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Department of Civil Engineering, DAU
Structural Analysis Lab.
This lecture note has been reproduced and distributed by courtesy by Prof. Hae Sung Lee, http://strana.snu.ac.kr
Review of Mathematics - Approximation of Functions -
≅ δ
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Structural Analysis Lab.
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� Discretization
)()(1
XX ∑=
=n
iii gaf υ∈∀ )(Xf where gi are the basis functions of υ.
� Approximations
)()()()(11
XXXX ∑∑==
=≈=
m
iii
hn
iii gafgaf where nm ≤
� Fundamental Questions
- What is the best approximation?
- How can we calculate ai that represents the best approximation?
� Summation Notation: Repeated indices denote summation ii
m
iii baba =∑
=1
.
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� Norms of Functions: A measure of a function space
A function space υ is said to be a normed space if to every υ∈f there associated a
nonnegative real number f , called the norm of f, in a such way that
- 0=f if and only if 0≡f
- ff || α=α for any real number α.
- gfgf +≤+
Every normed space may be regarded as a metric space, in which the distance between any
two elements in the space is measured by the defined norm. Various types of norm can be
defined for a function space. Among them the following norms are important.
- L1 norm: ∫=
VL
dVff1
- L2 norm: 2/12
)(2
∫=
VL
dVff
- H1 norm:
2/12))((1 ∫ ∇⋅∇+=
VH
dVffff
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� General Ideas for the Best Approximation
Let’s find out an approximate function that is closest to the given function by use of a norm
defined in the function space. If this is the case, the characteristics of an approximation
method depend on those of the norm used in the approximation.
� Least Square Error Minimization
Error: hffe −=
Minimize ∫ −=−==Π
V
h
L
h
LdVffffe
222
)(2
1
2
1
2
1
22
FKa ===−=−=
−=−=∂
∂−=
∂
Π∂
∑∫∑∫
∫∫ ∑∫∫
==
=
or ,,1for 0
)()(
11
1
mkFaKfdVgadVgg
fdVgdVaggdVgffdVa
fff
a
ki
m
iki
V
ki
m
i V
ik
V
k
V
m
iiik
V
kh
V k
hh
k
L
If the basis functions are orthogonal, K becomes diagonal.
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� Variation of a function
- The variation of a function means a possible change in the function for the fixed x.
� Variational Calculus
- if ii gaf = ii gaf δ=δ or i
i
aa
ff δ
∂
∂=δ .
- ff
Fa
a
f
f
Fa
a
FFfF i
i
i
i
δδδδ∂
∂=
∂
∂
∂
∂=
∂
∂= : )(
- hfhf δ+δ=+δ )( , hffhfh δδδ +=)(
- , )()(dx
fda
a
f
dx
da
dx
df
adx
dfi
i
i
i
δ=δ
∂
∂=δ
∂
∂=δ ∫∫∫∫ δ=δ
∂
∂=δ
∂
∂=δ fdxdxa
a
fafdx
afdx i
i
i
i
f
δf
Variation of a function
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� Minimization by Variational Calculus
Min ∫ −=Π
lhh
dxfff0
2)(2
1)(
k
k
k
l
k
hh
lhh
lh
lhh
aa
adxa
fff
dxfffdxffdxfff
δ∂
Π∂=δ
∂
∂−=
δ−=−δ=−δ=Πδ
∫
∫∫∫
0
00
2
0
2
)(
)()(2
1))(
2
1()(
Min k
lhh
adxfff δδ possible allfor 0)(2
1)(
0
2=Π⇔−=Π ∫
� Euler Equation
Min ka
dxxffFfk
l
allfor 0),,()(0
=∂
Π∂⇒′=Π ∫
∫
∫∫∫∫
′∂
∂−
∂
∂+
′∂
∂=
′′∂
∂+
∂
∂=
∂
′∂
′∂
∂+
∂
∂
∂
∂=
∂
∂=′
∂
∂=
∂
Π∂
l
kk
l
k
l
kk
l
kk
l
k
l
kk
dxgf
F
dx
dg
f
Fg
f
F
dxgf
Fg
f
Fdx
a
f
f
F
a
f
f
Fdx
a
FdxxffF
aa
00
0000
)(
)()(),,(
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In case the basis functions vanish at the boundary, then
0 allfor 0)(0
=′∂
∂−
∂
∂⇔=
′∂
∂−
∂
∂=
∂
Π∂∫
f
F
dx
d
f
Fkdxg
f
F
dx
d
f
F
a
l
k
k
∫∫∫ δ′∂
∂−δ
∂
∂+δ
′∂
∂=′δ
′∂
∂+δ
∂
∂=′δ=Πδ
llll
dxff
F
dx
df
f
Ff
f
Fdxf
f
Ff
f
FdxxffFf
0000
)()(),,()(
In case the variation vanishes at the boundaries, then
k
k
k
l
k
l
aa
adxgf
F
dx
d
f
Ffdx
f
F
dx
d
f
Fδ
∂
Π∂=δ
′∂
∂−
∂
∂=δ
′∂
∂−
∂
∂=Πδ ∫∫
00
)()(
Therefore,
Min 0=Πδ⇔Π
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� Example 1
Min ∫ ′+=Π
2
1
2)(1)(
x
x
dxyy subject to 11)( yxy = , 22 )( yxy =
0))(1( 2/12=′+′−=
′∂
∂−
∂
∂ −yy
dx
d
f
F
dx
d
f
F
0))(1())(1
)(1())(1(
))(1)(2
1())(1())(1(
2/32
2
22/12
2/322/122/12
=′+′′=′+
′−′+′′=
′′′′+−′+′+′′=′+′
−−
−−−
yyy
yyy
yyyyyyyydx
d
baxyy +=→=′′ 0 . By applying BC, 12
2112
12
120xx
yxyxx
xx
yyyy
−
−+
−
−=→=′′
� Example 2
Min ∫ −′=Π
l
dxufuu0
2))(
2
1()( subject to 0)0( =u , 0)( =lu
00))(2
1(
2=+′′→=′′−−=′−−=′
′∂
∂−−=
′∂
∂−
∂
∂fuufu
dx
dfu
udx
df
u
F
dx
d
u
F
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Principle of Minimum Potential Energy vs
Principle of Virtual Work
(1-D elliptic differential equation)
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� Problem Definition
0)()0( ,0 02
2
==<<=+ luulxfdx
ud
� Approximation – Discretization
∑=
=
m
iii
hgau
1
where 0)()0( == luuhh
� Residuals
Equation Residual : lxfdx
udR
h
E <<≠+= 0 02
2
Function Residual : lxuuRh
F <<≠−= 0 0
� Error Estimator :
∫∫ +−==Π
l hh
l
EFR
dxfdx
uduudxRR
02
2
0
))((
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� Least Square Error
))((2
1))((
2
1))((
2
1
))((2
1))((
2
1
000
02
2
2
2
02
2
∫∫
∫∫
−−=−−−−−=
−−=+−=Π
l hhl hhl
hh
l hh
l hhR
dxdx
du
dx
du
dx
du
dx
dudx
dx
du
dx
du
dx
du
dx
du
dx
du
dx
duuu
dxdx
ud
dx
uduudxf
dx
uduu
� Energy Functional – Total potential Energy
RRl
hl hhl
l hhl
hh
lh
lh
lhl
h
lh
hh
hl hhR
Cfdxudxdx
du
dx
duufdx
dxdx
du
dx
du
dx
duudxu
dx
udu
dx
du
dx
duudxfuuf
dxfudx
uduuf
dx
ududxf
dx
uduu
Π+=−+=
+−+−+−=
−−+=+−=Π
∫∫∫
∫∫∫
∫∫
)2
1(
2
1
})({2
1
)(2
1))((
2
1
000
0002
2
000
02
2
2
2
02
2
� Minimization Problems
RRLSRΠ↔Π⇔Π Min Min Min w.r.t.
hhu υ∈
ΠLS
−f
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� RR
ΠMin : Rayleigh-Ritz Method or Principle of Minimum Potential Energy
- 1st Order Necessary Condition of Minimization Problem
FKa =→==−=−=
−=
−=∂
Π∂
∑∫∑∫
∫ ∑∫ ∑∑
∫∫
==
===
mkFaKfdxgadxdx
dg
dx
dg
fdxgada
ddx
dx
dga
dx
dga
da
d
fdxda
dudx
dx
du
dx
du
da
d
a
m
ikiki
l
k
m
ii
lik
l m
iii
k
l m
i
ii
m
i
ii
k
l
k
hl hh
kk
RR
,,1for 0
)())((
)(
101 0
0 10 11
00
L
� 0=ΠδRR : Variational Principle or Principle of Virtual Work
�
0)()( 0)(
)(
)2
1(
1 1
1 01 0
000000
=−δ→=−δ=
−δ=
δ−δ
=δ−δ=−δ=Πδ
∑ ∑
∑ ∫∑∫
∫∫∫∫∫∫
= =
= =
FKaaT
m
k
m
ikikik
m
k
l
k
m
ii
lik
k
lh
l hhlh
l hhlh
l hhRR
FaKa
fdxgadxdx
dg
dx
dga
fdxudxdx
du
dx
udfdxudx
dx
du
dx
dufdxudx
dx
du
dx
du
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Finite Element Formulation & Constant
Strain Triangle (CST) Element
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� Interpolation and Shape Functions
To interpolate is to devise a continuous function that satisfies prescribed conditions at a finite
number of points.
in
i
i xa∑=
=
0
φ or xa=φ
in which ),,,,1( 2 nxxx L=x and T
naaaa ),,,,( 210 L=a .
where n = 1 for linear interpolation, n = 2 for quadratic interpolation, and so on.
The ai can be expressed in terms of nodal values of φ, which appear at known values of x as
follows:
Aa=eφφφφ
where each row of A is x evaluated at the appropriate nodal location.
eφφφφN=φ where ),,( 21
1LNN==
−xAN .
An individual Ni in matrix N is called a shape function or a basis function.
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� Example of Shape Functions (1-D)
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� Formulas for Element Stiffness Matrices Based on The Principle of Virtual Work
The principle of virtual work can be represented as the following equation:
∫∫∫ +=et
ee S
T
V
T
V
TdSdVdV tubu δδδ σσσσεεεε ( )
where ),,( wvuT
δδδδ =u and should be admissible virtual displacement. Here an admissible
displacement does not violate compatibility or displacement boundary conditions. b and t
denote body forces and tractions on a surface boundary.
� Interpolation of Displacement
Let displacements u be interpolated over an element such that
eNuu = where ),,( wvu=u
and eu represents the nodal displacement DOF of an element.
Strains can be expressed by differentiating the interpolated displacements as follows:
eBu=εεεε where NB ∂= (for differential operator ∂, refer to Eq.3.1-9 in the textbook)
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Then, the virtual displacements and strains can be represented as,
euNu δδ = and e
uBδδ =εεεε
Substituting the above equation into the Eq. ( ) representing the principle of virtual work, Eq.
( ) becomes
0)()( =
−− ∫∫∫
et
eeS
T
V
Te
V
TTedSdVdV tNbNuCBBuδ ( )
Eq.( ) must be satisfied for any admissible virtual displacement dd. Therefore, Eq.( )
yields the element stiffness equation as follows:
eeefuk =
where the element stiffness ∫=e
V
TedVCBBk and the consistent nodal force vector
∫∫ +=et
e S
T
V
T
e dSdVr tNbN
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Finite Element Procedure
1. Governing equations in the domain, boundary conditions on the boundary.
2. Derive weak form of the G.E. and B.C. by the variational principle or equivalent.
3. Descretize the given domain and boundary with finite elements.
S , 2121 nn
SSSVVVV ∪∪∪=∪∪∪= LL
4. Assume the displacement field by shape functions and nodal values within an element.
eNuu =
5. Calculate the element stiffness matrix and assemble it according to the computability.
dVe
V
Te
∫= CBBk , ∑=
e
ekK
6. Calculate the equivalent nodal force and assemble it according to the computability.
dSdVet
e S
T
V
Te
∫∫ += tNbNf , ∑=
e
eff
7. Apply the displacement boundary conditions and solve the stiffness equation.
8. Calculate strain, stress and reaction force.
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Finite Element Programming (Linear Static case)
Input
Preprocessing
Calculation of Element stiffness matrix and Load Vector
Assembling E.S.M and E.L.V.
Solve Global SE
Post-processing
Loop over all elements
Global Stiffness Matrix
and Load Vector
- Assemble Nodal Load Vector - Cal. of Destination Array - Cal. of Band width
Cal. Strain & Stress
Cal of Reaction Force
Display
Gauss Elimination Band solver Decomposition, etc
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� Data Structure
- Control Data : # of nodes, # of elements, # of support, # of forces applied at nodes …
- Geometry Data : Nodal Coordinates & Element information (Type, Material Properties,
Incidencies)
- Material Properties
- Boundary Condition : Traction BC & Displacement BC
- Miscellaneous options
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� Constant-Strain Triangle (CST)
In structural mechanics, CST element represents a triangular finite element of which the strain
field is constant all over the element. Therefore, the displacement field should be linear with
respect to the Cartesian coordinate x and y.
yxyxvyxyxuee
654321 ),( , ),( α+α+α=α+α+α=
33321333
23221222
13121111
),(
),(
),(
yxuyxu
yxuyxu
yxuyxu
e
e
e
α+α+α==
α+α+α==
α+α+α==
α
α
α
=
→
3
2
1
33
22
11
3
2
1
1
1
1
yx
yx
yx
u
u
u
1u
1v
2u
2v
3u
),( 11 yx
),( 22 yx
),( 33 yx
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∆=
=
α
α
α−
3
2
1
321
321
321
3
2
1
33
22
11
3
2
1
2
1
1
1
11
u
u
u
ccc
bbb
aaa
u
u
u
yx
yx
yx
where jmimjijmmji xxcyybyxyxa −=−=−= , ,
333322221111
333322221111
)(2
1)(
2
1 +)(
2
1),(
)(2
1)(
2
1 +)(
2
1),(
vycxbavycxbavycxbayxv
uycxbauycxbauycxbayxu
e
e
++∆
+++∆
++∆
=
++∆
+++∆
++∆
=
e
e
ee
e
e
e
e
e
e
v
u
v
u
v
u
bc
c
b
bc
c
b
bc
c
b
x
v
y
u
y
v
x
u
BU=
∆=
∂
∂+
∂
∂
∂
∂
∂
∂
=
γ
ε
ε
=
3
3
2
2
1
1
33
3
3
22
2
2
11
1
1
12
22
11
0
0
0
0
0
0
2
1εεεε
∫ ∆⋅⋅=⋅⋅=eA
eTTettdA BDBUBDBK
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� ELEMENT CHARACTERISTICS (CST)
The CST element was devised for plane stress analysis. However, a mesh of these elements is
undesirably stiff. Correct results are approached as a mesh is refined, but the convergence is slow.
1/4 of the correct σx
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Furthermore, in plane strain condition, a mesh can “lock” so that it cannot deform at all.
(Note) As commonly used in FEA, the term “locking” refers to excessive stiffness in one or more
deformation modes. Usually, locking does not imply complete rigidity. Thus, locking may not
preclude convergence with mesh refinement, but may preclude reasonable accuracy in coarse to
intermediate mesh densities.
Mesh layout under plane strain condition
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� CONVERGENCE OF SOLUTIONS (CST vs LST)
Mesh layout under plane strain condition
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� QUADRATIC TRIANGLE (LST)
Triangular element with side nodes in addition to vertex nodes, i.e. 12 DOFs per element. In
terms of generalized d.o.f. αi, the element displacement field is the complete quadratic:
2
1211
2
10987
2
65
2
4321
),(
),(
yxyxyxyxv
yxyxyxyxu
e
e
αααααα
αααααα
+++++=
+++++=
Element strains are represented as follows:
yxy
u
x
v
yxyy
vyx
x
u
eee
x
ee
y
ee
x
)2()2()(
22
11610583
12119542
ααααααγ
αααεαααε
+++++=∂
∂+
∂
∂=
++=∂
∂=++=
∂
∂=
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� BILINEAR RECTANGLE (Q4)
Four-node plane element having 8 DOF. The name ‘Q4’ identifies the element as a quadrilateral
having four nodes. In terms of generalized d.o.f. αi, the element displacement field and the
associated strain field are represented as follows:
xyyxyxv
xyyxyxu
e
e
8765
4321
),(
),(
αααα
αααα
+++=
+++= and
yxy
u
x
v
xy
v
yx
u
eee
x
ee
y
ee
x
8463
87
42
)( ααααγ
ααε
ααε
+++=∂
∂+
∂
∂=
+=∂
∂=
+=∂
∂=
In general, shape functions can be calculated by using the following relations:
=
⇔
=
−
4
3
2
1
1
4444
3333
2222
1111
4
3
2
1
4
3
2
1
4444
3333
2222
1111
4
3
2
1
1
1
1
1
1
1
1
1
u
u
u
u
yxyx
yxyx
yxyx
yxyx
yxyx
yxyx
yxyx
yxyx
u
u
u
u
α
α
α
α
α
α
α
α
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=
⇔
=
−
4
3
2
1
1
4444
3333
2222
1111
4
3
2
1
4
3
2
1
4444
3333
2222
1111
4
3
2
1
1
1
1
1
1
1
1
1
v
v
v
v
yxyx
yxyx
yxyx
yxyx
yxyx
yxyx
yxyx
yxyx
v
v
v
v
α
α
α
α
α
α
α
α
Like the CST, the Q4 element cannot exhibit pure bending. When bent, it displays shear strain as
well as the expected bending strain. This parasitic shear absorbs strain energy, so that if a given
bending deformation described, the bending moment needed to produce it is larger than the
correct value. In other words, the Q4 element exhibits shear locking behavior [Page 98 and 99 in
the textbook].
Derive the following equation:
})(2
)1(1{
)1(
)2()1(6
1
6
1
2
2
3
2
3
2
2
el
b
aab
a
bE
a
Eb
bν
ν
νν
θ
θ
−+
−=
+−
=
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Derive shape functions and element stiffness matrix of the Q4 element when the element has the
following geometry [Page 97 and 98 in the textbook].
44332211
44332211
),(
),(
vNvNvNvNyxu
uNuNuNuNyxu
e
e
+++=
+++=
ab
ybxaN
4
))((1
−−= ,
ab
ybxaN
4
))((2
−+= ,
ab
ybxaN
4
))((3
++= ,
ab
ybxaN
4
))((4
+−=
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� QUADRATIC RECTANGLE (Q8, Q9)
A quadratic rectangle is obtained by adding side nodes to the linear rectangle, much as side nodes
are added to the CST to obtain the LST triangular element. We use the name ‘Q8’ for this eight-
node quadrilateral.
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( )
dVE
dVE
dVE
dVE
dVU
V
xyyxyx
V
xyyyxxyx
Vxy
y
x
xyyxyx
Vxy
y
x
xyyx
V
T
el
∫
∫
∫
∫∫
−+++
−=
−++++
−=
−++
−=
−−==
}2
)1(2{
)1(2
1
}2
1)(){(
)1(2
1
2
1
)1(2
1
2
100
01
01
)1(2
1
2
1
222
2
2
2
2
2
γν
ενεεεν
γν
εενεενεεν
γ
ε
ε
γν
ενενεεν
γ
ε
ε
νν
ν
νγεεεεεεεεεε E
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Strain energy in case of pure bending:
23
2
2
2
22
2
222
2
6
1
42
1
)}2
)(2
(2)2
()2
{()1(2
1
}2
)1(2{
)1(2
1
b
a
a
b
b
b
a
a
b
b
bbbb
V
xyyxyxb
a
Eb
dxdya
Ey
dxdya
y
a
y
a
y
a
yE
dVE
U
θ
θ
θν
θν
θν
θ
ν
γν
ενεεεν
=
=
−++−−
=
−+++
−=
∫ ∫
∫ ∫
∫
− −
− −
Strain energy stored in the Q4 element under pure bending:
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2
el
3
2
2
el
3
2
3
2
2
el3
2
2
el
2
2
22
el
2
22
el
2
elel2el
2
222
2el
)2
)1((
)1(6
1}
3
2
3{
)1(2
1
}3
2)2(
42
)1(
3
2)2(
4{
)1(2
1
)42
)1(
4(
)1(2
1
)}2
)(2
(2
)1()
2{(
)1(2
1
}2
)1(2{
)1(2
1
θν
νθν
ν
θνθ
ν
θνθ
ν
θθνθ
ν
γν
ενεεεν
aba
bEab
a
bE
ab
a
ba
a
E
dxdya
x
a
yE
dxdya
x
a
x
a
yE
dVE
U
a
a
b
b
a
a
b
b
V
xyyxyx
−+
−=+
−=
−+
−=
−+
−=
−−−
+−−
=
−+++
−=
∫ ∫
∫ ∫
∫
− −
− −
})(2
)1(1{
)1(
)2()1(6
1
6
1
)2
)1((
)1(6
1
6
1
2
2
3
2
3
2
2
el
2
el
3
2
23
el
b
aab
a
bE
a
Eb
aba
bE
a
Eb
UU
b
b
b
ν
ν
νν
θ
θ
θν
νθ
−+
−=
+−
=
−+
−=
=
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Isoparametric Formulations
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Isoparametric Formulation Interpolation of Geometry
∑
∑
=
=
ηξ=ηξ++ηξ=
ηξ=ηξ++ηξ=
m
iiinm
m
iiinm
yNyNyNy
xNxNxNx
111
111
),(),(),(
),(),(),(
L
L
Natural (intrinsic) coordinate system
Actual coordinate system
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ee
e
m
m
m
me
ee
yx
yxyx
NN
NN
NN
yx
XNx =
⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
M
L 2
2
1
1
2
2
1
1
00
0
0
00
Interpolation of Displacement in a Parent Element
eee
ee
vu
UNu ),( ηξ=⎟⎟⎠
⎞⎜⎜⎝
⎛=
Derivatives of the Displacement Shape Functions
→⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂∂∂∂
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂η∂
∂η∂
∂ξ∂
∂ξ∂
=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂η∂∂ξ∂
→
⎪⎪⎭
⎪⎪⎬
⎫
∂η∂
∂∂
+∂η∂
∂∂
=∂η∂
∂ξ∂
∂∂
+∂ξ∂
∂∂
=∂ξ∂
yNx
N
yx
yx
N
N
yy
Nxx
NN
yy
Nxx
NN
i
i
i
i
iii
iii
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iixi
i
i
i
i
i
NNN
N
N
N
yx
yx
yNx
N
η−−
−
∇=∇⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂η∂∂ξ∂
=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂η∂∂ξ∂
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂η∂
∂η∂
∂ξ∂
∂ξ∂
=⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂∂∂∂
11
1
or JJ
XNJ ⋅∇=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂η∂∂ξ∂
∂η∂∂ξ∂
∂η∂∂ξ∂
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂η∂
∂η∂
∂ξ∂
∂ξ∂
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂η∂
∂η∂
∂ξ∂
∂ξ∂
= ξ
==
==
∑∑
∑∑
mm
m
m
m
ii
im
ii
i
m
ii
im
ii
i
yx
yxyx
N
N
N
N
N
N
yNxN
yNxN
yx
yx
22
11
2
2
1
1
11
11
ML
m > n NN ≠ : Superparametric element
m = n NN = : Isoparametric element
m < n NN ≠ : Subparametric element
∫ ∫∫− −
ηξηξ⋅⋅ηξ=⋅⋅1
1
1
1||),(),( ddJttdA T
A
T
e
BDBBDB
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Bilinear Isoparametric Element
Shape functions in the parent coordinate system.
)),(),,(( )),(),,((
8765
4321
ξηα+ηα+ξα+α=ηξηξξηα+ηα+ξα+α=ηξηξ
yxvyxu
4321444434214444444
4321334333213333333
4321224232212222222
4321114131211111111
)),(),,((),()),(),,((),()),(),,((),(
)),(),,((),(
α−α+α−α=ηξα+ηα+ξα+α=ηξηξ==α+α+α+α=ηξα+ηα+ξα+α=ηξηξ==α−α−α+α=ηξα+ηα+ξα+α=ηξηξ==
α+α−α−α=ηξα+ηα+ξα+α=ηξηξ==
yxuyxuuyxuyxuuyxuyxuu
yxuyxuu
ξ
η
η=0.5
η=−0.5
ξ=0.5ξ=−0.5ξ=−0.5 ξ=0.5
η=0.5
η=−0.5
1 2
3
1 2
3 4 4
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⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−−
−−−−
=
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
4
3
2
1
4
3
2
1
1111111111111111
αααα
uuuu
),( , ),( 4
43
32
21
14
43
32
21
1 vNvNvNvNyxvuNuNuNuNyxu ee +++=+++=
)1)(1(41 , )1)(1(
41 , )1)(1(
41 , )1)(1(
41
4321 η+ξ−=η+ξ+=η−ξ+=η−ξ−= NNNN
N1N2 N3 N4
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Triangular Isoparametric Element
Area Coordinate System
1 , , , 3213
32
21
1 =α+α+α=α=α=αAA
AA
AA
Shape functions
- CST Element
332211 , , α=α=α= NNN
- LST Element
A3
A1 A2
1 2
3
α1 constant line
α2 constant line
α3 constant line
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316325214
333222111
4 4 4)1(2 )1(2 )1(2
αα=αα=αα=−αα=−αα=−αα=
NNNNNN
Interpolation of Geometry
∑∑
∑∑
==
==
αα=ααα=
αα=ααα=
n
iii
n
iii
n
iii
n
iii
yNyNy
xNxNx
121
1321
121
1321
),(~),,(
),(~),,(
)(]~[~00~
~00~
)(1
1
1
1 ee
e
m
mn
ne
ee XN
yx
yx
NN
NN
yx
=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡=⎟
⎟⎠
⎞⎜⎜⎝
⎛= MLX
Interpolation of Displacement in a Parent Element
)()],(~[)()],,([)( 21321eeee
e
ee UNUN
vu
u αα=ααα=⎟⎟⎠
⎞⎜⎜⎝
⎛=
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Derivatives of the Displacement Shape Functions
ii
i
i
i
i
i
i
i
i
i
iii
iii
N
N
N
N
yx
yx
yNx
N
yNx
N
yx
yx
N
N
yy
Nxx
NN
yy
Nxx
NN
~
~
~
~
~
~
~
~
~
~
~~~
~~~
2
11
2
1
1
22
11
22
11
2
1
222
111
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂α∂∂α∂
=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂α∂∂α∂
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂α∂
∂α∂
∂α∂
∂α∂
=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂∂∂∂
→
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂∂∂∂
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂α∂
∂α∂
∂α∂
∂α∂
=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂α∂∂α∂
→
⎪⎪⎭
⎪⎪⎬
⎫
∂α∂
∂∂
+∂α∂
∂∂
=∂α∂
∂α∂
∂∂
+∂α∂
∂∂
=∂α∂
−
−
J
or
iix NN ~~ 1α
− ∇=∇ J
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XNJ ⋅∇=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂α∂∂α∂
∂α∂∂α∂
∂α∂∂α∂
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂α∂
∂α∂
∂α∂
∂α∂
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
∂α∂
∂α∂
∂α∂
∂α∂
= α
==
==
∑∑
∑∑ ~
~
~
~
~
~
~
~~
~~
22
11
2
1
2
2
1
2
2
1
1
1
1 21 2
1 11 1
22
11
mm
m
m
m
ii
im
ii
i
m
ii
im
ii
i
yx
yxyx
N
N
N
N
N
N
yNxN
yNxN
yx
yx
ML
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Numerical Integration
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Gauss Quadrature Rule
� One Dimension
∫ ∑− =
ξ≈ξξ
1
1 1
)()(n
iii fWdf
If the given function )(ξf is a polynomial, it is possible to construct the quadrature rule that
yields the exact integration.
- )(ξf is constant: 0)( af =ξ
2 1 2)( 1
1
1 100 =→==ξ∫ ∑
− =
W n=Waadxfn
ii
- )(ξf is first order: ξ+=ξ 10)( aaf One point rule is good enough.
- )(ξf is second order: 2
210)( ξ+ξ+=ξ aaaf
2 2 , 0 , 3
2
23
2)(
111
2
1
1 10
11
1
2202
=→==ξ=ξ
→+ξ+ξ=+=ξ
∑∑∑
∫ ∑∑∑
===
− ===
nWWW
WaWaWaaaf
n
ii
n
iii
n
iii
n
ii
n
iii
n
iii
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21211
1 , 0 ξ−=ξ=→=ξ∑=
WWWn
iii
3
1
3
22
222
222
222
211
2
1
2=ξ→=ξ=ξ+ξ=ξ∑
=
WWWWWi
ii
89626 02691 57735.01/3= 1 22 22221
2
1
=ξ→=→==+=∑=
WWWWWi
i
- )(ξf is third order: 3
32
210)( ξ+ξ+ξ+=ξ aaaaf Two point rule is enough.
- )(ξf is fourth order: 4
43
32
210)( ξ+ξ+ξ+ξ+=ξ aaaaaf
3 2 , 0 , 3
2 , 0 ,
5
2
23
2
5
2)(
111
2
1
3
1
4
10
11
1
22
1
33
1
44
1
1
024
=→==ξ=ξ=ξ=ξ
→+ξ+ξ+ξ+ξ=++=ξξ
∑∑∑∑∑
∑∑∑∑∑∫
=====
=====−
nWWWWW
WaWaWaWaWaaaadf
n
ii
n
iii
n
iii
n
iii
n
iii
n
ii
n
iii
n
iii
n
iii
n
iii
0 , , 0 , 0 2313111
3=ξξ−=ξ=→=ξ=ξ ∑∑
==
WWWWn
iii
n
iii
5
1
5
22 4
33433
433
411
3
1
4=ξ→=ξ=ξ+ξ=ξ∑
=
WWWWWi
ii
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55555 55555 55555.0 , 41483 66692 77459.0
9
5 = , 5/3
3
1
3
22
33
33233
233
233
211
3
1
2
==ξ
=ξ→=ξ→=ξ=ξ+ξ=ξ∑=
W
WWWWWWi
ii
88888 88888 0.88888= 9
8 22 223321
3
1
=→=+=++=∑=
WWWWWWWi
i
- Because of the symmetry condition, we need to decide only n unknowns for n-points G.Q..
- We can integrate 2n-1-th polynomials exactly with n-points G.Q. Since for 2m-th order
polynomials we have 2m conditions for G.Q. -which means we can determine (m+1)-point G.Q..
- Stiffness Equation
∑∫∫∫=−
ξ⋅⋅ξ=ξξ⋅⋅ξ=⋅⋅=⋅⋅
n
iiiiii
Ti
T
x
T
V
TJAWdJAAdxdV
ee 1
1
1
)()()()( BDBBDBBDBBDB
∑∫∫∫=−
ξ⋅ξ=ξξ⋅ξ=⋅=⋅
n
iiiii
Ti
T
x
T
V
TJAWdJAAdxdV
ee 1
1
1
||)()(||)()( bNbNbNbN
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� Two-Dimensional Case – Rectangular Elements
- Quadrature rule
∑∑∑∑∫∑∫ ∫= ===− =− −
ηξ=ηξ=ηηξ≈ηξηξ
m
j
n
ijiji
n
ijii
m
jj
n
iii fWWfWWdfWddf
1 111
1
1 1
1
1
1
1
)()()(),(
- Stiffness equation
∑∫∫
∑∑∫ ∫∫
∑∑∫ ∫∫
=−
= =− −
= =− −
⋅=⋅=⋅
⋅=⋅=⋅
⋅⋅=⋅⋅=⋅⋅
n
iiiipi
Tip
T
S
T
n
i
m
jijijijji
Tji
T
A
T
n
i
m
jijijjiijji
Tji
T
A
T
KtWdKttdS
JtWWddJttdA
JtWWddJttdA
e
e
e
1
1
1
1 1
1
1
1
1
1 1
1
1
1
1
||),(||),(
||),(||),(
||),(),(||),(),(
TNTNTN
bNbNbN
BDBBDBBDB
ηξξηξ
ηξηξηξ
ηξηξηξηξηξ
� Two-Dimensional Case – Triangular Elements
∑∫ ∫∫=
α−
αα⋅⋅αα=αααα⋅⋅αα=⋅⋅
n
iii
iiij
iiTi
T
A
TJtWddJttdA
e 12121
1
0
1
0
122121 ||),(),(2
1||),(),(
1
BDBBDBBDB
Department of Civil Engineering, DAU
Structural Analysis Lab.
This lecture note has been reproduced and distributed by courtesy by Prof. Hae Sung Lee, http://strana.snu.ac.kr
Reduced Integration � Q8 element
27
26
254
23210
27
26
254
23210
ξη+ηξ+η+ξη+ξ+η+ξ+=
ξη+ηξ+η+ξη+ξ+η+ξ+=
bbbbbbbbv
aaaaaaaau
276431
276431 22 , 22 η+ξη+η+ξ+=
ξ∂
∂η+ξη+η+ξ+=
ξ∂
∂bbbbb
vaaaaa
u
ξη+ξ+η+ξ+=η∂
∂ξη+ξ+η+ξ+=
η∂
∂7
265427
26542 22 , 22 bbbbb
vaaaaa
u
� Terms in stiffness matrix
- From complete polynomials: 22
, , , , , 1 ηξηξηξ
- From parasitic terms: 3322
, , , , ξηξηηξξη , 443322 , , , , ξηξηηξηξ
� Reduced Integration
Reduce the integration order by one to eliminate the effect of parasitic terms in the stiffness matrix.
Department of Civil Engineering, DAU
Structural Analysis Lab.
This lecture note has been reproduced and distributed by courtesy by Prof. Hae Sung Lee, http://strana.snu.ac.kr
Exact average shear stress
Nodal values extrapolated from Gauss point
Gauss point value by the reduced integration
Department of Civil Engineering, DAU
Structural Analysis Lab.
This lecture note has been reproduced and distributed by courtesy by Prof. Hae Sung Lee, http://strana.snu.ac.kr
Spurious Zero Energy mode
� Independent displacement modes of a bilinear element
Rigid Body motion – zero energy mode
Department of Civil Engineering, DAU
Structural Analysis Lab.
This lecture note has been reproduced and distributed by courtesy by Prof. Hae Sung Lee, http://strana.snu.ac.kr
� Spurious zero energy mode
� Zero energy mode of Q8 element
Hour glass mode
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