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The Finite Element Method (FEM) is a numerical technique used to perform Finite Element Analysis (FEA) of any given physical phenomenon.

##### Introduction

The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.

##### Basic Concepts

The finite element method(FEM), or finite element analysis (FEA), is based on the idea of building a complicated object with simple blocks, or, dividing a complicated object into small and manageable pieces. Application of this simple idea can be found everywhere in everyday life as well as in engineering.

Examples:
· Lego (kids’ play)
· Buildings
· Approximation of the area of a circle:

##### Why Finite Element Method?

· Design analysis: hand calculations, experiments, and computer simulations.
· FEM/FEA is the most widely applied computer simulation method in engineering.
· Closely integrated with CAD/CAM applications.

##### Applications of FEM in Engineering

· Mechanical/Aerospace/Civil/Automobile Engineering
· Structure analysis (static/dynamic, linear/nonlinear)
· Thermal/fluid flows
· Electromagnetics
· Geomechanics
· Biomechanics

##### FEM in Structural Analysis

Procedures:
· Divide structure into pieces (elements with nodes)
· Describe the behavior of the physical quantities on each element
· Connect (assemble) the elements at the nodes to form an approximate system of equations for the whole structure
· Solve the system of equations involving unknown quantities at the nodes (e.g., displacements)
· Calculate desired quantities (e.g., strains and stresses) at selected elements

Computer Implementations

· Preprocessing (build FE model, loads and constraints)
· FEA solver (assemble and solve the system of equations)
· Postprocessing (sort and display the results)

Available Commercial FEM Software Packages

· ANSYS(General purpose, PC and workstations)
· NASTRAN(General purpose FEA on mainframes)
· ABAQUS(Nonlinear and dynamic analyses)
· COSMOS(General purpose FEA)
· ALGOR (PC and workstations)
· PATRAN(Pre/Post Processor)
· HyperMesh(Pre/Post Processor)
· Dyna-3D(Crash/impact analysis)

##### Step by step process of FEM with the help of an Example click on the link below ##### Types of Finite Elements  We only consider linear problems in this introduction.

Consider the equilibrium of forces for the spring. At node i, we have where
k= (element) stiffness matrix
u= (element nodal) displacement vector
f= (element nodal) force vector
Note that k is symmetric. Is k singular or non-singular? That is, can we solve the equation? If not, why?  K is the stiffness matrix (structure matrix) for the spring system.
An alternative way of assembling the whole stiffness matrix: “ Enlarging” the stiffness matrices for elements 1 and 2, we have Adding the two matrix equations (superposition), we have This is the same equation we derived by using the force equilibrium concept.
Assuming, u1=0 and F1= F2= P Solving the equations, we obtain the displacements and the reaction force
F1=-2 P

Checking the Results
· Deformed shape of the structure
· Balance of the external forces
· Order of magnitudes of the numbers
· Suitable for stiffness analysis
· Not suitable for stress analysis of the spring itself
· Can have spring elements with stiffness in the lateral direction, spring elements for torsion, etc.

References

Lecture Notes: Introduction to Finite Element Method by Yijun Liu, University of Cincinnati

https://www.comsol.com/multiphysics/finite-element-method