Seminar - Enriched Multiple Scales for Strongly-Nonlinear Ordinary and Partial Differential Equations

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Topic: Enriched multiple scales (EMS) [1], differs from the conventional multiplescales approach by its incorporation of a homotopy parameter and dependent expansions. The central principle of the method, as applied to ODEs, requires that the undamped linear problem homotopic to the nonlinear problem be chosen to respond at the forcing frequency (or a sub- or super-harmonic thereof), leading to secular term cancellation using the excitation frequency. This results in an approach not strictly limited to weakly nonlinear systems and thus capable of capturing periodic, subharmonic, superharmonic, and quasiperiodic solutions at higher forcing and nonlinear strengths.

In this talk, we will first overview EMS for ordinary differential equations (ODEs) and its accuracy will be assessed by comparing its predictions to time-integration solutions for a Duffing system, a dry-friction damped system, and a forced van der Pol system. Ongoing research will then be discussed, including (i) extension of EMS to partial differential equations (PDEs), (ii) alignment of EMS with Complexification-Averaging (CX-A), and (iii) extensions to higher-orders.

[1] Cacan, M.R., Leadenham, S., Leamy, M.J., 2014, “An Enriched Multiple ScalesMethod for Harmonically Forced Nonlinear Systems,” Nonlinear Dynamics, 78 (2): 1205-1220.


Speaker: Dr. Michael J. Leamy - Mechanical Engineering Department Chair and Dorothean Chair of Engineering and Science at the University of Vermont. 


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  • University of Vermont
  • Burlington, Vermont
  • United States 05405
  • Building: Votey Hall
  • Room Number: 307
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  • Co-sponsored by UVM CREATE - Center for Resilient Energy & Autonomous Technologies in Engineering

  Speakers

Michael Leamy

Topic:

Enriched Multiple Scales for Strongly-Nonlinear Ordinary and Partial Differential Equations

Enriched multiple scales (EMS) [1], differs from the conventional multiplescales approach by its incorporation of a homotopy parameter and dependent expansions. The central principle of the method, as applied to ODEs, requires that the undamped linear problem homotopic to the nonlinear problem be chosen to respond at the forcing frequency (or a sub- or super-harmonic thereof), leading to secular term cancellation using the excitation frequency. This results in an approach not strictly limited to weakly nonlinear systems and thus capable of capturing periodic, subharmonic, superharmonic, and quasiperiodic solutions at higher forcing and nonlinear strengths.

In this talk, we will first overview EMS for ordinary differential equations (ODEs) and its accuracy will be assessed by comparing its predictions to time-integration solutions for a Duffing system, a dry-friction damped system, and a forced van der Pol system. Ongoing research will then be discussed, including (i) extension of EMS to partial differential equations (PDEs), (ii) alignment of EMS with Complexification-Averaging (CX-A), and (iii) extensions to higher-orders.

[1] Cacan, M.R., Leadenham, S., Leamy, M.J., 2014, “An Enriched Multiple ScalesMethod for Harmonically Forced Nonlinear Systems,” Nonlinear Dynamics, 78 (2): 1205-1220.

Biography:

Michael J. Leamy is the Mechanical Engineering Department Chair and Dorothean Chair of Engineering and Science at the University of Vermont. Immediately prior he was a Woodruff Endowed Professor in the George W. Woodruff School of Mechanical Engineering, Georgia Tech.  He received his B.S. from Clarkson University (1993), and his M.S. and Ph.D. (1995, 1998) from The University of Michigan, Ann Arbor, all in Mechanical Engineering. Professor Leamy’s research interests are in emerging and multidisciplinary areas of engineering science, with an emphasis on nonlinear dynamical behavior in structures, materials, and complex systems. Emerging engineering materials of particular interest include acoustic metamaterials, topological insulators, and reciprocity-breaking nonlinear lattices.

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