Page 105 - Process Modelling and Simulation With Finite Element Methods

92 Process Modelling and Simulation with Finite Element Methods

where u, are the unknown coefficients. The L,(x) is given by
n x-xM
(2.59a)
M=1,M#N xN - xM

The expansion generates the polynomials of desired order. Lagrange elements
are the most commonly used type in CFD. They provide the value of the
variable at nodes.
Hennite elements use the Hennite polynomials to interpolate the values of
the field variables. The main difference between Lagrange and Hermite elements
is the degrees of freedom (DOF) available. In the case of Lagrange elements
DOF are the values of the function at nodes (This consists values of all variables
at the node). However in Hermite elements other than the function values at
nodes, the first derivatives of the variables at corner points are available. Again,
suppose a variable u(x) to be determine over 1-D elements. Since the values at n
I li
th and n+lth nodes, u(xn) and U(X,+~) and the derivatives du dx , and
du/dxli+l are known a polynomial of four unknowns should be used to
approximate u(x).
u(x) = a, + a2x + a3x2 + a,x 3 (2.60)

Since xi and xi+! are known positions, one can easily write four equations: two
with function values at nodes and two with first derivatives at nodes.

where du denotes the derivative of u w.r.t. x at the nodes. By matrix inversion
ancan be expressed in terms of nodal values and values of derivatives. The
resulting equation is

where qi (x) are the Hermite interpolation functions (cubic functions in this
case, also known as cubic splines).