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Finite Element Method (3): 2D FEM
Lecture 12-13 Dr. Amr Bayoumi
Fall 2014 Advanced Engineering Mathematics (EC760)
Arab Academy for Science and Technology - Cairo
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
2
Outline
β’ 2D using Triangular Elements
β’ 2D using Rectangular Elements
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
3
References
β’ S. Chapra and R. Canale, βNumerical Methodβs for Engineersβ, McGraw-Hill, 5th Ed., 2006
β’ S. Moaveni, βFinite Element Analysis, Theory and Application with Ansysβ, Pearson Prentice Hall, 3rd Ed., 2008
β’ E. Thompson, βIntroduction to the Finite Element Method: Theory, Programming, and Applicationsβ, Wiley, 2005
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
4
Triangular Mesh
π’ π₯, π¦ = π0 + π1,1π₯ + π1,2π¦
1 π₯1 π¦11 π₯2 π¦21 π₯3 π¦3
π0π1,1π1,2=π’1π’2π’3
Find a0, a11, a12 (Cramerβs Rule, LU, GE,β¦)
π’ = π1π’1 +π2π’2 +π3π’3 Ae=Area of triangular element=(1/2) Det(A)
β’ π1 =1
2π΄π[π₯2π¦3β π₯3π¦2) + (π¦2β π¦3)π₯ + (π₯3 β π₯2)π¦]
β’ π2 =1
2π΄π[(π₯3π¦1β π₯1π¦3) + (π¦3β π¦1)π₯ + (π₯1 β π₯3)π¦]
β’ π3 =1
2π΄π[(π₯1π¦2β π₯2π¦1) + (π¦1β π¦2)π₯ + (π₯2 β π₯1)π¦]
x
y
1
2
3 u1
u2
u3
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
5
Rectangular Mesh (Local Coordinates)
π’ π₯, π¦ = π0 + π1π₯ + π2π¦ + π3π₯π¦
1 π₯1 π¦1 π₯1π¦11 π₯2 π¦2 π₯2π¦211
π₯3π₯4
π¦3π₯4
π₯3π¦3π₯4π¦4
π0π1π2π3
=
π’1π’2π’3π’4
π’ = π1π’1 +π2π’2 +π3π’3 + π4π’4
π1 = 1 βπ₯
π 1 β
π¦
π€,
π2 =π₯
π 1 β
π¦
π€,
π3 =π¦
π€1 βπ₯
π , π4 =
π₯π¦
ππ€
X
Y 4 3
1 2
u1 u2
u4 u3
π₯
π¦
π
π€
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
6
Example: 2D Potential Equation with Drichlet Boundary Conditions
I,
Y
X
100V
75V 50V
0V
4,4
0,0 10cm
10cm
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
7
Example: 2D Potential Equation with Drichlet Boundary Conditions
Element Equations:
π2π(π₯, π¦)
ππ₯2 + π2π(π₯, π¦)
ππ¦2 = βπ π₯
π π₯ = 0
ππ
ππ₯= βπΈπ₯
ππ
ππ¦= βπΈπ¦
X
Y 4 3
1 2
V1 V2
V4 V3
π₯
π¦
π
π€
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
8
Derivatives in 2D: ππ½
ππ₯
π = π1π£1 +π2π£2 +π3π£3 +π4π£4 = ππ π
ππ
ππ₯=π π π
ππ₯π =
π
ππ₯π1 π2 π3 π4 π
ππ1ππ₯=π
ππ₯1 βπ₯
π1 βπ¦
π€= 1 β
π¦
π€ β1
π =π¦ β π€
π€π
ππ2ππ₯=π
ππ₯
π₯
π 1 βπ¦
π€= 1 β
π¦
π€ 1
π =π€ β π¦
π€π
ππ3ππ₯=π
ππ₯
π¦
π€1 βπ₯
π =βπ¦
π€π,ππ4ππ₯=π
ππ₯
π₯π¦
ππ€=π¦
π€π
ππ
ππ₯=1
π€π(π¦ β π€) (π€ β π¦) (βπ¦) (π¦) π
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
9
Derivatives in 2D: ππ½
ππ
ππ1ππ¦=π
ππ¦1 βπ₯
π1 βπ¦
π€= 1 β
π₯
π β1
π€ =π₯ β π
π€π
ππ2ππ¦=π
ππ¦
π₯
π 1 βπ¦
π€=βπ₯
π€π
ππ3ππ¦=π
ππ¦
π¦
π€1 βπ₯
π =π β π₯
π€π,ππ4ππ₯=π
ππ₯
π₯π¦
ππ€=π₯
π€π
ππ
ππ¦=1
π€π(π₯ β π) (π β π₯) (βπ₯) (π₯) π
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
10
Residual Equations in 2D Rectangular F.E. Using Galerkinβs Method
π = π1π£1 +π2π£2 +π3π£3 +π4π£4 = ππ π
π2π(π₯, π¦)
ππ₯2 + π2π(π₯, π¦)
ππ¦2 = 0
π = π π4
π=1= ππ
π2π(π₯, π¦)
ππ₯2 + π2π(π₯, π¦)
ππ¦2ππ΄
π΄
4
π=1
= π ππ2π(π₯, π¦)
ππ₯2 + π2π(π₯, π¦)
ππ¦2ππ₯ππ¦
π΄
= 0
Where: π π = π1 π2 π3 π4
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
11
Greenβs Theory in 2D Finite Elements Using Galerkinβs Method (2)
Use: π
ππ₯π πππ
ππ₯= [π]π
π2π
ππ₯2+π π π
ππ₯
ππ
ππ₯
β [π]ππ2π
ππ₯2=π
ππ₯π πππ
ππ₯βπ π π
ππ₯
ππ
ππ₯
Similarly:
[π]ππ2π
ππ¦2=π
ππ¦π πππ
ππ¦βπ π π
ππ¦
ππ
ππ¦
By substituting: π = πΌ1 + πΌ2
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
12
2D Rectangular Finite Elements Using Galerkinβs Method (2)
The integral πΌ1 can be easily evaluated using derivatives of π π
πΌ1 = βπ π π
ππ₯
ππ
ππ₯ β π π π
ππ¦
ππ
ππ¦ππ₯ππ¦
π
π
π
π
π π π
ππ₯=1
π€π(π¦ β π€) (π€ β π¦) (βπ¦) (π¦) π
π π π
ππ¦=1
π€π(π₯ β π) (π β π₯) (βπ₯) (π₯)
ππ
ππ₯=π π π
ππ₯π ,
ππ
ππ¦=π π π
ππ¦π
Dr. Amr Bayoumi- Spring 2015 Lec. 13 EC760 Advanced Engineering Mathematics
13
2D Rectangular Finite Elements Using Galerkinβs Method (3)
The integral πΌ2 can be evaluated using Greenβs Theory (Next Lecture):
πΌ2 = π
ππ₯π πππ
ππ₯+π
ππ¦π πππ
ππ¦ππ₯ππ¦
π΄