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Hermite interpolation functions

ramdas chennamsetti's picture

Hi all!!!

In Finite Element Method (FEM), Hermite interpolation functions are used for interpolation of dependent variable and its derivative.

In FEM books, Hermite interpolation functions are directly written in terms of Lagrange interpolation functions. No derivations are given. I searched in Numerical methods books also for derivation of Hermite interpolation functions. I couldn't find.

I am looking for the origin (basically the derivation) of Hermite interpolation functions. Kindly help me.

Thanx in advance and regards,

- Ramdas

Comments

yawlou's picture

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.

ramdas chennamsetti's picture

Hi Louie,

Thank you. I am not looking for this.

I call this as a crude way of deriving interpolation functions. If you see some FE books (for ex. Asghar Bhatti's book on Advanced FE), direct expression for Hermite intepolation functions (interms of Lagrange interpolation functions) is available. I am looking for the origin of those expressions.

Hi Peter,

Could you please throw some light on "Sturm-Liouvilleboundary value problem"

Thanx and regards,

- Ramdas

Charles Augarde's picture

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

ramdas chennamsetti's picture

Dear Charles,

Thank you. I will search for these books in IIT Madras library.

Regards,

- Ramdas

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