Complexity and Spatio-Temporal Chaos in Material Failure: Analysis and Computation of Fiber Bundle Models

Wednesday, October 10, 2001 - 9:30am - 10:30am
Keller 3-180
William Newman (University of California, Los Angeles)
Problems manifesting complexity and spatio-temporal chaos are endemic in the physical sciences. These problems are often difficult to describe from first principles and generally beyond the reach of computational and analytic methods. For example, earthquakes show self-similar behavior in space and time and possess several power-law scalings valid over many orders of magnitudes, yet our knowledge of continuum mechanics is sufficiently primitive (and linear) that it offers no insight into the earthquake mechanism. Parallels are often made between earthquake activity and fluid turbulence; however, nothing paralleling the Navier-Stokes equations for earthquakes is known.

Remarkably, a variety of toy models have provided some important new insights into the problems. Some of these are reminiscent of the underlying simplicity (and emergent complexity) inherent in Feigenbaum's seminal work on deterministic chaos and scaling. Not only do these toy models deliver an improved understanding of complicated physical processes, they provide a rich set of problems that are ripe for mathematicians and computer scientists.

This lecture will focus on a class of cellular automata models---referred to as fiber bundles---developed to describe material failure and widely used in applications ranging from materials science to theoretical seismology. These models employ a probabilistic formulation applied to cellular automata organized geometrically according to the nature of the problem, and result in problems that have a hierarchical flavor, that is a functional iteration (in contrast with a simple function iteration).

An important ingredient in these investigations is the interplay between computation and analysis. Computation is often important to establishing the nature of large scale behavior and sometimes leads to theorems, in keeping with Von Neumann's dictum regarding computation and analysis in nonlinear problems. Sometimes, analysis is required to make the computation possible, owing to the large numbers of elements M required to show physical scalings---characterized by Avogadro's number or 1023 or greater---and restructuring of the problem, in the same spirit as the Fast Fourier Transform, can be used to render it computationally irreducible. [Typically, these problems require O(M*M) operations but can sometimes be reduced to some power of log(M).]