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EML 5526FINITE ELEMENT ANALYSISSections: 18EB, 4456, 4458

Class hour: MWF 6th period (12:50 – 13:40)

Class room: 201 NEB

Instructor: Nam-Ho Kim

Office: 210 MAE-A

Office hour: MWF 7th period (13:55 – 14:45)

Phone: 352-575-0665

E-mail: nkim@ufl.edu

http://www.mae.ufl.edu/nkim/

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SYLLABUS

• Teaching Assistants– TA1

• Office: TBD, Phone: TBD• Office hour: TBD, e-mail: TBD

– TA2• Office: TBD, Phone: TBD• Office hour: TBD, e-mail: TBD

• Textbook: – Concepts and applications of finite element analysis, by Cook, Malkus,

Plesha, and Witt, Wiley, 2002• Required. Available in the University Bookstore

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OFFICE HOURS

Period Mon Tue Wed Thu Fri

7:25 - 8:15

8:30 - 9:20

9:35 - 10:25

10:40 - 11:30

11:45 - 12:35 Class Prep Class Prep Class Prep

12:50 - 1:40 EML5526 EML5526 EML5526

1:55 - 2:45 Office hour Office hour Office hour

3:00 - 3:50

4:05 - 4:55

Instructor: 210 MAEA, 352-575-0665, nkim@ufl.edu TA1: TBDTA2: TBD

Class website (Sakai): https://lss.at.ufl.edu

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GRADES

• Homework– Homework problems will be assigned during every lecture, and

students are required to submit homework before starting Wednesday class of the following week. No late homeworks will be accepted.

• Exams– Three, equally contributing exams. Tentative schedules: Feb. 9th

(Exam1), March 20th (Exam2), Apr. 22nd (Exam3)

– Quiz: There will be unannounced pop quizzes during the class.

• Projects– Two projects in finite element analysis and design using Abaqus.

Formal report is required. All projects should be submitted to Sakai.

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GRADES

• Grading– Exams (50%), Projects (30%), Homework (15%), Quiz (5%)

– Opportunities for extra credits will be posted on Sakai time to time.

• Lectures– Lecture slides will be posted on Sakai 24 hours before the class

– Students are responsible to study the lecture before class starts

• Rules– Since this is a large class, it would be difficult to accommodate

individual excuses. Therefore, all the rules will be strictly kept

– Instead, the lowest 2 grades from HW + quizzes will be dropped in final grade calculation

A A- B+ B B- C+ C C- D+ D D- E

100 92.9 89.9 86.9 82.9 79.9 76.9 72.9 69.9 66.9 62.9 59.9

93 90 87 83 80 77 73 70 67 63 60

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Academic Honesty

• You have signed a statement of academic honesty

• You can discuss HWs and projects with classmates

• HWs: each problem must be solved by yourself

• Projects– All computer modeling simulation works should be done by yourself

– Report must be written by yourself (no copy and past from other sources)

– TA will check electronic version (model file & report)

• I will report all honor code violations to Student Conduct and Conflict Resolution

• Mishap during exam: Fail the course

• Mishap in submitted projects: Fail the course

• Mishap during quiz: One letter grade down

• Mishap in submitted HWs: One letter grade down

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SOFTWARE

• We will use a commercial finite element analysis software called “Abaqus” for homeworks and projects

• Download software from http://campus.3ds.com/simulia/freese– You will need an image file of your student ID.

– You need to install the software in a week

• 10 tutorials will be provided during the semester– Once a week (Time and classroom will be announced later)

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COURSE SCHEDULES

Ch1. Introduction

Ch2. 1D elements and computational procedures

Ch3. Basic elements

Ch4. Variational formulation

Ch5. Galerkin and weighted residual methods

Ch6. Isoparametric elements

Ch7. Isoparametric triangles and tetrahedra

Ch8. Coordinate transformation and analysis options

Ch9. Error, error estimation, and convergence

Ch10. Modeling considerations and software use

Ch11. Dynamics

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COURSE SCHEDULES cont.Wk Date Lect Class Reading

17-Jan 1 Syllabus and Introduction to FEA

9-Jan 2 1D system of springs 2.1, 2.2, 2.5, 2.7

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12-Jan 3 Bar element 2.4, 2.6, 2.10

14-Jan 4 Beam element 2.3

16-Jan 5 Sparsity, symmetry 2.8, 2.9, 2.11

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19-Jan No class, Martin Luther King Jr. Day

21-Jan 6 Stress-strain review 3.1

23-Jan 7 Stress-strain review

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26-Jan 8 Interpolation, element matrix formulation 3.2, 3.3

28-Jan 9 Plane elements (CST, LST, Q4, Q8, Q9) 3.4, 3.5, 3.6, 3.7

30-Jan 10 Plane elements

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2-Feb Exam1 review

4-Feb Exam1

6-Feb 11 Choice of interpolation, improved elements 3.8, 3.9, 3.10

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9-Feb 12 Equivalent nodal forces, stress calculation 3.11, 3.12, 3.13

11-Feb 13 Principle of stationary potential energy 4.1, 4.2, 4.3, 4.4

13-Feb 14Rayleigh-Ritz method, Strong & weak form, Project 1 discussion

4.5, 4.6, 4.7

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COURSE SCHEDULES cont.Wk Date Lect Class Reading

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16-Feb 15 FE form of R-R method, Convergence 4.8, 4.9

18-Feb 16 Galerkin method, Weighted residual 5.1, 5.2

20-Feb 17 1D Galerkin method, Integration by parts 5.3, 5.4

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23-Feb 18 2D Galerkin method 5.5

25-Feb 19 Isoparametric mapping, Quadrilateral element 6.1, 6.2

27-Feb 20 Quadrature, Q8, Q9, Hexahedral element 6.3, 6.4, 6.5

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2-Mar Spring break, No class

4-Mar Spring break, No class

6-Mar Spring break, No class

109-Mar

21Incompatible modes, static condensation, choice of integration

6.6, 6.7, 6.8

11-Mar 22 Distributed load, body force, stress calculation 6.9, 6.10

13-Mar 23 Element shape, mapping, patch test 6.11, 6.12, 6.13

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16-Mar 24 Iso-parametric triangles and tetrahedrons 7.1, 7.2, 7.3, 7.4

18-Mar Exam 2 review

20-Mar Exam 2

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COURSE SCHEDULES cont.Wk Date Lect Class Reading

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23-Mar 25 Coordinate transformation 8.1, 8.2, 8.3, 8.4

25-Mar 26 Dissimilar element, singularity, reanalysis, quality test 8.5 ~ 8.10

27-Mar 27 Source of error, ill-conditioning, condition number 9.1, 9.2, 9.3

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30-Mar 28 Diagonal decay test, residuals, convergence rate 9.4, 9.5, 9.6

1-Apr 29 Mesh revision, smoothing, error estimate 9.7, 9.8, 9.9

3-Apr30

Model simplification, element shape, material properties

10.2, 10.3, 10.4, 10.5

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6-Apr31

Model simplification, element shape, material properties

8-Apr32

Dynamic equation, mass lumping, Project 2 discussion

11.2, 11.3

10-Apr 33 Natural frequency, modes, damping, model reduction 11.4, 115., 11.6

1513-Apr

34 Modal method, Ritz vector, CMS, harmonic response11.7, 11.8, 11.9, 11.10

15-Apr 35 Direct integration, explicit and implicit, stability 11.11 ~ 11.14

17-Apr 36 Make-up class

1620-Apr Exam 3 review

22-Apr Exam 3

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TIPS FOR A

• Be patient and persistent– Read the text repeatedly until you understand it.

– If you don’t understand it, ask a question until you get answered.

• Follow equations– Do not just read the equation.

– You must follow all equations by HAND, not EYE.

• Try to understand the meaning of equations– If you memorize an equation that you don’t understand, you can’t solve

the problem. Math is a language.

• Follow the instruction carefully– Read carefully what is asked. If A is asked, then answer A not B.

– Do not submit a blank answer.

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Finite Element MethodA bridge between Mechanics of Materials and

real-world applications

Nam-Ho KimUniversity of Florida

Gainesville, FL 32611, USA

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Point of Departure

Let’s say that you have a cool idea of a new motorcycle

design

So, you decide to build

your own motorcycle

That means, you have to decide all parts, sizes, etc

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Design Trade-off

Need to remove material to have

a good acceleration

Need to add material for

safety Where can I remove material?Where should I

have to add material?

Use Mechanics of Materials!!!

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Gap

We learned Mechanics of

Materials, but …

How can I use it to solve real-world

applications?

Torsion

Bending

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Mechanics of Materials

But, how can I solve for mymotorcycle?

Equilibrium at every point is governed by differential equation

xyxxx

xy yyy

b 0x y

b 0x y

We know how to solve it in a simple domain

y

x

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Finite Element Method

• What is the finite element method (FEM)?– A technique for obtaining approximate solutions to boundary value

problems.

– Partition of the domain into a set of simple shapes (element)

– Approximate the solution using piecewise polynomials within an element

F

Structure

Element

u

x

Piecewise-Linear Approximation

xyxxx

xy yyy

b 0x y

b 0x y

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• How to discretize the domain?– Using simple shapes (element)

– All elements are connected using “nodes”.

– Solution at Element 1 is described using the values at Nodes 1, 2, 6, and 5 (Interpolation).

– Elements 1 and 2 share the solution at Nodes 2 and 6.

What Are Elements?

1 2 3

1 2 3 4

5 6 7 8

Elements

Nodes

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Interpolation

• Finite element analysis solves for Nodal Solutions.– All others can be calculated (or interpolated) from nodal solutions

– Displacement within the element

– Strain of the element

u1 u2

Lx

N1 N2

Interpolation (Shape) Function

2 11 1 2( )

u u L x xu x a bx u x u u

L L L

1 2

1 1( )

ux u u

x L L

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System of Matrix Equations

• How to calculate nodal solutions?– Construct a huge simultaneous system of equations and solve for

nodal solutions.

– Different physical problems have different matrices and vectors.

11 12 1 1 1

21 22 2 2 2

1 2

n

n

n n nn n n

K K K u F

K K K u F

K K K u F

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Example: Finite Elements

• Plastic Wheel Cover Model– 30,595 Nodes, 22,811 Elements

– Matrix size is larger than 150,000150,000.

– MSC/PATRAN (Graphic user interface)

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Numerical Models of Engineering Components

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X

Y

Z

A

A

A

B

B

A

B

C

A

C

AB

C

D

A

B

D

C

A

E

D

B

C

E

F

D

A

B

E

C

F

D

E

C

G

D

F

E

G

C

D

B

A

D

H

E

F

G

E

C

F

D

B

H

A

E

G

D

D

H

F

E

D

C

G

C

B

F

G

C

E

B

A

D

F

B

C

E

A

D

BC

E

A

D

A

B

C

BD

A

C

A

B

A

AB

C

AB

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Numerical Models of Engineering Components

3-d finite element mesh for analysis: head of femur for 2-day old infant (after T. Ribble)

~21,000 unknowns

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Computational Fluid Dynamics

Viscous flow past 2-d simulation of the forebody of a shuttle at Mach 2.

Initial Mesh

Mach Number contours

(after Zienkiewicz and Taylor, 1991).

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Crashworthiness Analysis

Crash of a SAAB 9000

17h CPU time on CRAY x-MP/48

(after J. Hallquist)

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Convergence Study

• How do you know the FEM solution is accurate?

• Convergence: the finite element solution converges to the exact solution as the size of elements decreases

u

x

Exact solution

Two elements

Four elements

Eight elements

uI

Number of Elements

Try with at least three different element sizesto determine convergence

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Element Selection

• What element should I have to use?– Element is mathematical representation

– Different elements behave differently

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Finite Element Procedure

Preliminary analysis

Preprocessing

Solving the problem

Postprocessing

Converged?

Stop

Correction/R

efinement

Yes

No

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Modeling Issues

• Common mistake: FE model is not a replication of CAD geometry

• Model: Mathematically identical to a purpose

• Simplification: delete unimportant small featuresseparate consideration of small holes

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Boundary Conditions

• Error in boundary conditions will not disappear no matter how much you refine the model!!

• Most assumptions are often made in BC

• Need to be careful in interpreting results near the boundary

(a) Concentrated force (b) Distributed forces

min = 0.973ave min = 0.668ave min = 0.198avemax = 1.027ave max = 1.387ave max = 2.575ave

0.25b0.5b

b

b

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Example: Automotive Door Panel Stamping

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Example: Airbag Deployment

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Forging

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Example: Vorticity

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Things to Remember

• Finite element method does not solve a problem, but it helps YOU to solve the problem

• It helps you to understand the mechanical system that you are working on

• Garbage inputs, garbage outputs

• Try to be an engineer, not a technician

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QUESTIONS?

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YouTube Videos

• Two vehicle impact NCAC FEA FEM– http://youtu.be/hrfcROMz2II

• Downmilling– http://youtu.be/pYCOkIVLA_c

• Rolling– http://youtu.be/E1d4WKkbtdY