Finite Element Analysis
(MCEN 4173/5173)
Instructor: Dr. H. Jerry Qi Fall, 2006
What is Finite Element Analysis (FEA)?
-- A numerical method. -- Traditionally, a branch of Solid Mechanics. -- Nowadays, a commonly used method for multiphysics problems.
What areas can FEA be applied?
-- Structure analysis: a cantilever, a bridge, an oil platform -- Solid mechanics: a gear, a automotive power train -- Dynamics: vibration of Sears Tower, earthquake, bullet impact -- Thermal analysis: heat radiation of finned surface, thermal stress brake disc -- Electrical analysis: piezo actuator, electrical signal propagation -- Biomaterials: human organs and tissues --
What is Finite Element Analysis (FEA)?
FEA is originally developed for solving solid mechanics problem. Solid Mechanics:
Object: A Solid with known mechanical properties. (Mr. Potato; a shaft; human tissue) Concepts:
Boundary: The surface enclosing the geometry Solid: Interior + Boundary Boundary conditions: Any prescribed quantities, such as prescribed displacements and prescribed tractions on the boundary
X1
P
undeformed
X2
Mr. Potato
What is Finite Element Analysis (FEA)?
Solid Mechanics:
P
undeformed
P
deformed Question: If we apply a force on a solid, what are the values of the displacements, stresses, and strains at EACH MATERIAL POINT?
X2
X2
X1
X1
Input
Boundary conditions (prescribed force; prescribed displacement )
Output Stresses, strains, displacements, at each material point (x1,x2,x3)
???
What is Finite Element Analysis (FEA)?
To answer this question, we need to solve the following equations:
u j u 1 eij = i + 2 X j X i
Equations for solving linear solid mechanics problem
ij = 2Geij + ekk ij
i1 i 2 i 3 + + + fi = 0 X 1 X 2 X 3
We need to solve a problem consisting of total 15 equations, among which 9 equations are partial differential equations!! Finding an exact solution:
Then:
MISSION IMPOSSIBLE !!!
Mission changes to find a solution that APPROXIMATES the exact solution.
FEA is a numerical method that offers a means to find this Approximate Solution.
How does FEA work?
Before we start to look at how FEA works, lets first review some calculus
1 1 1 xdx = x 2 = (12 0) = 0 2 0 2 2
1
2 0
sin xdx = cos x
2 0
= cos + cos(0) = 1 2
2 0
(sin x )2 dx
???
We can use numerical method to find the answer.
How does FEA work?
Integration using numerical methods: Example:
F=
(x
1 1
+ 6 dx
Exact solution:
F=
(
1 1
38 1 3 2 x + 6 dx = x + 6 x = 12.667 3 1 3
y
y = x2 + 6
x
The integration represents the area under the curve
How does FEA work?
Integration using numerical methods:
Numerical integration
Scheme 1: 1. Divide the interval of integration into N section; 2. Choose a function to approximate the variation of f (x) in each section; the simplest such function is a constant function that equals to the value of f (x) at the mid-point of each section. 3. The product of this constant function and the length of the section approximates the integration of f (x) over this section. 4. Summing the products for all sections gives an approximate answer to the integration of f (x) over (-1,1)
y = x2 + 6
y = x2 + 6
x
N=1, F=12, Error= -5.26% N=2, F=12.5, Error= -1.32%
How does FEA work?
Integration using numerical methods:
Numerical integration
y = x2 + 6
y = x2 + 6
x
N=4, F=12.625, Error=-0.33% N=8, F=12.656, Error=-0.08%
As the number of sections increases, the error decreases.
How does FEA work?
Integration using numerical methods:
Numerical integration
Scheme 2: Same as Scheme 1, except that we choose a linear function in each section to approximate the variation of f (x). This linear function takes the same value and slope of f (x) at the mid-point of that section.
y = x2 + 6
y = x2 + 6
x Different functions can be chosen to approximate f (x).
How does FEA work?
Integration using numerical methods: Two key steps: 1. 2. Divide the interval of integration. In each sub-interval, choose proper simple functions to approximate the true function.
Two key features: 1. 2. The numerical result is an approximation to exact solution. The accuracy of numerical result depends on the number of sub-interval and approximate function.
How does FEA work?
Two key steps in numerical integration: 1. 2. Divide the interval of integration. In each sub-interval, choose proper simple functions to approximate the true function.
Element
Discretize the solid
Node
Mesh of the 3D solid
A quarter of Mr. Potato
Solve linear equations
Formulate a set of linear equations with displacements at each node as unknowns
Using a simple function to approximate the displacements in each element
How does FEA work? General Procedure Physical model
Describe the problem: Simplifying a real engineering problem into a problem that can be solved by FEA
Pre-processing Fun! FEA model
Discretize/mesh the solid, define material properties, apply boundary conditions
Results
Obtain, visualize and explain the results and make your boss happy
FEA theory
Choose approximate functions, formulate linear equations, and solve equations
Post-processing Fun!
FEA core
Math!!
Examples of FEA
FEM simulation of the damage of San Francisco Oakland Bay Bridge caused by the 1989 Loma Prieta earthquake. (From Adina R & D, Inc.)
Examples of FEA
FEM simulation of crush of a car in roll-over situation. (From Adina R & D, Inc.)
Examples of FEA
FEA simulation of shape memory polymer tube
Examples of FEA
PC, 500m/s High Speed Bullet Impact
Courtesy of A. D. Mulliken, S. Sarva @ MIT
New Developments in FEA Integrating FEA into CAD design software
Do analysis as you design
Self-adaptive analysis
Change the mesh during the analysis
Analysis of problem of huge size
Analyze a model with millions of nodes; Parallel computing
Multi-scale analysis
Analyze physical problem ranging from atomistic level to macroscopic level; Combine FEA with molecular dynamics simulations
Multi-physics analysis
Mechano-electrical coupling (MEMS); mechano-chemical coupling (Chemical-mechanical polishing (CMP) )
What is this class about? Classroom Physical model
Describe the problem: Simplifying a real engineering problem into a problem that can be solved by FEA
Lab FEA model
Discretize the solid, define material properties, apply boundary conditions
Results
Obtain, visualize and explain the results and make your boss happy
FEA theory
Choose proper approximate functions, formulate linear equations, and solve equations
Things before we start
Website: http://www.colorado.edu/MCEN/MCEN4173/MCEN4173.html Mail list: mcen4173_06@lists.colorado.edu
How to subscribe the mail list: Send an e-mail to listproc@lists.colorado.edu with Subject line empty and SUBSCRIBE MCEN4173_06 Your Name as the content.
Things before we start
Instructor
Dr. H. Jerry Qi Office: ECME 124 Telephone: 2-1270 E-mail: qih@colorado.edu TA Mr. Kevin Long Office: E-mail: kevin.long@colorado.edu Mr. Philip Kao Office: E-mail: philip.kao@colorado.edu
Office Hours
Instructor: M: 4:00-6:00PM W: 4:00PM-6:00PM TA (KL about lecture): Tu: 3:00PM-4:00PM (GS), Th: 3:00PM-4:00PM (GS). TA (PK: @ about lab) Th: 10:00AM-11:00AM
For other time, by appointment.
Things before we start
Textbook
A first course in finite element method (3rd Edition). Daryl L. Logan. Brooks/Cole, 2002. ANSYS Tutorial, K.L. Lawrence. SDC Publications, 2003.
References
The finite element methods: Linear static and dynamic finite element analysis. T.J.R. Hughes. Dover Publications, 1987. Finite element procedures. K.J. Bathe. Prentice Hall, 1996.
Homework:
Discussions are encouraged but your work has to be finished by you own. Due on Wednesday before the class.
Be on time!!!
Things before we start Grade:
Your final grade depends on the overall performance of the class.
Homework Lab Report Exam 1: Exam 2: Total:
15% 25% 30% 30% 100%
Important Days: No Class Days:
09/04, Labor Day 11/20, 11/22, Fall Break 11/24, Thanksgiving
Exam Days (Tentative):
10/13 (F), First Exam 12/15 (F), Second Exam Exams will be in ECCE 141
Having fun for the semester!
