# The Concept of Finite Element Method (FEM) and Its Applications

Saurabha Palekar May 4, 2019
Finite Element Analysis or Finite Element Method (FEM) is a computer-based numerical method, for calculating the behavior and strength of engineering structures. It is also used to calculate deflection, vibration, buckling behavior, and stress.
The finite element method (FEM) was introduced in the late sixties in the aerospace industry and it was applied in dentistry in the early seventies. It is a method, in which computerized numerical iteration technique is used. Using this technique, stresses and displacements can be determined using a predetermined model.
FEM technique can be used to analyze small or even large-scale deflection, under applied or loading displacement. It is capable of analyzing plastic deformation of objects, which are permanently bent out of shape due to applied force; or elastic deformation.
As inconceivably large number of calculations are required to analyze a large structure, a computer is needed. Finite element method is easily available to many disciplines and companies, because of the power and low cost of modern computers.

### Basic Concept

Finite element method or FEM, solves a complex problem by redefining it as the summation of the solution of a series of interrelated simpler problems. In FEM, a complex structure is simplified by breaking it down into small elements. These elements are blocks, which form the structure.
Each of the geometrical shape formed by these elements has a specific strain function. They can form shapes of triangle, tetrahedron, square etc. depending upon that strain function. A relatively simple set of equations is used to describe the individual behavioral element. The whole structure is built using these set of elements.
Behavior of the whole structure is described through an extremely large set of equations, which is obtained by the joining of equations describing the behavior of the individual elements. The computer is capable of solving this large set of simultaneous equations.
The computer then extracts the behavior of the individual elements from the solution. After doing this, the computer gets the stress and deflection of all the parts of the structure. The strength of the structure is checked by comparing the stresses to its allowed values for the materials to be used.
FEM enables the computer to evaluate a detailed and complex structure, during the planning of the structure. It also helps in increasing the rating of structures that were significantly over-designed.
Generally two types of analysis are used in the industry, 2-D modeling and 3-D modeling. 2-D modeling is comparatively simple and it allows the analysis to be run on a relatively normal computer, but it also sometimes tends to yield less accurate results.
Whereas, 3-D modeling produces more accurate results, but it cannot effectively run only on normal computers. Numerous algorithms or functions can be inserted within each of these modeling schemes. These modeling schemes are responsible for the linear or non-linear behavior of the system.
Linear systems are less complex and effective in determining elastic deformation. Many of the non-linear systems are capable of testing a material all the way to fracture, and they do account for plastic deformation.

### Applications of FEM

✶ It is used for the description of form changes in biological structures (morphometrics), particularly in the area of growth and development.

✶ FEM and other related morphometric methods like the macro-element or the boundary integral equation method (BIE) are useful for assessment of complex shape changes.
✶ The knowledge of physiological values of alveolar stresses provides a guideline reference for the design of dental implants and it is also important for the understanding of stress related bone remodeling.

✶ It is useful with structures containing potentially complicated shapes like dental implants and inherent homogeneous material.
✶ It is useful for analysis of stresses produced in the periodontal ligament when subjected to orthodontic forces.

✶ It is also useful to study stress distribution in tooth in relation to different designs.

✶ It is used in the area of optimization of the design of dental restorations.
✶ It is used for investigation of stress distribution in tooth with cavity preparation.

✶ The type of predictive computer model described may be used to study the biomechanics of tooth movement, even though accurately assessing the effect of new appliance systems and materials without the need to go to animal or other less representative models.
✶ It is widely used in structural engineering.

✶ It is also used to predict and estimate the damages in electrical fields.

✶ It is also used in optimization of sheet metal blanking process.
FEM methods are widely applied in commercial fields because of its incredible precision in obtaining results. Its applications have also increased because of decreasing cost and increasing power of the computer.