Introduction on Introduction to Finite Elements And the Big Picture :

Application of physical principles, such as mass balance, energy conservation, and equilibrium, naturally leads many engineering analysis situations into differential equations. The exact solutions for differential equations are not applicable to many practical problems because either their governing differential equations does not have an exact solution or they involve complex geometries.

The finite element method is one of the numerical methods for obtaining approximate solution of ordinary and partial differential equations. It is especially powerful when dealing with boundary conditions defined over complex geometries that are common in practical applications.

Concepts and Formulas of Introduction to Finite Elements And the Big Picture:

Complete solution procedure:

The approximate solution of any problem governed by a differential equation can be obtained by using the element equations by following the usual finite element steps:

Development of element equations

Discretization of solution domain into a finite element mesh

Assembly of element equations

Introduction of boundary conditions

Solution for nodal unknowns

Computation of solution and related quantities over each element

Advantages of using Finite Element Method:

The subdivision of a whole domain into simpler parts has several advantages:

Accurate representation of complex geometry; while most analytical solutions are restricted to simplified geometries.

Inclusion of dissimilar material properties; while most analytical methods are applicable to homogeneous materials.