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INTRODUCTION TO FINITE ELEMENT METHOD (FEM)
1. INTRODUCTION
Preliminary design: concept
design
ENGINEERING DEPARTMENT: To develop new products and/or manufacturing processes
Product calculation: - Component dimensioning
- Design verification - Material selection
Process calculation: - Process parameters
- Tooling design
INDUSTRIALISATION: - Development of manufacturing
drawings - Manufacturing process
definition - Development of manufacturing
tools - Production
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1. INTRODUCTION: CALCULATION METHODS
- Analytic method:
- Consists on the use of analytic equation to represent the behaviour of a physical problem (As exercises solved by hand in previous chapters or other subjects, heat transfer, material resistance, dynamics, vibrations,…)
- Advantages: relatively fast to be solved
- Limitations: Hard to represent complex phenomena in real components, not always applicable.
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1. INTRODUCTION: CALCULATION METHODS
- Numeric methods (Finite Element Methods FEM)
- To divide a complex problem into many simple problems (elements)
- Problem solution by numeric methods (Newton-Raphson) by using iterations and increments.
- Advantages: Capability to solve complex problems.
- Limitations: Time expensive resolution method, the use of computers is required.
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1. INTRODUCTION: FORMULATION TYPES
- Implicit:
- Explicit:
- Tries to obtain the structural equilibrium for each time increment. - More sophisticate algorithms higher time increments (FASTER). - High precision - Convergence problems when solving non-linear phenomena: hard variations in boundary condition, material behaviour, loads, contacts,…
- Does not need iterations, just time increments (Does not try to get the exact solution) - No convergence problems - Utilizes constant time increments - High calculation time - Recommendable to solve non-linear problems.
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1. INTRODUCTION: APLICATIONS
Solid mechanics:
- Structural linear calculations (linear static, linear dynamics) IMPLICIT - Plasticity range calculation (no-linear quasi-static or dynamics) EXPLICIT
Fluid mechanics:
- Linear calculation (wind tunnel example) IMPLICIT
- Non-linear calculations (atmospheric phenomena, turbulence, wind,...) EXPLICIT
Thermodynamics: (linear problems-IMPLICIT)
Multiphysics: thermo mechanic, thermo fluidic, fluid structure interaction…
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1. INTRODUCTION: APLICATIONS
Solid mechanics: Structural static calculation
Set-up
Aluminium sheet bulge-test
Design
vs.
FEM
IMPLICIT
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1. INTRODUCTION: APLICATIONS
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Solid mechanics: eigenvalues
Trunk door
First mode17Hz
Solid mechanics: Forming processes
Punching
IMPLIT
EXPLICIT
1. INTRODUCTION: APLICATIONS
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Solid mechanics: Forming processes
1. INTRODUCTION: APLICATIONS
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Vc=300 m.min-1 Vc≥600 m.min-1
Solid mechanics: Machining process
EXPLICIT
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Fluid mechanics: linear and non-linear examples
Air flow simulation F1
Hurricane simulation
EXPLICIT IMPLICIT
1. INTRODUCTION: APLICATIONS
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Thermodynamics:
Turbine heat transfer simulation Tube and die temperature
pattern simulation IMPLICIT
IMPLICIT
1. INTRODUCTION: APLICATIONS
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
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- Definition of a finite element
Geometrical definition (element shape): - Composed by nodes Physic definition (element type): - Degrees of freedom (DOF) - Analytic formulation of the element (mechanical field resolution, thermal fields,…)
u v w θx θy θz
6 Degrees of Freedom
(DOF)
x y
z
u v
w
θx
θy
θz
u: Linear displacement in X v: Linear displacement in y w: Linear displacement in z θx: Rotation with respect X θy : Rotation with respect Y θz : Rotation with respect Z
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Interpolation order: linear (1st order) y quadratic (2nd order)
- Element types:
v1 v2 v1
v2 v3
Linear interpolation Quadratic interpolation
V(x)=mx+b V(x)=ax2+bx+c
x x v1 θ1 v2 θ2
δ= Nodal displacement vector of a first order beam element in 2D de
*
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
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- Mechanical field calculation: Motion differential equation
[M]{δ} + [C]{δ}+[k]{δ} ={Fext} . .. [M]: Mass matrix
[C]: Damping matrix [k]: Stiffness matrix {δ}: Displacement vector
{δ}: Velocity vector .
{δ}: Acceleration vector ..
{Fext}: External load vector
- Mechanical field: STATIC
Acceleration = 0 Velocity = 0
[M]{δ} + [C]{δ}+[k]{δ} ={Fext} . ..
[k]{δ} ={Fext}
5 unknowns and 5 equations
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
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- Stress field calculation through nodal displacement.
.
. ] N,....,N,N [=}{
n
1
n21e
δ
δ
δ
}{] [ = }{
w
v
u
.
x 0
z
y
z
0
0 x
y
z
0 0
0 y
0
0 0 x
=
zx
yz
xy
z
y
x
δε
γ
γ
γ
ε
ε
ε
∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
INTERPOLATION displacement at any point of the structure
Strain through local displacement
( ) ( )
( )( )
( )( )
( )
( )
−
−
−−
−−
+⋅⋅−=
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x
γγγεεε
ν
ν
νννν
νννννν
ννE
τττσσσ
22100000
02210000
00221000
000100010001
121
Generalised Hooke’s law Stress calculation through local strain ε1
ε3
ε2
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
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- INTERPOLATION FUNCTIONS: Determination of the displacement at any point of the structure.
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
{ } [ ]{ }e*
e δNδ =
{ } [ ]
=
1
,...,, 21
δ
δδ
nne NNN
So, the displacement at any point of a determined element is obtained:
Where [Nk] represents the contribution of node k’s displacement in the total displacement of any determined point.
{ } [ ]{ } [ ]{ } [ ]{ }nnNNN δδδδ +++= ...2211
= displacement vector at any point in a determined element
= nodal displacement vector of a determined element
= interpolation function matrix
{ }eδ
{ }e*δ
[ ]N
= interpolation function of the nodal displacement a determined node i.
= displacement vector at a determined node i.
{ }iδ
[ ]iN
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- STIFFNESS MATRIX:
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
In order to cause the nodal displacements and consequently deform the element it is necessary the presence of nodal forces
Definition: The stiffness coefficient Kij represents the necessary force to apply to a certain degree of freedom i to obtain an unitary displacement of the degree of freedom j being 0 the influence in the displacement of the rest of the degrees of freedom
{ }*δ{ }*f
j
n
jiji Kf δ⋅=∑
=1n= number of DOF
nnnnnn
nn
fKKK
fKKK
=⋅++⋅+⋅
=⋅++⋅+⋅
δδδ
δδδ
...
...
2211
11212111
Writing in matrix form: { } [ ] { }eee Kf ** δ=
[K]e = Stiffness matrix of the element
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2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
DETERMINATION OF THE STIFFNESS MATRIX OF A FINITE ELEMENT
Relation between nodal forces an nodal displacements:
Based on CAPLEYRON theory, the external work of the nodal forces is represented:
The internal deformation energy caused by the nodal displacements:
{ } [ ]{ }** δKf =
{ } { }**
21 fw Tδ=
{ } { } dvu T ⋅= ∫ σε21
{ } { } [ ]{ } [ ]{ }{ } [ ]{ } [ ][ ]{ }*
**
δεσ
δδδε
BDD
BN
==
=∂=∂=As: { } [ ]{ }
[ ][ ]{ }{ }
dvBDBuTTv
TT⋅= ∫
σε
δδ **
21
Being uw =
{ } [ ]{ }**
21 δδ Kw T
=
{ } [ ]{ } { } [ ] [ ][ ] { }****
21
21 δδδδ
⋅= ∫ dvBDBK
v
TTT
[ ] [ ] [ ][ ] dvBDBKv
T ⋅= ∫ STIFFNESS MATRIX
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2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
- DETERMINATION OF THE STIFFNESS MATRIX IN GLOBAL COORDINATES
Transformation matrix
[ ]
=
zyx
zyx
zyx
cccbbbaaa
T
From local coordinate system of the element
To global coordinate system
{ } [ ]{ }** δδ T= { } [ ]{ }** δKf = { } [ ]{ }** δKf =
{ } [ ] { } [ ] [ ]{ } [ ] [ ][ ]{ }**** δδ TKTKTfTf TTT ===
[K] in GLOBAL coord. system
{ } [ ]{ }** fTf =
{ } [ ] { }** fTf T= [ ] [ ] [ ][ ]TKTK T=
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- INTERPOLATION FUNCTION OF A TRUSS ELEMENT
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
Truss element:
- 2 node one-dimensional element (2DOF)
- Only allows to calculate tractive-compressive condition
Determination of the interpolation function 21 ,
,,,
uuji
zyxNodal displacement vector { } { }21,uuT =δ
2 G.D.L 1st order equation xaaxu .)( 10 +=
x
y
1 2
2u1u
L i j
)(xu
Local axis Element nodes Nodal displacements
1)0( uu =
2)( ulu = } laauau
102
01
+== }
=
1
0
2
1
101
aa
luu
211
10
11 ul
ul
a
ua
+−=
= }
−
=
2
1
1
0
1101
uul
laa [ ]
−=
−=
2
1
2
1 ,11101
,1uu
lx
lx
uu
llxu
[ ]
=2
121, u
uNNu
lxN −=11 l
xN =2
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- STIFFNESS MATRIX OF A TRUSS ELEMENT
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
Truss element: 2DOF
x
y
1 2
2u1u
L
{ }yx
yx uu
,2
1,
*
=δ
Local coordinate system
[ ] [ ] [ ][ ]∫=v
T dvBDBk .
[ ]
−=
=2
1
2
121, 1
uu
lx
lx
uu
NNu yx
[ ] [ ]21 ,NNN =
lxN
lxN
=
−=
2
1 1
Stiffness matrix obtaining formula:
[ ][ ] [ ]
−=
∂∂
∂∂
=
∂∂
=∂∂
=2
1
2
121
2
121
1,1,,uu
lluu
xN
xN
uu
NNxx
u
BB
x
ε
[ ] ∫∫
−
−=
−
−=
−
−=
l
v lSEdx
ll
llESdvll
E
l
lK0
22
22
1111..
11
11
..1,11
1
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- STIFFNESS MATRIX OF A TRUSS ELEMENT
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
[ ]
−
−=
1111.
eLSEk
Stiffness matrix of TRUSS element in local coordinate system
Displacement vector in global axis: y
x
x
y
1u
ϑ
- RIGIDITY MATRIX OF A TRUSS ELEMENT IN GLOBAL COORD. SYSTEM
yxYX
v
u
vuvu
,
1
1
,2
2
1
1
0
0
cossin00sincos0000cossin00sincos
−
−
=
θθθθ
θθθθ
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2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
Naming
==
ϑλϑµ
cossen
- STIFFNESS MATRIX OF A TRUSS ELEMENT IN GLOBAL COORD. SYSTEM
[ ] [ ] [ ] [ ]
[ ]
−
−
−
−
−
−
=
=
λµµλ
λµµλ
λµµλ
λµµλ
0000
0000
0000010100000101
0000
0000
.lSEK
TKTK
e
Te
[ ]
−−
−−
−−
−−
=
22
22
22
22
.
µµλµλµ
λµλλµλ
µλµµµλ
λµλλµλ
lSEK e
Stiffness matrix of a TRUSS element in GLOBAL coordinate system
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- INTERPOLATION FUNCTION OF A BEAM ELEMENT
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
2v1v
z
x
y
1ϑ 2ϑ
Beam element:
- 2 node unidimensional element (4DOF)
- Only allows to calculate behind loading condition
2
2
1
1
ϑ
ϑv
v
{ }=Tδ
=
2
2
1
1
4321
4321
ϑ
ϑϑ v
v
dxdN
dxdN
dxdN
dxdN
NNNNv
+−−+−+−
+−
−
+−
+
−
=
2
2
1
1
2
2
3
2
22
22
2
2
3232
2
3232
32,66,341,66
,23,2,231
ϑ
ϑϑ v
v
lx
lx
lx
lx
lx
lx
lx
lx
lx
lx
lx
lx
lx
lxx
lx
lx
v
Beam element interpolation function:
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- INTERPOLATION FUNCTION OF A COMPLETE BEAM ELEMENT
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
2v1v
z
x
y
1ϑ 2ϑ
Complete Beam element:
- 2 node one-dimensional element (6DOF)
- Only allows to calculate behind loading condition
2
2
2
1
1
1
ϑ
ϑ
vu
vu
{ }=Tδ
=
2
2
2
1
1
1
6543
6543
21
00
000000
ϑ
ϑ
ϑvu
vu
dxdN
dxdN
dxdN
dxdN
NNNNNN
vu
+−−+−+−
+−
−
+−
+
−
−
=
2
2
2
1
1
1
2
2
3
2
23
22
2
2
3232
2
3232
32660341660
23022310
00001
ϑ
ϑ
ϑvu
vu
lx
lx
lx
lx
lx
lx
lx
lx
lx
lx
lx
lx
lx
lxx
lx
lx
lx
lx
vu
Complete Beam element interpolation function:
2u1u
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- STIFFNESS MATRIX OF A BEAM ELEMENT
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
BEAM element: 4 DOF
{ }
yx
yx v
v
,2
2
1
1
,*
=
θ
θδ
Local coordinate system
[ ] [ ]4321 ,,, NNNNN =
Beam deflection 2
2
dd
dd
xvy
xyx −=−=
θε
[ ]{ }( ) [ ]{ }**2
2
2
2
dd
dd δδε BN
xy
xvyx =−=−=
[ ] [ ] [ ][ ] == ∫ vBDBkv
T d
∫∫
+−−+−+−
+−
−
+−
+−
=s
l
syxlx
llx
llx
llx
l
lx
l
lx
l
lx
l
lx
l
E dd62,126,64,126
62
126
64
126
2232232
0
2
32
2
32
xv
ddbeing =θ
[ ]
+−−+−+−−= 232232 62,126,64,126
lx
llx
llx
llx
lyB
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- STIFFNESS MATRIX OF A BEAM ELEMENT
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
Stiffness matrix of BEAM element in local coordinate system
Displacement vector in global axis: y
x
x
y
1u
ϑ
- STIFFNESS MATRIX OF A BEAM ELEMENT IN GLOBAL COORD. SYSTEM
yxYX v
uv
u
vuvu
,2
2
1
1
,2
2
1
1
cossin00sincos0000cossin00sincos
−
−
=
θθθθ
θθθθ
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2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
Naming
==
ϑλϑµ
cossen
- STIFFNESS MATRIX OF A BEAM ELEMENT IN GLOBAL COORD. SYSTEM
Stiffness matrix of a BEAM element in GLOBAL coordinate system
[ ]
−
−
−
−−−
−
−
−
−
=
1000000000000000010000000000
460260
61206120
000000
260460
61206120
000000
1000000000000000010000000000
.
22
2323
22
2323
λµµλ
λµµλ
λµµλ
λµµλ
llll
llll
llll
llll
IEk ze
[ ]
−−−
−−−
−
−
−
=
llllll
lllll
llll
lll
ll
l
IEk ze
466266
121261212
1261212
466
1212
12
.
223
2332
233
2323
23
22
233
23
µµµ
λλµλλλµ
µµλµµ
λµ
λλ
µ
30
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION
Naming
==
ϑλϑµ
cossen
- STIFFNESS MATRIX OF A COMPLETE BEAM ELEMENT IN GLOBAL COORD. SYSTEM
2 23
2 23 3
2 2
2 2 2 23 3 2 3
2 2 2 23 3 2 3 3
2 2
12
12 12 .
6 6 4
12 12 6 12
12 12 6 12 12
6 6 2
Te e
EI EAL L
EI EA EI EA syL L L L
EI EI EIL L L
EI EA EI EA EI EI EAL L L L L L L
EI EA EI EA EI EI EA EI EAL L L L L L L L L
EI EIL L
µ λ
µλ µλ λ µ
µ λ
µ λ µλ µλ µ µ λ
µλ µλ λ µ λ µλ µλ λ µ
µ λ
+
− + +
−
= =
− − − +
− − − − − + +
−
K T k T
2 2
6 6 4
cossin
EI EI EI EIL L L L
µ λ
λ θµ θ
−
==
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2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - DETERMINATION THE EQUIVALENT NODAL LOAD VECTOR
In a real problem different type of external loads can be found:
- Punctual forces
- Moments
- Distributed loads
f*
f
=
For FEM modelling all external load should be applied in the element nodes
- Punctual forces
- Moments
- Distributed loads
NECESITY TO OBTAIN AN EQUIVALENT SYSTEM BASED IN NODAL LOADS
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2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION - DETERMINATION THE EQUIVALENT NODAL LOAD VECTOR
{ } { }∫ ⋅=s
T dsfw δ21
1
The external work due to all the external load applied to the system is given by
By using the interpolation functions:
Thus the work of the equivalent system can be written as:
{ } [ ] { } { } [ ] { }∫∫ ⋅=⋅=s
TT
s
TT dsfNdsfNw **1 2
121 δδ
{ } { }**2 2
1 fw Tδ=
21 ww = { } [ ] { } { } { }***
21
21 fdsfN T
s
TTδδ =⋅∫
{ } [ ] { }∫ ⋅=s
T dsfNf *
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2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION DETERMINATION OF THE STRESS/STRAIN CONDITION:
Once, the nodal displacement vector of the studied system is solved the stress/strain condition at any point can be obtained.
STEP 1: STRAIN DETERMINATION AT A CERTAIN POINT
{ } [ ]{ }δε ∂=
.
. ] N,....,N,N [=}{
n
1
n21e
δ
δ
δ
∂∂
∂∂
∂∂
∂∂∂
∂∂
∂∂
∂∂
∂∂
∂
=
wvu
xz
yz
xy
z
y
x
zx
yz
xy
z
y
x
0
0
0
00
00
00
γγγεεε
Determination of the elongation at the selected point Strain vector determination
{ } [ ]{ } [ ]{ }** δδε BN =∂=
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2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION DETERMINATION OF THE STRESS/STRAIN CONDITION:
STEP 2: STRESS DETERMINATION AT A CERTAIN POINT The relation between the strain and the stress in the linear elastic domain is given by the generalised Hooke’s law:
( )[ ]
( )[ ]
( )[ ]yxzz
xzyy
zyxx
E
E
E
σσυσε
σσυσε
σσυσε
+−=
+−=
+−=
1
1
1 ( )
( )
( )zx
zxzx
yzyz
yz
xyxy
xy
EG
EG
EG
τυτγ
τυτγ
τυτγ
+==
+==
+==
12
12
12
Generalized Hooke’s law
( )υ+=
12:_' EGlawsLAMÉ For isotropic materials
35
2. FEM THEORY: STATIC IMPLICIT LINEAR CALCULATION DETERMINATION OF THE STRESS/STRAIN CONDITION:
STEP 2: STRESS DETERMINATION AT A CERTAIN POINT The relation between the strain and the stress in the linear elastic domain is given by the generalized Hooke’s law:
( )( )
( )( )
( )
−
−
−
−−
−
+−=
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x
E
γ
γ
γε
εε
υ
υ
υυυυ
υυυυυυ
υυ
τ
τ
τσ
σσ
22100000
02210000
00221000
000100010001
121
{ } [ ]{ }εσ D=