0
Research Papers

Direct and Adjoint Sensitivity Analysis of Ordinary Differential Equation Multibody Formulations

[+] Author and Article Information
Daniel Dopico

Advanced Vehicle Dynamics Laboratory
and Computational Science Laboratory,
Departments of Mechanical Engineering
and Computer Science,
Virginia Tech,
Blacksburg, VA 24061
e-mail: ddopico@udc.es

Yitao Zhu

Advanced Vehicle Dynamics Laboratory
and Computational Science Laboratory,
Departments of Mechanical Engineering
and Computer Science,
Virginia Tech,
Blacksburg, VA 24061
e-mail: yitao7@vt.edu

Adrian Sandu

Computational Science Laboratory,
Department of Computer Science,
Virginia Tech,
Blacksburg, VA 24061
e-mail: asandu7@vt.edu

Corina Sandu

Advanced Vehicle Dynamics Laboratory,
Department of Mechanical Engineering,
Virginia Tech,
Blacksburg, VA 24061
e-mail: csandu@vt.edu

1Address all correspondence to this author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received July 18, 2013; final manuscript received January 10, 2014; published online September 12, 2014. Assoc. Editor: Parviz Nikravesh.

J. Comput. Nonlinear Dynam 10(1), 011012 (Sep 12, 2014) (7 pages) Paper No: CND-13-1184; doi: 10.1115/1.4026492 History: Received July 18, 2013; Revised January 10, 2014

Sensitivity analysis of multibody systems is essential for several applications, such as dynamics-based design optimization. Dynamic sensitivities, when needed, are often calculated by means of finite differences. This procedure is computationally expensive when the number of parameters is large, and numerical errors can severely limit its accuracy. This paper explores several analytical approaches to perform sensitivity analysis of multibody systems. Direct and adjoint sensitivity equations are developed in the context of Maggi's formulation of multibody dynamics equations. The approach can be generalized to other formulations of multibody dynamics as systems of ordinary differential equations (ODEs). The sensitivity equations are validated numerically against the third party code fatode and against finite difference solutions with real and complex perturbations.

FIGURES IN THIS ARTICLE
<>
Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.

References

Brenan, K., Campbell, S., and Petzold, L., 1989, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland, New York.
Ascher, U., and Petzold, L., 1998, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Society for Industrial and Applied Mathematics, Philadelphia, PA.
Serna, M., Aviles, R., and García de Jalon, J., 1982, “Dynamic Analysis of Plane Mechanisms With Lower Pairs in Basic Coordinates,” Mech. Mach. Theory, 17(6), pp. 397–403. [CrossRef]
Wehage, R., and Haug, E., 1982, “Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Mechanical Systems,” J. Mech. Des., 104, pp. 247–255. [CrossRef]
Garcia de Jalon, J., and Bayo, E., 1994, Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge, Springer-Verlag, New York.
Haug, E. J., and Arora, J. S., 1979, Applied Optimal Design: Mechanical and Structural Systems, Wiley, New York.
Cao, Y., Li, S., and Petzold, L., 2002, “Adjoint Sensitivity Analysis for Differential-Algebraic Equations: Algorithms and Software,” J. Comput. Appl. Math., 149(1), pp. 171–191. [CrossRef]
Cao, Y., Li, S., Petzold, L., and Serban, R., 2003, “Adjoint Sensitivity Analysis for Differential-Algebraic Equations: The Adjoint DAE System and Its Numerical Solution,” SIAM J. Sci. Comput., 24(3), pp. 1076–1089. [CrossRef]
Chang, C. O., and Nikravesh, P. E., 1985, “Optimal Design of Mechanical Systems With Constraint Violation Stabilization Method,” J. Mech., Trans. Autom. Des., 107(4), pp. 493–498. [CrossRef]
Haug, E., 1987, “Design Sensitivity Analysis of Dynamic Systems,” Computer Aided Optimal Design: Structural and Mechanical Systems, (NATO ASI Series F: Computer and Systems Sciences), Springer-Verlag, Berlin.
Bestle, D., and Seybold, J., 1992, “Sensitivity Analysis of Constrained Multibody Systems,” Arch. Appl. Mech., 62, pp. 181–190. [CrossRef]
Bestle, D., and Eberhard, P., 1992, “Analyzing and Optimizing Multibody Systems,” Mech. Struct. Mach., 20(1), pp. 67–92. [CrossRef]
Pagalday, J., and Avello, A., 1997, “Optimization of Multibody Dynamics Using Object Oriented Programming and a Mixed Numerical-Symbolic Penalty Formulation,” Mech. Mach. Theory, 32(2), pp. 161–174. [CrossRef]
Dias, J., and Pereira, M., 1997, “Sensitivity Analysis of Rigid-Flexible Multibody Systems,” Multibody Syst. Dyn., 1, pp. 303–322. [CrossRef]
Feehery, W. F., Tolsma, J. E., and Barton, P. I., 1997, “Efficient Sensitivity Analysis of Large-Scale Differential-Algebraic Systems,” Appl. Numer. Math., 25(1), pp. 41–54. [CrossRef]
Anderson, K. S., and Hsu, Y., 2002, “Analytical Fully-Recursive Sensitivity Analysis for Multibody Dynamic Chain Systems,” Multibody Syst. Dyn., 8, pp. 1–27. [CrossRef]
Anderson, K., and Hsu, Y., 2004, “Order-(n+m) Direct Differentiation Determination of Design Sensitivity for Constrained Multibody Dynamic Systems,” Struct. Multidisc. Optim., 26(3-4), pp. 171–182. [CrossRef]
Ding, J.-Y., Pan, Z.-K., and Chen, L.-Q., 2007, “Second Order Adjoint Sensitivity Analysis of Multibody Systems Described by Differential-Algebraic Equations,” Multibody Syst. Dyn., 18, pp. 599–617. [CrossRef]
Schaffer, A., 2006, “Stabilized Index-1 Differential-Algebraic Formulations for Sensitivity Analysis of Multi-body Dynamics,” Proc. Inst. Mech. Eng., Part K, 220(3), pp. 141–156. [CrossRef]
Neto, M. A., Ambrosio, J. A. C., and Leal, R. P., 2009, “Sensitivity Analysis of Flexible Multibody Systems Using Composite Materials Components,” Int. J. Numer. Methods Eng., 77(3), pp. 386–413. [CrossRef]
Bhalerao, K., Poursina, M., and Anderson, K., 2010, “An Efficient Direct Differentiation Approach for Sensitivity Analysis of Flexible Multibody Systems,” Multibody Syst. Dyn., 23, pp. 121–140. [CrossRef]
Banerjee, J. M., and McPhee, J., 2013, “Symbolic Sensitivity Analysis of Multibody Systems,” Multibody Dynamics. Computational Methods and Applications, (Computational Methods in Applied Sciences), Vol. 28, Springer, New York, pp. 123–146.
Zhang, H., and Sandu, A., 2012, “fatode: A Library for Forward, Adjoint, and Tangent Linear Integration of ODEs.” Available at: http://people.cs.vt.edu/asandu/Software/FATODE/index.html
Garcia de Jalon, J., Callejo, A., and Hidalgo, A. F., 2011, “Efficient Solution of Maggi's Equations,” ASME J. Comput. Nonlinear Dyn., 7(2), pp. 021003. [CrossRef]
Bayo, E., García de Jalon, J., and Serna, M., 1988, “A Modified Lagrangian Formulation for the Dynamic Analysis of Constrained Mechanical Systems,” Comput. Methods Appl. Mech. Eng., 71(2), pp. 183–195. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Five-Bar Mechanism

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In