Concepts of FEM

The basic concept of FEM is that a body or a structure may be divided into smaller elements of finite dimensions called finite elements. The original body or the structure is then considered as an assemblage of these elements connected at a finite number of joints called nodes or nodal points. The properties of the elements are formulated and combined to obtain the solution for the entire body or structure. For example, in the displacement formulation widely adopted in finite element analysis, simple functions known as shape functions are chosen to approximate the variation of the displacement within an element in terms of the displacements at the nodes of the elements. This follows the concept used in the Rayleigh-Ritz procedure of functional approximation method but the difference is that the approximation to field variable is made at the element level. The strains and stresses within an element will also be expressed in terms of the nodal displacements. Then the principle of virtual displacement or the minimum potential energy is used to derive the equation of equilibrium  for the element and the nodal displacements will be the unknowns in the equations. The equations of equilibrium for the entire structure or body are then obtained by combining the equilibrium equation of each element such that the continuity of displacement is ensured at each node where the elements are connected. The necessary boundary conditions are imposed and the equations of equilibrium are solved for the nodal displacements. Having thus obtained the values of displacements at the nodes of each element, the stains and stresses are evaluated using the element properties derived earlier.

Consider the volume of some material having known physical properties. The volume represents the domain of a boundary value problem to be solved

The entire domain is discretized into finite number of sub domains called elements. These elements are bot a differential element of size dx X dy. Hence the finite element. Every element is connected to other element at specified joints called nodes.

A node is a specified point in the finite element at which the value of the field variable is to be calculated. Exterior nodes are located on the boundaries of the finite element and may be used to connect an element to the adjacent finite elements. Nodes that do not lie on element boundaries are interior nodes and cannot be connected to any other elements.

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