The Casimir force has first been predicted by Casimir in 1948 [1]. It exists among charge-neutral bodies due to the quantum fluctuation of the elecromagnetic fields in vacuum. Although it is tiny when the objects have large separations, it becomes the dominant force between charge-neutral objects when the separation is below micron meters. 

Micro-electromechanical systems (MEMS) are micron-sized devices in which mechanical elements and moving parts, such as tiny sensors and actuators are carved into a silicon substrate. As further miniaturization takes place, the devices may reduce to nano-scale and become nanoelectromechanical systems (NEMS). They have a wide class of applications. For example, the release of the airbag in cars is controlled by a MEMS-based accelerometer. One of the principal causes of malfunctioning in MEMS is stiction, i.e., the collapse of movable elements into nearby surfaces, resulting in their permanent adhesion. Casimir effect is often an important underlying mechanism causing this phenomenon [2]. On the other hand, this effect could be put into good use. Capasso and his group showed that the force can be used to control the actuation of a micromachined torsional device [3]. This group also showed that the Casimir attraction can be used to make a nonlinear oscillator. The force influences the dynamical properties of a micromachined device, changing its resonance frequency, hysteretic behavior, and bistability in its frequency response to an AC excitation. They proposed that this device could serve as a position sensor working in nanometer scale [4, 5]. 

Since Casimir effect is rather significant in MEMS and NEMS devices, it is important to be able to predict it accurately before the devices going into manufacturing. The geometry of the MEMS and NEMS devices may be complicated; thus approximation methods, such as the proximity force approximation (PFA) [6], and optical approach in terms of virtual photons moving along ray optical paths [7], are no longer sufficient. Instead, an exact and general numerical method that can handle arbitrary geometry is desired. With the numerical method, we can get the desired accuracy by giving enough computational resource. Recently, major progress has been made in the development of numerical methods for Casimir force. In general, there are two popular approaches: the path integral approach (also known as EGJK method) and the Maxwell stress-tensor approach. 

Schwinger is the first one to attribute the Casimir interaction to fluctuating current and charge densities inside the objects. With path integral method, the Casimir energy is expressed as the logarithm of a functional integral over all allowed configurations of the fluctuating currents, weighted by the appropriate action [8]. It has been generalized in [9] 

