 # Finite Element Method

In this section, the finite element method (FEM) has been discussed where the nodes can be spaced in any desired orientation so that a region of any shape can be accommodated. Particularly, closely spaced nodes are used for approximations of curved boundaries. It is not such a difficult task to assign nodes closer together in subregions, where the function is changing rapidly, so it improves the accuracy. Due to the wide adaptability of this method, it is very popular. The complicated application problems are investigated through various finite element analysis (FEA) software in which it has to define the region, set up the equations for all types of boundary conditions and then get the solution.

The basis of FEA is to break up the regions (or domains) of the physical problem into small subregions that are called elements. For example, triangles or rectangles are used to approximate a 2D (two-dimensional) region as well as the curved sides. Whereas in a 3D (three-dimensional) region, pyramids or bricks elements are used. First, the region (2D or 3D ) and its elements (triangular or pyramidal) are defined, then the equations for the system are set up and finally investigate the system of equations. The equations are provided some boundary conditions, which are incorporated and solved.

FEA can be used to investigate all three types (parabolic, elliptic and hyperbolic) of partial-differential equations and other eigenvalue problems. A general treatment of FEA to solve ordinary differential equation has been discussed here to give a clear picture.