Bistable Lattice Research Articles

Perception of multi-stable dot lattices in the visual periphery: An effect of internal positional noise

Dot lattices are very simple multi-stable images where the dots can be perceived as being grouped in different ways. The probabilities of grouping along different orientations as dependent on inter-dot distances along these orientations can be predicted by a simple quantitative model. L. Bleumers, P. De Graef, K. Verfaillie, and J. Wagemans (2008) found that for peripheral presentation, this model should be combined with random guesses on a proportion of trials. The present study shows that the probability of random responses decreases with decreasing ambiguity of lattices and is different for bi-stable and tri-stable lattices. With central presentation, similar effects can be produced by adding positional noise to the dots. The results suggest that different levels of internal positional noise might explain the differences between peripheral and central proximity grouping.

Still states of bistable lattices, compatibility, and phase transition

We study a two-dimensional triangular lattice made of bistable rods. Each rod has two equilibrium lengths, and thus its energy has two equal minima. A rod undergoes a phase transition when its elongation exceeds a critical value. The lattice is subject to a homogeneous strain and is periodic with a sufficiently large period. The effective strain of a periodic element is defined. After phase transitions, the lattice rods are in two different states and lattice strain is inhomogeneous, the Cauchy–Born rule is not applicable. We show that the lattice has a number of deformed still states that carry no stresses. These states densely cover a neutral region in the space of entries of effective strains. In this region, the minimal energy of the periodic lattice is asymptotically close to zero. When the period goes to infinity, the effective energy of such lattices has the “flat bottom” which we explicitly describe. The compatibility of the partially transited lattice is studied. We derive compatibility conditions for lattices and demonstrate a family of compatible lattices (strips) that densely covers the flat bottom region. Under an additional assumption of the small difference of two equilibrium lengths, we demonstrate that the still structures continuously vary with the effective strain and prove a linear dependence of the average strain on the concentration of transited rods.

Analysis of Newton’s Method to Compute Travelling Waves in Discrete Media

We present a variant of Newton’s method for computing travelling wave solutions to scalar bistable lattice differential equations. We prove that the method converges to a solution, obtain existence and uniqueness of solutions to such equations with a small second order term and study the limiting behaviour of such solutions as this second order term tends to zero. The robustness of the algorithm will be discussed using numerical examples. These results will also be used to illustrate phenomena like propagation failure, which are encountered when studying lattice differential equations. We finish by discussing the broad application range of the method and illustrate that higher dimensional systems exhibit richer behaviour than their scalar counterparts.

Kinetics of Martensitic Phase Transitions: Lattice model

Martensitic phase transitions are often modeled by mixed-type hyperbolic-elliptic systems. Such systems lead to ill-posed initial-value problems unless they are supplemented by an additional kinetic relation. In this paper we explicitly compute an appropriate closing relation by replacing the continuum model with its natural discrete prototype. The procedure can be viewed as either regularization by discretization or a physically motivated account of underlying discrete microstructure. We model phase boundaries by traveling wave solutions of a fully inertial discrete model for a bi-stable lattice with harmonic long-range interactions. Although the microscopic model is Hamiltonian, it generates macroscopic dissipation which can be specified in the form of a relation between the velocity of the discontinuity and the conjugate configurational force. This kinetic relation respects entropy inequality but is not a consequence of the usual Rankine--Hugoniot jump conditions. According to the constructed solution, the dissipation at the macrolevel is due to the induced radiation of lattice waves carrying energy away from the propagating front. We show that sufficiently strong nonlocality of the lattice model may be responsible for the multivaluedness of the kinetic relation and can quantitatively affect kinetics in the near-sonic region. Direct numerical simulations of the transient dynamics suggest stability of at least some of the computed traveling waves.

Dynamic copying in multi-layered bistable lattices

Dynamic copying in multi-layered bistable lattices