Hybrid Numerical Modelling and Simulation of Electrostatic Force Microscope

This work presents hybrid numerical modelling and simulation of the electrostatic force microscope (EFM) which is one of the most important tools for the investigation of electric field at a nanometer scale. Basically the EFM is an atomic force microscope (AFM) which is operated in a noncontact mode.
In order to develop a simulation model for the EFM, some problems need to be overcome. Due to the enormous difference in size between the tip and the cantilever, the set up has to be treated as a multiscale problem. In addition electrical and mechanical properties have to be considered and coupled in the simulation. Furthermore it may be necessary to include nonlinearities, depending on the material of the sample under investigation. In order to solve these problems efficiently several numerical methods are implemented and coupled in this work. These numerical methods include the finite element method (FEM) and the boundary element method (BEM) which are implemented in 2D and 3D. Since the scanning process of the EFM is dynamic, the FE mesh has to be adapted to the changing geometry during the simulation. This is done by using the arbitrary Lagrangian Eulerian Method (ALE).
Due to the densely populated BEM matrices which arise from the boundary integral operators in 3D, the solution of the system of equations is expensive in terms of computation time. In order to save computational time the data-sparse BEM is implemented in 3D case. At the same time a specific preconditioner and a fast solver are implemented for the fast solution of this huge matrix system.