On bubble dynamics and gas dynamics in open tubes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997.Includes bibliographical references (p. 88-91).by Guillermo H. Golsztein.Ph.D

Mathematical modeling of heterogeneous media

Issued as final reportNational Science Foundation (U.S.

The effective energy and laminated microstructures in martensitic phase transformations

We study the behavior of crystals that undergo martensitic transformations. On cooling, the high-temperature phase (austenite) transforms to the martensite phase changing its crystalline symmetry. The lower crystalline symmetry of the martensite gives rise to several variants of martensite. Each variant has an associated transformation strain. These variants accommodate themselves (according to the boundary conditions) forming a microstructure that minimizes the elastic energy. This minimum value of the energy is called the effective energy. We assume that all the material is in the martensite phase (i.e. the material is at low temperatures). We show that, assuming the geometrically linear approximation, the maximum of the effective energy restricted to applied strains in the convex hull of the transformation strains is attained by an applied strain that is a convex combination of only two transformation strains. We derive a recurrence relation to compute the energy corresponding to laminated microstructures of arbitrary rank, under the assumption that the variants of martensite are linearly elastic and their elastic moduli are isotropic. We use this recurrence relation to develop an algorithm that minimizes the energy over microstructures that belong to the class of rank-r laminates. We apply our methods to the case in which the transformation is cubic to monoclinic (corresponding to TiNi). We conclude with some comments on the possible implications of our calculations on the behavior of this shape-memory alloy

A Mathematical Model of the Formation of Lanes in Crowds of Pedestrians Moving in Opposite Directions

In crowded environments, pedestrians moving in opposite directions segregate into lanes of individuals moving in the same direction. It is believed that this formation of lanes that facilitates the flow results from the individuals acting on their behalf, responding to local stimuli, without the intention of benefiting the crowd as a whole. We give evidence that this is true by developing and analyzing a simple mathematical model. Our results suggest that the simple behavior of moving out of the way to avoid imminent collisions leads to the formation of lanes of individuals moving in the same direction

Lateral oscillations of the center of mass of bipeds as they walk. Inverted pendulum model with two degrees of freedom

The use of inverted pendulum models to study the bio-mechanics of biped walkers is a common practice. In its simplest form, the inverted pendulum consists of a point mass, which models the center of mass of the biped, attached to two straight mass-less legs. Most works using the simplest inverted pendulum model constrain the mass and the legs to the sagittal plane (the plane that contains the direction perpendicular to the ground and the direction toward the biped is walking). In this article, we remove this constrain and use this unconstrained inverted pendulum model to study the oscillations the mass experiences in the direction perpendicular to the sagittal plane as the biped walks. While small, these oscillations are unavoidable and of importance in the understanding of balance and stability of walkers, as well as walkers induced oscillations in pedestrian bridges