Applied Computational Fluid Dynamics Techniques - Wiley Episode 1 Part 6 ppt

112 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES

4.1.3. LEAST-SQUARES FORMULATION

If we consider the least-squares problem

Ils =

(h)

2 d =

(uh − u)2 d =

(Nkak − u)2 d → min, (4.15)

then it is known from the calculus of variations that the functional Ils is minimized for

δIls = δak

Nk(Nl

al − u) d = 0, (4.16)

i.e. for each variable ak

NkNl d al =

Nku d . (4.17)

But this is the same as the Galerkin WRM! This implies that, of all possible choices for Wi

,

the Galerkin choice Wi = Ni yields the best results for the least-squares norm Ils. For other

norms, other choices of Wi will be optimal. However, for the approximation problem, the

norm given by Ils seems the natural one (try to produce a counterexample).

4.2. Choice of trial functions

So far, we have dealt with general, global trial functions. Examples of this type of function

family, or expansions, were the Fourier (sin-, cos-) and Legendre polynomials. For general

geometries and applications, however, these functions suffer from the following drawbacks.

(a) Determining an appropriate set of trial functions is difficult for all but the simplest

geometries in two and three dimensions.

(b) The resulting matrix K is full.

(c) The matrix K can become ill-conditioned, even for simple problems. A way around this

problem is the use of strongly orthogonal polynomials. As a matter of fact, most of the

global expansions used in engineering practice (Fourier, Legendre, etc.) are strongly

orthogonal.

(d) The resulting coefficients aj have no physical significance.

The way to circumvent all of these difficulties is to go from global trial functions to local

trial functions. The domain on which u(x) is to be approximated is subdivided into a set

of non-overlapping sub-domains el called elements. The approximation function uh is then

defined in each sub-domain separately. The situation is shown in Figure 4.3.

In what follows, we consider several possible choices for local trial functions.

4.2.1. CONSTANT TRIAL FUNCTIONS IN ONE DIMENSION

Consider the piecewise constant function, shown in Figure 4.4:

P E =

1 in element E,

0 in all other elements. (4.18)

APPROXIMATION THEORY 113

￾￾￾￾￾￾￾

￾￾￾￾￾￾￾

￾￾￾￾￾￾￾

￾￾￾￾￾￾￾

￾￾￾￾￾￾￾

￾￾￾￾￾￾￾

￾￾￾￾￾￾￾

￾￾￾￾￾￾￾

node

element

Figure 4.3. Subdivision of a domain into elements

Then, globally, we have

u ≈ uh = P EuE, (4.19)

and locally, on each element el,

u ≈ uh = uel. (4.20)

u

x

u N

1

x x x 1 [

P

1 2

e

Figure 4.4. Piecewise constant trial function in one dimension

4.2.2. LINEAR TRIAL FUNCTIONS IN ONE DIMENSION

A better approximation is obtained by letting uh vary linearly in each element. This is

accomplished by placing nodes