract :
[en] The equivalent static load (ESL) method is a powerful approach to solve dynamic response structural optimization problems. The method transforms the dynamic response optimization into a static response optimization under multiple load cases. The ESL cases are defined based on the transient analysis response whereupon all the standard techniques of static response optimization can be used. In the last decade, the ESL method has been applied to perform the structural optimization of flexible components of mechanical systems modeled as multibody systems (MBS). The ESL evaluation strongly depends on the adopted formulation to describe the MBS and has been initially derived based on a floating frame of reference formulation. In this paper, we propose a method to derive the ESL adapted to a nonlinear finite element approach based on a Lie group formalism for two main reasons. Firstly, the finite element approach is completely general to analyze complex MBS and is suitable to perform more advanced optimization problems like topology optimization. Secondly, the selected Lie group formalism leads to a formulation of the equations of motion in the local frame, that turns out to be a strong practical advantage for the ESL evaluation. Examples are provided to validate the proposed method.
Disciplines :
Mechanical engineering
Author, co-author :
Tromme, Emmanuel ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Ingénierie des véhicules terrestres
Sonneville, Valentin ; Université de Liège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Bruls, Olivier ; Université de Liège > Département d'aérospatiale et mécanique > Laboratoire des Systèmes Multicorps et Mécatroniques
Duysinx, Pierre ; Université de Liège > Département d'aérospatiale et mécanique > Ingénierie des véhicules terrestres
Language :
English
Title :
On the equivalent static load method for flexible multibody systems described with a nonlinear finite element formalism
Publication date :
February 2016
Journal title :
International Journal for Numerical Methods in Engineering
ISSN :
0029-5981
eISSN :
1097-0207
Publisher :
Wiley, Chichester, United Kingdom
Volume :
108
Issue :
6
Pages :
646-664
Peer reviewed :
Peer Reviewed verified by ORBi
Name of the research project :
CIMEDE 2 Project
Funders :
The pole of competitiveness GreenWin and the Walloon Region of Belgium (Contract RW-7179)
Kang B, Park G, Arora J. A review of optimization of structures subjected to transient loads. Structural and Multidisciplinary Optimization 2006; 31(2):81–95.
Fox R, Kapoor M. Structural optimization in the dynamic regime: a computational approach. AIAA Journal 1970; 8:1798–1804.
Park GJ. Analytic Methods for Design Practice. Springer-Verlag: London, 2007.
Bruns T, Tortorelli D. Computer-aided optimal design of flexible mechanisms. Proceedings of the Twelfth Conference of the Irish Manufacturing Commitee, IMC12, Competitive Manufacturing, University College Cork, Ireland, 1995; 29–36.
Choi W, Park G. Transformation of dynamic loads into equivalent static loads based on modal analysis. International Journal for Numerical Methods in Engineering 1999; 46(1):29–43.
Kang B, Park G, Arora J. Optimization of flexible multibody dynamic systems using the equivalent static load method. AIAA Journal 2005; 43(4):846–852.
Häussler P, Minx J, Emmrich D. Topology optimization of dynamically loaded parts in mechanical systems: coupling of MBS, FEM and structural optimization. Proceedings of NAFEMS Seminar Analysis of Multi-Body Systems Using FEM and MBS, Wiesbaden, Germany, 2004; 1–11.
Hong E, You B, Kim C, Park G. Optimization of flexible components of multibody systems via equivalent static loads. Structural Multidisciplinary Optimization 2010; 40:549–562.
Sherif K, Irschik H. Efficient topology optimization of large dynamic finite element systems using fatigue. AIAA Journal 2010; 48(7):1339–1347.
Oral S, Kemal Ider S. Optimum design of high-speed flexible robotic arms with dynamic behavior constraints. Computers & Structures 1997; 65(2):255–259.
Etman L, Van Campen D, Schoofs A. Design optimization of multibody systems by sequential approximation. Multibody System Dynamics 1998; 2(4):393–415.
Brüls O, Lemaire E, Duysinx P, Eberhard P. Optimization of multibody systems and their structural components. In Multibody dynamics: Computational Methods and Applications, Vol. 23. Springer: Springer Netherlands, 2011; 49–68.
Tromme E, Tortorelli D, Brüls O, Duysinx P. Structural optimization of multibody system components described using level set techniques. Structural Multidisciplinary Optimization 2015; 52:959–971.
Bauchau O. Flexible Multibody Dynamics. Springer: Netherlands, 2011.
Géradin M, Cardona A. Flexible Multibody Dynamics: A Finite Element Approach. John Wiley & Sons: New York, 2001.
Brüls O, Cardona A. On the use of Lie group time integrators in multibody dynamics. Journal of Computational and Nonlinear Dynamics 2010; 5(3):031002–1–031002–13.
Brüls O, Cardona A, Arnold M. Lie group generalized-α time integration of constrained flexible multibody systems. Mechanism and Machine Theory 2012; 48:121–137.
Paraskevopoulos E, Natsiavas S. A new look into the kinematics and dynamics of finite rigid body rotations using Lie group theory. International Journal of Solids and Structures 2013; 50(1):57–72.
Sonneville V, Brüls O. A formulation on the special Euclidean group for dynamic analysis of multibody systems. Journal of Computational and Nonlinear Dynamics 2014; 9(4):041002–1–041002–8.
Sonneville V, Cardona A, Brüls O. Geometrically exact beam finite element formulated on the special Euclidean group SE(3). Computer Methods in Applied Mechanics and Engineering 2014; 268:451–474.
Brüls O, Arnold M, Cardona A. Two Lie group formulations for dynamic multibody systems with large rotations. Proceedings of the IDETC/MSNDC Conference, Washington D.C., USA, 2011; 85–94.
Holm D, Schmah T, Stoica C, Ellis D. Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford University Press: London, 2009.
Boothby W. An Introduction to Differentiable Manifolds and Riemannian Geometry (Second edn.) Academic Press: San Diego, 2009.
Wen H, Reddy T, Reid S. Deformation and failure of clamped beams under low speed impact loading. International Journal of Impact Engineering 1995; 16(3):435–454.
Zhang Q, Vrouwenvelder A, Wardenier J. Dynamic amplification factors and {EUDL} of bridges under random traffic flows. Engineering Structures 2001; 23(6):663–672.
Choe U, Gang S, Sin M, Park G. Transformation of a dynamic load into an equivalent static load and shape optimization of the road arm in self-propelled howitzer. The Korean Society of Mechanical Engineers 1996; 20(12):3767–3781.
Shabana A. Dynamics of Multibody Systems (Fourth edn.) Cambridge University Press: England, 2013.
Haftka R, Gürdal Z. Elements of Structural Optimization (Third edn.) Springer: Netherlands, 1992.
Boyd S, Vandenberghe L. Convex Optimization. Cambridge University Press: Cambridge, United-Kingdom, 2004.
Held A. On structural optimization of flexible multibody systems. Ph.D. Thesis, Institut für Technische und Numerische Mechanik, Universität Stuttgart, 2014.
Choi W, Park G. Structural optimization using equivalent static loads at all time intervals. Computer Methods in Applied Mechanics and Engineering 2002; 191(19-20):2105–2122.
Jung U, Park G. A new method for simultaneous optimum design of structural and control systems. Computers & Structures 2015; 160:90–99.
Lee H, Park G. Nonlinear dynamic response topology optimization using the equivalent static loads method. Computer Methods in Applied Mechanics and Engineering 2015; 283:956–970.
Stolpe M. On the equivalent static loads approach for dynamic response structural optimization. Structural and Multidisciplinary Optimization 2014; 50:921–926.
Cauchy A. Méthodes générales pour la résolution des systèmes d'équations simultanées. Comptes rendus de l'Académie des Sciences de Paris 1847; 25:536–538.
Sigmund O, Maute K. Topology optimization approaches: a comparative review. Structural and Multidisciplinary Optimization 2013; 48(6):1031–1055.
Deaton J, Grandhi R. A survey of structural and multidisciplinary continuum topology optimization: post 2000. Structural and Multidisciplinary Optimization 2014; 49(1):1–38.
Schittkowski K. NLPQL: A fortran subroutine solving constrained nonlinear programming problems. Annals of Operation Research 1986; 5(2):485–500.
Fleury C, Braibant V. Structural optimization: a new dual method using mixed variables. International Journal for Numerical Methods in Engineering 1986; 23:409–428.
Svanberg K. The method of moving asymptotes – a new method for structural optimization. International Journal for Numerical Methods in Engineering 1987; 24:359–373.
Svanberg K. A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM Journal on Optimization 2002; 12(2):555–573.
Bruyneel M, Duysinx P, Fleury C. A family of MMA approximations for structural optimization. Structural and Multidisciplinary Optimization 2002; 24:263–276.
Sohoni V, Haug E. A state space technique for optimal design of mechanisms. Journal of Mechanical Design 1982; 104(4):792–798.