1. OVERVIEW OF FINITE ELEMENT METHOD

2. COMPUTER-AIDED ENGINEERING (CAE)
The use of computers to analyze and simulate the function (structural, motion, etc.) of mechanical, electronic or electromechanical systems.
Computer-Aided Design (CAD)
Drafting
Solid modeling
Animation & visualization
Dimensioning & tolerancing
Engineering Analyses
- Analytical & numerical methods

3. FINITE ELEMENT METHOD (FEM)
A numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems.
The basic premise of the method is that a solution region can be analytically modeled or approximated by replacing it with an assemblage of discrete elements.

4. Steps in solving a continuum problem by FEM Select the solution domain
Select the solution region for analysis.
Discretize the continuum
Divide the solution region into finite number of elements, connected to each other at specified points / nodes.
Select interpolation functions
Choose the type of interpolation function to represent the
variation of the field variables over the element.
Derive element characteristic matrices and vectors
Employ direct, variational, weighted residual or energy balance approach.
[k](e) {}(e) = {f}(e)

5. Assemble the element characteristic matrices and vectors
Steps … (Continued)
Assemble the element characteristic matrices and vectors
Combine the element matrix equations and form the matrix equations expressing the behavior of the entire solution region / system.
Modify the system equations to account for the boundary conditions of the problem.
Solve the system equations
Solve the set of simultaneous equations to obtain the unknown nodal values of the field variables.
Make additional computations, if desired
Use the resulting nodal values to calculate other important parameters.

6. FE procedures: FE software user steps:
Select the solution domain
Discretize the continuum
Choose inerpolation functions
Derive element characteristic matrices and vectors
Assemble element characteristic matrices and vectors
Solve the system equations
Make additional computations, if desired
Draw the model geometry
Mesh the model geometry
Select element type
(The FEA software was written to do this) Input material properties
(The computer will assemble it) Input specified load and boundary conditions
(The compiler will solve it) Request for output
Post-process the result file

7. EXAMPLE: Stresses in a C(T) specimen
Draw the model geometry
Mesh the model geometry
Select element type [20-node Hexahedral elements]
Input material properties [Elastic modulus, Poisson’s ratio]
Input specified load and BC [Pin loading, displacement rate, zero displacement at lower pin]
Request for output [Displacement, strain and stress components]
Post-process the result file

8. EXAMPLE: Stresses in a C(T) specimen
Check for obvious mistakes/errors
Extract useful information
Explain the physics/ mechanics

9. MODELING CAPABILITIES – Microelectronic device reliability
- Prediction of spatial distribution of physical parameters
Silicon Die
Substrate
Heat sink
Motherboard PCB
Solder joints
Solder
mask
-40
125
183 25
Re-flow T
Time
- Prediction of damage evolution characteristics

10. MODELING CAPABILITIES – Fatigue life of compressor components
- Identify critical locations at early stage of design

11. MODELING CAPABILITIES – Prediction of composite behavior
SiO2-filled epoxy
Single Maxwell model for viscoelasticity
Epoxy
Silica Filler

12. MODELING CAPABILITIES – Deflection of composite laminates beamw Al
Comp-0º
Comp-90º E
E11
E22
70 GPa
44.74 GPa
12.46 GPa
14 layers [0/45/90/-45/45/-45/02]s
Layer thickness = mm

13. MODELING CAPABILITIES – Deformation of composite panel
[0/45/90/-45/45/-45/02]SYM
Uni-directional

14. MODELING CAPABILITIES – Failure investigation
F.S. Silva, Engineering Failure Analysis 13 (2006)

15. FINITE DIFFERENCE VERSUS FINITE ELEMENT METHOD
envisions the solution region as built up of many small interconnected sub-regions or elements.
gives a piecewise approximation to the governing equations.
Finite difference scheme:
envisions the solution region as an array of grid points.
gives a pointwise approximation to the governing equations.

16. Finite difference scheme:
envisions the solution region as an array of grid points.
gives a pointwise approximation to the governing equations.
For node (1,1):