Iterative spectral method for solving electrostatic or magnetostatic problems in complex and evolving heterostructures

Electrostatic or magnetostatic problems involving complex heterogeneity are nontrivial for modeling and simulation. Most existing numerical methods focus on sharp interface models and the computational cost increases with increasing complexity of the geometry. Here we develop an iterative spectral method, the bound charge successive approximation algorithm, to solve electrostatic or magnetostatic heterogeneity problems in the context of diffuse-interface modeling. As tests and verifications, this algorithm is applied to calculation of the depolarization factor of an ellipsoid, and simulation of random dielectric mixtures and the dielectophoretic motion of multiple particles. The algorithm shows excellent efficiency and the computational cost mainly depends on the permittivity or permeability contrast in the whole system, regardless of the complexity of the geometry. In particular, for evolving heterostructures the solution of bound charge in one time step can be used as input for the next, which could further significantly shorten the iteration (approximation) process, making it practical to simulate the long-range electrostatic or magnetostatic interaction in complex and evolving heterostructures.

Figure 1

(a) Schematic of a circular particle in vacuum modeled by diffuse-interface approach (the color denotes relative permittivity); (b) distribution of $\chi \left(\mathbf{r}\right)$ along the path A-B in (a); (c) 3D visualization of the permittivity (set as height) and mesh in (a).

Figure 2

Calculated bound charge density distribution as a circular dielectric inclusion is placed in a uniform external electric field $\left(|{\mathbf{E}}^{ex}|=0.1E_0\right)$, the length and direction of the arrows represent the magnitude and direction of the local electric field.

Figure 3

Calculated depolarization factors for ellipses of different aspect ratios as compared with the results by Mejdoubi and Brosseau [16], and the analytical solution of Osborn [38].

Figure 4

Solution of electrostatics in a random dielectric heterostructure subject to a uniform external electric field. (a) Distribution of local relative permittivity; (b) distribution of the $y$ component of electric field, normalized by the magnitude of external electric field. The direction and length of arrows indicate the direction and magnitude of local electric field.

Figure 5

Study on the performance of the BCSA algorithm. (a) Simulations with different permittivity contrast; (b) simulation with different target precision.

Figure 6

Simulation of colloidal particles movement under electrostatic forces by using diffuse-interface field approach (DIFA) and the BCSA algorithm. Randomly distributed circular particles are attracted by two fixed oppositely charged poles. The dashed line around each particle indicates the halo of that particle. The signs of charge on the poles are manually added for visual aid.

Figure 7

(a) Electric field distribution inside the dash-dot box in Fig. 6. The color represents the local magnitude of electric field (normalized by ${\rho }_0{l}_0/{\varepsilon }_0$) and the arrows indicate the local direction. (b) External electric field in the same zone as (a), calculated by removing the particles.

Figure 8

The real-time iterations needed (averaged per 10 time steps) in the BCSA algorithm corresponding to Fig. 6 (solid blue line), with the red dotted line denotes the maximum particle displacement in a time step during evolution. By applying the RLTI scheme the overall mean required iterations in BCSA is only 3.2 (dashed green line).