1. FEM
: Finite Element Method
2017

2. Handout Instructor Hyukchun Noh ChungMoo 714 cpebach@Sejong.ac.kr
Prerequisites
Basic Statics and Mechanics
Textbook
Lecture Note

3. Handout
References

4. Handout
Syllabus

5. Handout
Grading related

6. Handout
NOTICE:
Time is precious, and is related with the credit of a person
To me, time must be a kind of ruler to grade a person
So, please be on time at class
If you do not present on time, even just a minute, you will be counted as absent: for Starting time & After short break
No excuse for this

7. Introduction to FEM(Finite Element Method)
- Brief concept and Flow of FEM-

8. FEM ? - An approximate analysis scheme for natural phenomenon
e.g. Solids, Electromagnetic field, Thermal effect and so on
- FEM always gives WRONG answers
Degree of accuracy is of importance
- Do not believe FEM if you don’t know exact or analytical solutions
Governing Equations
Solution Domain
Boundary conditions

9. Physical Phenomena are expressed by
- PDE(partial differential equation)
- Boundary conditions or Initial conditions
Examples:
Mechanical deformation of solids
Transfer of thermal effects
Flow of fluid
Distribution of electric voltage
ets.
Boundary Value Problems
(경계치 문제)
FEM is just one of approximate Numerical schemes
for solving above problems in nature

10. Properties Behavior Action: source
FEM approximates the PDE in the form of
Linear Algebraic Equations (선형대수방정식)
instead of solving directly the PDE
: Linear Algebraic Equation
Properties
Behavior
(response)
Action: source
(Applied forces)

11. The properties, behavior and action are determined differently
depending on the types of problem under consideration

12. The unknowns in the LAE(Linear Algebraic Equation) : Action, Behavior
The unknowns are determined at nodes (discretized, pointwise)
The nodes are located arbitrarily in the domain (depending on the analyzer)
Nodes(절점)
(Red dots)
Red dots

13. (unknowns)
Field of variable U
interpolated by means of Shape Function
and nodal values ui , in the figure below
thus
Continuous field is obtained
Linear (1st order)
Shape function
Quadratic (2nd order)
Shape function
Nodal values
Nodes
Interpolated field using

14. Not-possible in general
Representation by one equation ?
Not-possible in general
Divide the entire domain into sub-domains (i.e, finite elements)
Example of
Solid Element
Elements
(요소)
Sharing the nodes between adjacent elements
Shared nodes

15. The governing equation of an element is equal to that of entire domain
Element equation:
Single LAE, easily obtained by means of an approximation process
Numerical integrations are used: e.g. Gauss-quadrature rule
[Example]
(Element equations)

16. After obtain all the element equations
For from 1 to N number of FEs in the domain
We can assemble them
for the total domain
leading to global equations
By using Connectivity
Global LAE

17. The global equation is solved by
Nodal point displacement U: continuous (throughout the domain)
Stresses can or cannot be continuous depending on element type
Example of stress or displacement contour

18. Geometry (nodal coord.s)
General Flow of FE Analysis
Start
Various schemes
- Direct Inverse
- Cholesky decomp.
- Band
- Skyline
- Frontal
- Iterative solvers
and so on
Form
Load vector
Read data
Geometry (nodal coord.s)
Connectivity
Boundary condition
Materials
Analyze
To obtain U
Construction of
Element Stiffness
Determine
Stresses U=De
Assemblage to
Global stiffness
End
Read data
Actions

19. End of introduction to FEM