Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 129

Equation (5.59) represents the continuous equation of the Upper Differencing
scheme where the first term,

φ is the convective transport of ϕ and the remain-
ing terms resembles a diffusion term in the convection-diffusion equation, despite
the initial equation being pure convection. The discretisation results in a solution
that produces artificial diffusion from the numerical scheme and not real diffusion
from molecular viscosity. This is most problematic in flows that exhibit small Peclet
numbers. However the solution improves and false diffusion reduces for large Peclet
numbers in Upper Differencing.
The Central Differencing also suffers from false diffusion but to a lesser extent
at low Pe numbers. At higher Pe numbers Central Differencing produces unrealistic
results while using the QUICK scheme produces undershoots and overshoots. To
overcome the false diffusion problem, the following can be performed: refining the
mesh; aligning grid lines more in the direction of flow; and including more neigh-
bour nodes in the discretisaiton.

d

u

dx

5.4.4 Finite Element Method (FEM)

Although the finite element method can be used to discretise the fluid flow equa-
tions, its use for structural analysis is more widespread. This method considers the
solution region comprised of smaller, interconnected elements. Each element ap-
proximates the governing equations by applying interpolation functions that de-
scribe its behaviour between nodal points. The continuum problem which has an
infinite number of unknowns is reduced to one with a finite number of unknowns at
specified locations called nodes .
Each element produces a matrix relation called the stiffness matrix, k that relates
the applied force to its displacement. The local stiffness matrix corresponds to an
individual element while the global stiffness matrix defines the stiffness of the en-
tire domain from assembling of all local stiffness matrices. In this introduction we
present the mathematical steps leading to the local stiffness matrix.
The main steps involved with FEM are summarized below:

I� Discretisation —A physical domain is discretized by dividing the region into
non-overlapping elements or sub-regions defined by nodes, and linked together
through a shape function. The elements are formed by joining its nodal points
which are also shared with neighbouring connected elements.
II� Interpolation —Shape functions are used to interpolate field variables across
the element from node to node, and are typically denoted by the letter N which
appear as coefficients in the interpolation polynomial. A shape function is writ-
ten for each individual node and has the property that its magnitude is 1 at that
node and 0 for all other nodes in that element
III� Assembling the equations— The nodal values of the unknown field variable are
related to other parameters to establish a matrix equation. The element equa-
tions are then assembled together to form a global equation system for the