Energy and Force Stepping Integrators in Lagrangian Mechanics

Author: Gonzalez, Marcial

Year: 2011

Degree: Dissertation (Ph.D.)

Advisor: Ortiz, Michael

Committee Members: Ravichandran, Guruswami; Daraio, Chiara; Lapusta, Nadia; Ortiz, Michael

Option: Aeronautics; Materials Science

DOI: 10.7907/SP10-A207

Abstract

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.

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