José Daniel Muñoz

Three-dimensional lattice Boltzmann model for electrodynamics

In this paper we introduce a three-dimensional Lattice-Boltzmann model that recovers in the continuous limit the Maxwell equations in materials. In order to build conservation equations with antisymmetric tensors, like the Faraday law, the model assigns four auxiliary vectors to each velocity vector. These auxiliary vectors, when combined with the distribution functions, give the electromagnetic fields. The evolution is driven by the usual Bhatnager-Gross-Krook (BGK) collision rule, but with a different form for the equilibrium distribution functions. This lattice Bhatnager-Gross-Krook (LBGK) model allows us to consider for both dielectrics and conductors with realistic parameters, and therefore it is adequate to simulate the most diverse electromagnetic problems, like the propagation of electromagnetic waves (both in dielectric media and in waveguides), the skin effect, the radiation pattern of a small dipole antenna and the natural frequencies of a resonant cavity, all with 2% accuracy. Actually, it shows to be one order of magnitude faster than the original Finite-difference time-domain (FDTD) formulation by Yee to reach the same accuracy. It is, therefore, a valuable alternative to simulate electromagnetic fields and opens lattice Boltzmann for a broad spectrum of new applications in electrodynamics.

Three-dimensional lattice Boltzmann model for magnetic reconnection

We develop a three-dimensional (3D) lattice Boltzmann model that recovers in the continuous limit the two-fluids theory for plasmas, and consequently includes the generalized Ohms law. The model reproduces the magnetic reconnection process just by giving the right initial equilibrium conditions in the magnetotail, without any assumption on the resistivity in the diffusive region. In this model, the plasma is handled similar to two fluids with an interaction term, each one with distribution functions associated to a cubic lattice with 19 velocities (D3Q19). The electromagnetic fields are considered as a third fluid with an external force on a cubic lattice with 13 velocities (D3Q13). The model can simulate either viscous fluids in the incompressible limit or nonviscous compressible fluids, and successfully reproduces both the Hartmann flow and the magnetic reconnection in the magnetotail. The reconnection rate in the magnetotail obtained with this model lies between R=0.062 and R=0.073, in good agreement with the observations.

Broad histogram method: extension and efficiency test

A method is presented which allows for a tremendous speed-up of computer simulations of statistical systems by orders of magnitude. This speed-up is achieved by means of a new observable, while the algorithm of the simulation remains unchanged.

Oedometric test, Bauer's law and the micro-macro connection for a dry sand

Abstract What is the relationship between the macroscopic parameters of the constitutive equation for a granular soil and the microscopic forces between grains? In order to investigate this connection, we have simulated by molecular dynamics the oedometric compression of a granular soil (a dry and bad-graded sand) and computed the hypoplastic parameters h s (the granular skeleton hardness) and η (the exponent in the compression law) by following the same procedure than in experiments, that is by fitting the Bauers law e / e 0 = exp ( − ( 3 p / h s ) n ) , where p is the pressure and e 0 and e are the initial and present void ratios. The micro-mechanical simulation includes elastic and dissipative normal forces plus slip, rolling and static friction between grains. By this way we have explored how the macroscopic parameters change by modifying the grains stiffness, V ; the dissipation coefficient, γ n ; the static friction coefficient, μ s ; and the dynamic friction coefficient, μ k . Cumulating all simulations, we obtained an unexpected result: the two macroscopic parameters seems to be related by a power law, h s = 0.068 ( 4 ) η 9.88 ( 3 ) . Moreover, the experimental result for a Guamo sand with the same granulometry fits perfectly into this power law. Is this relation real? What is the final ground of the Bauers Law? We conclude by exploring some hypothesis.

A statistical model of fracture for a 2D hexagonal mesh: The Cell Network Model of Fracture for the bamboo Guadua angustifolia

Abstract A 2D, hexagonal in geometry, statistical model of fracture is proposed. The model is based on the drying fracture process of the bamboo Guadua angustifolia . A network of flexible cells are joined by brittle junctures of fixed Young moduli that break at a certain thresholds in tensile force. The system is solved by means of the Finite Element Method (FEM). The distribution of avalanche breakings exhibits a power law with exponent − 2.93 ( 9 ) , in agreement with the random fuse model (Bhattacharyya and Chakrabarti, 2006) [1] .

Effect of disorder on temporal fluctuations in drying-induced cracking

We investigate by means of computer simulations the effect of structural disorder on the statistics of cracking for a thin layer of material under uniform and isotropic drying. For this purpose, the layer is discretized into a triangular lattice of springs with a slightly randomized arrangement. The drying process is captured by reducing the natural length of all springs by the same factor, and the amount of quenched disorder is controlled by varying the width ξ of the distribution of the random breaking thresholds for the springs. Once a spring breaks, the redistribution of the load may trigger an avalanche of breaks, not necessarily as part of the same crack. Our computer simulations revealed that the system exhibits a phase transition with the amount of disorder as control parameter: at low disorders, the breaking process is dominated by a macroscopic crack at the beginning, and the size distribution of the subsequent breaking avalanches shows an exponential form. At high disorders, the fracturing proceeds in small-sized avalanches with an exponential distribution, generating a large number of microcracks, which eventually merge and break the layer. Between both phases, a sharp transition occurs at a critical amount of disorder ξ(c)=0.40±0.01, where the avalanche size distribution becomes a power law with exponent τ=2.6±0.08, in agreement with the mean-field value τ=5/2 of the fiber bundle model. Moreover, good quality data collapses from the finite-size scaling analysis show that the average value of the largest burst ⟨Δ(max)⟩ can be identified as the order parameter, with β/ν=1.4 and 1/ν≃1.0, and that the average ratio ⟨m(2)/m(1)⟩ of the second m(2) and first moments m(1) of the avalanche size distribution shows similar behavior to the susceptibility of a continuous transition, with γ/ν=1, 1/ν≃0.9. These results suggest that the disorder-induced transition of the breakup of thin layers is analogous to a continuous phase transition.

Influence of rotations on the critical state of soil mechanics

Abstract The ability of grains to rotate can play a crucial role on the collective behavior of granular media. It has been observed in computer simulations that imposing a torque at the contacts modifies the force chains, making support chains less important. In this work we investigate the effect of a gradual hindering of the grains rotations on the so-called critical state of soil mechanics. The critical state is an asymptotic state independent of the initial solid fraction where deformations occur at a constant shear strength and compactness. We quantify the difficulty to rotate by a friction coefficient at the level of particles, acting like a threshold. We explore the effect of this particle-level friction coefficient on the critical state by means of molecular dynamics simulations of a simple shear test on a poly-disperse sphere packing. We found that the larger the difficulty to rotate, the larger the final shear strength of the sample. Other micro-mechanical variables, like the structural anisotropy and the distribution of forces, are also influenced by the threshold. These results reveal the key role of rotations on the critical behavior of soils and suggest the inclusion of rotational variables into their constitutive equations.

Traffic Flow in Bogotá

We introduce cellular automaton models for both cars alone [1] and mixed traffic (cars and buses) on motorways in Bogota. Our model includes three elements: hysteresis between acceleration and braking gaps, a delay time in the acceleration, and instantaneous braking. In addition, we include a lane changing rule and the disordered behavior of Bogotan bus drivers. The parameters of our model were obtained from direct measurements on a car and a bus in this city. We use this model to simulate the flux-density fundamental diagram for a singlelane road with car traffic and a two-lane road with mixed traffic, and compare the results with experimental data. Our simulations are in very good agreement with experimental measurements, and reproduce both the shape and the value of the maximal flux. Moreover, they show that the causes of the measured high fluxes are the short gaps that the Bogotan drivers are used to maintain to the car ahead (the agressive driving that is typical for this city).

Percolation study for the capillary ascent of a liquid through a granular soil

Capillary rise plays a crucial role in the construction of road embankments in flood zones, where hydrophobic compounds are added to the soil to suppress the rising of water and avoid possible damage of the pavement. Water rises through liquid bridges, menisci and trimers, whose width and connectivity depends on the maximal half-length {\lambda} of the capillary bridges among grains. Low {\lambda} generate a disconnect structure, with small clusters everywhere. On the contrary, for high {\lambda}, create a percolating cluster of trimers and enclosed volumes that form a natural path for capillary rise. Hereby, we study the percolation transition of this geometric structure as a function of {\lambda} on a granular media of monodisperse spheres in a random close packing. We determine both the percolating threshold {\lambda}_{c} = (0.049 \pm 0.004)R (with R the radius of the granular spheres), and the critical exponent of the correlation length {\nu} = (0.830 \pm 0.051), suggesting that the percolation transition falls into the universality class of ordinary percolation.

A Simple Statistical Method for Reproducing the Highway Traffic

Some of the most important questions concerning the traffic flow theory are focused on the correct functional form of the empirical flow-density fundamental diagram. Although most cellular automata intend to reproduce this diagram by measuring the limit steady-states from the dynamic simulation, real roads are constantly perturbed by external factors, driving the system to explore a much broader phase space. Hereby, we show that a Monte Carlo sampling of all states compatible with a driving rule (previously derived for Bogota) actually reproduces the measured fundamental diagram, both in mean values and dispersion, when all such states are assumed equally probable. Even more, by using the Wardrop’s relation, the same gathered data also approximates the general form of the time-mean fundamental diagrams. These results suggest that driving rules are much richer in information than usually expected and, that the assumption of equally probable states plus a finite length of road may be a first model for the statistical description of highways.