HIGHER ORDER INTERPOLATION, NUMERICAL INTEGRATION

1. Introduction

One of the important point of finite element formulation is the approximate of the

different fields i.e., displacement, temperature, velocity and pressure etc. Already we

have seen in the previous discussions that the simplest approximation of the fields

with the help of linear interpolation yields the derivative (i.e., strain, temperature

gradient, velocity gradient) piecewise constant which in tern makes the element

matrices partially explicit. Unfortunately these linear elements are not always useful

particularly for the problems with rapid change of the gradients within a small zone.

Higher order interpolation for the fields, thus, is essential for such type of problems.

Although the higher order interpolation makes the element matrices complicated but

these elements even with a course mesh are able to capture the behavior with

sufficient accuracy. But the evaluation of the element matrices requires the

integration over one, two or three dimensions. Since, the explicit integration is

tedious and complicated almost every where, the numerical integration is essential.

Thus to illustrate the different steps, we discuss the following parts associated to the

higher order element.

• Natural Coordinates

• Higher order interpolation

• Numerical Quadrature

Natural Coordinates

Of all the quardature formula, as will be discussed in the subsequent discussions

require the domain of the integration to be [-1, 1] . This require the transformation of

the problem coordinate x to a local coordinate ξ such that

when 1 and when 1A Bx x x xξ ξ= = − = = (1) The transformation between x and ξ can be represented by the linear ‘stretch’ transformation x a b ξ= + (2)

���� ���

�xx

xA

B

A B

BA

x

x x

x

1 2

3

r

s

tA

A

A

1

2

3

Where a,b are constants to be determined to satisfy Eq, (1). Thus, the transformation

in this situation take the form

( ) (1 ) (1 )(1 )2 2 2

B AA A B

x xx x x xξ ξξ− − += + + = + (3)

Figure 1: Geometric link between the natural coordinate and the true coordinate

The local coordinate ξ is called the normalized coordinate or natural coordinate and

its value always lie between -1 and 1 with the origin at the center of the elements.

For the triangular elements the natural coordinate is related to the area ratio as shown

in the following figure:

Figure 2: Master triangle with area coordinates r,s,t

The local coordinate ξ is useful in two ways:

(i) it is convenient in constructing the interpolation function

(ii) it is required in numerical integration

Higher order interpolation

In this section the higher order interpolation for 1D and 2D elements are summarized.

For the 1D and 2D quadrilateral element, the Lagrange formula is useful to construct

the interpolation formula. The following conditions are essential to conduct the

interpolation formula

1 if( ) (Kroneker Delta property)

0 ifi i

i ji j

ξ=⎧

Ψ = ⎨ ≠⎩ (4)

���� ���

�

���� ������

�

1( ) 1 (Partion of unity property)

Node

ii

ξ=

Ψ =∑ (5)

Lagrange Formula for 1D: The interpolation associated with node ‘i’ of any n-noded

element is

1 1 2 1 1

1 2 1 1

( )( )...( )( ).....( )( )( )( )...( )( ).....( )

n i i ni

i i i i i i i n

l ξ ξ ξ ξ ξ ξ ξ ξ ξ ξξξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

− − +

− +

− − − − −=

− − − − − (6)

Example 1: 2- noded element

11 1

12 2

(1 )( ) ( )2

(1 )( ) ( )2

l

l

ξξ ξ

ξξ ξ

−Ψ = =

+Ψ = =

Figure 3: Linear interpolation function in 1D

Example 1: 3-noded element

21 1

2 22 2

23 3

(1 )( ) ( )2

( ) ( ) (1 )(1 )( ) ( )

2

l

l

l

ξ ξξ ξ

ξ ξ ξξ ξξ ξ

− −Ψ = =

Ψ = = −+

Ψ = =

Figure 4: Quadratic interpolation function in 1D

Lagrange Interpolation for 2D Quadrilaterals: Using the Lagranges interpolation

formula for 1D, the interpolation function for the 2D uadrilateral may be construct

easily as:

( , ) 1( , ) ( ) ( ) for node ( , ) ( , ) ( 1) ( 2)a i j i j a i j where a i j j n iξ η ξ ηΨ = Ψ Ψ = − ∗ + +

Higher Order Triangles: From the higher order triangular element with k nodes

(equally spaced per side, the total of nodes per element is

( 1)1 2 ......( 1)2

k kn k k += + + − + = and interpolation (Lagrange type) degree will be

1

2

3

4

s

s

s

s

s

1

2

3

4=1

=2/3

=1/3

(k-1). For example the quadratic element k-1=2 or k=3 will have

( 1) 3 4 62 2

k k + ×= = nodes. Let the corner (vertex nodes be denoted by I, J, K and let

ih be the perpendicular distance of the node I from the side connectivity J and K.

Then the area coordinates S to the p-th row parallel to the side J-K (under the

assumption that the nodes are equally spaced along the sides and the rows) is given in

non-dimensional form by

11 , 0 11p i k

pS S and S SK−

= = = =−

Figure 5: Higher order master Triangle with node arrangements

The interpolation function iΨ should be zero at the nodes on the lines

1 2 20, , , ... and 1 1 1i i

ksk k k

−= Ψ

− − −should be unity at i k= . Thus we have the

necessary information for the construction of the interpolation function iΨ

1

( , , ) ( ) ( ) ( )2( 1) 1

( )1 1

a I I I

II

II

r s t T r T s T trl for I

rwhere rfor I

−

Ψ =

⎧ − ≠⎪Τ = ⎨⎪ ≠⎩

1

2

3

r

s

(1,1,2)(2,1,1)

(1,2,1)

(1,1,2)=>( )r ,s ,t1 21

1

2

3

r

s

(1,1,3)(3,1,1)

(1,3,1)

(2,1,2)

(2,2,1)(1,2,2)

45

6

Example: 3-noded triangle

( ) ( )( ) ( )

( )1

2 1

1 0 01 2 1 1 2 1 1

2 1 1

12

2 1 2 1 3 1 1 2

2 2 2( , , ) ( ) ( ) ( ) ( 1) ( 1) ( 1)

2 1 2 1 2 1 ( 1)(2 1)

(1) ( 1)2 1 2 1

( , , ) ( ) ( ) ( ) ; ( , , ) ( ) ( ) ( )

r r

r r r r

r s tr s t T r T s T t l l lr s t

r r rl r r

r r

r s t T r T s T t s r s t T r T s T t t

ϕ

ϕ ϕ

=

= =

= = − − −

− − − − − −= − = = =

− −− − −

= = = =

Example:6-noded triangle:

( ) ( )( ) ( )

( ) ( )( ) ( )

1 2

3 1 3 2

2 0 01 3 1 1 3 1 1

2 1 1

23

2 1 3 1

3 1 1 3

2 2 2( , , ) ( ) ( ) ( ) ( 1) ( 1) ( 1)

2 1 2 1 2 1 2 1(2 1) (2 1)

2 1 2 1 2 1 2 1

( , , ) ( ) ( ) ( ) (2 1)( , , ) ( ) ( ) ( ) (2

r r r r

r r r r r r r r

r s tr s t T r T s T t l l lr s t

r r r rl r r r

r r r r

r s t T r T s T t s sr s t T r T s T t t

ϕ

ϕϕ

= =

= = = =

= = − − −

− − − − − −= − = = −

− − − − − −

= = −= =

4 2 2 1

5 1 2 2

6 2 1 2

1)( , , ) ( ) ( ) ( ) 4( , , ) ( ) ( ) ( ) 4( , , ) ( ) ( ) ( ) 4

tr s t T r T s T t rsr s t T r T s T t str s t T r T s T t rt

ϕϕϕ

−

= == =

= =

Figure 6: A 3-nodfed and a 6-noded triangle with local natural co-ordinates

Brick/solid and tetrahedral elements: The higher order interpolation for bricks and

tetrahedral elements may be constructed in a similar manner and may be written

explicitly as 31 2 11 1

( , , )

( , , )

( , , ) ( ) ( ) ( ) for bricks

( , , , ) ( ) ( ) ( ) ( ) for tetrahedron

nn na i j k i i i

a i j k I J K L

l l l

r s t u T r T s T t T u

ϕ ξ η ζ ξ η ζ

ϕ

++ +=

=

Interpolation of the Geometry and transformation of the integrand:

Accurate representation of domains with curved boundaries can be accomplished with the

aid of the higher order interpolation. For example, for a 3 noded 1D element the

geometry may be represented as:

1 1 2 2 3 3

1 1 2 2 3 3

( ) ( ) ( )( ) ( ) ( )

x x x xy y y y

ϕ ξ ϕ ξ ϕ ξϕ ξ ϕ ξ ϕ ξ

= + += + +

Jacobian of the transformation becomes 2 2 2 2 2 2 2

, ,

2 2, , 1D

or ( ) ( )

or J

ds dx dy ds x y d

ds x y d d

ξ ξ

ξ ξ

ξ

ξ ξ

⎡ ⎤= + = +⎣ ⎦

= + =

�

(x ,y )

(x ,y )

(x ,y )

1

2

3 3

2

1

1

2

3

1 2 3

x

y

�

Figure 7: A curved 3-noded 1D element

Similarly for 2D 9-noded elements the geometry may be represented as:

91 21 2 9

91 2

, ,

, ,

( , ) ( , ) ........ ( , )xx xxyy yy

x xdx dor

y ydy dξ η

ξ η

ϕ ξ η ϕ ξ η ϕ ξ η

ξη

⎧ ⎫⎧ ⎫ ⎧ ⎫⎧ ⎫= + + +⎨ ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬

⎩ ⎭ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭⎡ ⎤⎧ ⎫ ⎧ ⎫

=⎨ ⎬ ⎨ ⎬⎢ ⎥⎩ ⎭ ⎩ ⎭⎣ ⎦

Thus, if the integral is defined over the element domain (1D or 2D or 3D) may easily be

transformed and be represented over the mapped domain as: 1

1

1 1, ,

, ,1 1

( , ) ( , )

( , ) ( , )det

e

e

L

dsf ds f dd

x xf dxdy f d d

y yξ η

ξ η

ξ η ξ η ξξ

ξ η ξ η ξ η

−

− −Ω

=

⎡ ⎤= ⎢ ⎥

⎣ ⎦

∫ ∫

∫ ∫ ∫

Numerical Integration or Quadrature:

Exact and explicit evaluation of the integral associated to the element matrices and the

source/loading vector is not always possible because of the algebraic complexity of the

coefficient of the different equation (i.e., the stiffness influence coefficients, elasticity

matrix loadng functions). In such cases, it is convenient to seek for the numerical

evaluation of these integral expressions. Numerical evaluations of the coefficient are

some times very useful in problem involving constraints (i.e., Timoshenko beam). The

key idea of numerical evaluation of integral, called numerical integration or numerical

quadrature, rests on the polynomial expansion of the integrated upto a sufficient degree,

since the integral of a polynomial can be evaluated exactly. To make the numerical

quadrature easy with respect to the computer implementations, it is convenient to

represent the different types of quadrature into a similar format. To illustrate, let us

consider the following 1D integral 11 1 2 3 2 3

1 1

1 1 1

1 ..... 2.35 1 .....2! 3! 2! 3!

2 2 22.3504 2 ........ 2.3504 (upto 6th degree)6 120 5040

x x x x xe dx e e x dx x

or

−

− − −

⎛ ⎞ ⎛ ⎞= − ≅ + + + + ⇒ ≅ + + + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

≅ + + + + =

∫ ∫

Approximation integration with Newton-Coates: The :well known trapezoidal and the Simpson’s rule are the special cases of the more

generalized ‘Newton Coates’ formula. The integral is approximated as: 1

01

( ) 2n

ni i n

i

f x dx c f R=−

= +∑∫

Where nR is the reminder term and nic are the Newton Coates constant which are

tabulated as:

No of interval nic n

ic nic n

ic nic n

ic nic nR

Trapezoidal 1 1 1 1 3 ''10 2 ( )f x− ×

Simpsons 2 1/3 4/3 1/3 3 5 410 2 ( )f x− ×

3 1/4 3/4 3/4 1/4 3 5 410 2 ( )f x− ×

4 7/45 32/45 12/45 32/45 7/45 6 7 610 2 ( )f x− ×

Note: It may noted that formulas for n=3 and n=5 have the same order of accuracy as the

formulas for n=2 and n=4 respective. For this reason, the even order formulas (i.e., are

used in practice). The function is evaluated at the equal spaced intervals 1

1 11

1 0 1

1 11 13 3

2.718 0.368 3.086

1 4 1 2.718 4 0.3681 2.3623 3 3 3 3 31 3 3 14 4 4 4

0.368 3 3 2.7180.716 1.396 2.35534 4 4 4

x x x

x x

x x x

x x x

x x x x

x x x x

e dx e e

e e e

e e e e

= =−−

=− = =

=− =− = =

≅ + = + =

≅ × + × + × = + × + =

≅ × + × + × + ×

= + × + × + =

∫

The Gauss-Legendre Quadrature: In the Newton Coates quadrature the base point locations have been specified. If the

sampling locations are not specified then there will be 2r+2 undetermined parameters, the

weights iw and the base points ix which define a polynomial of degree 2rH. The Gauss-

legendre and the gauss quadrature are based on the same idea. The base points ix and the

weights iw are chosen so that the sum of the r+1 appropriately weighted value of the

function yields the integral exactly when f(x) is a polynomial of degree 2r+1 or less. The

weight and the sampling points/base points are given in the following table

One point 0

2 2 1 2x

xe

=× = × =

Two point 1 13 3

1.7813 0.56138 2.3427x x

x xe e

= − =+ = + =

4 point 0.86113 0.86113

0.339981 0.339981

0.347855 0.347855

0.652145 ( ) 2.3504

x x

x x

x x

x x

e e

e e= =−

= =−

× + × +

× + =

Figure 8: Gauss points for triangle

References K. J. Bathe, Finite element procedures in engineering analysis, Prentice-Hall Englewood Cliffs, NJ, 1982. O.C. Zeinkiewicz and R. Taylor, The finite element method: volume I; The Basis, Butterworth Heinemann, Oxford, 2000 R D Cook, D S Malkus, M E Plesha and R J Witt, Concepts and applications of finite element analysis, John Wiley and Sons, Inc, Singapore,2002 B Szabo and I Babuska, Finite Element Analysis, John Wiley and Sons, Inc, Canada, 1991 C A Fellippa, Intruduction to Finite Elements, <web page >

Table 1: Gauss-Legendra Quadrature rules

N Base points Weights N Base points Weights

3 0.7745966692414834 0.0000000000000000

0.5555555555555556 0.8888888888888889 7

0.9491079123427585 0.7415311855993944 0.4058451513773972 0.0000000000000000

0.12948496616886969 0.27970539148927667 0.38183005050511895 0.41795918367346939

4 0.8611363115940526 0.3399810435848563

0.3478548451374539 0.6521451548625461 8

0.9602898564975362 0.7966664774136267 0.5255324099163290 0.1834346424956498

0.10122853629037626 0.22238103445337447 0.31370664587788729 0.36268378337836198

5 0.9061798459386640 0.5384693101056831 0.0000000000000000

0.2369268850561891 0.4786286704993665 0.5688888888888889

9

0.9681602395076261 0.8360311073266358 0.6133714327005904 0.3242534234038089 0.0000000000000000

0.08127438836157441 0.18064816069485740 0.26061069640293546 0.31234707704000284 0.33023935500125976

6 0.9324695142031520 0.6612093864662645 0.2386191860831969

0.17132449237917034504 0.36076157304813860757 0.46791393457269104739

10

0.9739065285171717 0.8650633666889845 0.6794095682990244 0.4333953941292472 0.1488743389816312

0.06667134430868814 0.1494513491505806 0.2190863625159820 0.2692667193099964 0.2955242247147529

Table 2: Gauss-Lobatto Quadrature rules

N Base points Weights N Base points Weights

Table 3: Quadrature rules for triangle