Model order reduction for dynamic simulation of beams with forcing and geometric nonlinearities
Pages 50 - 62
Abstract
The objective of the paper is to investigate the applicability of a model order reduction technique for dynamic simulation of beams with forcing and geometric nonlinearities. A cantilever and a doubly clamped beams actuated by an electrostatic force are considered in the paper. The governing partial differential equations for the two cases which account for the nonlinearities are presented. These equations are spatially discretized using the Galerkin finite element method (FEM). The resulting system of nonlinear ordinary differential equations is reduced using the trajectory piecewise linear model order reduction (TPWLMOR) method. Simulation indicates that the use of the original TPWLMOR method leads to the presence of a phase error in the long term dynamic simulation of the models. To improve the accuracy of the dynamic response, a modification to the original TPWLMOR based on minimization of residual at linearization point is proposed. Further, the parameters affecting the accuracy of the modified TPWLMOR are studied.
References
[1]
Antoulas, A., Approximation of Large Scale Dynamical Systems. 2005. SIAM.
[2]
Bathe, K.J., Finite Element Procedures. 2009. Phi Learning.
[3]
Qu, Z., Model Order Reduction Techniques: With Applications in Finite Element analysis. . 2004. Springer.
[4]
Guyan, R.J., Reduction of stiffness and mass matrices. AIAA Journal. v3 i2. 380
[5]
Hurty, W., Dynamic analysis of structural systems using component modes. AIAA Journal. v3 i4. 678-685.
[6]
Craig, R. and Bampton, M., Coupling of substructures for dynamic analysis. AIAA Journal. v6 i7. 1313-1319.
[7]
Bechtold, T., Rudnyi, E.B. and Korvink, J.G., Fast Simulation of Electro-Thermal MEMS: Efficient Dynamic Compact Models, Microtechnology and MEMS. . 2006. Springer Verlag Heidelberg.
[8]
Han, J.S., Rudnyi, E.B. and Korvink, J.G., Efficient optimisation of transient dynamic problems in MEMS devices using model order reduction. Journal of Micromechanics and Microengineering. v15 i4. 822-832.
[9]
Eid, R., Salimbahrami, B., Lohmann, B., Rudnyi, E.B. and Korvink, J.G., Parametric order reduction of proportionally damped second-order systems. Sensors and Materials. v19 i3. 149-164.
[10]
Nickell, R.E., Nonlinear dynamics by mode superposition. Computer Methods in Applied Mechanics and Engineering. v7 i1. 107-129.
[11]
Bathe, K.J. and Gracewski, S., On nonlinear dynamic analysis using substructuring and mode superposition. Computers and Structures. v13 i5-6. 699-707.
[12]
Clough, R.W. and Wilson, E.L., Dynamic analysis of large structural systems with local nonlinearities. Computer Methods in Applied Mechanics and Engineering. v17-18 iPART 1. 107-129.
[13]
I.R. Praveen Krishna, C. Padmanabhan, Improved reduced order solution techniques for nonlinear systems with localized nonlinearities, Nonlinear Dynamics 63 (4) (2011) 561-586
[14]
Noor, A.K., On making large nonlinear problems small. Computer Methods in Applied Mechanics and Engineering. v34 i1-3. 955-985.
[15]
Idelsohn, S.R. and Cardona, A., A reduction method for nonlinear structural dynamic analysis. Computer Methods in Applied Mechanics and Engineering. v49 i3. 253-279.
[16]
A load-dependent basis for reduced nonlinear structural dynamics. Computers and Structures. v20 i1-3. 203-210.
[17]
Kosambi, D., Statistics in function space. Journal of the Indian Mathematical Society. v7. 76-78.
[18]
Holmes, P., Lumley, J., Berkooz, G., Mattingly, J. and Wittenberg, R., Low-dimensional models of coherent structures in turbulence. Physics Reports. v287 i4. 337-384.
[19]
Lin, W., Lee, K., Lim, S. and Lu, P., A model reduction method for the dynamic analysis of microelectromechanical systems. International Journal of Nonlinear Sciences and Numerical Simulation. v2 i2. 89-100.
[20]
Binion, D. and Chen, X., A Krylov enhanced proper orthogonal decomposition method for efficient nonlinear model reduction. Finite Elements in Analysis and Design. v47 i7. 728-738.
[21]
Phillips, J.R., Projection-based approaches for model reduction of weakly nonlinear time-varying systems. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. v22 i2. 171-187.
[22]
Y. Chen, J. White, An algorithm for automatic model-order reduction of nonlinear MEMS devices, in: Proceedings of the International Conference on Modeling and Simulation of Microsystems, 2000, pp. 477-480.
[23]
Bai, Z. and Skoogh, D., A projection method for model reduction of bilinear dynamical systems. Linear Algebra and its Applications. v415 i2-3. 406-425.
[24]
Bai, Z., Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Applied Numerical Mathematics. v43 i1-2. 9-44.
[25]
M. Rewienski, J. White, A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices, in: IEEE/ACM International Conference on Computer-Aided Design, Digest of Technical Papers, 2001, pp. 252-257.
[26]
Younis, M.I., Abdel-Rahman, E.M. and Nayfeh, A.H., A reduced-order model for electrically actuated microbeam-based MEMS. Journal of Microelectromechanical Systems. v12 i5. 673-680.
[27]
Nayfeh, A.H., Younis, M.I. and Abdel-Rahman, E.M., Reduced-order models for MEMS applications. Nonlinear Dynamics. v41. 211-236.
[28]
Hung, E.S. and Senturia, S.D., Generating efficient dynamical models for microelectromechanical systems from a few finite-element simulation runs. Journal of Microelectromechanical Systems. v8 i3. 280-289.
[29]
Reduced-order modeling of weakly nonlinear MEMS devices with Taylor-series expansion and Arnoldi approach. Journal of Microelectromechanical Systems. v13 i3. 441-451.
[30]
Rewienski, M. and White, J., A trajectory piecewise-linear approach to model order reduction and fast simulation of nonlinear circuits and micromachined devices. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems. v22 i2. 155-170.
[31]
Ouakad, H.M. and Younis, M.I., Nonlinear dynamics of electrically actuated carbon nanotube resonators. Journal of Computational and Nonlinear Dynamics. v5 i1. 1-13.
[32]
Batra, R.C., Porfiri, M. and Spinello, D., Vibrations of narrow microbeams predeformed by an electric field. Journal of Sound and Vibration. v309 i3-5. 600-612.
[33]
Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics. v5 i3. 283-292.
[34]
Hughes, T.J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. . 2000. Dover Publication.
[35]
Bathe, K.J., Conserving energy and momentum in nonlinear dynamics: a simple implicit time integration scheme. . Computers and Structures. v85 i7-8. 437-445.
[36]
Bathe, K.J. and Baig, M.M.I., On a composite implicit time integration procedure for nonlinear dynamics. Computers and Structures. v83 i31-32. 2513-2524.
[37]
Bathe, K.J. and Noh, G., Insight into an implicit time integration scheme for structural dynamics. Computers and Structures. v98-99. 1-6.
[38]
Li, G. and Aluru, N.R., Linear, nonlinear and mixed-regime analysis of electrostatic MEMS. Sensors and Actuators A. v90 i3. 278-291.
[39]
Hu, Y., Chang, C. and Huang, S., Some design considerations on the electrostatically actuated microstructures. Sensors and Actuators A. v112 i1. 155-161.
[40]
Li, H., Wang, Q.X. and Lam, K.Y., A variation of local point interpolation method (¿ LPIM) for analysis of microelectromechanical systems (MEMS) device. Engineering Analysis with Boundary Elements. v28. 1261-1270.
[41]
Plesha, Witt, Concepts and Applications of Finite Element Analysis. 2007. Wiley.
[42]
Reddy, J.N., An Introduction to Nonlinear Finite Element Analysis. 2004. Oxford University Press.
[43]
N. Dong, J. Roychowdhury, Piecewise polynomial nonlinear model reduction, in: Proceedings of the 40th Annual Design Automation Conference, DAC '03, 2003, pp. 484-489.
[44]
S.K. Tiwary, R.A. Rutenbar, Faster, parametric trajectory-based macromodels via localized linear reductions, in: Proceedings of the 2006 IEEE/ACM International Conference on Computer-Aided Design, ICCAD '06, 2006, pp. 876-883.
[45]
J.A. Martinez, Model Order Reduction of Nonlinear Dynamic Systems Using Multiple Projection Bases and Optimized State-space Sampling, Ph.D. Thesis, University of Pittsburgh, 2009.
[46]
Yang, Y.J. and Shen, K.Y., Nonlinear heat-transfer macromodeling for MEMS thermal devices. Journal of Micromechanics and Microengineering. v15 i2. 408-418.
[47]
D. Gratton, Reduced-order, Trajectory Piecewise-Linear Models for Nonlinear Computational Fluid Dynamics, Ph.D. Thesis, MIT, Cambridge, 2004.
[48]
T. Bechtold, Model Order Reduction of Electro-Thermal MEMS, Ph.D. Thesis, Albert-Ludwigs-Universität Freiburg, 2005.
[49]
De, S.K. and Aluru, N.R., Full-lagrangian schemes for dynamic analysis of electrostatic MEMS. Journal of Microelectromechanical Systems. v13 i5. 737-758.
[50]
Krylov-subspace methods for reduced-order modeling in circuit simulation. Journal of Computational and Applied Mathematics. v123 i1-2. 395-421.
- Model order reduction for dynamic simulation of beams with forcing and geometric nonlinearities
Recommendations
Model order reduction for nonlinear Schrödinger equation
We apply the proper orthogonal decomposition (POD) to the nonlinear Schrödinger (NLS) equation to derive a reduced order model. The NLS equation is discretized in space by finite differences and is solved in time by structure preserving symplectic mid-...
Model order reduction for dynamic simulation of slender beams undergoing large rotations
In this paper a model order reduction technique for the dynamic simulation of beams undergoing large rotations is presented. The finite element model for the motion of such beams is based on the corotational formulation. The trajectory piecewise linear ...
Index-Aware Model Order Reduction for Linear Index-2 DAEs with Constant Coefficients
A model order reduction method for index-2 differential-algebraic equations (DAEs) is introduced, which is based on the intrinsic differential equations and on the remaining algebraic constraints. This extends the method introduced in a previous paper ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
Copyright © Elsevier B.V. © 2013.
Publisher
Elsevier Science Publishers B. V.
Netherlands
Publication History
Published: 01 November 2013
Author Tags
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 13 Nov 2024
Other Metrics
Citations
View Options
View options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in