1

FINITE ELEMENT METHODS FOR MECHANICAL ENGINEERING

MME3360b Winter2012

Prof. Paul M. Kurowski

2

MME3360b web site

http://www.eng.uwo.ca/MME3360b/2012/

3

Design Center web site

http://www.eng.uwo.ca/designcentre/

4

SOFTWARE INSTALLATION

Questions? Ask Yara Hosein your TA

5

Download for Thursday lab

Download and read it for Thursday lab if you don’t have the book

LAB THIS WEEK

6http://www.solidworks.com/sw/support/810_ENU_HTML.htm

3% BONUS

7

MONKEY CLIMBING TWO ROPES

8

TWO ROPES

9

RequiredRecommended

TEXT BOOKS

10

FUNDAMENTAL CONCEPTS OF

FINITE ELEMENT ANALYSIS

11

DESIGN ANALYSIS

REAL OBJECTS

MODELS

PHYSICAL MODELS

MATHEMATICAL MODELS

NUMERICAL

FINITE DIFFERENCE METHOD

BOUNDARY ELEMENT METHOD

ANALYTICAL

FINITE ELEMENT METHOD

TOOLS OF DESIGN ANALYSIS

12

Finite Element Method

Based on the variational formulation of a boundary value problem. In the FEM the

unknowns are approximated by functions generated from polynomials. The polynomials

are defined on standard elements (such as triangles and rectangles) which are then

mapped onto elements with (possibly) curved sides or faces and the continuity of the

mapped polynomials is enforced. These functions are effective for the reasons of

numerical efficiency. The solution domain must be divided (meshed) into elements that

can be mapped from the standard elements using mapping functions.

For reasons of numerical efficiency and versatility, most commercial analysis

systems are based on the Finite Element Method commonly called Finite Element

Analysis (FEA)

Finite Difference Method

Based on the differential formulation of a boundary value problem. This results in a

densely populated, often ill-conditioned matrix leading to numerical difficulties.

Boundary Element Method

Based on the integral equation formulation of a boundary value problem. This also results

in densely populated, non-symmetric matrix. Boundary Element Methods are efficient only

for “compact” 3D shapes.

NUMERICAL TOOLS OF DESIGN ANALYSIS

Structural design analysis problems are described by a set of partial differential equations and belong to

the class called boundary value field problems. Such problems can be solved approximately by different

numerical methods.

13

Step 1 Creation of a mathematical model

An idealization of a real object accounting for geometry, loads, supports and material properties leads to a

formulation of a boundary value problem described by a set of governing partial differential equations. Most often

these equations are impossible to solve analytically and an approximate numerical method must be used.

Step 2 Deciding on the solution method

For reasons of numerical efficiency and generality we select the Finite Element Method.

Step 3 Approximating solution with piecewise polynomials

In order to represent solution with piecewise polynomials, we divide the body into simple shape sub domains

(elements) and define our polynomials (also called shape functions), with yet unknown factors a i , bi ,ci (also

called nodal degrees of freedom) in each of element separately.

BASIC STEPS IN THE FINITE ELEMENT ANALYSIS

N

iixixpau

1

N

iiyiypbu

1

N

iizizpcu

1

In the finite element method, nodal degrees of freedom are nodal displacements or temperatures.

Notice that by selecting certain polynomial order, we impose displacement pattern in each element. Working

with the first order polynomial (linear) we agree on linear displacement field, while second order polynomial will

return second order displacement field etc.

14

Step 4 Finding nodal displacements

Now we use the principle of minimum total potential energy (the state of minimum total potential energy is

also the state of equilibrium) to find this set of a i , bi ,ci factors that minimizes the total potential energy of the

body. This is also the new state of equilibrium under load. Knowing a i , bi ,ci we can now calculate discertized

displacement anywhere in the body. Notice that displacements are primary unknowns and are calculated first.

Of course the accuracy of the results will depend on how well the exact solution can be approximated by the

particular design of the mesh and selection of the polynomial degrees.

Step 5 Finding strains and stresses

Once displacements have been found, we calculate strains as derivatives of displacements. Knowing strains

and material properties we can now find stresses.

BASIC STEPS IN THE FINITE ELEMENT ANALYSIS

Continuous body - mathematical model Discretized body – finite element model

15

Idealization of geometry (if necessary)

CAD geometry Simplified geometry

CAD FEA Pre-processing

BASIC STEPS IN THE FINITE ELEMENT ANALYSIS

Restraints

Material properties

Type of analysis Loads

MATHEMATICAL

MODEL

16

FEA model FEA results

Discretization Numerical solver

FEA Pre-processing FEA Solution FEA Post-processing

BASIC STEPS IN THE FINITE ELEMENT ANALYSIS

MATHEMATICAL

MODEL

17

FEA EQUATIONS

[ F ] = [ K ] * [ d ]

[ F ] vector of nodal loads known

[ K ] stiffness matrix known

[ d ] vector of nodal displacements unknown

18

CAD GEOMETRY AND FINITE ELEMENTS

GEOMETRYGEOMETRIC

ENTITY MESHED

ELEMENTS

CREATED

Volume Solids

2D Plane Beams

3D Surface Shells

Curve Beams

19

Before deformation

Before deformation

After deformation

1st order tetrahedral element

2nd order tetrahedral element

DEGREES OF FREEDOM, SHAPE FUNCTIONS

Degrees of freedomEverything there is to know about the behaviour of this element under load can be calculated as soon as x, y and z displacements of all nodes defining that element are found. x, y and z displacements components fully describe node displacement for these 3D tetrahedral elements. x, y and z displacements are the three degrees of freedom of each node.

Shape functionsThe displacement at any point within the element is a function of nodal displacements. This function is called shape function. In the first order element the shape function is a linear combination of nodal displacement, in the second order element this a second order function etc.

After deformation

20

DEGREES OF FREEDOM

With only one node restrained the element spins in three directions.

With two nodes restrained the element spins about the line connecting two nodes.

With three nodes restrained the element won’t move.

Nodes of solid elements do not have rotational degrees of freedom.

DOF.SLDASM

21

Triangular shell element6 D.O.F. per node

Tetrahedral solid element3 D.O.F. per node

First order elements

Linear displacementConstant stress

Second order elements

Second order displacementLinear stress

Most often used element

DEGREES OF FREEDOM

22

2D plane stress, plane strain, axi-symmetric

x and y displacement fully describe behavior of each node.

Each node has two degrees of freedom.

TYPES OF ELEMENTS AND DEGREES OF FREEDOM

Solids

x, y and z nodal displacement components fully describe

behavior of each node. Each node has 3 D.O.F

Shells and beams

x, y and z displacements are not sufficient to describe what

is happening to each node while element deforms. Also

needed are rotations about x, y and z axis so each node

has 6 D.O.F.

23

TYPICAL ANALYSIS ASSUMPTIONS: LINEAR MATERIAL MODEL

STRAIN

ST

RE

SS

Linear material model

Non-linear material model

The linear material behavior complies with Hooke’s law:

= E in tension

= G in shear

Linearrange

[K] = const

[K] const

24

To comply with assumptions of small displacements theory, the displacement must not change the stiffness in a significant way.

Note that displacements don’t have to be large to significantly change the stiffness.

TYPICAL ANALYSIS ASSUMPTIONS: SMALL DISPLACEMENTS

[K] = const

[K] const

25

3D STATE OF STRESS

State of stress expressed by six stress components.

State of stress expressed by three principal stresses.

2 2 2 2 2 2

2 2 2

* ( ) ( ) ( ) *von Mises

* ( ) ( ) ( )von Mises 1 2 2 3 3 1

0.5 [ ] 3 ( )

0.5 [ ]

x y y z z x xy yz zx

VON MISES STRESS CRITERION

The maximum von Mises stress criterion is based on the von Mises-Hencky theory, also known as the

shear-energy theory or the maximum distortion energy theory. The theory states that a ductile material

starts to yield at a location when the von Mises stress becomes equal to the stress limit. In most cases,

the yield strength is used as the stress limit.

Factor of safety (FOS) = limit  / von Mises

 

Also known as Tresca yield criterion, is based on the Maximum Shear stress theory. This theory

predicts failure of a material to occur when the absolute maximum shear stress (max) reaches the stress

that causes the material to yield in a simple tension test. The Maximum shear stress criterion is used for

ductile materials. 

 max is the greatest of ,

 Where: 12 = (1 – 2)/2; 3 = (- )/2; 13 = (1- )/2

 Hence: Factor of safety (FOS) =  limit /(2*max)

 

THE MAXIMUM SHEAR STRESS CRITERION

28

Also known as Coulomb’s criterion is based on the Maximum normal stress theory. According to this

theory failure occurs when the maximum principal stress reaches the ultimate strength of the material for

simple tension.

This criterion is used for brittle materials. It assumes that the ultimate strength of the material in tension

and compression is the same. This assumption is not valid in all cases. For example, cracks decrease the

strength of the material in tension considerably while their effect is smaller in compression because the

cracks tend to close.

Brittle materials do not have a specific yield point and hence it is not recommended to use the yield

strength to define the limit stress for this criterion.

This theory predicts failure to occur when: 1 ≥ limit

 where 1 is the maximum principal stress. Hence:

 Factor of safety (FOS) = limit  / 1

 

THE MAXIMUM NORMAL STRESS CRITERION

Is based on the Mohr-Coulomb theory also known as the Internal Friction theory. This criterion is used for brittle materials with different tensile and compressive properties. Brittle materials do not have a specific yield point and hence it is not recommended to use the yield strength to define the limit stress for this criterion.This theory predicts failure to occur when: 1 ≥ TensileLimit  if 1 > 0 and > 0

   ≥ - CompressiveLimit  if 1  < 0 and < 0

 1  / TensileLimit +  / CompressiveLimit < 1  if 1 ≥ 0 and  ≤ 0

 The factor of safety is given by: Factor of Safety (FOS) = {1 / TensileLimit +  / CompressiveLimit }

(-1)

 

THE MOHR-COULOMB STRESS CRITERION

30

COMMON TYPES OF ANALYSES

STRUCTURAL

Linear static

Nonlinear static

Modal (frequency)

Linear buckling

THERMAL

Steady state thermal

Transient thermal

31

FINITE ELEMENT MESH

32

MESH COMPATIBILITY

Compatible elements

The same displacement shape function along edge 1 and edge 2

Incompatible elements

Different displacement shape function along edge 1 and edge 2

There is only one node here

There is only one node here

There is only one node here

There is only one node here

33

Model of flat bar under tension. There is a mesh incompatibility along the mid-line between left and right side of the model.

The same model after analysis. Due to mesh incompatibility a gap has formed along the mid-line.

MESH COMPATIBILITY

34

Shell elements and solid elements combined in one model.

Shell elements are attached to solid elements by links constraining their translational D.O.F. to D.O.F. of solid elements and suppressing their rotational D.O.F. This way nodal rotations of shells are eliminated and nodal translations have to follow nodal translations of solids.

Unintentional hinge will form along connection to solids if rotational D.O.F. of shells are not suppressed.

MESH COMPATIBILITY

Shell elements

Solidelements

Hinge

35

Elements before mapping Elements after mapping

MESH QUALITY

36

MESH QUALITYaspect ratio

angular distortion ( skew )

angular distortion ( taper )

curvature distortion

midsize node position

warpage

37

Element distortion: aspect ratio

Element distortion: warping

MESH QUALITY

38

Element distortion: tangent edges

MESH QUALITY

39

MESH ADEQUACY

This stress distribution needs to be modeled

This stress distribution is modeled with one layer of first order elements

Support Load

40

cantilever beam, model 1 terribly bad cantilever beam, model 2 also terribly bad cantilever beam model 3 a good beginning ! cantilever beam, model 4 an acceptable model

cantilever beam size: 10" x 1" x 0.1"modulus of elasticity: 30,000,000psiload: 150 lbfbeam theory maximal deflection: f = 0.2"beam theory maximal stress: = 90,000psi

our definition of the discretization error : ( beam theory result - FEA result ) / beam theory result

model #

FEA deflection

[in]

deflection error [%]

FEA stress [ PSI ]

stress error [%]

1 0.1358 32 1,500 98

2 0.1791 10 39,713 56

3 0.1950 2.5 65,275 27

4 0.1996 0.2 80,687 10

MESH ADEQUACY

41

MESH ADEQUACY

Two layers of second order solid elements are generally recommended for modeling bending.

Shell elements adequately model bending.

42

CONVERGENCE PROCESSCONTROL OF DISCRETIZATION ERROR

43

DISCRETIZATION OF STRESS DISTRIBUTION

Mesh built with first order triangular elements called constant stress triangles

First order element assumes linear distribution of displacements within each element. Strain, being derivative of

displacement, is constant within each element. Stress is also constant because it is calculated based on strain.

Discrete stress distribution in constant stress triangles

44

Tensile hollow strip modelled with a coarse mesh of 2D plate elements.

An isometric view of von Mises effective stress distribution in

the upper right quarter of the model shown above. The

height of bars represents the magnitude of stress. Notice

that stresses are constant within each element.

DISCRETIZATION OF STRESS DISTRIBUTION

45

CONVERGENCE ANALYSIS BY MESH REFINEMENT

The same tensile strip modelled three times with increasingly refined meshes. The

process of progressive mesh refinement is called h convergence.

46

CHARACTERISTIC ELEMENT SIZE

The process of progressive mesh refinement is called h convergence because

characteristic element size h is modified during this process

47

Discretization errors

Discretization error is an inherent part of FEA. It is the price we pay for discretization of a continuous structure.

Discretization error can be defined either as solution error or convergence error.

Convergence error

Convergence error is the difference between two consecutive mesh refinements and/or element order upgrade. Let’s

say convergence error is 10%. If convergence takes place, then the next refinement and/or element order upgrade will

produce results that will be different from the current one by less than 10%.

Solution error

The solution error is the difference between the results produced by a discrete model with a finite number of elements

and the results that would be produced by a hypothetical model with an infinite number of infinitesimal elements. To

estimate the solution error, one has to assess the rate of convergence and predict changes in results within the next few

iterations as if they were performed.

CONVERGENCE CURVE

1 2 3

Mesh refinement and / or element order upgrade number

Con

verg

ence

crit

erio

n

Solution of the hypothetical “infinite” Finite element model (unknown)

Solution error for model # 3

Convergence error for model # 3

# of D.O.F.

48

h - elements

The name h comes from characteristic element size usually denoted as h.

That characteristic element size is reduced during h convergence process.

p - elements

The name p comes from polynomial function describing displacement field in the element.

The order of polynomial function is increased during p convergence process.

COMPARISON BETWEEN h ELEMENTS AND p ELEMENTS

49

h - elements p - elements

Element shape: tetrahedral, wedge, hexahedralElement shape: tetrahedral, wedge, hexahedral

Mapping allows for only little deviation from the ideal shape.

Displacement field mapped by lower order polynomials (1st or 2nd), polynomial order does not change during solution

Mapping allows for higher deviation from the ideal shape but may introduce errors on highly curved edges and surfaces

Displacement field described by mapped higher order polynomials, polynomial order adjusted automatically to meet user’s accuracy requirements.

COMPARISON BETWEEN h ELEMENTS AND p ELEMENTS

results are produced in the iterative process that continues until the known, user specified accuracy, has been obtained

results are produced in one single run with unknown accuracy

fewer large elements typically 500 – 10,000many small elements typically 5,000 – 500,000

Only tetrahedral elements can be reliably created with the available auto-meshers

Only tetrahedral elements can be reliably created with the available auto-meshers

50

HOLLOW PLATE

Model file HOLLOW PLATE.sldprt

Model type solid

Material Alloy Steel

Restraints fixed to left end face

Load 100000N tensile load to right end face

Objectives

• meshing solid CAD geometry

• using solid elements

• demonstrating h convergence process

100,000 Ntensile load

Fixedrestraint