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JN Reddy
The Finite Element Method
Read: Chapter 14
3D Problems Heat Transfer and Elasticity
CONTENTS
Finite element models of 3-D Heat Transfer
Finite element modelof 3-D Elasticity
Typical 3-D Finite Elements
JN Reddy
3-D HEAT TRANSFER
T = T on Γ1,
kx∂T
∂xnx + ky
∂T
∂yny + kz
∂T
∂znz + β(T − T∞) = q on Γ2( )
where kx, ky and kz are conductivities of an or-
thotropic solid in the three coordinate directions,
g is the internal heat generation per unit volume
in a three-dimensional domain Ω, and T and q
are specified functions of position on the portions
Γ1 and Γ2, respectively, of the surface Γ of the
domain (see Fig.1); β is the convection coefficient
and T∞ is the ambient temperature.
− ∂
∂xkx
∂T
∂x− ∂
∂yky
∂T
∂y− ∂
∂zkz
∂T
∂z= g in Ω
Boundary Conditions
Governing Equation
3-D Problems 2
JN Reddy
z
x
y
n
xn
ynzn
ds
ΓSurface
ΩDomain
A six-face 3-D finite element
eΩ
eΓParts of the boundary
3-D HEAT TRANSFER (continued)
Ωe
w − ∂
∂xkx
∂T
∂x− ∂
∂yky
∂T
∂y− ∂
∂zkz
∂T
∂z− g dx
Ωe
kx∂w
∂x
∂T
∂x+ ky
∂w
∂y
∂T
∂y+ kz
∂w
∂z
∂T
∂z−wg dx
+Γe
βwT ds−Γew (qn + βT∞) ds (3)
Weak Form
0=
=
3-D Problems 3
JN Reddy
KeTe = fe +Qe
T =n
j=1
Tjψej (x, y, z)
3-D HEAT TRANSFER (continued)
Finite element approximation
Finite element model
where
Keij =
Ωe
kx∂ψei∂x
∂ψej∂x
+ ky∂ψei∂y
∂ψej∂y
+ kz∂ψei∂z
∂ψej∂z
dx
+Γe
βψeiψj ds
fei =Ωe
fψei dx, Qei =Γe(qn + βT∞)ψei ds
( )
3-D Problems 4
JN Reddy
3-D ELASTICITY
Equations of Motion
Strain-Displacement Relations
εxx =∂ux∂x
, εyy =∂uy∂y
, εzz =∂uz∂z
2εxy =∂ux∂y
+∂uy∂x
, 2εxz =∂ux∂z
+∂uz∂x
2εyz =∂uy∂z
+∂uz∂y
3-D Problems 5
∂σxx∂x
+∂σxy∂y
+∂σxz∂z
+ fx = ρ∂ux∂t2
∂σxy∂x
+∂σyy∂y
+∂σyz∂z
+ fy = ρ∂uy∂t2
∂σxz∂x
+∂σyz∂y
+∂σzz∂z
+ fz = ρ∂uz∂t2
2
2
2
JN Reddy
3-D ELASTICITY (continued)
Constitutive Relations⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
σxxσyyσzzσxzσyzσxy
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭=
⎡⎢⎢⎢⎢⎢⎣c11 c12 c13 0 0 0c12 c22 c23 0 0 0c13 c23 c33 0 0 00 0 0 c44 0 00 0 0 0 c55 00 0 0 0 0 c66
⎤⎥⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
εxxεyyεzz2εxz2εyz2εxy
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭( )
tx ≡ σxxnx + σxyny + σxznz = txty ≡ σxynx + σyyny + σyznz = tytz ≡ σxznx + σyzny + σzznz = tz
⎫⎬⎭ on Γσ u = u on Γuor
Boundary Conditions
The material axes are assumed coincide with the global axes and the material is orthotropic with respect to the global axes.
3-D Problems 6
JN Reddy
3-D ELASTICITY (continued)
MATRIX FORM OF THE GOVERNING EQUATIONS
ε =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
εxxεyyεzz2εxz2εyz2εxy
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭,
DT =
⎡⎣∂/∂x 0 0 ∂/∂z 0 ∂/∂y0 ∂/∂y 0 0 ∂/∂z ∂/∂x0 0 ∂/∂z ∂/∂x ∂/∂y 0
⎤⎦
σ =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
σxxσyyσzzσxyσxzσyz
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭, f =
⎧⎨⎩ fxfyfz
⎫⎬⎭ , u =⎧⎨⎩uxuyuz
⎫⎬⎭
DTσ + f = ρu σ = Cεε = Du,
Notation
Governing equations
3-D Problems 7
JN Reddy 3-D Problems 8
3-D ELASTICITY (continued)
u =
⎧⎨⎩uxuyuz
⎫⎬⎭ =Ψ∆, w = δu =
⎧⎨⎩ δuxδuyδuz
⎫⎬⎭ =Ψδ∆
0 =Ωe
(Dδu)TC (Du) + ρuTu dx −Ωe
(δu)Tf dx−Γe
(δu)Tt ds
Principle of virtual displacements (in matrix form)
Finite element approximation (in matrix form)
Ψ =
⎡⎢⎣ψ1 0 0 ψ2 0 0 . . . ψn 0 00 ψ1 0 0 ψ2 0 . . . ψn 00 0 ψ1 0 0 ψ2 0 . . . 0 ψn
⎤⎥⎦∆ = u1x u1y u1z u2x u2y u2z . . . unx uny unz T
d
JN Reddy 3-D Problems 9
3-D ELASTICITY (continued)
Finite Element Model
Me∆e +Ke∆e = Fe +Qe
At each node ( , , )u v w
•
••
••
1
2
35
64
••
•
1
2
3
4
•
•
where
Ke =Ωe
heBTCB dx, Me =
Ωe
ρheΨTΨe dx
Fe =Ωe
ΨTf dx, Qe =Γe
ΨTt ds
( )
JN Reddy 3-D Problems 10
TYPICAL 3-D FINITE ELEMENTS
L1 = 0
••
1
2
3
4
•L3 = 0
L4 = 0
•
u = a0 + a1x+ a2y + a3z
Ψe =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
L1(2L1 − 1)L2(2L2 − 1)L3(2L3 − 1)L4(2L4 − 1)4L1L24L2L34L3L14L1L44L2L44L3L4
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
Ψe =
⎧⎪⎨⎪⎩L1L2L3L4
⎫⎪⎬⎪⎭
••
1 7•
•
•
•
25
4•
••
3
6
8
910
•
Quadratic tetrahedral elementLinear tetrahedral element
JN Reddy 3-D Problems 11
TYPICAL 3-D FINITE ELEMENTS
•
••
15
1
10
11
121314
•
•
•
•
•
•••
•
• •
•2
3
4
5
6
7
9
8
L1 = 0
•
••
••
1
2
35
64
•L3 = 0
L2 = 0
ξ
ζη
1+=ζ
1−=ζ
u = a0 + a1x+ a2y + a3z + a4xz + a5yz
Ψe = 1
2
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
L1(1− ζ)L2(1− ζ)L3(1− ζ)L1(1 + ζ)L2(1 + ζ)L3(1 + ζ)
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
Linear prism element Quadratic prism element
JN Reddy
Quadratic Prism Element
Ψe = 1
2
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
L1[(2L1 − 1)(1− ζ)− (1− ζ2)]L2[(2L2 − 1)(1− ζ)− (1− ζ2)]L3[(2L3 − 1)(1− ζ)− (1− ζ2)]L1[(2L1 − 1)(1 + ζ)− (1− ζ2)]L2[(2L2 − 1)(1 + ζ)− (1− ζ2)]L3[(2L3 − 1)(1 + ζ)− (1− ζ2)]
4L1L2(1− ζ)4L2L3(1− ζ)4L3L1(1− ζ)2L1(1− ζ2)2L2(1− ζ2)2L3(1− ζ2)4L1L2(1 + ζ)4L2L3(1 + ζ)4L3L1(1 + ζ)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
TYPICAL 3-D FINITE ELEMENTS (cont…)
3-D Problems 12
•
••
15
1
10
11
121314
•
•
•
•
•
•••
•
• •
•2
3
4
5
6
7
9
8
JN Reddy
•
•
•• •
••
•
•• • •
••
••
••
•
•
ζ
ξη
15
1
2
3
5
6
7
8
9
10 11
12
14
4
1316
17
18
19
20
ξ = +1
ξ = −1η = −1
η = +1
ζ = +1
ζ = −1
••
••
••
nodes••1
2
3
4
56 8
7ξ
η
ζ
TYPICAL 3-D FINITE ELEMENTS (cont…)
Linear brick element Quadratic brick element
Ψe = 1
8
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(1− ξ)(1− η)(1− ζ)(1 + ξ)(1− η)(1− ζ)(1 + ξ)(1 + η)(1− ζ)(1− ξ)(1 + η)(1− ζ)(1− ξ)(1− η)(1 + ζ)(1 + ξ)(1− η)(1 + ζ)(1 + ξ)(1 + η)(1 + ζ)(1− ξ)(1 + η)(1 + ζ)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
u = a0 + a1x+ a2y + a3z + a4yz + a5xz+ a6xy + a7xyz
3-D Problems 13
JN Reddy
TYPICAL 3-D FINITE ELEMENTS (cont…)
Ψe = 1
8
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
(1− ξ)(1− η)(1− ζ)(−ξ − η − ζ − 2)(1 + ξ)(1− η)(1− ζ)(ξ − η − ζ − 2)(1 + ξ)(1 + η)(1− ζ)(ξ + η − ζ − 2)(1− ξ)(1 + η)(1− ζ)(−ξ + η − ζ − 2)(1− ξ)(1− η)(1 + ζ)(−ξ − η + ζ − 2)(1 + ξ)(1− η)(1 + ζ)(ξ − η + ζ − 2)(1 + ξ)(1 + η)(1 + ζ)(ξ + η + ζ − 2)(1− ξ)(1 + η)(1 + ζ)(−ξ + η + ζ − 2)
2(1− ξ2)(1− η)(1− ζ)2(1 + ξ)(1− η2)(1− ζ)2(1− ξ2)(1 + η)(1− ζ)2(1− ξ)(1− η2)(1− ζ)2(1− ξ)(1− η)(1− ζ2)2(1 + ξ)(1− η)(1− ζ2)2(1 + ξ)(1 + η)(1− ζ2)2(1− ξ)(1 + η)(1− ζ2)2(1− ξ2)(1− η)(1 + ζ)2(1 + ξ)(1− η2)(1 + ζ)2(1− ξ2)(1 + η)(1 + ζ)2(1− ξ)(1− η2)(1 + ζ)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
•
•
•• •
••
•
•• • •
••
••
••
•
•
ζ
ξη
15
1
2
3
5
6
7
8
9
10 11
12
14
4
1316
17
18
19
20
Quadratic Brick Element
JN Reddy 2-D Problems: 15
TYPICAL 3-D or SHELL FINITE ELEMENT MESHES
JN Reddy 2-D Problems: 16
TYPICAL 3-D FINITE ELEMENT MESHES
