Finite element method, FEM

"Finite element method (FEM)" is one of numerical analytical methods to obtain an approximate solution of partial differential equations that are difficult to solve analytically. First, an object of interest is divided into elements that each has a simple shape and a finite size. Next, physical quantities (temperature, stress, etc.) of each element are approximated by a simpler equation, and then the equations for the elements are combined to construct simultaneous equations. By solving the obtained simultaneous equation under the boundary conditions of the physical quantities at surfaces of the elements, the distribution of the physical quantities over the object are obtained. Since an object is subdivided to polyhedrons, FEM can be conveniently applied to complicated-shape objects. In electron microscopy, the method is used for calculation of mechanical strength and thermal distribution, calculation of distributions of magnetic fields and electrostatic fields of magnetic lenses and electrostatic lenses, etc. In the development of lens polepieces, aberration coefficients are obtained by calculation of electron trajectories using the magnetic field distributions obtained by FEM, and then the shapes of magnetic poles are optimized.