Finite Element Method January 12, ... You are free to make modifications to your original ... Finite Element Method Boundary Element Method Finite Dif...

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Finite Element Method Instructor(s) Prof. Olivier de Weck [email protected]

Dr. Il Yong Kim [email protected]

January 12, 2004

Plan for Today

FEM Lecture (ca. 50 min)

Cosmos Introduction (ca. 30 min)

FEM fundamental concepts, analysis procedure Errors, Mistakes, and Accuracy

Follow along step-by-step

Conduct FEA of your part (ca. 90 min)

Work in teams of two First conduct an analysis of your CAD design You are free to make modifications to your original model

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Course Concept today

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Course Flow Diagram Learning/Review

Problem statement

Deliverables

Design Intro

Hand sketching

Design Sketch v1

CAD/CAM/CAE Intro

CAD design

Drawing v1

FEM/Solid Mechanics Overview

FEM analysis

Manufacturing Training Structural Test “Training” Design Optimization

Produce Part 1

Part v1

Test

Experiment data v1

Optimization

Design/Analysis output v2

Produce Part 2

Part v2

Test

Experiment data v2

today

due today

Analysis output v1

Wednesday

Final Review 16.810 (16.682)

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Numerical Method Finite Element Method Boundary Element Method Finite Difference Method Finite Volume Method Meshless Method

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What is the FEM? FEM: Method for numerical solution of field problems. Description - FEM cuts a structure into several elements (pieces of the structure). - Then reconnects elements at “nodes” as if nodes were pins or drops of glue that hold elements together. - This process results in a set of simultaneous algebraic equations. Number of degrees-of-freedom (DOF) Continuum: Infinite FEM: Finite (This is the origin of the name, Finite Element Method)

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Fundamental Concepts (1) Many engineering phenomena can be expressed by “governing equations” and “boundary conditions” Elastic problems Thermal problems

Governing Equation (Differential equation)

L(φ ) + f = 0

Fluid flow Electrostatics etc.

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Boundary Conditions

B (φ ) + g = 0

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Fundamental Concepts (2) Example: Vertical machining center Geometry is very complex!

Elastic deformation Thermal behavior etc.

Governing Equation:

L(φ ) + f = 0

Boundary Conditions:

B(φ ) + g = 0

FEM

Approximate!

A set of simultaneous algebraic equations

[K ]{u} = {F}

You know all the equations, but you cannot solve it by hand

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Fundamental Concepts (3) [K ]{u} = {F} Property

{u} = [K ]−1{F}

Action Behavior

Unknown Property

[K ]

Behavior

{u}

Action

{F}

Elastic

stiffness

displacement

Thermal

conductivity

temperature

heat source

Fluid

viscosity

velocity

body force

Electrostatic

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dialectri permittivity

electric potential

force

charge

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Fundamental Concepts (4) It is very difficult to make the algebraic equations for the entire domain Divide the domain into a number of small, simple elements A field quantity is interpolated by a polynomial over an element Adjacent elements share the DOF at connecting nodes

Finite element: Small piece of structure

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Fundamental Concepts (5) Obtain the algebraic equations for each element (this is easy!) Put all the element equations together [K E ]{u E } = {F E }

[K E ]{u E } = {F E }

[K E ]{u E } = {F E }

[K E ]{u E } = {F E }

[K E ]{u E } = {F E }

[K E ]{u E } = {F E }

[K E ]{u E } = {F E }

[K E ]{u E } = {F E }

[K E ]{u E } = {F E }

[K ]{u} = {F} 16.810 (16.682)

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Fundamental Concepts (6) Solve the equations, obtaining unknown variabless at nodes.

[K ]{u} = {F}

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{u} = [K ]−1{F}

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Concepts - Summary - FEM uses the concept of piecewise polynomial interpolation. - By connecting elements together, the field quantity becomes interpolated over the entire structure in piecewise fashion. - A set of simultaneous algebraic equations at nodes.

[K ]{u} = {F}

Kx = F

K: Stiffness matrix x: Displacement

Property

Action

K

F: Load

Behavior

x F

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Brief History - The term finite element was first coined by clough in 1960. In the early 1960s, engineers used the method for approximate solutions of problems in stress analysis, fluid flow, heat transfer, and other areas. - The first book on the FEM by Zienkiewicz and Chung was published in 1967. - In the late 1960s and early 1970s, the FEM was applied to a wide variety of engineering problems. - Most commercial FEM software packages originated in the 1970s. (Abaqus, Adina, Ansys, etc.) - Klaus-Jurgen Bathe in ME at MIT

Reference [2] 16.810 (16.682)

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Advantages of the FEM Can readily handle very complex geometry: - The heart and power of the FEM

Can handle a wide variety of engineering problems - Solid mechanics - Fluids

- Dynamics - Heat problems - Electrostatic problems

Can handle complex restraints - Indeterminate structures can be solved.

Can handle complex loading - Nodal load (point loads) - Element load (pressure, thermal, inertial forces) - Time or frequency dependent loading

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Disadvantages of the FEM A general closed-form solution, which would permit one to examine system response to changes in various parameters, is not produced. The FEM obtains only "approximate" solutions. The FEM has "inherent" errors. Mistakes by users can be fatal.

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Typical FEA Procedure by Commercial Software User

Computer

User

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Preprocess

Process

Postprocess

Build a FE model

Conduct numerical analysis

See results

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Preprocess (1) [1] Select analysis type

[2] Select element type

- Structural Static Analysis - Modal Analysis - Transient Dynamic Analysis - Buckling Analysis - Contact - Steady-state Thermal Analysis - Transient Thermal Analysis

2-D

Linear

Truss

3-D

Quadratic

Beam Shell Plate

[3] Material properties

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E, ν , ρ , α , "

Solid

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Preprocess (2) [4] Make nodes

[5] Build elements by assigning connectivity

[6] Apply boundary conditions and loads

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Process and Postprocess [7] Process - Solve the boundary value problem

[8] Postprocess - See the results

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Displacement Stress Strain Natural frequency Temperature Time history

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Responsibility of the user Fancy, colorful contours can be produced by any model, good or bad!!

200 mm

BC: Hinged supports Load: Pressure pulse Unknown: Lateral mid point displacement in the time domain

Displacement (mm)

1 ms pressure pulse

Results obtained from ten reputable FEM codes and by users regarded as expert.* Time (ms) * R. D. Cook, Finite Element Modeling for Stress Analysis, John Wiley & Sons, 1995

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Errors Inherent in FEM Formulation Approximated domain

Domain

- Geometry is simplified.

FEM

- Field quantity is assumed to be a polynomial over an element. (which is not true) True deformation

Linear element

Quadratic element

Cubic element

FEM

- Use very simple integration techniques (Gauss Quadrature) f(x)

Area: -1

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x

1 1 f ( x ) dx f f ≈ + − ∫−1 3 3 1

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Errors Inherent in Computing - The computer carries only a finite number of digits.

2 = 1.41421356,

e.g.)

π = 3.14159265

- Numerical Difficulties e.g.) Very large stiffness difference

k1 k2 , k2 ≈ 0

[(k1 + k2 ) − k2 ]u2 = P ⇒ u2 =

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P P ≈ k2 0

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Mistakes by Users - Elements are of the wrong type e.g) Shell elements are used where solid elements are needed - Distorted elements - Supports are insufficient to prevent all rigid-body motions - Inconsistent units (e.g. E=200 GPa, Force = 100 lbs) - Too large stiffness differences Æ Numerical difficulties

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Plan for Today

FEM Lecture (ca. 50 min)

Cosmos Introduction (ca. 30 min)

FEM fundamental concepts, analysis procedure Errors, Mistakes, and Accuracy

Follow along step-by-step

Conduct FEA of your part (ca. 90 min)

Work in teams of two First conduct an analysis of your CAD design You are free to make modifications to your original model

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References Glaucio H. Paulino, Introduction to FEM (History, Advantages and Disadvantages), http://cee.ce.uiuc.edu/paulino Robert Cook et al., Concepts and Applications of Finite Element Analysis, John Wiley & Sons, 1989 Robert Cook, Finite Element Modeling For Stress Analysis, John Wiley & Sons, 1995 Introduction to Finite Element Method, http://210.17.155.47 (in Korean) J. Tinsley Oden et al., Finite Elements – An Introduction, Prentice Hall, 1981

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