The Finite Element Method

Introduction to the WHY?

FE Commercial Software
“ANSYS”
ANSYS”
WHAT?

by
WHERE?
Professor Shaker A. Meguid
School of Mechanical and Production Engineering
Nanyang Technological University
Singapore HOW?

Introduction to Finite Element Method Discretization of Continuum

Numerical Methods in Continuum Mechanics

Introduction
Numerical techniques in continuum mechanics are based on the

Without numerical techniques, it would be almost principle that a continuum can be divided into an equivalent
impossible to solve practical engineering problems system of smaller bodies.

Finite Element Method (FEM) is a numerical method for solving
These bodies are connected at points (nodes) common to the
engineering problems. sub-
sub-regions (smaller bodies called elements).

The finite element method has been employed in:
As the size of these small bodies gets smaller, the numerical
(i) structural analysis solution becomes more accurate.
(ii) fluid flow
(iii) heat transfer
The cost of computation time may become prohibitive.
prohibitive.

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 Numerical Methods in Continuum
Mechanics Numerical Methods in PDEs
Numerical Methods in Continuum Mechanics
Numerical methods in
continuum mechanics

(i) Finite Difference (FDM),

Finite Boundary Finite
element (FE) element (BE) difference (FD)
methods methods methods (ii)Finite Element (FEM), and

(iii) Boundary Element(BEM).

Domain Boundary (surface) Internal

elements elements elements

The Finite Element Method

(i) The steps involved in the finite element analysis:
• Assemble element equations to obtain the equilibrium
• Divide continuum into a collection of pre-
pre-selected elements equation of the whole problem
of simple geometries (triangles, rectangles and quadrilateral
elements)
[k ] g {u}g = {F }g
• Derive element equation for all types of elements involved in
the mesh such that
– equilibrium and compatibility are enforced • Impose boundary conditions
– assumed displacement within each element is dependent • Solve the equilibrium equations for the nodal displacement
upon nodal values • Calculate stresses and strains and post-
post-process results
– equivalent nodal loads are established using principle of
virtual work

[ K ]e [u ]e = [ F ]e

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 Block-
Block-diagram of a commercial FE code
(ii) Features of FEM include Inputfrom
fromother
other
Input
FE/CADcode
FE/CAD code
• based on variational methods or weighted residual,
Pre-processor
Pre-processor
• have no limitations with regards to geometry, physical
composition of domain and nature of loading,
Solver
Solver
• involves a systematic procedure that can be automated for
use with digital computers, and
Post-processor
Post-processor
• yields approximate analysis by assuming a displacement
field (or a stress field)
Outputto
Output toother
other
CAD/Presentationprogram
CAD/Presentation program

Pre-
Pre-Processing Post-
Post-Processing

Geometrical Features
Nodes, Elements, Topology
Co-ordinate Axes, etc.
Printed
Results
Stresses
Material Properties
Yield Stress, Density,
Coefficient of Thermal Contours and
Appropriate Pressure Hard Copy Plots
Expansion, etc. Input General Purpose
Pre-Processor Data File FE Software Displacement
Boundary Conditions General Purpose Post-Processor Appropriate Tabulated
FE Software Result Files Results
Imposed, Mechanical and
Velocity
Thermal Restraints
Graphical
Display
Eigen Values
Applied Loads

Temperatures

3
 Usage of FE Packages

APPLICATIONS

The Commercial
Finite Element Code
AEROSPACE
ANSYS
AUTOMOTIVE POWER GENERATION

BIOMECHANICS

Preferences-
Preferences- Define Type of Problem Structural Problems
1- Define type of problem Analysis types Force

Is the problem structural, thermal, vibration, Static Dynamic (transient) Time

electromechanical or the combination between them ?
Response
Response

M=0 M#0 F(t)

 Structural: all mechanical loadings (static or dynamic); such as Time Time

in the modeling of aero-engine compressor disc. Natural Frequency Harmonic response

 Thermal: thermal loadings F=0 F(t)=F0sin(t)
Response

Response

 Combined structural and thermal: where you have both

Time
mechanical and thermal loadings; such as in the modeling of Time

aeroengine turbine disc.

Transient Harmonic

 Electromechanical: combined structural and electrical effects; Random vibration Buckling

such as in the modeling of pizoelectric sensors and actuators
F F
Response

 Combined Structural and CFD: such as in modeling pumps.

Time

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 Structural Problems (Cont.) Preprocessor-
Preprocessor- Define Element Type

Analysis types (continue) 2- Define element type

In this section, we select the type of element that we are going to use.
We can define more than one element depending on the nature of the
Fatigue Crack propagation problem, e.g. (plane element and beam element and then assign those
elements, later on, to the proper parts of the model).
S-N curve
Cyclic stress

Different types of elements are supported in Commercial software

1- Solid elements (plane and 3-D elements)
2- Beam and shell elements
Number of cycles
3- Contact elements
4- Coupled field elements
5- Hyperelastic and viscoelastic elements etc..

Examples of FE in ANSYS Examples of FE in ANSYS

 Line FE  Solid 2D FE (plain stress/strain/axisymmetric)
Additional Features:

2-nodesbeam
beaminin2D
2Dspace
space 4-nodessolid
4-nodes solid 3-nodessolid
3-nodes solid 6-nodessolid
6-nodes solid 8-nodessolid
8-nodes solid
2-nodesspar
sparinin2D
2Dspace
space 2-nodes
2-nodes DOF:UX,
UX,UY,
UY,ROTZ
ROTZ 3-nodesbeam
3-nodes beaminin3D
3Dspace
space DOF:UX,
DOF: UX,UY UY DOF:UX,
DOF: UX,UY UY DOF:UX,
DOF: UX,UY UY DOF:UX,
DOF: UX,UY UY
DOF:UX,
UX,UYUY DOF:
DOF: 2-nodesbeam
beaminin3D
3Dspace
space DOF:UX,
DOF: UX,UY,
UY,UZ UZ
2-nodes
2-nodes spar in 3D space
2-nodes spar in 3D space DOF:UX,
UX,UY,
UY,UZ UZ ROTX, ROTY,
ROTX, ROTY, ROTZ ROTZ
DOF:
DOF:UX,
DOF: UX,UY,
UY,UZ UZ ROTX,ROTY,
ROTY,ROTZROTZ
ROTX,

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 Examples of FE in ANSYS Examples of FE in ANSYS
 Solid 2D axisymmetric FE with non-
non-axisymmetric loading  Solid 3D FE
Examples: bending, shear, torsion

4-nodes
4-nodes 8-nodes
8-nodes
DOF:UX,
UX,UY,
UY,UZ
UZ 20-nodes
20-nodes 10-nodes
10-nodes
DOF:UX,
DOF: UX,UY,
UY,UZ
UZ DOF: DOF:UX,
DOF: UX,UY,
UY,UZ
UZ DOF:UX,
DOF: UX,UY,
UY,UZ
UZ

Examples of FE in ANSYS Examples of FE in ANSYS

 Solid p – elements with variable order of polynomial shape  Contact FE between deformable bodies
functions interpolation n =[1 – 8]
• User specified degree of accuracy
• Linear elastostatic analysis only

6-nodes2D
6-nodes 2D 8-nodes2D
8-nodes 2D 10-nodes3D
10-nodes 3D 20-nodes3D
20-nodes 3D
DOF:UX,
DOF: UX,UYUY DOF:UX,
DOF: UX,UYUY DOF:UX,
DOF: UX,UY,
UY,UZ
UZ DOF:UX,
DOF: UX,UY,
UY,UZ
UZ Node-to-surface2D
2D Node-to-ground2D
2D
Node-to-surface Node-to-ground
DOF:UX,
DOF: UX,UY,
UY, DOF:UX,
DOF: UX,UYUY

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 Preprocessor-
Preprocessor- Define
Material Properties

3- Define material properties

Define the material properties for the different parts of the model

Different material models are supported depending on the used

code. Some of the models that are supported by Ansys are:

1- Elastic Isotropic
2- Elastic Orthotropic
3- Temperature dependent material models
4- Material models that you can input the their behavior through a
polynomial or input data; such as in elastoplastic material
5- Mooney-Rivlin (hyperelastic model)

Examples of Material Models in ANSYS Examples of Material Models in ANSYS

Materials models Materials models

• Linear elastic isotropic/anisotropic • Plasticity
 • kinematic hardening isotropic hardening

E,  or Cij, i,j=1…6

• Nonlinear elastic
 • Druger – Prager

E,  or Cij, i,j=1…6

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 Examples of Material Models in ANSYS Examples of Material Models in ANSYS

Materials models (continue) Materials models (continue)

• Hyperelasticity • Temperature dependent elasticity
• For almost incompressible (=0.5) rubber like materials • For plastic materials
• Different models have different expression for E(t), (T) or Cij(T), i,j=1…6
strain energy function (elastic potential ) W
• Basic equations:

• Viscoelasticity
• For glass and polymer materials with
time dependent relaxation
• Basic equations:

Preprocessor-
Preprocessor- Define Geometry FE Meshing
3- Define geometry FE meshing is dependent upon several aspects:
Geometry: curves, stress concentration feature.
The model geometry can be represented in many different ways.
Loading: pressure, point load, contact.
Some of the basic methods are described as follows:
Material behavior: elastic, elastoplastic.
Problem type: small deformation, large strain.
1- Define keypoints, which can be connected by lines/surfaces
and then create areas/volumes surrounded by those lines/surfaces. sharp edges and cracks

2- Define simple areas/volumes directly for different sections of Ansys does support different types of meshing tools such as:
the model and do operations on those areas/volumes by adding, Manual meshing (manually divide lines or define element size and
subtracting, intersecting, dividing etc.. number at each edge or area)

3- The combination of the above methods Automatic meshing where we use a smart size of the mesh and the code
generate the mesh depending on the geometry of the model
4- Create fillets
Adaptive meshing
Special focus mesh around crack tips

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 FE Meshing Boundary Conditions & Loads
Meshing
•Lines Boundary Conditions & Loads
•Areas •On FE nodes
•Volumes •On FE elements
•On solid entities

Key points Lines (Areas) Volumes

F
g
p

Freemesh
Free mesh Mappedmesh
Mapped mesh

Other ANSYS Features Loads and Boundary Conditions

Restart
Element birth and death The main types of loadings that are supported by
Submodeling ANSYS are:
Substructuring
High-
High-level language for parametric input Structural : point force, pressure, body force such gravity
Interface with other FE and CAD/CAM codes and centrifugal force, temperature for thermal strain,
displacement.
Thermal: temperatures, heat flow rate, convection.
Magnetic: magnetic potential, magnetic flux, magnetic
current segments.
Electric: electric potential (voltage), electric current.
Fluid: pressures, velocities.

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 Solvers in ANSYS FE Solvers

ANSYS uses a number of solvers for different applications. Some of Linear equation solvers
theses solvers can be listed below: • Direct methods
• Sparse matrix
1- Frontal Solver (direct elimination solver)
• Frontal wavefront
2- Sparse Direct Solver (direct elimination solver) • Iterative methods
• Preconditioned conjugate gradient
Both of the above solvers can be used when memory is limited.
• Jacobi conjugate gradient
However, it requires large disk space. It can be used for problems
• Cholesky conjugate gradient
with low convergence rates.

3- PCG (Preconditioned Conjugate Gradient Method):Its an iterative Non-

Non-linear equation solvers
method
• Newton-
Newton-Raphson
• Modified Newton-
Newton-Raphson
It requires high memory but low disk requirement. It can be used
for large analysis of large models • Newton-
Newton-Raphson + arc-
arc-length

FE Solvers Postprocessor in ANSYS

Eigenvalue solvers
• Lanczos In general ANSYS has a very good Preprocessor and Postprocessor
• Subspace
• Reduced Some of the many features that ANSYS postprocessor support are:
Transient dynamic solvers
• Implicit time integration 1- List all results (stresses, strains, deformations, nodal and
• Newmark method element solution) and they are relatively easy to handle
• Wilson method
• Mode superposition 2- Plot all results (stresses, strains, deformations, nodal and
element solution).
Buckling solvers 3- Both List and Plot of the results can be mapped to graphs and
• Applying gradually increasing loads geometry
• Eigenvalue analysis (Euler method)
Fracture solvers 4- Support vector results
• Static analysis using special crack-
crack-tip FE 5- Calculate reactions forces

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 Solved Example
ANSYS Interactive Window (Linear Elastic Problem)
60

R = 10 Thickness = 1

20
r=4

40
Fillet R= 4
fixed pin hole R = 10

Pressure p = 200 20

Preferences-
Preferences- Define Type of Problem Preprocessor-
Preprocessor- Define Element Type
2
1- Click on Preferences 1 – Preprocessor 1
2- Choose structural 3
2 – element type
3- Press ok 3 – Add/edit/delete 9
4 – Add
1 5 – Choose Solid
2
6 – Quad – 4 node 42 10
7 – Press ok
8 – Press Options to
select the type of 6
analysis that will be 5
performed on elements
4 8
9 – Select Plane stress 7
11
3 10 – Press ok
11 – Close

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 Preprocessor-
Preprocessor- Define Preprocessor-
Preprocessor- Define Geometry
Material Properties Create Keypoints
1 5
2
4 2
3
3
1
6
1- Create keypoints, which
define the object.
1– Preprocessor
2– Material properties 4 1 – Preprocessor - Modeling -
3– Select Isotropic Create
4– Press ok 2– Keypoints
5– Input material properties 3– On Working Plane
Ex = 200e9 4– Input keypoint coordinates
6– NUXY = 0.3 in ANSYS Input as follows: 5
0,0 enter 0,-10 enter 40,-10
(Poisson’s ratio (minor) it’s the
enter 40,-30 enter 50,-30 enter
same as major in isotropic materials) 60,-30 enter 60,10 enter 0,10
7– Press ok 7
enter
5– Press ok

Preprocessor-
Preprocessor- Define Geometry Preprocessor-
Preprocessor- Define Geometry
Create Keypoints (Cont.) Create Lines (Cont.)
At this stage, the main geometry is defined by the keypoints

2 3
1

2- Create lines connecting

the keypoints

1 – Preprocessor – Modeling
Create
2– Lines - Lines
3– Straight Lines
4– Use mouse to connect
keypoints 2-3, 3-4, 6-7,
5
and 7-8
5– Press ok

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 Preprocessor-
Preprocessor- Define Geometry Preprocessor-
Preprocessor- Define Geometry
Create Lines (Cont.) Create Arcs
1
In cases where we want to
3
create arcs with angle
equals 1800, such as those
Zoom out the at the ends of this bracket,
drawing we have to create another
1- Go to PlotCtrls in the keypoints shifted towards
ANSYS Utility Menu and 2 the inside of the arcs.
select Pan,Zoom,Rotate
This is because there are
2- Click twice to zoom out two possibilities for
twice creating the arcs (inward
3- In case that you cannot and outward)
see lines or keypoints
after zooming, you can
always go to Plot in the
the ANSYS Utility Menu
and select e.g lines or
keypoints to plot them

Preprocessor-
Preprocessor- Define Geometry Preprocessor-
Preprocessor- Define Geometry
Create Arcs (Cont.) Create Arcs (Cont.)
1- Create arcs at the ends
of the bracket 8
4 2
3 1 2 3
1 1 – Preprocessor – Modeling-
Create
1- Create keypoints, in 2– Lines - Arcs 9
order to be able to create
arcs at the ends of the 3– By End KPs & Rad
bracket 4– Using mouse, pick the end
keypoints 2 and 8
1 – Preprocessor – Modeling-
Create 5– Press ok
2– Keypoints 6– Then pick, using mouse, the
center of the arc, which is in this
3– On Working Plane
case the shifted keypoint 9
4– Input keypoint coordinates
in ANSYS Input as follows: 7– Press ok again
5 5 7
-5,0 enter 50,-35 enter 8– Input the radius of the
arc as 10
5– Press ok
9– Press ok
10– Repeat for the other end

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 Preprocessor-
Preprocessor- Define Geometry Preprocessor-
Preprocessor- Define Geometry
Create Arcs (Cont.) Create Circles (Cont.)
Create a circle to represent
the pinhole located at (0,0)
4 1 2 In order to repeat the previous
1 – Preprocessor – Modeling- 3 operation for the other hole 1
Create where its center is located at
(50,-30), we need to relocate
2– Lines - Arcs
the working plane to the new
3– Full Circle center. Otherwise, the code will
4– Input the coordinates select a radius such that it
of center of the pin hole in intersects with a point at
ANSYS Input as 0,0 enter distance of 4 from the old
and then the radius as 4 enter working plane.
5– Press ok
1- From ANSYS Utility Menu,
select WorkPlane – Offset WP to- 3
Keypoints
5
2- Select,using mouse, keypoint
which is the center of the pinhole
that is needed to be created
3- Press ok

Preprocessor-
Preprocessor- Define Geometry Preprocessor-
Preprocessor- Define Geometry
Create Circles (Cont.) Create Fillets (Cont.)
Create a circle to represent
the pinhole located at (50,-30)
5 1 2 1
4
1 – Preprocessor – Modeling- 3
Create In order to create
2– Lines - Arcs fillet at the lower
3– Full Circle corner of the bracket 5
4– Select WP Coordinates to
1- Go to Create-Line Fillet
switch to the newly defined
working plane 2- Select the intersecting
5– Input the coordinates lines that we need to put a
of center of the pin hole in fillet between them
4
ANSYS Input as 0,0 enter 3- Press ok
and then the radius as 4 enter
6 4- Input the radius of 3
6– Press ok the fillet as 4
5- Press ok

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 Preprocessor-
Preprocessor- Define Geometry Preprocessor-
Preprocessor- Define Geometry
Create Areas Area Operations
At this stage, the geometry is completely 3
defined by lines. The next step is to create 2
area surrounded by those lines. All areas We do now the subtraction of
1 2
should be closed. areas
1
Here, we are going to create three areas: 1- Go to Operate
the bracket and the two holes and then we
do subtraction 2- Subtract
3- Areas
1- Go to Create- Areas- Arbitrary 4- Using mouse, pick areas from
2- By Lines which to subtract, which is in
3- Pick using mouse all the lines defining the this case the bracket
first pinhole 5- Press ok
4- Press ok 6- Using mouse, pick areas to be
subtracted, which are the
5- Repeat the above steps to create area 2 & 3 4 pinholes 5 7
for the other pinhole and bracket
7- Press ok again
6- Go to the ANSYS Utility Menu and select
area to plot them

Apply Loads and Boundary Conditions

FE Meshing (Cont.)
2
3 - In this example, the upper 3
In this example, we are going
left pinhole is rigidly fixed at 1
to use meshing with a smart
its inner circumference
size of element
1- Go to Solution
1- Go to Preprocessor- Meshtool
2- Apply
2- Check Smart Size 3- Displacement 2
3- Set size at 5 4- On Keypoints
1
4- Select under Mesh Area 4 5- Using mouse, select the four
5 keypoints, which connects the four 4
5- Check Shape of element as Quad 7
6 arcs of the circle. (Those keypoints
6- Set Mesher to Free. Since the were generated automatically when
geometry is not simple, we cannot creating a circle).
guarantee only Quad elements 7 8
6- Press ok 6 9
7- Press Mesh 7- Select All DOF
9 10
8- Select, using mouse, area to 8- Input Displacement value as 0
mesh. In this example we only one
9- Check Yes for Expand disp to nodes
area after we did the subtraction.
10- Press ok
9- Press ok

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 Apply Loads and Transfer all Loads and Constraints
Boundary Conditions (Cont.) to the FE Nodes
- Apply pressure at the other The loads and constraints at this stage
pinhole to represent a tensile 1 are applied to the solid model. We
3
load transmitted by a pin have to transfer them to FE nodes
1- Go to Solution 2
2- Apply 1- Go to Solution- Loads- Operate
3- Pressure 2 2- Go to Transfer to FE- and select
4- On Line All Solid Lds 1
5- Using mouse, select the two 3- Press ok
lower arcs that form the lower 4
pinhole) 7
6- Press ok
7- Input pressure value as 200
8 3
8- Press ok 6

Solve Postprocessing
At this stage, the model is Getting result (list or plot of
finished and ready for solve. results) for the nodal
stresses, displacement and 1 3
1- Go to Solution- Solve-Current LS other result, can be achieved
through postprocessing
2- You will get a summary of the
analysis type and loading. You 1- Go to General Postprocessor 2
can close this window 4
2- Plot Results for example 5
3- Go to Solve Current Load Step 3- Nodal Solution for example
and press ok
4- Select Stress

5- von Mises SEQV

6- Press ok
1
The solution is finished 3 6

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