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MECH 5390/6390/6396 Fundamentals of Finite Element Method

(Professor Lall)MECH 5390/6390/6396 FUNDAMENTALS OF FINITE ELEMENT METHOD

Department of Mechanical Engineering

Auburn UniversitySpring Semester 2013

Instructor: Professor Lall

Office: 1438 Wiggins Hall

E-mail: [email protected]

Office Hours: 11am -12 pm, Tuesday and Thursday (TR)

Lecture:

Shelby 1122 2:00-3:15 am, Tuesday and Thursday (TR)

Labs:

ANSYS/LS-DYNA and MATLAB are available in Computer Labs, Wiggins Hall.

Lab Teaching Assistants:

None

Textbook: Concepts and Application of Finite Element Analysis (4th Edition)

by Robert D. Cook, David S. Malkus, Michael E. Plesha, Robert J. Witt

John Wiley & Sons, ISBN 0-471-35605-0, 2002

References: Finite Element Procedures, Klaus-Jrgen BathePrentice Hall, ISBN 0-13-301458-4, 1996

An Introduction to the Finite Element Method," J. N. Reddy

McGraw-Hill, NY, 2nd Ed., 1993

Grade Determination: 15% - Mid-Term Exam 1 (February 5, 2013)

15% - Mid-Term Exam 2 (March 5, 2013)15% - Mid-Term Exam 3 (April 2, 2013)

30% - Final Examination (May 1, 2013, 4-6:30 pm)

25% - MATLAB and ANSYS/LS-DYNA Analysis Projects

Grading Curve: 90-100 A

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MECH 5390/6390/6396 Fundamentals of Finite Element Method

(Professor Lall)MECH 5390/6390/6396 FUNDAMENTALS OF FINITE ELEMENT METHOD

Department of Mechanical Engineering

Auburn UniversitySpring Semester 2013

Catalog Data:MECH 6390 FUNDAMENTALS OF FINITE ELEMENT METHOD (3). LEC. 2, LAB. 3.

Pre-requisitesMATH 2660 Topics in Linear AlgebraMECH 3040 Heat TransferMECH 3130 Mechanics-of-Materials

Introduction to the fundamentals of finite element method.

Accommodation Policy for Students with Disabilities (Cited from Schedule of Classes)It is the policy of Auburn University to provide accessibility to its program and activitiesand reasonable accommodation for persons defined as having disabilities under Section504 of the Rehabilitation Act of 1973, as amended, and the Americans with Disabilities

Act of 1990. Students with disabilities desiring additional information should contact theProgram for Students with Disabilities, 1244 Haley Center, (334) 844-2096 (Voice/TT).

Course Goals:1. Demonstrate an understanding of the fundamental concepts of the finite element

method by forming the stiffness, displacement, load matrix equations for simplestructures, solving for displacement, and then computing strains and stresses.

2. Be able to use finite element methods to solve some of the most common types ofproblems in mechanical engineering. Develop ability to select appropriate FE elementtype for the physical model desired.

3. Understand how a finite element method approximates the solution to a partialdifferential equation, understand how different types of boundary conditions arehandled, and have a general understanding of how the error in the approximation canbe estimated and reduced by an adaptive method

4. Describe fundamental properties of the method in terms of compatibility, convergence,error sources, error estimation, orthogonality, energy minimization, etc

5. Develop familiarity with numerical methods and techniques closely related to thefinite element method, such as variational techniques, numerical integration,

isoparametric mappings, and solving large systems of equations6. Understand the importance of checking the finite element solutions and models with"back-of-the-envelope" solutions and engineering judgment.

Assigned ReadingThe assigned readings are listed for each lecture. The student is encouraged to read theassigned material prior to class lecture

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MECH 5390/6390/6396 Fundamentals of Finite-Element Method(Professor Lall)

2013 Auburn UniversityDepartment of Mechanical Engineering page 1 of 6

LECTURES

Week Lecture Date COURSE TOPIC Assigned

Reading

Assigned

Problems

Project

Assignmentfor Week

1 1 Jan 10 Finite Elements: An Overview

Modeling and Discretization

Interpolation, Elements, Nodes, DOF

Solving a problem using FEA

Matrix Algebra

Definitions and Manipulations

Introduction to MATLAB

Chapter 1

Section

1.1to 1.6Appendix A

1.3-2

1.3-3

1.3-4

2 2 Jan 15 One-Dimensional Elements

Bar Element

Beam Element

Chapter 2

Sections

2.1 to 2.3

2.2-1

2.2-3

2.2-5

2.3-1

2.3-3

3 Jan 17 One-Dimensional Elements

Bar and Beam of Arbitrary OrientationDirect Stiffness Method

Assembly of Elements(2.5)

Properties of Stiffness Matrices (2.6)

Solutions of Equations

Boundary Conditions (2.7)

Chapter 2

Sections

2.4 to 2.7

2.4-2

2.5-12.5-4

2.6-2

2.6-3

2.7-1

2.7-2

3 Jan 21 M. L. King Day Holiday

3 4 Jan 22 Solutions of Equations

Exploiting Sparsity (2.8)

Stress Analysis with 1-D Elements

Mechanical Loads

Thermal Loads

Structural Symmetry

Chapter 2

Sections

2.8 to 2.13

2.8-1

2.8-4

2.8-6

2.9-2

2.9-5

2.10-2

2.11-1

2.11-2

5 Jan 24 Elasticity Chapter 3 3.1-2

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Reading

Assigned

Problems

Project

Assignment

for WeekStress-Strain Relations

Strain-Displacement Relations

Compatibility, Equilibrium Equations

Exact and approximate solutions.

Interpolation Functions

Degree of Continuity

Section

3.1 to 3.2

3.1-3

4 6 Jan 29 Interpolation Functions

C0 InterpolationC1 Interpolation

2D and 3D Interpolation

Formulas for Element Matrices

Bar Element

Beam Element

Chapter 3

Section

3.2 to 3.3

3.2-1

3.2-33.3-1

3.3-3

7 Jan 31 Stiffness Matrices for 2D Problems

Constant Strain Triangle (CST or T3)Quadratic Triangle (LST)

Chapter 3

Section

3.4 to 3.5

3.4-1

3.4-33.5-1

5 Feb 5 Mid-Term 1

8 Feb 7 Element Derivations: T6, Q4, Q8 Elements

Bilinear Rectangle (Q4)

Quadratic Rectangle (Q8, Q9)

Rectangular Solid Elements

Element Performance

Choice of Interpolation Functions

Chapter 3

Sections

3.6 to 3.9

3.6-2

3.6-3

3.6-7

3.7-1

3.8-1

3.9-1

3.9-2

3.9-3

6 9 Feb 12 Element Performance

Improved Triangles and Quadrilaterals

Work Equivalence of Loads

Equivalent Nodal Loads

Chapter 3

Sections

3.10 to 3.14

3.10-2

3.11-1

3.11-4

3.11-5

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Reading

Assigned

Problems

Project

Assignment

for Week

Stress Calculation: Examples

10 Feb 14 Introduction to ANSYS/LS-DYNA Handout on

WebCT

None Project 1

ANSYS/LS-

DYNA Analysis.

7 11 Feb 19 Error, Error Estimation and Convergence

Sources of Error

Ill-Conditioning, Condition Number

Diagonal Decay TestResiduals

Chapter 9

Sections

9.1 to 9.5

9.2-1

9.2-3

9.3-1

9.3-29.4-4

9.5-1

12 Feb 21 Error, Error Estimation and Convergence

Discretization Error, Convergence Rate

Multi-Mesh Extrapolation

Mesh Revision Methods

Gradient Recovery and SmoothingA-Posteriori Error Estimate

Chapter 9

Sections

9.6 to 9.11

9.7-1

9.7-6

9.7-7

9.9-2

9.10-1

8 13 Feb 26 Modeling Considerations and Software Use

Physical Behavior VS Element Behavior

Element Shapes and Interconnection

Material Properties, Loads and Reactions

Repetitive Symmetry

Chapter 10

Sections

10.1 to 10.9

10.9-1

10.9-2

14 Feb 28 Modeling Considerations and Software Use

Sub-Models and Sub-structures

Planning an Analysis

Checking the Model

Critique of Computed Results

Variational Methods

Principle of Stationary Potential Energy

Problems with many D.O.F

Potential Energy of an Elastic Body

Chapter 10

Sections

10.10 to

10.16

Chapter 4

Sections

4.2-1

4.2-2

4.2-5

4.3-1

4.3-2

4.4-1

4.4-2

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Reading

Assigned

Problems

Project

Assignment

for Week

4.1 to 4.4

Mar 3rd Mid-Semester

9 Mar 5 Mid-Term 2

15 Mar 7 Variational Methods

Rayleigh-Ritz Method

Strong and Weak Form

Finite-Element Form of Rayleigh-Ritz

Method

Chapter 4

Sections

4.5 to 4.8

4.5-4

4.5-5

4.5-11

4.6-1

4.7-14.7-2

4.8-1

March 11-15, Spring Break

10 16 Mar 19 Convergence, Additional Formulations

Convergence of Finite Element Solutions

Hybrid Elements

Introduction to Galerkin Method

Chapter 4

Sections

4.9 to 4.10

Chapter 5

Sections

5.1

4.9-1

4.9-2

4.9-4

4.10-15.1-2

17 Mar 21 Weighted Residual Methods

Methods of Weighted Residuals (MWR)

Galerkin Finite Element Method in 1D

and 2D

Mixed Formulation

Chapter 5

Sections

5.2 to 5.6

5.2-3

5.2-4

5.3-1

5.3-2

5.5-4

5.5-6

11 18 Mar 26 Isoparametric Elements

Bilinear Quadrilateral (Q4)

Stiffness by Numerical Integration

Quadratic Quadrilateral (Q8 and Q9)

Chapter 6

Sections

6.1 to 6.4

6.1-3

6.2-6

6.2-7

6.3-5

Project 1

Due Today

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Reading

Assigned

Problems

Project

Assignment

for Week

6.3-6

6.4-1

6.4-2

19 Mar 28 Isoparametric Elements

Hexahedral Isoparametric Elements

Incompatible Modes, Nodeless D.O.F

Static Condensation

Choices in Numerical Integration

Chapter 6

Sections

6.5 to 6.8

6.7-2

6.7-3

6.8-1

6.8-2

6.8-9

Project 2

ANSYS/LS-

DYNA Analysis.

Assigned

12 Apr 2 Mid-Term 3

20 Apr 4 Isoparametric Elements

Load Considerations

Stress Calculation

Effect of Element Geometry

Validity of Isoparametric Elements

Patch Test

Chapter 6

Sections

6.9 to 6.13

6.9-1

6.10-2

6.10-3

6.10-9

6.11.2

6.12-26.13-1

13 21 Apr 9 Isoparametric Triangles and Tetrahedra

Shape Functions

Element Characteristic Matrices

Analytical Integration

Numerical Integration

Chapter 7

Sections

7.1 to 7.4

7.1-3

7.2-3

7.3-2

7.3-5

7.4-1

22 Apr 11 Coordinate Transformation

Stress, Strain, Property Transformation

Transformation of Characteristic Matrix

Changing Direction of Restraints

Chapter 8

Sections

8.1 to 8.4

8.1-1

8.2-2

8.3-1

8.3-2

8.4-3

14 23 Apr 16 Plate Theory

Kirchoff Plate Theory

Mindlin Plate Theory

Chapter 15

Sections

15.1 to 15.3

15.1-1

15.1-3

15.1-4

15.2-2

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Reading

Assigned

Problems

Project

Assignment

for Week

15.2-3

15.3-3

15.3-4

24 Apr 18 Structural Dynamics and Vibrations

Mass and Damping Matrices

Particle Mass Lumping

Consistent Mass Matrix

HRZ LumpingNatural Frequencies and Modes

Chapter 11

Sections

11.1 to 11.4

11.2-1

11.3-1

11.3-3

11.3-5

11.4-111.4-5

11.4-7

15 25 Apr 23 Structural Dynamics and Vibrations

Damping

Reduction of Number of D.O.F

Response History, Modal Methods

Ritz Vectors

Chapter 11

Sections

11.5 to 11.8

11.5-1

11.6-1

11.6-2

11.6-5

11.7-311.7-4

11.8-1

Project 2

Due Today

26 Apr 25 Review

Apr 26 Classes End

May 1 Final Exam (4 pm - 6:30 pm)

SCHEDULE MAY CHANGE BASED ON NEEDS OF CLASS.

PLEASE CHECK BLACKBOARD FOR UPDATES FREQUENTLY.

mech6390_spring2013_jan10

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