Higher Order Elements - Introduction to Finite Elements - Lecture Slides, Slides for Finite Element Method. Shree Ram Swarup College of Engineering & Management

Finite Element Method

Description: The lecture slides of the Introduction to Finite Elements are very helpful and interesting the main points are:Higher Order Elements, Properties of Shape Functions, Triangular Elements, Area Coordinates, Lagrange Family, Rectangular Elements, Serendipity Family, Kronecker Delta Property, Polynomial Completeness, Boundary Conditions
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1
MANE 4240 & CIVL 4240
Introduction to Finite Elements
Higher order elements
Reading assignment:
Lecture notes
Summary:
• Properties of shape functions
• Higher order elements in 1D
• Higher order triangular elements (using area coordinates)
• Higher order rectangular elements
Lagrange family
Serendipity family
Recall that the finite element shape functions need to satisfy the
following properties
1. Kronecker delta property
=nodesotherallat
inodeat
Ni0
1
...
2211 ++= uNuNu
Inside an element
At node 1, N1=1, N2=N3=…=0, hence
1
1uu node =
Facilitates the imposition of boundary conditions
2. Polynomial completeness
yyN
xxN
N
i
ii
i
ii
ii
=
=
=
1
yxuIf 321
α
α
α
+
+
Then
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2
Higher order elements in 1D
2-noded (linear) element:
12
x2
x1
12
1
2
21
2
1
xx
xx
N
xx
xx
N
=
=
In “local” coordinate system (shifted to center of element)
12
x
aa a
xa
N
a
xa
N
2
2
2
1
+
=
=
x
3-noded (quadratic) element:
12
x2
x1
()
(
)
()()
()()
()()
()()
()()
2313
21
3
3212
31
2
3121
32
1
xxxx
xxxx
N
xxxx
xxxx
N
xxxx
xxxx
N
=
=
=
In “local” coordinate system (shifted to center of element)
12
x
aa
()
()
2
22
3
2
2
2
1
2
2
a
xa
N
a
xax
N
a
xax
N
=
+
=
=
3
x3x
axxax
=
== 321 ;0;
3
4-noded (cubic) element:
12
x2
x1
()()()
()()()
()()()
()()()
()()()
()()()
()()()
()()()
342414
321
4
432313
421
3
423212
431
2
413121
432
1
xxxxxx
xxxxxx
N
xxxxxx
xxxxxx
N
xxxxxx
xxxxxx
N
xxxxxx
xxxxxx
N
=
=
=
=
In “local” coordinate system (shifted to center of element)
12
2a/3 x
aa
)3/)(3/)((
16
27
))(3/)((
16
27
)3/)(3/)((
16
9
)3/)(3/)((
16
9
3
4
3
3
3
2
3
1
axaxax
a
N
xaxaxa
a
N
axaxax
a
N
axaxax
a
N
+=
+=
++=
+=
3
x3x
x4
4
34
2a/3
2a/3
Polynomial completeness
#
4
3
2
1
x
x
x
x2 node; k=1; p=2
3 node; k=2; p=3
4 node; k=3; p=4
Convergence
rate (displacement)
1;
0+=kpChuu p
h
Recall that the convergence in displacements
k=order of complete polynomial
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