Papers by Guillermo Goldsztein
SIAM Journal on Applied Mathematics
Chaos: An Interdisciplinary Journal of Nonlinear Science
Using a combination of theory, experiment, and simulation, we revisit the dynamics of two coupled... more Using a combination of theory, experiment, and simulation, we revisit the dynamics of two coupled metronomes on a moving platform. Our experiments show that the platform’s motion is damped by a dry friction force of Coulomb type, not the viscous linear friction force that has often been assumed in the past. Prompted by this result, we develop a new mathematical model that builds on previously introduced models but departs from them in its treatment of friction on the platform. We analyze the model by a two-timescale analysis and derive the slow-flow equations that determine its long-term dynamics. The derivation of the slow flow is challenging due to the stick-slip motion of the platform in some parameter regimes. Simulations of the slow flow reveal various kinds of long-term behavior including in-phase and antiphase synchronization of identical metronomes, phase locking and phase drift of non-identical metronomes, and metronome suppression and death. In these latter two states, one...
Bulletin of the American Physical Society, 2020
The research objectives are the development of new mathematical tools for the study of the influe... more The research objectives are the development of new mathematical tools for the study of the influence of the microgemoetry of heterogeneous materials on their macroscopic properties. With those goals in mind, three projects were selected to study: 1) Mechanical properties of composites and polycrystals, 2) Solute transport in porous media and 3) Small particles migration and clogging. The broad educational objectives are the exposure to students to interdisciplinary work, the development of activities to help students become good modelers, and the participation of students in the research projects of the PI. The educational activities proposed were the development of a course in mathematical modeling of materials and a working seminar in interdisciplinary mathematics. During the length of the grant (three years) we have obtained very satisfying results in all three projects. As a result, we have published five articles in first rate scientific peer reviewed journals, a sixth article has been submitted for reviewed, three more articles are in preparation and, equally importantly, we have generated several new promising ideas and research directions that we plan pursue. We have presented our results in several conferences and colloquia.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2021
In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronize... more In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they often tend to synchronize in phase, not antiphase. Here, we study both in-phase and antiphase synchronization in a model of pendulum clocks and metronomes and analyze their long-term dynamics with the tools of perturbation theory. Specifically, we exploit the separation of timescales between the fast oscillations of the individual pendulums and the much slower adjustments of their amplitudes and phases. By scaling the equations appropriately and applying the method of multiple timescales, we derive explicit formulas for the regimes in the parameter space where either antiphase or in-phase synchronization is stable or where both are stable. Although this sort of perturbative analysis is standard in other parts of nonlinear science, surprisingly it has ...
Nature Physics, 2017
Nematic order on curved surfaces is often disrupted by the presence of topological defects, which... more Nematic order on curved surfaces is often disrupted by the presence of topological defects, which are singular regions in which the orientational order is undefined. In the presence of force-generating active materials, these defects are able to migrate through space like swimming microorganisms. We use toroidal surfaces to show that despite their highly chaotic and non-equilibrium dynamics, pairs of defects unbind and segregate in regions of opposite Gaussian curvature. Using numerical simulations, we find that the degree of defect unbinding can be controlled by tuning the system activity, and even suppressed in strongly active systems. Furthermore, by using the defects as active microrheological tracers and quantitatively comparing our experimental and theoretical results, we are able to determine material properties of the active nematic. Our results illustrate how topology and geometry can be used to control the behaviour of active materials, and introduce a new avenue for the quantitative mechanical characterization of active fluids.
Quarterly of Applied Mathematics, 2017
We consider fiber reinforced ideally plastic composites. We analyze a mathematical model valid fo... more We consider fiber reinforced ideally plastic composites. We analyze a mathematical model valid for microstructures and applied stresses that lead to both microscopic and macroscopic anti-plane shear deformations. We obtain a bound on the yield set of the reinforced material in terms of the shapes of the cross section of the fibers, their volume fraction, and the yield stresses of the matrix. We construct examples showing that our bound is sharp.
Physica A: Statistical Mechanics and its Applications, 2017
h i g h l i g h t s • We analyze a mathematical model of the formation of lanes in crowds. • We p... more h i g h l i g h t s • We analyze a mathematical model of the formation of lanes in crowds. • We prove that the system self-organizes. • We compute the time required for self-organization.
Discrete Dynamics in Nature and Society, 2015
In crowded environments, pedestrians moving in opposite directions segregate into lanes of indivi... more 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.
SIAM Journal on Applied Mathematics, 1997
We study dynamical hysteresis in a simple class of nonlinear ordinary differential equations, nam... more We study dynamical hysteresis in a simple class of nonlinear ordinary differential equations, namely first-order equations subject to sinusoidal forcing. The assumed nonlinearities are such that the area of the hysteresis loop vanishes as the forcing frequency tends to zero; in other words, there is no static hysteresis. Using regular and singular perturbation techniques, we derive the first term in the asymptotic expansion for the loop area as a function of the driving frequency, in the limit of both large and small frequency. Although the theory was originally motivated by experiments on bistable semiconductor lasers, it is applied here to explain (and in some cases, to correct) the scaling laws that were recently reported in numerical studies of mean-field kinetic Ising models of ferromagnets.
SIAM Journal on Applied Mathematics, 2009
In this paper we model filters as networks of channels. As suspension (fluid with particles) flow... more In this paper we model filters as networks of channels. As suspension (fluid with particles) flows through the filter, particles are trapped and clog channels. We assume there is no flow through clogged channels. The filter becomes impermeable after not all, but only a portion, of the channels clog. In this paper we compute an upper bound on the number of channels that clog. This bound is a function of properties of the network. Our results provide an understanding of the relationship between the filter pore space geometry and the filter efficiency.
SIAM Journal on Applied Mathematics, 2012
We consider fiber reinforced composites where both the matrix and the fibers are ideally plastic ... more We consider fiber reinforced composites where both the matrix and the fibers are ideally plastic materials. We restrict our attention to microstructures and applied stresses that lead to both microscopic and macroscopic antiplane shear deformations. We describe a bound we have recently obtained on the yield set of the composite in terms of the shape of the fibers, their volume fraction, and the yield set of the matrix. We construct examples of composites showing that our bound is essentially optimal.
SIAM Journal on Applied Mathematics, 2005
Lacunar-canalicular systems are networks of pores (lacunae) interconnected by thin channels (cana... more Lacunar-canalicular systems are networks of pores (lacunae) interconnected by thin channels (canaliculi) that are embedded in bones. The efficient transport of nutrients within lacunarcanalicular systems is necessary to keep bones healthy. Several theories have been proposed to identify the physical phenomena responsible for this efficient transport. In this paper, we develop and study a mathematical model motivated by one of those theories.
SIAM Journal on Applied Mathematics, 2005
We consider the following experiment. A container is filled with a suspension consisting of parti... more We consider the following experiment. A container is filled with a suspension consisting of particles immersed in an incompressible liquid. An opening is made on the container wall and the suspension flows through the opening. We develop a mathematical model to compute the expected volume of suspension extracted before particles clog the opening. Our studies are relevant to the understanding of clogging of pore throats in porous media, which plays an important role in geomaterials, biological systems, and industrial applications.
SIAM Journal on Applied Mathematics, 2011
Fluid in porous media flows through tortuous paths. When the fluid velocities are large enough, t... more Fluid in porous media flows through tortuous paths. When the fluid velocities are large enough, this tortuosity and inertial effects cause suspended particles to collide with pore walls. After each collision, a particle loses momentum and needs to be accelerated again by hydrodynamic forces. As a result, the average velocity of suspended particles may be smaller than that of the fluid. In addition, if the fluid velocity field is not macroscopically homogeneous, the retardation of the particles with respect to the fluid leads to an increase of the concentration of particles in certain regions which may eventually result in the clogging of the porous medium. This effect is of importance in flows near wells where the flow has circular symmetry and thus it is not macroscopically homogeneous. In this paper, we study the physical effect described above. We first develop a mathematical model that takes into account tortuosity and inertial effects at the pore scale. This model provides us with the average particle velocities as a function of the fluid velocity. We use this function in a second model, a macroscopic scale model, to study the evolution of concentration of particles and the development of clogging when the flow has circular symmetry.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2004
We present a fast numerical method for the simulation of martensitic transformations in three-dim... more We present a fast numerical method for the simulation of martensitic transformations in three-dimensional polycrystals. To produce the relevant overall elastic energy arising from given boundary conditions, this method proceeds by reducing the corresponding non-convex minimization problem to minimization of a certain quadratic form-over the set of arrays of transformation strains which are compatible with a given distribution of crystallite orientations. The evaluation of this quadratic form for a given array of transformation strains requires solution of certain linear elasticity problems. An acceleration strategy we use, which for a polycrystal containing N grains reduces the complexity of the algorithm from O(N 2) to O(N) operations, results from a formulation of the minimization problem which takes advantage of certain decorrelations present in the minimizing arrays of transformation strains. We illustrate our presentation with a number of examples involving cubic-to-monoclinic and cubic-to-orthorhombic polycrystalline phase transitions, such as those arising in the TiNi and CuAl shape-memory alloys. In particular, our study quantifies the effects of texture on the overall properties of such polycrystalline shape-memory alloys.
Physical Review Letters, 1995
We examine scaling laws for dynamical hysteresis in an optically bistable semiconductor laser. An... more We examine scaling laws for dynamical hysteresis in an optically bistable semiconductor laser. An analytic derivation of these laws from multidimensional laser equations is outlined and they are expected to be universal for systems that exhibit a cusp catastrophe. The scaling laws for the hysteresis loop area or width are numerically verified and experimentally measured for operation of the bistable laser above and below threshold. Excellent agreement with theory is obtained in the limit of low switching frequencies.
Journal of the Mechanics and Physics of Solids, 2001
We study the behavior of crystals that undergo martensitic transformations. On cooling, the high-... more 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 e ective 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 e ective 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.
Journal of Applied Physics, 1952
The velocity and pressure field in a slightly compressible liquid resulting from the collapse of ... more The velocity and pressure field in a slightly compressible liquid resulting from the collapse of a spherical bubble is computed as a function of the pressure at the bubble wall, using the acoustic approximation. The results are accurate as long as the liquid velocities are small compared to the sonic velocity in the liquid; they agree with the results previously obtained by C. Herring. The following bubble model is then investigated. The bubble is supposed filled with inviscid perfect non-conducting gas; special emphasis is given to the gas motion which involves a series of shock waves. A fraction of the energy of compression is thus degraded so that the radius of the bubble after the first rebound is somewhat less than the original radius. The pressure variation at the bubble wall is virtually the same as if the gas were compressed uniformly and isentropically.
International Journal of Bifurcation and Chaos, 1995
We analyze the linear stability of the synchronized state in networks of N identical digital phas... more We analyze the linear stability of the synchronized state in networks of N identical digital phase-locked loops. These are pulse-coupled oscillator arrays in which the frequency (rather than the phase) of each oscillator is updated discontinuously whenever that oscillator reaches a specific phase in its cycle. Three different coupling configurations are studied: one-way rings, two-way rings, and globally coupled arrays. In each case we obtain explicit formulas for the transient time to lock, the critical gain at which the synchronized state loses stability, and the period of the bifurcating solution at the onset of instability. Our results explain the numerical observations of de Sousa Vieira, Lichtenberg, and Lieberman.
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Papers by Guillermo Goldsztein