Technical NotesPhysics- and Engineering

Knowledge-Based Repair of

Computer-Aided Design

Parametric Geometries

Dong Li,∗ András Sóbester,† and Andy J. Keane‡

University of Southampton,

Hampshire, England SO17 1BJ, United Kingdom

DOI: 10.2514/1.J050761

I. Introduction

B ECAUSE of the highly global nature of conceptual designsearch, the geometry engine in the optimization framework

should be able to deliver a variety of different geometries defined by awide range of design variable configurations without difficulty, i.e.,the geometry engine should be flexible as well as robust. However, aflawless coverage of the design space is very difficult to realize. Anexpediential measure for ensuring robustness is to place sufficientlytight bound limits on design variables so that any combination of thevariables in the trimmed design space leads to a feasible design.However, infeasible regions for complex geometry models are oftenirregular and hard to avoid. The preceding measure will then eitherleave a very limited design space to be explored or leave someinfeasible region(s) remaining in the design space, which will makethe model-generation process fail from time to time. So far, there isrelatively little work that has addressed this problem.

Following on from Sóbester and Keane [1], we propose anautomatic geometry repair system to handle this problem. Thesystem aims to repair geometrically or physically flawed geometriesbased on an engineering knowledge base and, therefore, to assist thegeometry engine to generate robust models without limiting itsflexibility. The system aims to reduce reliance on human designexperts in the conceptual design phase and to improve the stability ofthe optimization cycle. It also helps speed up the design process byreducing the time and computational power that could be wasted onflawed geometries or frequent human interventions. The prototypesystem aims to provide the following capabilities: capturing andstoring the knowledge of a design engineer as well as physics-basedempirical data; synthesizing the knowledge into a general knowledgebase; deploying the knowledge automatically to recommend arepaired geometry alternative when and as required; and producinginferences that the human expert may not be able to devise in areasonable amount of time. In a nutshell, the system aims to be anefficiency improvement tool in automated design optimization.

We next discuss the basis of the proposed knowledge-based repairsystem; this is followed by the descriptions of results obtained on asimple application: a two-dimensional (2-D) intake duct design.

II. Methodology

A. Knowledge Representation

The knowledge of the feasibility of a geometrical model can bedrawn from various sources, such as 1) explicit rules, extracted fromengineering or geometrical judgment. These rules could be equalityor inequality constraints or requirements on certain geometricparameters; 2) physical properties of the model, discovered by com-putational analysis results [computational fluid dynamics (CFD),finite element analysis, etc.] or engineering laws; or 3) assessment ofindividual design cases by an expert engineer.

There are no universally applicable schemes for incorporatingsuch heterogeneous engineering knowledge into a design system. Inthis work, we first show the possibility of converting knowledge intoexplicit penalty functions and incorporate this knowledge intosurrogate models by constructing four penalty functions [2] resultingfrom practical considerations and physics-based analysis. Thepenalty functions take a set of designvariables as input and generate anumerical penalty value for this set as output. The resulting data arecollected, stored, and used to train a surrogate. As a result, thesurrogate learns from the penalty functions, which are repre-sentations of engineering knowledge.

When knowledge is drawn from the physical properties of thegeometric model in question (for example, the aerodynamics or themechanics), these can be calculated from engineering laws. Morecommonly in optimization design practice, they are predicted bynumerical analysis codes. These codes can be viewed as black boxfunctions, whose input (parametric geometries defined by designvariables) and output (the physical properties in question) are known.Once the data have been collected, they can be used in the same waythe data from penalty functions were used to train the surrogate.

Furthermore, knowledge of specific designs can be captureddirectly from case-by-case assessment by engineers. Experiencedengineers can exercise their own judgement and score a design basedon its feasibility, for example, from 0 to 10. The score and itscorresponding design variable set can be stored and used as trainingdata for the surrogate. This processwas demonstrated in Sóbester andKeane [1].

B. Surrogate Modelling

After the engineering knowledge has been translated into mathe-matical equations or numerical mappings, various surrogatemodelling techniques can be deployed to set up a meta-model ofthe knowledge base, which can be used for storing knowledge in aconcise format, judging new geometries, and providing guidance onrepairing faulty geometries. Here, we make use of a surrogatemodelling technique, support vector regression (SVR) [3–5].

C. Geometry Repair

After the knowledge surrogate is properly trained with a set oftraining data, typically obtained by evaluating the feasibility-relatedpenalty functions at a set of designs included in a sampling plan, thesurrogate can be used to predict the feasibility of unknowngeometries by giving a numerical value, which can be regarded as apredictor of the feasibility of the unknown geometry. To draw a linebetween feasible and infeasible designs, a feasibility threshold pth isdefined. The unknown geometries that have a value higher than thethreshold will be regarded as infeasible. Then, when an infeasiblegeometry is detected by the surrogate, the repair process identifies the

Presented as Paper 2010-1508 at the 48th AIAA Aerospace SciencesMeeting, Orlando, FL, 4–7 January 2010; received 9 August 2010; revisionreceived 15 September 2011; accepted for publication 6 December 2011.Copyright © 2011 by the authors. Published by the American Institute ofAeronautics andAstronautics, Inc., with permission. Copies of this papermaybe made for personal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 0001-1452/12 and $10.00 incorrespondence with the CCC.

∗Postgraduate Research Student, Computational Engineering and DesignGroup, Faculty of Engineering and the Environment; d.li@soton.ac.uk.Student Member AIAA (Corresponding Author).

†Lecturer, Computational Engineering and Design Group, Faculty ofEngineering and the Environment. Member AIAA.

‡Professor, Faculty of Engineering and the Environment.

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design alternative with the smallest possible repair alteration(SPRA). To be precise, the SPRA is the set of design variableincrements that will make the design feasible while keeping changesto a minimum, thus retaining the original design intent to the greatestpossible extent.

In essence, the search for the SPRA is a single-objectiveconstrained optimization problem:

minimize SPRA�Xni�1

����������������������xi � xri �2

qsubject to f̂�x� � pth (1)

where xi is the ith design variable of the unknown geometry x, xri is

the ith design variable of the proposed repair alternative xr, and f̂�x�is the feasibility predictor of x, given by the surrogate f̂.

Avariable resolution evolutionary operation (EVOP) approach [6]has been implemented to find the SPRA. The optimization processstarts with an initial global search, which covers the design space.Further local searches are then repetitively performed within thehypersphere, which is centered on the original design point with aradius equal to the distance between nearest feasible design found sofar. In each repetition, or “generation” in EVOP terminology, a seriesof offspring are obtained by using the full factorial samplingtechnique. If a whole generation fails to produce a better designalternative, the sampling densitywill be increased to allow for amoreintense search. After a successful round of searching, the designvariable sets that fulfil the quality constraint are listed; the one that isclosest to the original design is picked and used as the benchmark forthe next round of searching. The search terminates when either theoptimized design reaches a predetermined penalty value belowwhich the design is deemed satisfactory or the limit of the computingbudget is reached.

III. Case Study: A Two-DimensionalAeroengine Intake Design

A. Model Setup

In this section, we use a simple 2-D parameterized inlet model as atest case, as shown in Fig. 1. The external shape of the airframe, theposition of the engine, and rear bulkhead are fixed. The intake shapeis defined by its center axis (the dashed curve in the figure), which is aB-spline with seven control points (C1–C7, as noted in the figure).The horizontal positions of these control points are held constant.The vertical positions of C1 and C2 are kept the same so that theentrance of the intake duct stays horizontal. Additionally, the verticalposition of C6 and C7 are on the engine centreline to ensure that theoutlet of the intake is level and connects smoothly with the engine.The cross-sectional area of the intake duct equals that of the engine

face and is held constant along the intake duct center axis. Thevertical positions of other points are left free as design variables.Horizontal positions, acceptable vertical position ranges, andcorresponding design variables for each control point are listed inTable 1. The design variable set x comprises four design variables inour test case:

x � fx1; x2; x3; x4g

For example, the design variable set x� f359:5; 359:9; 165:8;116:1g results in the design shown in Fig. 1.

B. Explicit Knowledge Representation

As mentioned in Section II.A, engineering knowledge in the formof explicit rules can be transformed into penalty functions to makethem optimizable objectives. In this section, the idea is illustrated bythree examples.

1. Knowledge Type 1: Knowledge Based on a Single Parameter

First of all, an intake position penalty function P1 is related to thevertical distance D between the aircraft fuselage (upper boundary)and the intake lower boundary at the air intake entrance position, asshown in Fig. 1. A positive D corresponds to a protruding intakedesign, whichwill increase the aerodynamic drag. On the other hand,negative distancewill result in an intake design where the entrance ispartially submerged into the fuselage. Because the air capture area isreduced, the capture/throat area ratio is reduced, which is undesirablefrom an intake aerodynamics point of view. Such engineeringconsiderations are incorporated into P1, in which both unfavorablescenarios receive a penalty. We define P1 as having the form

P1 ��D3=2 if D � 0

�100D if D< 0(2)

−100 0 100 200 300 400 5000

50

100

150

200

250

300

350

400

Engine entry

Rear Pressure

Bulkhead

C1

C2 C

3

C4

C5

C6

C7

Aircraft fuselage

Air

D

Fig. 1 A simplified 2-D intake duct model.

Table 1 List of the horizontal positions, acceptable vertical

position ranges, and corresponding design variables

for each control point

Control point Horizontal position Vertical range Design variable

C1 �100 245–385 x1C2 �65 same as C1 same as C1

C3 0 175–455 x2C4 100 70–350 x3C5 200 35–315 x4

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Because the airframe external shape is fixed and the vertical positionof the intake entrance is defined by design variable x1, D is solelydependent upon x1. Here, x1 and D are related by

D� x1 � 306:25 (3)

so that P1 can be rewritten as a function of the design variables

P1 � P1�x� ���x1 � 306:25�3=2 if x1 � 306:25 � 0

�100�x1 � 306:25� if x1 � 306:25< 0(4)

2. Knowledge Type 2: Knowledge Based on Geometry Analysis

A second penalty function P2 is related to the curvature of theintake duct. It can be seen that certain design variable combinationscan render the overall shape of the intake duct excessivelyconvoluted. From an aerodynamic engineer’s point of view, designswith sharp bends are unfavorable because the airflow can separateand cause distorted pressure fields on the engine face. Furthermore,too high a curvature of the center line can cause a loop in its upper andlower offset curves and render the design impractical. Such a faileddesign is shown in Fig. 2, in which the design variables are set to bex� f360; 300; 50; 250g. Therefore, to represent such engineeringconcerns, a penalty function P2 is set up to penalize geometries withexcessive curvature. The radius of curvature R�u� of the center B-spline axis can be computed as R�u� � jf0�u�j3=jf0�u� � f00�u�j,where the domain of the curve u is [0, 1]. A loop will occur in theoffset curve ifR�u� is less than the radius of the engine face, which is43.75 in this intake design case. P2 is set up as a sum of R�u�swhenever its value is less than 43.75, that is:

P2 �X1u�0

R�u�; 8 R�u�< 43:75 (5)

3. Knowledge Type 3: Knowledge Based on the Interaction of Different

Geometries

To set up a third penalty function P3, it is noted that those intakedesigns that interfere with the rear pressure bulkhead areundesirable because precious space in the fuselage is occupied andthe bulkhead may become structurally inefficient (Fig. 2). A penaltyfunction

P3 � Total area of interference

is set up to penalize interference between the two parts. A moresevere interference will incur a higher penalty value of P3.

So far, three penalty functions have been set up for three explicitrules, each of which represents some form of engineeringknowledge. Compared with direct consultation with a human expertor evaluation of CFD codes, explicit rules are relatively cheap tocalculate and, thus, more favorable in the process of generatingtraining data. However, in contrast to the four design variables in ourtest case, many more design variables may exist in real engineeringapplications with underlying interactions of which the designer maybe unaware.

C. Incorporation of Flow Simulation

To illustrate the use of a penalty based on computational analysis,flow simulation through the duct is considered next. Here, the flowthrough the intake duct is meshed in GAMBIT, and the flow ismodeled in FLUENT. The purpose is to predict the fan-face pressuredistortion at the outlet of the intake duct (on the fan face) and,therefore, make it possible to add fan-face pressure distortion as anadditional metric in the knowledge base. Geometries that areunphysical, for example, those that exhibit loops in the lower orupper walls as shown in Fig. 2, would not be meshed. It would,therefore, not be possible to predict their flow behavior or find theirfan-face distortion.

If the geometry in question is successfully meshed, the corre-sponding mesh file can be read into FLUENT, and a flow model canbe set up and solved. A �-� turbulence model is used to take viscousboundary layer effects into consideration.

After the flow is computed, the values of the total pressure at eachnode of the outlet mesh are recorded. The standard deviation of thepressure is calculated and used as a metric of the fan-face distortion.Figure 3 shows the total pressure distribution and velocity contour,andFig. 4 shows the fan-face pressure for this typical sensible design,respectively.

On the other hand, some unfavorable designs can lead to reverseflow and severe fan-face pressure distortion. Figure 5 shows the totalpressure distribution and velocity contour, and Fig. 4 shows the fan-face pressure for this typical distorted design, respectively.

−100 0 100 200 300 400 5000

50

100

150

200

250

300

350

400

EngineRear PressureBulkhead

Unphysical loop in offset splines

Sharp bend

1

Interference with rear pressure bulkhead

Fig. 2 Failed design example.

Fig. 3 Typical pressure and velocity contour plots.

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Figure 4 also illustrates the range of total pressure distributionsat the outlet of the intake ducts, which are in the samplingplan established in Section III.D. Only the total pressure distri-bution profiles of those designs that can be solved in FLUENT arepresented.

D. Sampling Plan, Penalty Table, and Surrogate Modelling

After the three penalty equations based on the geometry and theone penalty based on the flow modelling have been set up, 200sample runs have been made to obtain the training data, which areused to build up the knowledge surrogate model.

1. Sampling Plan

Agood sampling plan should cover the design space in a thoroughand uniform fashion. The requirement of a uniform projection ontoeach dimension of the design space leads to the idea of Latin

−4 −3 −2 −1 0 1 2

x 104

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

0.18

Total pressure, Pa

Pos

ition

, m

A typical normal pressure distributionA typical distorted pressure distributionAll pressure distributions

Fig. 4 Total pressure distribution profiles at the outlet of the intake

ducts.

Fig. 5 Typical distorted pressure and velocity contour plots.

Table 2 Penalties and their statistics

Sample Penalty 1 Penalty 2 Penalty 3 Standard deviation (std.) Modified std.

1 373.2 1541.6 0 NaN 73666.92 21.6 2665.9 31887.9 NaN 73666.93 467.4 0 0 33117.5 33117.5� � � � � � � � � � � � � � � � � �198 628.6 0 0 22695.3 22695.3199 5319.7 1982.1 29462.3 NaN 73666.9200 13.6 2917.2 0 NaN 73666.9Average 1495.1 1093.8 9443.8 49303.3Std 1818.6 1745.4 14826.4 22676.0

Table 3 Normalized penalty value table

Normalized SumSample Penalty 1 Penalty 2 Penalty 3 Penalty 41 �0:6169 0.2565 �0:6370 1.0744 0.07712 �0:8103 0.9007 1.5138 1.0744 2.67863 �0:5651 �0:6267 �0:6370 �0:7138 �2:5426� � � � � � � � � � � � � � � � � �199 2.1031 0.5089 1.3502 1.0744 5.0367200 �0:8147 1.0447 �0:6370 1.0744 0.6675

−100 0 100 200 300 400 5000

50

100

150

200

250

300

350

400

← Engine Intake

Unphysical loops

Interference with rear pressure bulkhead

Fig. 6 A design candidate with multiple flaws.

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Table 4 Repair suggestions based on the SVR surrogate (� � 0:5)

−100 0 100 200 300 400 5000

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150

200

250

300

350

400

← Engine Intake

−100 0 100 200 300 400 5000

50

100

150

200

250

300

350

400

← Engine Intake

−100 0 100 200 300 400 5000

50

100

150

200

250

300

350

400

← Engine Intake

Reduced interference area

Smooth corners

−100 0 100 200 300 400 5000

50

100

150

200

250

300

350

400

← Engine Intake

−100 0 100 200 300 400 5000

50

100

150

200

250

300

350

400

← Engine Intake

−100 0 100 200 300 400 5000

50

100

150

200

250

300

350

400

← Engine Intake

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hypercube sampling [7], in which the sample points align in such away that there is only one sample in each row and each column. It isalso desired that the sampling points spread in the design space asuniformly as possible. The maximin metric, originally introduced inJohnson et al. [8] and further elaborated in Morris and Mitchell [9]and Ye et al. [10], is widely used to evaluate the uniformity of asampling plan. Forrester et al. [5] illustrates how optimized space-filling Latin hypercube can be obtained by evolutionary operationusing themaximin criterion. Here, the method illustrated in Forresteret al. [5] was adopted, and a four-dimensional sampling plan of 200points was generated.

2. Penalty Table

For each of the samples, penalty values based on the penaltyfunctions that have been introduced in Sections III.B and III.C arecalculated and assembled in a penalty value table as presented inTable 2.

Note that, for the fourth column, which is the standard deviation ofthe total pressure distribution on the fan face, some values are notavailable because the corresponding designs are not meshable inGAMBITor solvable in FLUENT. For the meshable geometries, thelargest value of standard deviation found was 7:37 � 104, whichindicated a relatively severe fan-face distortion. A meshing or flow-solution failure indicates an even less suitable design (perhaps highlyunphysical) than one with a high standard deviation. Therefore, weendeavour to penalize those samples that failed to be meshed orsolved with a large value of 7:37 � 104, which equals the largeststandard deviation value from those samples that can be solved. Theresult is presented in the last columnof Table 2 as “Modified std.”Theaverages and standard deviations of each penalty are also shown inTable 2.

To give each penalty a fair and equal consideration, the penaltyvalues are normalized by

pn �pi � �pispi

(6)

where pn is the normalized penalty value, pi is the originalpenalty value, �pi is the average of the penalty in question, and spi isthe standard deviation of the penalty. The resulting normalizedpenalties and the sum of the four normalized penalties are presentedin Table 3.

3. Surrogates

After the 200 sample designs were generated and their corre-sponding total penalty values were determined (Table 3), SVR wasused to generate the knowledge surrogate models. The surrogateswere built and tuned based on the training data set, which representsthe knowledge we have at hand.

E. Repair

To investigate the proposed method’s ability to evaluate anunknown design candidate and provide repair suggestions, a designwith multiple flaws is picked (Fig. 6). The intake face of the design isslightly submerged in the fuselage, and the intake itself is snaky andinterfereswith the aft bulkhead. The centerline is so convoluted that itbecomes unphysical to the extent that it fails to be meshed. Thedesign variables of the geometry shown in Fig. 6 are

x � 0:45; 0:35; 0:05; 0:65

The feasibility thresholds used for the repair and results based onthe SVR surrogate (� � 0:5) are listed in Table 4 alongside theresulting geometries.

As expected, the repair suggestions all look similar to the originaldesign (Fig. 6), which is a sign of the preservation of the originaldesign intent. However, subtle changes do occur, which correct themeshing failure and improve the feasibility of the geometry. For

example, if we compare Fig. 6 and the figures in Table 4, it is notdifficult to notice that 1) the unphysical loop and the first-orderdiscontinuities, which occurred on the intake wall between x� 100and x� 200 have been eliminated and 2) the area of interferencewiththe rear pressure bulkhead gradually diminishes and eventuallydisappears altogether.

Additionally, the geometry has now become solvable from a flow-simulation point of view (starting from a penalty threshold value of�0:7).

IV. Conclusions

The computational framework presented here enables, on theevidence of a simple, illustrative example, the repair of a parametricgeometry (in terms of its parameters). The process can be overseenand guided by the engineer (manual control of acceptable constraintviolation) or can be fully automatic (in the context of an automaticoptimization process).

Although, in the case of the 2-D intake design repair example, therepair direction can be predicted by an observant reader, the proposedmethod has the potential to greatly reduce the engineer’s workloadand increase design efficiency and robustness when it is applied onreal-scale engineering design problems, where the best repairalteration is by no means evident.

Acknowledgments

The work of postgraduate student Dong Li has been supported bythe Dorothy Hodgkin Postgraduate Awards scheme, which isadministered by the Engineering and Physical Sciences ResearchCouncil (EPSRC), United Kingdom, with financial support fromRolls-Royce plc. The Royal Academy of Engineering and theEPSRC supported the work of A. Sóbester. The authors alsoacknowledge ANSYS for facilitating on-campus use of theirsoftware FLUENT.

References

[1] Sóbester, A., and Keane, A., “Supervised Learning Approach toParametric Computer-Aided Design Geometry Repair,” AIAA Journal,Vol. 44, No. 2, Feb. 2006, pp. 282–289.doi:10.2514/1.17193

[2] Keane, A. J., and Nair, P. B.,Computational Approaches for AerospaceDesign, Wiley, Hoboken, NJ, 2005, Chap. 3, p. 147.

[3] Smola, A., and Schölkopf, B., “A Tutorial on Support VectorRegression,” Statistics and Computing, Vol. 14, No. 3, 2004,pp. 199–222.doi:10.1023/B:STCO.0000035301.49549.88

[4] Clarke, S., Griebsch, J., and Simpson, T., “Analysis of Support VectorRegression for Approximation of Complex Engineering Analyses,”Journal of Mechanical Design, Vol. 127, No. 6, Nov. 2005,pp. 1077–87.doi: 10.1115/1.1897403.

[5] Forrester, A., Sóbester, A., and Keane, A. J., Engineering Design

via SurrogateModeling: APractical Guide,Wiley,Hoboken,NJ, 2008.[6] Beyer, H., and Schwefel, H., “Evolution Strategies: A Comprehensive

Introduction,” Natural Computing, Vol. 1, No. 1, 2002, pp. 3–52.doi:10.1023/A:1015059928466

[7] McKay, M. D., Beckman, R. J., and Conover, W. J., “AComparison ofThree Methods for Selecting Values of Input Variables in the AnalysisofOutput from aComputerCode,”Technometrics, Vol. 21,No. 2, 1979,pp. 239–245.doi:10.2307/1268522

[8] Johnson, M., Moore, L., and Ylvisaker, D., “Minimax and MaximinDistance Designs,” Journal of Statistical Planning and Inference,Vol. 26, No. 2, 1990, pp. 131–148.

[9] Morris, M. D., and Mitchell, T. J., “Exploratory Designs forComputational Experiments,” Journal of Statistical Planning and

Inference, Vol. 43, No. 3, 1995, pp. 381–402.[10] Ye,K., Li,W., and Sudjianto, A., “Algorithmic Construction ofOptimal

Symmetric Latin Hypercube Designs,” Journal of Statistical Planningand Inference, Vol. 90, No. 1, 2000, pp. 145–159.

R. KapaniaAssociate Editor

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