FEM--Lecture 5 CEE6504-Interpolation.pdf

FEM

1. Interpolation

2. Interpolation and Shape Functions
Interpolation
To interpolateis to device a continuous function that
satisfies prescribed conditions at a finite number of
points.
In FEA, the interpolating function is almost always a
polynomial which provides a single-valued and
continuous field.

3. Interpolation and Shape Functions
In terms of generalized DOF a
i,an interpolating
polynomialwith dependent variable ϕand
independent variable xcan be written in the form:
a
i: generalized DOF
x: independent variable
ϕ: dependent variable2
1 2 3
()
n
m
x a a x a x a x= + + + +

4. Interpolation and Shape Functions
or
The a
ican be expressed in terms of nodal values of 
at known values of x. The relation between nodal
values of {ϕ
e }and a
iis given as}]{[ax= ]1[][
2 n
xxxx = 1 2 3
{ } [ ]
T
m
a a a a a= }]{[}{ aA
e
=

5. Interpolation and Shape Functions2
11 1 1 1
2
22 2 2 2
2
33 3 3 3
2
1
1
{} 1
1
n
n
n
e
n
mm m m m
ax x x
ax x x
ax x x
ax x x




   
   
   
   
==   
   
   
       }]{[}{ aA
e
=

6. Interpolation and Shape Functions
Recalling}{][}{
1
e
Aa 
−
= }{]][[
1
e
Ax 
−
= }]{[
e
N= }]{[}{ aA
e
= }]{[ax=

7. Interpolation and Shape Functions
And
??????=????????????
−1
=??????
1??????
2??????
3⋯??????
??????
An individual N
iin matrix [N]is called a shape
function. The name basis function sometimes used
instead. }]{[
e
N=

8. Interpolation and Shape Functions
In General
If a polynomial type of variation is assumed
for the field variable ϕ(x)in one-dimensional
element, ϕ(x)can be expressed as 2
1 2 3
()
n
m
x a a x a x a x= + + + +

9. Interpolation and Shape Functions
Similarly, in two –and three-dimensional finite elements the
polynomial form of interpolation functions can be expressed
as22
1 2 3 4 5 6
( , )
n
m
x y a a x a y a x a y a xy a y = + + + + + + 2 2 2
1 2 3 4 5 6 7
8 9 10
( , , )

n
m
x y z a a x a y a z a x a y a z
a xy a yz a zx a z
 = + + + + + +
+ + + +

10. Interpolation and Shape Functions
Where a
1, a
1, …., a
m are the coefficients of the polynomial; n
is the degree of the polynomial; and the number of the
polynomial coefficients is given byfor one-dimensional elements1 mn=+ 1
1
for two-dimensional elements
n
j
mj
+
=
= 1
1
for three-dimensional elements( 2 )
n
j
m j n j
+
=
= + −

11. Interpolation and Shape Functions
In most practical applications, the order of the polynomial in
the interpolation functions is taken as one, two, or three.
For n= 1 (linear model)
One-dimensional
Two-dimensional
Three-dimensional12
()x a a x=+ 1 2 3
( , )x y a a x a y = + + 1 2 3 4
( , , )x y z a a x a y a z = + + +

12. Interpolation and Shape Functions
For n= 2 (quadratic model)
One-dimensional
Two-dimensional
Three-dimensional2
1 2 3
()x a a x a x= + + 22
1 2 3 4 5 6
( , )x y a a x a y a x a y a xy = + + + + + 2 2 2
1 2 3 4 5 6 7 8 9 10
( , , )x y z a a x a y a z a x a y a z a xy a yz a zx = + + + + + + + + +

13. Interpolation and Shape Functions
For n= 3 (cubic model)
One-dimensional
Two-dimensional23
1 2 3 4
()x a a x a x a x= + + + 22
1 2 3 4 5 6
3 3 2 2
7 8 9 10
( , )

x y a a x a y a x a y a xy
a x a y a x y a xy
 = + + + + + +
+ + +

14. Interpolation and Shape Functions
For n= 3 (cubic model)
Three-dimensional2 2 2
1 2 3 4 5 6 7
3 3 3
8 9 10 11 12 13
2 2 2 2
14 15 16 17
22
18 19 20
( , , )



x y z a a x a y a z a x a y a z
a xy a yz a zx a x a y a z
a x y a x z a y x a y z
a z x a z y a x
 = + + + + + + +
+ + + + + +
+ + + +
++


yz

15. Interpolation and Shape Functions
Pascal Triangle –complete polynomials in 2 D

16. Interpolation and Shape Functions
Pascal Tetrahedron–complete polynomials in 3 D

17. Interpolation and Shape Functions
Degree of Continuity
Field quantity ϕis interpolated in piecewise fashion
over each element. So while ϕcan be guaranteed to
vary smoothly within each element, the transition
between elements may not be smooth. The symbol
C
m
is used to describe the continuity of a piecewise
field between elements.
A field is C
m
continuous if its derivatives up to and
including degree mare inter-element continuous.

18. Interpolation and Shape Functions
Thus, in one dimension, ϕ=ϕ(x) is C
0
continuous
if ϕis continuous but ϕ
,xis not, and ϕ= ϕ(x) is C
1
continuous if both ϕandϕ
,xare continuous but ϕ
,xx
is not.

19. Interpolation and Shape Functions
The C
m
terminology is also applied to element types.
For example, the bar element and the beam element
are called “C
0
element” and “C
1
element”,
respectively.
Usually, C
0
elements are used to model plane and
solid bodies. C
1
elements are used to model beams,
plates and shells, thus providing inter-element
continuity of the slope.

20. Interpolation and Shape Functions
Convergence Requirements
Since the finite element method is a numerical
technique, we obtain a sequence of approximate
solutions as the element size is reduced successively.
This sequence will converge to exact solution if the
interpolation polynomial satisfies the following
requirements:

21. Interpolation and Shape Functions
Convergence Requirements
1.The field variable must be continuous within the element.
2.All uniform states of the field variable ϕand its partial
derivatives up to the highest order appearing in the
functional I(ϕ)must have representation in the
interpolation polynomial when, in the limit, the element
size reduces to zero. Rigid body (zero strain) and constant
strain states of the element.
????????????=න
�
�
??????(??????,??????,??????
′
)????????????

22. Interpolation and Shape Functions
Convergence Requirements
3.The field variable ϕand its partial derivative up
to one order less the highest order derivative
appearing in the functional I(ϕ)must be
continuous at the element boundaries or
interfaces.

23. Interpolation and Shape Functions
Convergence Requirements
The elements whose interpolation polynomials satisfy
1and 3are called compatible or conforming elements and
those satisfying 2are called complete.
If the r-thderivative of the field variable ϕis continuous, then
ϕis said to have C
r
continuity.
In terms of notation, the completenessrequirement implies
that ϕmust have C
r
continuity within the element, whereas
the compatibilityrequirement implies that ϕmust have C
r-1
continuity at element interface.

24. Interpolation and Shape Functions
A.Linear Interpolation
1.One-dimensional
In FEA, with ϕ= u, the displacement field in the
linear case can be given as
or the displacement field can be given in the form







=
2
1
21
][


 NN 







=
2
1
21
][
u
u
NNu

25. Interpolation and Shape Functions
We begin with linear interpolation between points (x
1, ϕ
1) and
(x
2, ϕ
2) for which [x] = [1 x]. Evaluating at points 1and 2,
we obtain
Inverting [A]and using we obtain][]][[][
21
1
NNAxN ==
− 
11
22
a
A
a


   
=   
   

26. Interpolation and Shape Functions
The two shape functions are shown in the figure
Linear interpolation and shape functions

27. Interpolation and Shape Functions
A.Linear Interpolation
2.Two-dimensional (linear triangle-Constant-Strain
Triangle (CST).
A linear triangle is a plane triangle whose field
quantity varies linearly with Cartesian coordinates x
and y. In stress analysis, a linear displacement field
produces a constant strain field, so the element may
be called a Constant-Strain Triangle (CST).

28. Interpolation and Shape Functions
The two-dimensional element is a straight-sided
triangle with three nodes, one at each corner, as
indicated in the figure. Let the nodes be labeled i, j,
andkby proceeding counterclockwise from i, which
arbitrarily specified. Let the global coordinates of
nodes i, j, and kbe given by (x
i, y
i), (x
j, y
j), and (x
k,
y
k) and nodal values of the field variable ϕ(x, y) by
ϕ
i, ϕ
j , ϕ
k respectively. The variation of ϕinside the
element is assumed to be linear as

29. Interpolation and Shape Functions
The nodal conditions
Two-dimensional element

30. Interpolation and Shape Functions
Lead to the system of equations
The solution of this system yields

31. Interpolation and Shape Functions
where Ais the area of the triangle ijkgiven by
and

32. Interpolation and Shape Functions
Substituting the values of α
1, α
2, and α
3in the field
variable ϕ(x, y) and rearrange yields the equation
Where

33. Interpolation and Shape Functions
Stress analysis Element
The DOF’s at each node for this element are uand v. The
same linear interpolation is used for both dependent variables.
CST element for 2-D stress
analysis









=
3
2
1
]1[
a
a
a
yxu 









=
6
5
4
]1[
a
a
a
yxv

34. Interpolation and Shape Functions
Stress analysis Element 
 

2 1 2 1 2 1
2 2 2
2 1 2 2 1 2 2
( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , )
( , )
( , ) ( , )
( , )
( , ) 0 ( , ) 0 ( , ) 0
( , )
0 ( , ) 0 ( , ) 0 ( , )
i i j j k k
i i j j k k
i j k
i j k
T
i i j j k
u x y N x y N x y N x y
v x y N x y N x y N x y
u x y
x y N x y
v x y
N x y N x y N x y
N x y
N x y N x y N x y
  
  

     
− − −
−−
= + +
= + +

==



=


=
12 k

−



35. Interpolation and Shape Functions
A.Linear Interpolation
3.Three-dimensional Element
The three-dimensional element is a flat-faced tetrahedron
with four nodes, one at each corner, as shown in the figure.
Let the nodes be labeled as i, j, k,andl, wherei, j, andkare
labeled in a counterclockwise sequence on any face as viewed
from the vertex opposite this face, which is labeled as l.

36. Interpolation and Shape Functions
Three-dimensional Element
Let nodal values of the field variable ϕ(x, y, z) be Φ
i
, Φ
j , Φ
k and Φ
l and the global coordinates be (x
i, y
i,
z
i), (x
j, y
j, z
j), (x
k, y
k, z
k) and (x
l, y
l, z
L) at nodes i, j, k,
and l, respectively. The variation of ϕ(x, y, z)inside
the element is assumed to be linear as

37. Interpolation and Shape Functions
The nodal conditions lead
to the system of equations
Three-dimensional Element

38. Interpolation and Shape Functions
Solving the system of equations for α
i(i= 1,2, 3, 4)

39. Interpolation and Shape Functions
where the volume of the tetrahedron ijklgiven by
With the other constants defined by cyclic interchange of the
subscripts in the order i, j, k, and l. The signs in front of a, b,
c, and dare to be reversed when generating a
j, b
j, c
j, d
janda
l,
b
l, c
l, d
l

40. Interpolation and Shape Functions
By substituting in we get

41. Interpolation and Shape Functions
B. Quadratic Interpolation
1. One-dimensional. Quadratic interpolation fits a parabola to
the points (x
1, ϕ
1), (x
2, ϕ
2), and (x
3, ϕ
3). These points need not
be equidistant. Matrix [x] = [1 xx
2
] and
This equation yields the shape
functions][]][[][
321
1
NNNAxN ==
− 
11
22
33
a
Aa
a



   
   
=   
   
   

42. Interpolation and Shape Functions
which are given in the figure.
Quadratic interpolation and shape functions

43. Interpolation and Shape Functions
or more generally
in which the bracketed terms are omitted to obtain the kth
shape function. For linear interpolation, N’sand x’shaving
subscripts greater than 2do not appear; for quadratic
interpolation, N’sand x’shaving subscripts greater than 3 do
not appear; and so on. This is called the Lagrange’s
interpolation formula, which provides the shape functions for
a curve fitted ordinates at npoints.

44. Interpolation and Shape Functions
The foregoing shape functions have the following
characteristics:
◼All shape functions N
i, along with function ϕitself, are
polynomial of the same degree.
◼For any shape function N
i, N
i= 1whenx = x
iand N
i= 0
whenx = x
jfor any integer j ≠ i. That is N
iis unity at its own
node but is zero at other nodes.
◼C
0
shape functions sum to unity; that is, ∑N
i= 1. This
conclusion is implied by ϕ= [N]{ϕ
e}, because we must obtain
ϕ= 1when {ϕ
e}is a column of 1’s.

45. Interpolation and Shape Functions
Lagrange’s interpolation formula uses only ordinates ϕ
iin
fitting a curve. Slope information is not used, so Lagrange
interpolation may display slopes at nodes other than those
desired.

46. Interpolation and Shape Functions
C
1
Interpolation-One-dimensional
Consider a cubic curve ϕ= ϕ(x), whose shape is determined
by four data items. We take these items to be ordinates ϕ
iand
small slopes (dϕ/dx)
iat either end of the line length L, as in
the figure

47. Interpolation and Shape Functions
Now [x] = [1 xx
2
x
3
], and upon evaluating ϕand ϕ
,xat x
= 0 and x= L, the equation {ϕ
e}= [A]{a} becomes
The obtained shape function are nothing but the four lateral
displacements and rotations of beam nodes.
1 1
,1 2
2 3
,2 4
x
x
a
a
A
a
a




 
 
   
=   
   
   


48. Interpolation and Shape Functions
Beam shape functions
Shape functions of a cubic curved fitted to ordinates and
slopes at x= 0 and x= L.

49. Interpolation and Shape Functions
2-D and 3-D Interpolation
In two-or-three-dimensional problems, two or three
independent variables are needed. These interpolations are
extensions of one-dimensional interpolations. When there are
two or three dependent variables, such as displacements in 2-
D or 3-D problems, usually all components are interpolated
using the same shape functions.