Hermite Interpolations

These definitions result in the Hermite s interpolation formula,... 

Hermite interpolation for the velocity and velocity gradient of a power law fluid in a narow slit. Consider the analytical expressions for the velocity and the velocity gradient for the flow of an incompressible power law fluid through a narrow slit due to apressure gradient to represent velocity measurements. With eqns. (7.17) and (7.18), the velocity and its gradient are evaluated in 10 equally spaced points through the thickness. An expression to interpolate both, the velocity and its gradient, is required for a point i + 1/2 located between point i and + 1. 

In order to use Hermite interpolation, we must first chosse the order for the interpolation of hi(x) and ht(x). For simplicity, let s use a first order interpolation, n = 2, for the Lagrange polynomials involved in these two terms. Using Hermite interpolation formula (eqn. (7.31)) and eqns. (7.32) and (7.33) we obtain... 

Therefore, one has recourse to other interpolation polynomials associated with the names of Lagrange, Newton, Stirling, Hermite, etc. Let us give the following formulae, for equally spaced points [136]. 

S. D. Capper and D. R. Moore, On high orderMIRK schemes and Hermite-Birkhoff interpolants, JNAIAM J. Numer. Anal. Indust. Appl. Math., 2006, 1(1), 27 7. 

This is to evaluate the limit curve points corresponding to the control points, using the row eigenvector of unit eigenvalue, and also the first derivatives, using the next row eigenvector. These are then used to make a Hermite cubic interpolant, which converges as the fourth power of the number of refinements. 

Each of the methods described under rendering above can be applied directly to evaluation. The third, Hermite, form is probably most relevant to applications requiring high accuracy. In fact where the second derivative can also be evaluated exactly at dyadic points, a quintic Hermite interpolant can be used to give an even higher rate of approximation. 

J.-L.Merrien A family of Hermite interpolants by bisection algorithms. Numerical Algorithms 2, ppl87-200, 1992... 

B. Jiittler, U.Schwaneke Analysis and Design of Hermite subdivision schemes. The Visual Computer 18, pp326-342, 2002 M.F.Hassan, I.P.Ivrissimtzis, N.A.Dodgson and M.A.Sabin An interpolating 4-point C2 ternary stationary subdivision scheme. CAGD 19(1), ppl-18, 2002... 

L.Romani A circle-preserving C2 Hermite interpolatory subdivision scheme with tension control. CAGD 27(1), pp36-47, 2010 C.Deng and G.Wang Incenter subdivision scheme for curve interpolation. CAGD 27(1), pp48-59, 2010... 

Up to now the basis functions Ni x) are still arbitrary and not restricted to a finite element approach. In the finite element frame a suitable approximation is given by Lagrange- or Hermite-interpolation polynomials. 

In order to be able to combine the working equations of 18 and 20, first order derivatives of pressure (i.e., pressure gradients) should be directly calculated as independent degrees of freedom in the numerical scheme. This is only possible if Hermite elements which incorporate the first order derivatives of interpolated functions as the nodal unknowns are used. 

An essential feature of the grid adaptation procedure is its smooth behavior. After each integration step some nodes are deleted and inserted, but there is no generation of a totally new mesh. Therefore the problem of indroducing errors from interpolation is drastically reduced. We use the piecewise cubic monotone Hermite interpolation due to [6] to get solution approximations at new nodes. In addition to the space discretization error the interpolation error is estimated and controlled also. 

At least the graphical solution display requires a cheap solution representation not only at the integration points and the computational grid. Our global solution representation and evaluation is done by means of two quite different Hermite interpolation variants. We found that doing first the interpolation in time then the interpolation in space gives better results than vice versa. 

Interpolation in Space. The monotone Hermite interpolation due to [6] is used also to interpolate for output purposes. 

Low-Order Hermite Interpolation in Direct Dynamics Calculations of Vibrational Energies Using the Code MULTIMODE . 

We have interpolated a function based on its values at die support points however, we may wish to include as well information about the leading order derivatives at some or aU of the support points. In Hermite interpolation, we find the polynomial / (x) of degree N that satisfies tiie following AT -I-1 conditions at the iff - -1 points xo < xi < < xm. 

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