Modelo tridimensional de Lattice Boltzmann para potenciales electrodinámicos
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Vargas arboleda, Fabian Ricardo
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Este trabajo construye e implementa un modelo de lattice Boltzmann (LBM) para si- mular la evolución de los potenciales electromagnéticos producidos por cualquier distribución de corriente dada. El modelo reproduce en el límite continuo las ecuaciones de onda con fuente que gobiernan el comportamiento de los potenciales electromagnéticos en el gauge de Lorentz, junto con la ecuación de conservación de la carga, que calcula la densidad de carga para la distribución de corriente. La estrategia consiste en modificar el modelo de lattice Boltzmann para ondas de manera que se puedan incluir fuentes, y luego utilizar cuatro de estos modelos para simular la evolución del potencial escalar φ y de las tres componentes del potencial vector ~ A . Finalmente, un LBM adicional se encarga de calcular la densidad de carga de manera que se cumpla la ecuación de conservación. El conjunto reproduce exitosamente los potenciales y los campos electro- magnéticos generados por un dipolo eléctrico oscilante, y sus resultados coinciden tanto con las expresiones analíticas para el campo magnético como con los resultados obtenidos por el método FDTD ( Finite Difference Time Domain ). El modelo propuesto extiende las ventajas de los LBM al cálculo de potenciales electromagnéticos, con muchas posibles aplicaciones en el futuro. Por lo tanto, constituye una alternativa valiosa para la simulación numérica en electrodinámica y un aporte en el desarrollo de estos métodos de simulación.
Abstract: This work develops and implements a lattice Boltzmann model (LBM) to simulate the evolution of the electromagnetic potentials produced by any given distribution of current density. The model reproduces in the continuous limit the wave equations with sources driving the behavior of the electromagnetic potentials in the Lorentz’s gauge, together with the conservation of charge equation, which computes the density of charge for the distribution of current density. The strategy consisted in modifying a lattice Boltzmann for waves, so that sources could be included and, then, in using four of these models to simulate the evolution of the scalar potential φ and the three components of the potential vector ~ A . Finally, an additional LBM computes the density of charge so that the equation of conservation is fulfilled. The set successfully reproduces the potentials and the electromagnetic fields generated by an electrical oscillating dipole, and its results agree very well with the analytical expressions for the magnetic field and with the results obtained by the FDTD (Finite Difference Time Domain). The proposed model extends the advantages of the LBM to the simulation of electromagnetic potentials and makes possible many alternatives of future development. Therefore, it constitutes a valuable alternative for the numerical simulation in electrodynamics and a contribution in the development of lattice Boltzmann models.
Abstract: This work develops and implements a lattice Boltzmann model (LBM) to simulate the evolution of the electromagnetic potentials produced by any given distribution of current density. The model reproduces in the continuous limit the wave equations with sources driving the behavior of the electromagnetic potentials in the Lorentz’s gauge, together with the conservation of charge equation, which computes the density of charge for the distribution of current density. The strategy consisted in modifying a lattice Boltzmann for waves, so that sources could be included and, then, in using four of these models to simulate the evolution of the scalar potential φ and the three components of the potential vector ~ A . Finally, an additional LBM computes the density of charge so that the equation of conservation is fulfilled. The set successfully reproduces the potentials and the electromagnetic fields generated by an electrical oscillating dipole, and its results agree very well with the analytical expressions for the magnetic field and with the results obtained by the FDTD (Finite Difference Time Domain). The proposed model extends the advantages of the LBM to the simulation of electromagnetic potentials and makes possible many alternatives of future development. Therefore, it constitutes a valuable alternative for the numerical simulation in electrodynamics and a contribution in the development of lattice Boltzmann models.