Overview of Finite Element Method (FEM)

This article explains the Finite Element Analysis or Method, and it provides the complete steps or procedures to solve any problems of structure or thermal physics. This article also provides advantages of FEM (Finite Element Method) or FEA (Finite Element Analysis) over other analysis or methods for a detailed study of any engineering problems.

Our life does not survive without nature. One has to follow nature's laws like gravitational laws, energy conservation law or the first law of thermodynamics or the second law of thermodynamics or Faraday's law, etc. Every physical phenomenon is normally explained by partial differential equations. To have complete knowledge about a phenomenon, one needs to look into the problem. The given problem is represented by the governing equation (An equation that totally governs the physical phenomenon). This governing equation is normally a partial differential or simple differential one. Numerical methods are used to solve the governing equations by converting the equations into different sets of algebraic equations depending on the mesh and number of nodes used.

Steps of Finite Element Method (FEM)

The finite element method or analysis consists of three main steps in which any problem can be solved with systematic manner. These involve:
  1. Pre-Processing: This process needs or prerequisites a geometry or domain, material property data and boundary conditions to solve a problem.
  2. Solution: Solution of the problem is done by numerical methods either by hand calculation or by programming in computer.
  3. Post Processing: After the solution, one has to analyses the problem and take the decision whether to accept the material, reduce or increase the loading, or any combination which solves the problem or satisfies the objective of the problem

In order to solve any problem with FEM, one needs to first identify the problem and its nature and the laws. Governing equation is required to apply to each cell of mesh and this process is called Descretization. This process involves the determination of Finite element development. This includes the main three steps:
  • Construction of the weak form of the differential equation that governs the problem
  • Assume the form of the approximate solution over a typical finite element or cell
  • Derivation of finite element equations by substituting the approximate solution into the weak form developed in step no. 1

The approximate solution must fulfill following certain conditions in order to be convergent to the actual solution. They are:
  • It should be continuous over the element and differentiable as required by weak form
  • It should be a complete polynomial
  • The interpolation of primary variable at the nodes of the finite element should be possible.

Now all the cells or elements are having their own finite model equation and they are required to be assembled to solve at once. Once the solutions is obtained by numerical methods with certain assumptions, as well as consideration of convergence and error estimation, a final solution can be obtained.

Advantages of Finite Element Method (FEM)

  • FEM can model irregular shaped bodies quite easily by mapped meshing with higher order element size
  • FEM can model the system composed of different materials
  • FEM can handle general load condition without any difficulty
  • This method or FEM can handle unlimited numbers and different kinds of boundary conditions to have best solutions possible
  • This method also handles non-linear behavior associated with large deformation and non linear material
  • Element size can be varied to make it possible to use small sized elements whenever it requires
  • Finite Element Method also includes dynamics effects with ease

Applications of Finite Element Method

There are numerous applications of FEM. This includes - Analysis of structure composed of linear as well as non-linear behavior of physics. The typical structural analysis includes:
Stress analysis including truss and frame analysis and also Stress concentration problem typically associated with holes, fillets or change in geometries of anybody. FEM also helps to do buckling analysis of columns or a member having more Length/Diameter ratio. Buckling analysis is critical for member or body with low diameter but high length.Vibration analysis is also performed with the help of FEM. FEM also helps to solve problems associated with Biomedical engineering which includes analysis of the human spine, jaws, tooth implants, heart, eye or skull, etc.

Thus, FEM is a powerful tool that can be helpful to analyze complex problems with greater ease and efficiency. Students in Mechanical Engineering can use this FEM as tool for analysis of final year projects components.


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