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

Representation of surface changes The representation of surface changes has a direct effect on the number of design iterationsrequired, as well as the shape that may be obtained through

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 iterationsrequired, 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 thepossibility of ‘noisy designs’ due to a richer surface representation space

One can broadly differentiate two classes of surface change representation: direct andindirect In the first case, the design changes are directly computed at the nodes representingthe surface (line points, Bezier points, Hicks–Henne (Hicks and Henne (1978)) functions, aset 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 isthen defined on this set of patches, and subsequently translated to the CAD representation ofthe surface (see Figure 20.9)

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 buildingprocess During the optimization process, more and more information is gained about thedesign space and the consequences design changes have on the objective function Considerthe case of a wing for a commercial airplane At the start, only global objectives, measures andconstraints 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 determinethe lift and drag A neural net or a lifting line theory yield sufficient flow-related informationfor these global objectives, measures and constraints As the design matures, the informationshifts to more local measures: thickness, chamber, twist angle, local wing stiffness, etc Theflowfield prediction needs to be upgraded to either lifting line theory, potential flow or Eulersolvers During the final stages, the information reaches the highest precision It is here thatRANS or LES solvers need to be employed for the flow analysis/prediction This simple wingdesign 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 473Hierarchical in this context means that the addition of further degrees of freedom does notaffect in a major way the preceding ones A typical case of a hierarchical representation isthe Fourier series for the approximation of a function The addition of further terms in theseries does not affect the previous ones Due to the nonlinearity of the physics, such perfectorthogonality is difficult to achieve for the components of optimal shape design (designvariables, physical description, discretization used).

In order to carry out a hierarchical design, all the components required for optimal shapedesign: design variables, physical description and discretization used must be organizedhierarchically

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 fidelcomplex-ity 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 fidelityincrease 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, coarsergrids 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 donot carry any significant loads (i.e are stress-free), until the minimum weight under stressconstraints is reached It so happens that one of the most common design objectives instructural mechanics is the minimization of weight, making topological design an attractiveprocedure 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 residencetime 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 costfunction(s) to a small change in volume For classic optimal shape design, applying thisderivative 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

474 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES

an increase in the porosity π in the flowfield:

Note that no flow will occur in regions of large porosities π If we consider the porosity as

the design variable, we have, from (20.35), after solution for the adjoint,

Observe, however, that analytically and to first order,

If we furthermore assume that the objective function involves boundary integrals (lift, drag,

total loss between in- and outlets, etc.) then I ,π= 0 and we have

This implies that if we desire to remove the volume regions that will have the least (negative)effect on the objective function, we should proceed from those regions where either theadjoint velocity vanishes or where the velocity vanishes If the adjoint is unavailable,then we can only proceed with the second alternative This simplified explanation at leastmakes plausible the fact that the strategy of removing from the flowfield the regions ofvanishing velocity by increasing the porosity there can work Indeed, a much more detailed

mathematical analysis (Hassine et al (2004)) reveals that this is indeed the case for the Stokes limit (Re→ 0)

20.10 Examples

20.10.1 DAMAGE ASSESSMENT FOR CONTAMINANT RELEASE

The intentional or unintentional release of hazardous materials can lead to devastatingconsequences Assuming that the amount of contaminant is finite and that the populationdensity in the region of interest is given, for any given meteorological condition the location

of the release becomes the main input variable Damage as a function of release locationcan have many local extrema, as pockets of high concentration can linger in recirculationzones or diffuse slowly while being transported along street canyons For this reason, geneticalgorithms offer a viable optimization tool for this process design The present example, takenfrom Camelli and Löhner (2004), considers the release in an area representative of an innercity composed of three by two blocks The geometry definition and the surface mesh areshown in Figure 20.10(a) Each of the fitness/damage evaluations (i.e dispersion simulationruns) took approximately 80 minutes using a PC with Intel P4 chip running at 2.53 GHz with

1 Gbyte RAM, Linux OS and Intel compiler

Three areas of release were studied (see Figure 20.10(b)): the upwind zone of the complex

of buildings, the street level and the core of one of the blocks In all cases, the height of

release was set to z = 1.5 m The genetic optimization was carried out for 20 generations,

with two chromosomes (x/y location of release point) and 10 individuals in the population.

The location and associated damage function for each of the releases computed duringthe optimization process are shown in Figures 20.11(a)–(d) Interestingly, the maximum

OPTIMAL SHAPE AND PROCESS DESIGN 475

100m 50m

20m

Wind Direction

Source Location

(a)

UPWIND ZONE ZONE

CORE

ZONE STREET

(b) Figure 20.10 (a) Problem definition; (b) zones considered for release;

damage is produced in the street area close to one of the side corners of the blocks.The cloud (Figures 20.11(e)–(i)) shows the suction effect produced by the streets that are

in the direction of the wind, allowing for a very long residence time of contaminants close tothe ground for this particular release location

20.10.2 EXTERNAL NOZZLE

This example, taken from Soto et al (2004), considers an external nozzle, typical of those

envisioned for the X-34 airplane There are no constraints on the shape The objective is tomaximize the thrust of the nozzle The flow conditions are as follows:

- inflow (external flowfield): Ma = 3.00, ρ = 0.5, v = 1.944, pr = 0.15, α = 0.0◦;

- nozzle exit: Ma = 1.01, ρ = 1.0, v = 1.000, pr = 0.18, α = −45.0◦.

476 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES

OPTIMAL SHAPE AND PROCESS DESIGN 477Although this is in principle a 2-D problem, the case was run in three dimensions The meshhad approximately 51 000 points and 267 000 elements The total number of design variableswas 918 Thus, the only viable choice was given by the adjoint methodology outlined above.Figures 20.12(a)–(f) show the initial and final mach numbers and pressures, as well as theevolution of the shape and the thrust.

0.004 0.0042 0.0044 0.0046 0.0048 0.005

Design Iteration Thrust

Figure 20.12 External nozzle: Mach numbers for (a) first and (b) last shape; pressures for (c) first and

(d) last shape; evolution of (e) shape and (f) thrust

20.10.3 WIGLEY HULL

This example, taken from Yang and Löhner (2004), shows the use of an indirect surfacerepresentation as well as the finite difference evaluation of gradients via reduced complexitymodels The geometry considered is a Wigley hull The hull length and displacement arefixed while the wave drag is minimized during the optimization process The water line

is represented by a six-point B-spline Four of the six spline points are allowed to changeand they are chosen as design variables The hull surface, given by a triangulation, is thenmodified according to these (few) variables The gradients of the objective function with

478 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES

-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

(a)

(b) Figure 20.13 Wigley hull: (a) comparison of frame lines (solid, original; dashed, optimized); (b) com-

parison of wave patterns generated (left, original; right, optimized); (c) comparison of wave profiles(solid, original; dashed, optimized)

OPTIMAL SHAPE AND PROCESS DESIGN 479

-0.01 -0.005 0 0.005 0.01 0.015

X-Coordinates

original optimized-H2O_3

(c) Figure 20.13 Continued

Figure 20.14 KCS: (a) surface mesh; (b) wave pattern; pressure contours of (c) the original hull; and

(d) the modified hull

480 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUESrespect to the design variables are obtained via finite differences using a fast potential flowsolver Whenever the design variables are updated, the cost function is re-evaluated using anEuler solver with free surface fitting Given that the potential flow solver is two orders ofmagnitude faster than the Euler solver, the evaluation of gradients via finite differences doesnot add a significant computational cost to the overall design process Figures 20.13(a)–(c)compare the frame lines and wave patterns generated for the original and optimized hulls at

a Froude number of Fr = 0.289 While the wetted surface and displacement remain almost

unchanged, more than 50% wave drag reduction is achieved with the optimized hull Theoptimized hull has a reduced displacement in both the bow and stern regions and an increaseddisplacement in the middle of the hull

20.10.4 KRISO CONTAINER SHIP (KCS)

This example, taken from Löhner et al (2003), considers a modern container ship with bulb

bow and stern The objective is to modify the shape of the bulb bow in order to reduce

the wave drag The Froude number was set to Fr = 0.25, and no constraints were imposed

on the shape The volume mesh had approximately 100 000 points and 500 000 tetrahedra,and the free surface had approximately 10 000 points and 20 000 triangles The number ofdesign variables was in excess of 200, making the adjoint approach the only one viable.Figure 20.14(a) shows the surface mesh in the bulb region for the original and final designsobtained Figure 20.14(b) shows the comparison of wave patterns generated by the originaland final hull forms The wave drag reduction was of the order of 4% Figures 20.14(c) and(d) show the pressure contours on the original and modified hulls

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