Subroutine SHAPE

This subroutine calculates the values of shape functions N_i  and their derivatives at the given value of natural coordinate for the given order (p) of approximation. Since is a natural coordinate its value lies between -1 to 1. Further, we restrict the value of p to only 1, 2 and 3. Note that when the order of approximation is p , the number of nodes per element is p +1. This is also the number of degrees of freedom per element or the number of shape functions. We use the following notation for the variables.

• Scalar Variables:

• Array Variables :

dshap (ndofel) : values of the shape function derivatives at given
psin (ndofel) : natural coordinates of nodes (real)
shap (ndofel) : shape function values at given x

• INPUT Variables : p, psi

• OUTPUT Variables : ndofel, shap (ndofel), dshap (ndofel)

Now, the subroutine SHAPE can be written as follows.

Subroutine SHAPE (p, psi, ndofel, shap, dshap)

ndofel = p+1

do i =1, ndofel

psin( i)  = -1 + (i-1) * 2/p

enddo

If p = 1, then

shap (1) = (psi - psin(2))/(psin(1) - psin(2))

shap( 2) = (psi - psin(1)) / (psin(2) - psin(1))

dshap (1) = 1 / (psin (1) - p sin(2))

dshap (2) =1/(psin(2) - psin(1))

end if

If p = 2, then

shap (1) = (psi -psin(2)) * (psi - psin(3))/((psin(1)-psin(2)) * (psin(1) - psin(3)))

shap (2) = (psi -psin(3)) * (psi - psin(1))/((psin(2)-psin(3)) * (psin(2) - psin(1)))

shap (3) = (psi -psin(1)) * (psi - psin(2))/((psin(3)-psin(1)) * (psin(3) - psin(2)))

dshap (1) = (psi -psin(3))+(psi - psin(2))/((psin(1)-psin(2)) * (psin(1) - psin(3)))

dshap (2) = (psi -psin(1))+(psi - psin(3))/((psin(2)-psin(3)) * (psin(2) - psin(1)))

dshap (3) = (psi -psin(2))+ (psi - psin(1))/((psin(3)-psin(1)) * (psin(3) - psin(2)))

end if

If p = 3, then

deno 1 = (psin(1) - psin(2)) * (psin(1) - psin(3)) * (psin(1) - psin(4)

deno 2 = (psin(2) - psin(3)) * (psin(2) - psin(4)) * (psin(2) - psin(1)

deno 3 = (psin(3) - psin(4)) * (psin(3) - psin(1)) * (psin(3) - psin(2)

deno 4 = (psin(4) - psin(1)) * (psin(4) - psin(2)) * (psin(4) - psin(3)

shap (1) = (psi -psin(2)) * (psi - psin(3))*(psi-psin(4)) /deno 1

shap (2) = (psi -psin(3)) * (psi - psin(4))*(psi-psin(1)) /deno 2

shap (3) = (psi -psin(4)) * (psi - psin(1))*(psi-psin(2)) /deno 3

shap (4) = (psi -psin(1)) * (psi - psin(2))*(psi-psin(3)) /deno 4

dshap (1) = ((psi -psin(3))*(psi - psin(4))+(psi-psin(2))*(psi-psin(4))
                    +(psi-psin(2))*(psi-psin(3)))/deno 1

dshap (2) = ((psi -psin(4))*(psi - psin(1))+(psi-psin(3))*(psi-psin(1))
                    +(psi-psin(3))*(psi-psin(4)))/deno 2

dshap (3) = ((psi -psin(1))*(psi - psin(2))+(psi-psin(4))*(psi-psin(2))
                    +(psi-psin(4))*(psi-psin(1)))/deno 3

dshap (4) = ((psi -psin(2))*(psi - psin(3))+(psi-psin(1))*(psi-psin(3))
                    +(psi-psin(1))*(psi-psin(2)))/deno 4

end if

Note that the following things should be kept in mind while writing the computer codes :

• Meaningful symbols should be used to denote the variables. For example ndofel for denoting the number of degrees of freedom per element

• A group of statements should be indented to identify various steps and sub steps.

• A generous amount of comment statements should be inserted to provide explanation for the following group of statements.