Hermite interpolation functions

As I recall, J. N. Reddy's book(introduction to finite element method) has the derivation for hermite polynomials for beam elements

The book by Michael Gosz (finite element method) also shows the derivation roughly as follows:

Outline of the procedure (for hermite polynomials for beam element)

1. The lowest order polynomial that assures continuity of both v and v,x is

v=a+bX+cX^2+dX^3  (a cubic polynomial, a, b, c, d are constants to be determined)

2.   To get the hermite shape functions it is necessary to solve for the constants in terms of the nodal quantities.  Ie, boundary conditions for the beam element, they are

v(X=0)=v1

v,x(X=0)=theta1

v(X=L)=v2

v,x(X=L)=theta2

,where L is the length of the beam element.

The above boundary conditons applied to the cubic polynomial give 4 equations and four unknowns (a,b,c,d)

3.  It is possible to solve for the 4 unknowns algebraically and substitute the result for each back into the cubic equation.

4.  Then by algebra the cubic equation can be expressed as

v=N1(X)v1+N2(X)theta1+N3(X)v2+N4(X)theta2

5.  The coefficients(or shape functions) Ni(X) are the C1 continuous hermite polynomials for a beam element, they are

N1=1-3q^2+2q^3

N2=Lq(1-2q+q^2)

N3=q^2(3-2q)

N4=Lq^2(q-1)

where q=X/L

 I Hope that helps.

regards,

 Louie

Not sure if this is what you are asking but Hermite polynomials are the eigenfunctions of a Sturm-Liouvilleboundary value problem. Any good book on applied mathematics should have a discussion.

Dear Ramdas

I did some work on this and published a paper in Computers and Structures in 1998. Looking back at this I referenced three books: Jacques & Judd "Numerical Analysis"; Morris "Computational methods in elementary numerical analysis" and Spanier and Oldham "An atlas of functions" . Now I do not have these to hand but they may contain the origin of the expressions linking Lagrange to Hermite functions.

Regards

Charles