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Energy and Force Stepping Integrators in Lagrangian Mechanics


Gonzalez, Marcial (2011) Energy and Force Stepping Integrators in Lagrangian Mechanics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/SP10-A207.


The overarching goal of this thesis is to develop new numerical time integration schemes for Lagrangian mechanics that better cope with the challenges of understanding the dynamic behavior of materials. We specifically address the formulation of convergent time integration schemes that exhibit good long-term behavior---such as conferred by symplecticity and exact conservation properties---and that have the ability to automatically and asynchronously modulate the time step in different regions of the domain. We achieve these properties in a progression of three developments: (i) energy-stepping, (ii) force-stepping, and (iii) asynchronous energy-stepping integrators. These developments are based on a new method of approximation for Lagrangian mechanics, proposed in this thesis, that consists of replacing the Lagrangian of the system by a sequence of approximate Lagrangians that can be solved exactly. Then, energy-stepping integrators result from replacing the potential energy by a piecewise constant approximation, force-stepping integrators result from replacing the potential energy by a piecewise affine approximation, and asynchronous energy-stepping integrators result from replacing localized potential energies by piecewise constant approximations. Throughout the dissertation, the properties of these time integrators are theoretically predicted and born out by a number of selected examples of application. Furthermore, we address the challenges of understanding the propagation of solitary waves in granular crystals at low impact velocity conditions by investigating the role of energy-trapping effects with the numerical time integration schemes developed in this work.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Lagrangian mechanics; time-integration; symplectic integrators; energy-momentum integrators; granular crystals
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Aeronautics
Minor Option:Materials Science
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Ortiz, Michael
Thesis Committee:
  • Ravichandran, Guruswami (chair)
  • Daraio, Chiara
  • Lapusta, Nadia
  • Ortiz, Michael
Defense Date:20 August 2010
Record Number:CaltechTHESIS:10052010-230939247
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6103
Deposited By: Marcial Gonzalez
Deposited On:01 Nov 2010 21:08
Last Modified:09 Oct 2019 17:06

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