Applied Computational Fluid Dynamics Techniques - Wiley Episode 2 Part 10 doc

472 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES

20.7. Representation of surface changes

The representation of surface changes has a direct effect on the number of design iterations

required, as well as the shape that may be obtained through optimal shape design. In general,

the number of design iterations increases with the number of design variables, as does the

possibility of ‘noisy designs’ due to a richer surface representation space.

One can broadly differentiate two classes of surface change representation: direct and

indirect. In the first case, the design changes are directly computed at the nodes representing

the surface (line points, Bezier points, Hicks–Henne (Hicks and Henne (1978)) functions, a

set of known airfoils/wings/hulls, discrete surface points, etc.). In the second case, a (coarse)

set of patches is superimposed on the surface (or embedded in space). The surface change is

then defined on this set of patches, and subsequently translated to the CAD representation of

the surface (see Figure 20.9).

CFD Domain Surface New CFD Domain Surface

Movement Surface Deformation of Movement Surface

Figure 20.9. Indirect surface change representation

20.8. Hierarchical design procedures

Optimal shape design (and optimization in general) may be viewed as an information building

process. During the optimization process, more and more information is gained about the

design space and the consequences design changes have on the objective function. Consider

the case of a wing for a commercial airplane. At the start, only global objectives, measures and

constraints are meaningful: take-off weight, fuel capacity, overall measures, sweep angle, etc.

(Raymer (1999)). At this stage, it would be foolish to use a RANS or LES solver to determine

the lift and drag. A neural net or a lifting line theory yield sufficient flow-related information

for these global objectives, measures and constraints. As the design matures, the information

shifts to more local measures: thickness, chamber, twist angle, local wing stiffness, etc. The

flowfield prediction needs to be upgraded to either lifting line theory, potential flow or Euler

solvers. During the final stages, the information reaches the highest precision. It is here that

RANS or LES solvers need to be employed for the flow analysis/prediction. This simple wing

design case illustrates the basic principle of hierarchical design. Summarizing, the key idea

of hierarchical design procedures is to match the available information of the design space

to:

- the number of design variables;

- the sophistication of the physical description; and

- the discretization used.

OPTIMAL SHAPE AND PROCESS DESIGN 473

Hierarchical in this context means that the addition of further degrees of freedom does not

affect in a major way the preceding ones. A typical case of a hierarchical representation is

the Fourier series for the approximation of a function. The addition of further terms in the

series does not affect the previous ones. Due to the nonlinearity of the physics, such perfect

orthogonality is difficult to achieve for the components of optimal shape design (design

variables, physical description, discretization used).

In order to carry out a hierarchical design, all the components required for optimal shape

design: design variables, physical description and discretization used must be organized

hierarchically.

The number of design variables has to increase from a few to possibly many (>103), and

in such a way that the addition of more degrees of freedom do not affect the previous ones.

A very impressive demonstration of this concept was shown by Marco and Beux (1993) and

Kuruvila et al. (1995). The hierarchical storage of analytical or discrete surface data has been

studied in Popovic and Hoppe (1997).

The physical representation is perhaps the easiest to organize. For the flowfield, complex￾ity and fidelity increase in the following order: lifting line, potential, potential with boundary

layer, Euler, Euler with boundary layer, RANS, LES/DNS, etc. (Alexandrov et al. (2000),

Peri and Campana (2003), Yang and Löhner (2004)). For the structure, complexity and fidelity

increase in the following order: beam theory, shells and beams, solids.

The discretization used is changed from coarse to fine for each one of the physical

representations chosen as the design matures. During the initial design iterations, coarser

grids and/or simplified gradient evaluations (Dadone and Grossman (2003), Peri and Cam￾pana (2003)) can be employed to obtain trends. As the design matures for a given physical

representation, the grids are progressively refined (Kuruvila et al. (1995), Dadone and

Grossman (2000), Dadone et al. (2000), Dadone (2003)).

20.9. Topological optimization via porosities

A formal way of translating the considerable theoretical and empirical legacy of topological

optimization found in structural mechanics (Bendsoe and Kikuchi (1988), Jakiela et al.

(2000), Kicinger et al. (2005)) is via the concept of porosities. Let us recall the basic

topological design procedure employed in structural dynamics: starting from a ‘design space’

or ‘design volume’ and a set of applied loads, remove the parts (regions, volumes) that do

not carry any significant loads (i.e. are stress-free), until the minimum weight under stress

constraints is reached. It so happens that one of the most common design objectives in

structural mechanics is the minimization of weight, making topological design an attractive

procedure for preliminary design. Similar ideas have been put forward for fluid dynamics

(Borrvall and Peterson (2003), Moos et al. (2004), Hassine et al. (2004), Guest and Prévost

(2006), Othmer et al. (2006)) as well as heat transfer (Hassine et al. (2004)). The idea is to

remove, from the flowfield, regions where the velocity is very low, or where the residence

time of particles is considerable (recirculation zones). The mathematical foundation of all

of these methods is the so-called topological derivative, which relates the change in the cost

function(s) to a small change in volume. For classic optimal shape design, applying this

derivative at the boundary yields the desired gradient with respect to the design variables.

The removal of available volume or ‘design space’ from the flowfield can be re-interpreted as