the square of the radius ( F ∗ R $^{2}$/ ℏ c 0 ) is a dimensionless and scale-invariant quantity. So the result is given by their product ( F ∗ R $^{2}$/ ℏ c 0 ) versus the normalized separation ( Z/R ). In the numerical process, first the Casimir force per unit area (pressure) is calculated at different points on the surface, and then the total force is obtained by integrating the pressure over the entire surface. The analytical results are interpolated from the data given in ref. [9]. Good agreement has been observed. 0.5 1 1.5 2 2.5 3 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Z/R F*R*R/hbar*c0 analytical numerical Z R FIG. 1: Casimir force between two spheres of radius R and a separation of Z . The geometry of the second example contains two identical capsules. A capsule is a cylin- der of radius R with hemispherical end caps. Its total length is denoted by L . Experiment had been carried out to measure the interaction between two crossed cylindrical capsules [23] so a numerical study of the geometry would be interesting to experimentalist. It has also been simulated by using EGJK method in ref. [11] and reference data is available to make comparison. The force between two capsules is evaluated for two cases: they are parallel or they are perpendicular. Similar to the 2D geometries [20] case, the integral equation method could reduced the computation cost significantly since it involves only surface unknowns. Moreover, both the coordinate space integration (1) and the spectral space integration (8) to (10) are smooth and independent of the number of unknowns. As we know, both our surface integral method following the stress tensor approach and Reid et al numerical method following the path integral approach [11] are effective numerical methods for calculating the Casimir force among arbitrary 3D objects. Though they start from completely different description of the Casimir energy and force, they are connected 7