Finite element method

Finite Element Method (FEM)

Finite element method is modern method to solve Physical problems in Aeronautical, Automobile, Bio mechanical, Civil, Electromagnetic, Mechanical, Mining, Nuclear, Ocean….Engineering and the relevant field. But, Finite element method is widely used for Structural engineering problems.

The finite element method involves modeling the structure using small interconnected elements called finite elements. This process is called discretization or meshing.

Finite element Method has three steps to solve any Physical problem

Viz

1. Preprocessing
2. Processing
3. Post processing

or

1. Defining problem by Finite element model
2. Solving or solution
3. Interpreting the results or reviewing the results

Pre Processing

1. Model Designing
2. Discretization or Meshing
3. Selection of elements
4. Geometrical properties (For 1-D, 2-d)
5. Material selection
6. Constrains or Boundary conditions
7. Loads
8. Type of analysis

Processing or solution

There are two general direct approaches used in the finite element method as applied to structural mechanics problems to get Solution. i.e.

1. Force or flexibility method
2. Displacement or stiffness method

Force or flexibility method

Internal forces as the unknowns of the problem. To obtain the governing equations, first the equilibrium equations are used. Then necessary Additional equations are found by introducing compatibility equations. The Result is a set of algebraic equations for determining the redundant or unknown forces.

Displacement or stiffness method

Displacements of the nodes as the unknowns of the problem. For instance, compatibility conditions requiring that elements connected at a common node, along a common edge, or on a common surface before loading remain connected at that node, edge, or surface after deformation takes place are initially satisfied. Then the governing equations are expressed in terms of nodal displacements using the equations of equilibrium and an applicable law relating forces to displacements.

Most of the Finite element software programs uses the Displacement or stiffness method to solve the Problems.

During the solution phase, finite element software assembles the governing algebraic equations in matrix form and computes the unknown values of the primary field variable(s).

The equation is

[k] [u] = [F]

Where

K- Stiffness matrix from material property and geometrical shape

u- Results needs to be calculated (Unknown)

F- Applied force

S.no	Domain	Property [K]	Result [u]	External action [F]
1	Structural	Stiffness	Displacement	Force
2	Thermal	Conductivity	Temperature	Heat source
3	Fluid flow	Viscosity	Velocity	Body force

The computed values are then used by back substitution to compute additional, derived variables.

• Reaction forces
• Element stresses
• Heat flow

As it is common for a finite element model to be represented by tens of thousands of equations, special solution techniques are used to reduce data storage requirements and computation time.

Post processing

Evaluation of the solution results is referred to as post processing. Postprocessor software contains sophisticated routines used for sorting, printing, and plotting selected results from a finite element solution.
Various operations in the post processing

• Visualization of Deformed structure
• Displacements magnitude in three directions
• Nodal stress magnitude
• Element stress magnitude
• Check equilibrium
• Factor of safety
• Strain energy
• Natural frequency
• Amplitude
• Time history
• Thermal stress and strain
• Plot deformed structural shape
• Animate dynamic model behavior
• Produce color-coded temperature plots

While solution data can be manipulated many ways in post processing, the most important objective is to apply sound engineering judgment in determining whether the solution results are physically reasonable.

Generally in FEA industry two terms are in normal usage about the analysis results:
Garbage in and garbage out. The finite element method software program will solve the given input. But, it can’t justify the results.

List of popular FEA software’s according to its purposes

Preprocessing tools

• ANSA
• HyperMesh
• Patran

Processing tools

• Abaqus
• ADINA
• Algor
• ANSYS
• COSMOS
• LS-Dyna
• MSC/NASTRAN
• NISA
• Optistruct
• RADIOSS
• SAP
• STARDYNE

Postprocessing tools

• Hyperview
• µETA