FEM--Lecture 5 CEE6504-Interpolation.pdf

Interpolation and Shape Functions
Interpolation
To interpolate is 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.

In terms of generalized DOF ai , an interpolating
polynomial with dependent variable ϕ and
independent variable x can be written in the form:
ai : generalized DOF
x : independent variable
ϕ : dependent variable
2
1 2 3
( ) n
m
x a a x a x a x
 = + + + +

or
The ai can be expressed in terms of nodal values of 
at known values of x. The relation between nodal
values of {ϕe }and ai is given as
}
]{
[ a
x
=

]
1
[
]
[ 2 n
x
x
x
x 
=
1 2 3
{ } [ ]
T
m
a a a a a
=
}
]{
[
}
{ a
A
e =


2
1 1
1 1 1
2
2 2
2 2 2
2
3 3
3 3 3
2
1
1
{ } 1
1
n
n
n
e
n
m m
m m m
a
x x x
a
x x x
a
x x x
a
x x x


 

 
   
 
   
 
   
   
 
= =
   
 
   
 
   
 
   
   
 
}
]{
[
}
{ a
A
e =


Recalling
}
{
]
[
}
{ 1
e
A
a 
−
=
}
{
]
][
[ 1
e
A
x 
 −
=
}
]{
[ e
N 
 =
}
]{
[
}
{ a
A
e =

}
]{
[ a
x
=


And
𝑁 = 𝑥 𝐴 −1 = 𝑁1 𝑁2 𝑁3 ⋯ 𝑁𝑚
An individual Ni in matrix [N] is called a shape
function. The name basis function sometimes used
instead.
}
]{
[ e
N 
 =

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
 = + + + +

Similarly, in two – and three-dimensional finite elements the
polynomial form of interpolation functions can be expressed
as
2 2
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
 = + + + + + +
+ + + +

Where a1, a1, …., am are the coefficients of the polynomial; n
is the degree of the polynomial; and the number of the
polynomial coefficients is given by
for one-dimensional elements
1
m n
= +
1
1
for two-dimensional elements
n
j
m j
+
=
= 
1
1
for three-dimensional elements
( 2 )
n
j
m j n j
+
=
= + −


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-dimensional
1 2
( )
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
 = + + +

For n = 2 (quadratic model)
One-dimensional
Two-dimensional
Three-dimensional
2
1 2 3
( )
x a a x a x
 = + +
2 2
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
 = + + + + + + + + +

For n = 3 (cubic model)
One-dimensional
Two-dimensional
2 3
1 2 3 4
( )
x a a x a x a x
 = + + +
2 2
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
 = + + + + + +
+ + +

For n = 3 (cubic model)
Three-dimensional
2 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
2 2
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

Pascal Triangle – complete polynomials in 2 D

Pascal Tetrahedron– complete polynomials in 3 D

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

Thus, in one dimension, ϕ = ϕ (x) is C 0 continuous
if ϕ is continuous but ϕ ,x is not, and ϕ = ϕ (x) is C 1
continuous if both ϕ and ϕ ,x are continuous but ϕ ,xx
is not.

The Cm 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.

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:

1. The field variable must be continuous within the element.
2. All uniform states of the field variable ϕ and 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.
𝐼 𝜙 = න
𝑎
𝑏
𝐹(𝑥, 𝜙, 𝜙′
)𝑑𝑥

3. The field variable ϕ and 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.

The elements whose interpolation polynomials satisfy
1 and 3 are called compatible or conforming elements and
those satisfying 2 are called complete.
If the r-th derivative of the field variable ϕ is continuous, then
ϕ is said to have C r continuity.
In terms of notation, the completeness requirement implies
that ϕ must have C r continuity within the element, whereas
the compatibility requirement implies that ϕ must have C r-1
continuity at element interface.

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
2
1 ]
[


 N
N








=
2
1
2
1 ]
[
u
u
N
N
u

We begin with linear interpolation between points (x1, ϕ1) and
(x2, ϕ2) for which [x] = [1 x]. Evaluating at points 1 and 2,
we obtain
Inverting [A] and using we obtain
]
[
]
][
[
]
[ 2
1
1
N
N
A
x
N =
= −
 
1 1
2 2
a
A
a


   
=
   
   

The two shape functions are shown in the figure
Linear interpolation and shape functions

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).

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,
and k by proceeding counterclockwise from i, which
arbitrarily specified. Let the global coordinates of
nodes i, j, and k be given by (xi, yi), (xj, yj), and (xk,
yk) and nodal values of the field variable ϕ (x, y) by
ϕ i , ϕ j , ϕ k respectively. The variation of ϕ inside the
element is assumed to be linear as

The nodal conditions
Two-dimensional element

Lead to the system of equations
The solution of this system yields

where A is the area of the triangle ijk given by
and

Substituting the values of α1, α2, and α3 in the field
variable ϕ (x, y) and rearrange yields the equation
Where

Stress analysis Element
The DOF’s at each node for this element are u and 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
y
x
u










=
6
5
4
]
1
[
a
a
a
y
x
v

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
  
  
 
     
− − −
− −
= + +
= + +
 
= =
 
 
 
=  
 
= 1 2k

−
 
 

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, and l, where i, j, and k are
labeled in a counterclockwise sequence on any face as viewed
from the vertex opposite this face, which is labeled as l.

Three-dimensional Element
Let nodal values of the field variable ϕ (x, y, z) be Φi
, Φ j , Φ k and Φ l and the global coordinates be (xi, yi,
zi), (xj, yj, zj), (xk, yk, zk) and (xl, yl, zL) at nodes i, j, k,
and l, respectively. The variation of ϕ (x, y, z) inside
the element is assumed to be linear as

The nodal conditions lead
to the system of equations
Three-dimensional Element

Solving the system of equations for αi (i = 1,2, 3, 4)

where the volume of the tetrahedron ijkl given 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 d are to be reversed when generating aj, bj, cj, dj and al,
bl, cl, dl

By substituting in we get

B. Quadratic Interpolation
1. One-dimensional. Quadratic interpolation fits a parabola to
the points (x1, ϕ1), (x2, ϕ2), and (x3, ϕ3). These points need not
be equidistant. Matrix [x] = [1 x x2] and
This equation yields the shape
functions
]
[
]
][
[
]
[ 3
2
1
1
N
N
N
A
x
N =
= −
 
1 1
2 2
3 3
a
A a
a



   
   
=
   
   
   

which are given in the figure.
Quadratic interpolation and shape functions

or more generally
in which the bracketed terms are omitted to obtain the kth
shape function. For linear interpolation, N’s and x’s having
subscripts greater than 2 do not appear; for quadratic
interpolation, N’s and x’s having 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 n points.

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

Lagrange’s interpolation formula uses only ordinates ϕi in
fitting a curve. Slope information is not used, so Lagrange
interpolation may display slopes at nodes other than those
desired.

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 ϕi and
small slopes (dϕ/dx)i at either end of the line length L, as in
the figure

Now [x] = [1 x x2 x3], and upon evaluating ϕ and ϕ ,x at 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




   
   
   
=
   
   
   
 
 

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

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.

FEM--Lecture 5 CEE6504-Interpolation.pdf

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