Variational Time Integrators in Computational Solid Mechanics

Citation

Lew, Adrián José (2003) Variational Time Integrators in Computational Solid Mechanics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/6C74-GC16. https://resolver.caltech.edu/CaltechETD:etd-05262003-200254

Abstract

This thesis develops the theory and implementation of variational integrators for computational solid mechanics problems, and to some extent, for fluid mechanics problems as well. Variational integrators for finite dimensional mechanical systems are succinctly reviewed, and used as the foundations for the extension to continuum systems. The latter is accomplished by way of a space-time formulation for Lagrangian continuum mechanics that unifies the derivation of the balance of linear momentum, energy and configurational forces, all of them as Euler-Lagrange equations of an extended Hamilton's principle. In this formulation, energy conservation and the path independence of the J- and L-integrals are conserved quantities emanating from Noether's theorem. Variational integrators for continuum mechanics are constructed by mimicking this variational structure, and a discrete Noether's theorem for rather general space-time discretizations is presented. Additionally, the algorithms are automatically (multi)symplectic, and the (multi)symplectic form is uniquely defined by the theory. For instance, in nonlinear elastodynamics the algorithms exactly preserve linear and angular momenta, whenever the continuous system does.

A class of variational algorithms is constructed, termed asynchronous variational integrators (AVI), which permit the selection of independent time steps in each element of a finite element mesh, and the local time steps need not bear an integral relation to each other. The conservation properties of both synchronous and asynchronous variational integrators are discussed in detail. In particular, AVI are found to nearly conserve energy both locally and globally, a distinguishing feature of variational integrators. The possibility of adapting the elemental time step to exactly satisfy the local energy balance equation, obtained from the extended variational principle, is analyzed. The AVI are also extended to include dissipative systems. The excellent accuracy, conservation and convergence characteristics of AVI are demonstrated via selected numerical examples, both for conservative and dissipative systems. In these tests AVI are found to result in substantial speedups, at equal accuracy, relative to explicit Newmark.

In elastostatics, the variational structure leads to the formulation of discrete path-independent integrals and a characterization of the configurational forces acting in discrete systems. A notable example is a discrete, path-independent J-integral at the tip of a crack in a finite element mesh.

Thesis Files