The development of nanoscience and nanotechnology has important implications for advances of electronics, biology, medicine, photonics, and other areas. The growing knowledge in this field will lead to profound progress in the ways that materials, devices, and systems are understood and created. Numerical simulation is an indispensabe tool for understanding nanoscale systems, as our usual intuition may be misleading at the nanoscale.
This dissertation focuses on two classes of numerical methods: the finite element method (FEM) and finite difference (FD) methods with their generalization known as the flexible local approximation method (FLAME). FEM is a versatile numerical method that is widely applied in all areas of engineering analysis. This method remains powerful for many physical nanoscale models, especially problems invloving complex geometries and inhomogeneous media, provided that the required number of finite elements is not too large. However, for a large number of objects, the complexity and the computational overhead of FE meshes and the related data structures become too high.
Based on the simple Taylor expansions, FD method has significant advantage for geometrically simple problems. However, the accuracy of FD deteriorates for problems with geometrically complex boundaries and material interfaces not conforming to the FD grid lines. The Taylor expansion breaks down at material interface boundaries because the solution is not sufficiently smooth for such problems. FLAME is a generalized FD calculus recently developed. It replaces the Taylor expansion with a physically and mathematically more accurate local approximation. By this way, this method reduces or even eliminates the “staircase” noise at slanted or curved material interfaces. FLAME is first applied in the simulations of electrostatic and magnetostatic multiparticle problems. It shows higher accuracy both in two dimensions (2D) and three dimensions (3D) compared with the finite difference (FD) method and FEM. FLAME also exhibits flexibility in the interpolation of the potential, electric field, and the calculation of the force. For the problems in which components are in close proximity to each other, analytical/numerical bases and adaptive mesh algorithms are developed based on FLAME for better accuracy without increasing the complexity of the calculation.
The FLAME method, including analytical/numerical bases and adaptive mesh algorithms, is also applied to wave scattering problems. The computational cost of FLAME in many cases is much lower than that of other methods at comparable levels of numerical accuracy.
As a novel application of FLAME, this method is used to explore electrostatic interactions for macromolecules (e.g. protein molecules) in electrolytes. In the conventional model, the whole domain is divided into two layers: the inner macromolecular core and the outer solvent. The inner layer is governed by the Poisson equation with the existance of point charge, and the outer one is governed by the Poisson-Boltzmann equation due to the Boltzmann-like distribution of ions. Results show that this model has great accuracy for short-distance interaction. However, the accuracy for long-distance interaction is not as good as for short-distance interaction. To improve the whole accuracy, an interim layer with a low dielectric permittivity is introduced to simulate the region between macromolecular core and solvent. The simulation based on FLAME shows significant accuracy improvement compared with that of the conventional FD method. The accuracy in FLAME is high even for the area around point charge singularities.
FEM is applied to a ferrofluid model that is of interest in magneticly driven assembly of micro- and nanoparticles. The ferrofluid particles are characterized by their volume density with a Boltzmann-like distribution function in the magnetic field. The problem is formulated in terms of the scalar , rather than vector, magnetic potential, which significantly reduces the computational cost.
FEM is used for the problem of nano-focusing of light by a self-similar cascade of silver nanoparticles. The goal is to explore the electrodynamic effects affecting the very high local field enhancement. The results lead to appreciable corrections of field enhancement in real applications.