1 / 42

ME280 “Fractional Order Mechanics” Variable Order Mechanics and Distributed Order Mechanics

ME280 “Fractional Order Mechanics” Variable Order Mechanics and Distributed Order Mechanics. YangQuan Chen, Ph.D., Director, MESA (Mechatronics, Embedded Systems and Automation) Lab ME/EECS/SNRI/ UCSolar , School of Engineering, University of California, Merced

val
Download Presentation

ME280 “Fractional Order Mechanics” Variable Order Mechanics and Distributed Order Mechanics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ME280 “Fractional Order Mechanics”Variable Order Mechanics andDistributed Order Mechanics YangQuan Chen, Ph.D., Director, MESA (Mechatronics, Embedded Systems and Automation)Lab ME/EECS/SNRI/UCSolar, School of Engineering, University of California, Merced E: yqchen@ieee.org; or, yangquan.chen@ucmerced.edu T: (209)228-4672; O: SE1-254; Lab: Castle #22 (T: 228-4398) 11/05/2013 and 11/07/2013. Tue. Thu. 09:00-10:15, KL217 (Week-11)

  2. Part 1: Variable-Order Mechanics • The first paper – 1993 Samko and Ross • Motivations • Definitions • Computational Tools ME280 "Fractional Order Mechanics" @ UC Merced

  3. ME280 "Fractional Order Mechanics" @ UC Merced

  4. Two key references • Carl F. Lorenzo and Tom T. Hartley. Variable Order and Distributed Order Fractional Operators. Nonlinear Dynamics 29: 57–98, 2002. • Lynnette E. S. Ramirez and Carlos F. M. Coimbra. On the Selection and Meaning of Variable Order Operators for Dynamic Modeling. International Journal of Differential Equations. Volume 2010, Article ID 846107, 16 pages. doi:10.1155/2010/846107 ME280 "Fractional Order Mechanics" @ UC Merced

  5. Simple Viscoelastic Equation(Phenomenological Approach) Combine viscous (dashpot) and elastic (spring) elements Serial combination – Maxwell unit Cannot describe creep stress σ, strain ε E is a modulus of elasticity and ηis the viscosity. ME280 "Fractional Order Mechanics" @ UC Merced

  6. Simple Viscoelastic Equation(Phenomenological Approach) Combine viscous (dashpot) and elastic (spring) elements Parallel combination – Voigt unit No stress relaxation ME280 "Fractional Order Mechanics" @ UC Merced

  7. Constant Order Models General Integer Order Model General Fractional Order Model Four-parameter model: ME280 "Fractional Order Mechanics" @ UC Merced

  8. Variable order derivative VO operator (Coimbra, 2003) • Initial condition term accounts for behavior of a system when it departs from dynamic equilibrium • Returns the corresponding nth order fractional derivative of x(t) when q(t) = n • Allows for a continuous transition from 0-order to 1st-order derivatives (elastic – viscous) ME280 "Fractional Order Mechanics" @ UC Merced

  9. Variable order constitutive model General model Simple model for 1-D linear viscoelasticity ME280 "Fractional Order Mechanics" @ UC Merced

  10. Physical ModelDisorder Imposing stress on a material increases disorder Increasing total disorder

  11. VO Definitions • S. G. Samko and B. Ross, “Integration and differentiation to a variable fractional order,” Integral Transforms and Special Functions, vol. 1, no. 4, pp. 277–300, 1993. ME280 "Fractional Order Mechanics" @ UC Merced

  12. VO Definitions • C. F. Lorenzo and T. T. Hartley, “Variable order and distributed order fractional operators,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 57–98, 2002. • C. F. Lorenzo and T. T. Hartley, “Initialization, conceptualization, and application in the generalized fractional calculus,” NASA Technical Publication 98-208415, Lewis Research Center, NASA, Cleveland, Ohio, USA, 1998. ME280 "Fractional Order Mechanics" @ UC Merced

  13. 9 VO Definitions ME280 "Fractional Order Mechanics" @ UC Merced

  14. VO Simulink Toolkit • http://www.mathworks.com/matlabcentral/fileexchange/38801-fractional-variable-order-derivative-simulink-toolkit • by DominikSierociuk • 26 Oct 2012 (Updated 01 Nov 2012) ME280 "Fractional Order Mechanics" @ UC Merced

  15. VO via GL definition (3 types) ME280 "Fractional Order Mechanics" @ UC Merced

  16. VO via GL definition (3 types) Duarte Valerio and Jose Sa da Costa. Variable-order fractional derivatives and their numerical approximations Signal Processing, Vol. 91, Number 3, pp. 470-483, 2011. ME280 "Fractional Order Mechanics" @ UC Merced

  17. VO (Anomalous) Diffusion • Extensively observed, e.g., polluter diffusion in soil, oil seepage, nuclear waste disposal, turbulence, etc. • Characteristics: scale effect, heavy tail, history dependence, long-range correlation, skewness characteristics, etc. • Drawbacks of classical non-linear models: unclear interpretation, obscure parameters, computationally expensive, sometimes unfeasible, etc. ME280 "Fractional Order Mechanics" @ UC Merced

  18. Open issues in anomalous diffusion modeling • How to deal with the diffusion process which changes with time evolution, spatial variation or system parameters? • Which are the best differential equation models for the description of multi-scale diffusion or multi-field coupling transport system? • How to deal with the diffusion process in the inhomogeneous or anisotropic medium? • The link in the differential equation, statistical methods and continuous time random walk model (CTRW)? ME280 "Fractional Order Mechanics" @ UC Merced

  19. Big picture of anomalous diffusion modeling Statistical methods CTRW Fractional differential equations Real world diffusion phenomena Engineering application ME280 "Fractional Order Mechanics" @ UC Merced

  20. Variable Oder operator (VO) Why select variable order operator? • Some diffusion behaviors change with time, space or system parameters • To depict the accelerating/decelerating diffusion process • Variable order can well represent the change of the diffusion rate ME280 "Fractional Order Mechanics" @ UC Merced

  21. Four types of variable order models Time dependent: Space dependent: Concentration dependent: System parameter dependent: ME280 "Fractional Order Mechanics" @ UC Merced

  22. Time dependent VO model Concentration dependent VO model P=0.1 X=0.6 ME280 "Fractional Order Mechanics" @ UC Merced

  23. Remarks VODO is a natural candidate to offer an effective mathematical framework for the description of some complex anomalous diffusion processes Different forms of the variable-order fractional equations can depict and settle different diffusion processes. The mathematical foundations of VODO are still immature; the theoretical analysis and numerical schemes for VODO are still underway The classification of VODO model, properly speaking, is not strict. ME280 "Fractional Order Mechanics" @ UC Merced

  24. What next? Variable order model CTRW Variable order time derivative Variable waiting time PDF Variable order space derivative Variable jump length PDF ME280 "Fractional Order Mechanics" @ UC Merced

  25. Distributed order diffusion Time distributed order (DO) model Discrete form Multi-power law Multi-order model Multi-power law ME280 "Fractional Order Mechanics" @ UC Merced

  26. Some remarks • The DO model belong to the constant order models • In the long time behavior, it can depict the decelerating/accelerating diffusion process ME280 "Fractional Order Mechanics" @ UC Merced

  27. Why propose Random Order Diffusion model? • The considered diffusion or transport system is interfered by some noises (e.g. oscillating external field or unstable system parameters), these noises inevitably cause the fluctuation of the whole system. • The random order model is a right choice to depict the diffusion process in perturbation field. http://www.mathworks.com/matlabcentral/fileexchange/26407-predictor-corrector-method-for-constant-variable-and-random-fractional-order-relaxation-equation ME280 "Fractional Order Mechanics" @ UC Merced

  28. Here is the reason Anomalous Diffusion in Purkinje Cell Dendrites Caused by Spines. Santamaria, F., Wils, S., Schutter, E. D., and Augustine, G. J., 2006. Neuron, 52, 635–648 ME280 "Fractional Order Mechanics" @ UC Merced

  29. Here is the reason A.M. Reynolds , Physica A 340 (2004) 298 – 308 ME280 "Fractional Order Mechanics" @ UC Merced

  30. http://www.mathworks.com/matlabcentral/fileexchange/26407-predictor-corrector-method-for-constant-variable-and-random-fractional-order-relaxation-equationhttp://www.mathworks.com/matlabcentral/fileexchange/26407-predictor-corrector-method-for-constant-variable-and-random-fractional-order-relaxation-equation ME280 "Fractional Order Mechanics" @ UC Merced

  31. RO model of diffusion process Time fractional diffusion equation The expression in the Fourier-Laplace domain (iterated Laplace transform) ME280 "Fractional Order Mechanics" @ UC Merced

  32. The mean square displacement (MSD) On mean square displacement behaviors of anomalous diffusions with variable and random orders HongGuangSun, Wen Chen, Hu Sheng, YangQuan Chen Physics Letters A 374 (2010) 906–910 doi:10.1016/j.physleta.2009.12.021 ME280 "Fractional Order Mechanics" @ UC Merced

  33. a0=0.5 ME280 "Fractional Order Mechanics" @ UC Merced

  34. DO model Vs. VO model The DO model and VO model can not be changed into each other The DO model depict the coupling process The VO model depict the evolution process The VO model is more suitable to depict the decelerating/accelerating diffusion process ME280 "Fractional Order Mechanics" @ UC Merced

  35. Generalized Distributed-Variable order model This generalized model including all the potential models: Variable order model, distributed order model, random order model, variable coefficient integer derivative model ME280 "Fractional Order Mechanics" @ UC Merced

  36. Applications of VO fractional models • Experimental data of bacterial chemotaxis An example of bacteria migrate with chemo attractant Tracking of the bacteria ME280 "Fractional Order Mechanics" @ UC Merced

  37. Fuzzy order diffusion model Cooperation with XiaoNa Song ME280 "Fractional Order Mechanics" @ UC Merced

  38. Fractional order dynamic systems of dynamic order a1(t) a2(y1(t)) y2d(t) y2(t) u2 u1 e C1 + system1 y1(t) system2 y2d(t) + C2 Dynamic order system model and control ME280 "Fractional Order Mechanics" @ UC Merced

  39. Fractional systems of dynamic order a1(t) u1 a2(y1(t)) system1 y1(t) u2 y2(t) system2 http://arxiv.org/pdf/1103.0082.pdf A Dynamic-Order Fractional Dynamic SystemS Hong-Guang, S Hu, C Yang-Quan, C Wen, Y Zhong-BoChinese Physics Letters 30 (4), 046601. 2013. ME280 "Fractional Order Mechanics" @ UC Merced

  40. Part 2: Distributed-Order Mechanics • Mainardi FCDay@UCMerced Lecture • DO Relaxation • DO Diffusion • Book “Distributed-Order Dynamic Systems” (Jiao, Chen, Podlubny, 2012) – SpringerBrief. • Stability condition of DO-LTI systems • MATLAB Codes (Podlubny’s Matrix Approach) • Igor Podlubny’s UIC 2013 Talk ME280 "Fractional Order Mechanics" @ UC Merced

  41. DODS book http://rd.springer.com/book/10.1007%2F978-1-4471-2852-6 http://www.mathworks.com/matlabcentral/fileexchange/36574-demos-for-investigating-distributed-order-linear-time-invariant-systems ME280 "Fractional Order Mechanics" @ UC Merced

  42. Acknowledgements • Prof. Carlos Coimbra • Prof. Igor Podlubny • Prof. Francesco Mainardi • Prof. Tom Hartley and Carl Lorenzo • Dr. Hongguang Sun • Dr. Zhuang Jiao ME280 "Fractional Order Mechanics" @ UC Merced

More Related