Turbine Blade Fir Tree FEA

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Turbine blade fir-tree root design optimisation using

intelligent CAD and finite element analysis

Wenbin Song a,*, Andy Keane a, Janet Rees b, Atul Bhaskar a, Steven Bagnall b

a Department of Mechanical Engineering, University of Southampton, Highfield, Southampton, SO17 1BJ, UKb Rolls-Royce Plc., Bristol, BS34 7QE, UK

Received 16 January 2001; accepted 26 May 2002

Abstract

This paper is concerned with automation and optimisation of the design of a turbine blade fir-tree root by incor-

porating a knowledge based intelligent computer-aided design system (ICAD) and finite element analysis. Various

optimisation algorithms have been applied in an effort to optimise the shape against a large number of geometric and

mechanical constraints drawn from industrial experience in the development of such a structure. Attention is devoted to

examining the effects of critical geometric features on the stress distribution at the interface between the blade and disk

using a feature-based geometry modelling tool and the optimisation techniques. Various aspects of this problem are

presented: (1) geometry representation using ICAD and transfer of the geometry to a finite element analysis code, (2)

application of boundary conditions/loads and retrieval of analysis results, (3) exploration of various optimisation

methods and strategies including gradient-based and modern stochastic methods. A product model from Rolls-Royce

is used as a base design in the optimisation.

2002 Elsevier Science Ltd. All rights reserved.

Keywords:Intelligent CAD; Design optimisation; Finite element analysis; Genetic algorithms

1. Introduction

Over the last several decades, the engineering design

world has been transformed by the introduction of

massive computational power. Computers are now the

principal tools for conceptual design, analysis, mock-up,

and manufacture. CAD, analysis, mock-up and assem-

bly checking are now completely carried out in a virtualenvironment and this process is being continuously up-

dated [1]. In the engineering design community, the most

notable changes lie in two aspects: CAD and analysis.

Most companies use CAD and analysis software to de-

liver more reliable products in increasingly reduced time

scales, but the pursuit of lower cost and shorter devel-

opment time never stops because of the competitive

world market.

The design process is a recursive one in which changes

are often required during the later stages due to various

factors arising as the design progresses, and design re-

quirements often not only involve structure and func-

tionality, but also cost, manufacture and environmental

aspects. Traditional design processes typically treat these

areas using different models and requirements are often

described in different formats and different documents.The use of finite element analysis has long become a

common practice during the product development pro-

cess, especially in the aerospace industry, where a large

number of FE codes, including commercial packages

and in-house codes are currently in use. The capability

covers stress analysis, thermal and vibration analysis, as

well as fatigue life estimates. Although most of these

tools have user-friendly interfaces and powerful pre- and

post-processing capabilities, the successful use of such

tools requires experience in the field. These tools are now

used at all stages of the design process from concept to

detailed design.

Computers and Structures 80 (2002) 18531867

www.elsevier.com/locate/compstruc

* Corresponding author. Fax: +44-23-8059-4813.

E-mail address: [email protected](W. Song).

0045-7949/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.

PII: S0 0 4 5 -7 9 4 9 (0 2 )0 0 2 2 5 -0
http://mail%20to:%[email protected]/http://mail%20to:%[email protected]/


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The study and development of computer programs

on mathematical optimisation algorithms has been a

long term effort since the early 1960s and a large number

of numerical algorithms are now available in the form of

standard function library APIs together with some de-

sign exploration systems with built-in algorithms. OP-

TIONS [2] is such a system which provides a flexibleframework for incorporating user codes ranging from

very simple ones to complicated external software pack-

ages as well as more than 40 search algorithms and can

be used either interactively or in batch mode.

The optimisation process typically starts with the

parameterisation of the model and is then followed by a

search process using different algorithms and strategies

based on the evaluation of a measure of merit. Basically,

there are three different types of parameterisation ap-

proach currently being developed for design optimisa-

tion in the engineering community [3]. The first is based

on using the coordinates of the boundary grid points ina discretised domain as design variables, which is rela-

tively easy to implement but makes it difficult to main-

tain a smooth geometry and the resulting optimised

designs may be impractical for manufacture. The second

is based on using a CAD system. Although the advan-

tages of using a CAD system is obvious: smooth ge-

ometry, relatively small number of design variables, etc.,

it is difficult for traditional CAD systems to handle the

variations of topology due to large perturbations in

some dimensions. In addition, the underlying parame-

ters describing the geometry are not normally available

to calculate the sensitivities analytically. The third ap-

proach is similar to the morphing techniques used in

computer sciences, and is termed free-form deformation,

which is based on the idea of modelling the deformation

rather than the base geometry. In this work, the second

approach is adopted but the parametric modelling is

performed using the design automation tool ICAD

from KTI [4]. The difference between ICAD and tradi-

tional CAD systems is that it can be used to generate

parametric models based on various geometric features.

Furthermore, non-geometric information can also be

integrated into the model and used in the following

analysis stages. The underlying geometric modelling

engine used by ICAD makes possible the smoothtransfer of geometry to analysis codes. The incorpora-

tion of ICAD and finite element codes into the search

process gives the ability to carry out search based on

high fidelity analysis results and to explore different

geometric features using a relatively small set of pa-

rameters.

2. Overview

A fir-tree joint may be used in turbine structures to

attach a blade to the rotating disk. Mechanical loads are

transferred through the joint from the blade to the disk

[5]. The overall aim of fir-tree structure optimisation

program is to enable the designer to explore different

candidate geometries at the preliminary stage, ranging

from relatively simple designs to rather complex ones, at

a reduced cost compared to using previous manual

methods (such costs have often prevented the explora-tion of more complex profiles and different size and

shape teeth). This process involves the use of feature-

based parametric geometry construction tools such as

ICAD and large-scale structural finite analysis packages,

along with an optimisation program implementing var-

ious search strategies. Although this problem is basically

a structural analysis problem, the strategy employed

here is rather different. The use of the ICAD system and

finite element analysis software together gives it the ca-

pability to model the variations parametrically both for

dimensions and in the topology used, and to analyse the

effect of geometric features on the stress distributionbased on the finite element analysis results. Further-

more, the above process may be incorporated into a

search loop to automatically find the best solution

against pre-defined constraints. As considered here, this

is a 2D problem nested in the overall 3D blade/disc

optimisation procedure and uses well established 2D

criteria modified from 3D analysis.

The overall architecture of the ICAD based design

optimisation used is illustrated in Fig. 1. In this struc-

ture, ICAD is used to generate the model definition

based on rules which may be stored in a knowledge

database. The model is defined in a descriptive form

using the ICAD design language which is a derivation of

and extension to common LISP, aimed at geometric

modelling capabilities. This model is used to produce

geometry and related information. The geometry is then

passed to the analysis code along with any geometry

dependent properties to evaluate the design perfor-

mance. The optimiser is used to find the best solution for

a particular set of design parameters. Parallel processing

can be achieved by using facilities provided by the

OPTIONS search systems.

3. Geometry representation using ICAD

The product design process normally begins with the

definition of requirements based on customer demands

and goes from the conceptual design stage to the de-

tailed design stage. Once the design specification is de-

termined, the conceptual design stage starts with the

implementation of the design in a computer environ-

ment followed by analysis to explore the feasibilities of

candidates. Analysis plays an important role in investi-

gating various design alternatives to determine the best

design. It is, however, very difficult, for traditional CAD

systems to capture design intent and to describe the

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designs from the same template. In addition, multi-modality and backward compatibility can be achieved

by incorporating different behaviours into one model

with a single interface while only the internal imple-

mentation is modified.

A single basic tooth geometry is illustrated in Fig. 2,

which is defined in such a way as to allow the designer to

explicitly control the non-contact clearance and to avoid

duplicate entities in the model. This latter feature eases

the application of boundary conditions and loads during

analysis.

The acceptability of any fir-tree geometry needs to be

checked since some particular combination of parame-

ters may result in unacceptable features such as inter-

sections between entities or the collapse of very short

entities. The handling of unacceptable features is im-

portant to the optimisation process as well as to the

analysis code. Using ICAD, geometry features can be

checked within the modelling process, as part of the

whole model and appropriate actions can then be taken

using preset default values, while signalling which pa-

rameter is causing the problem. Taking the modelling of

the base tooth as an example, in every step in the

modelling process acceptability is checked to make sure

an acceptable geometry can be produced, otherwise, a

geometry failure is signalled to the optimiser to cancelthe analysis. An example of unacceptable base tooth

geometry, in which two circular arcs are intersected, is

also illustrated in Fig. 2.

Here, the blade root and disc head geometry are

defined in the same way as the basic tooth, with further

parameters and rules being needed. Details are covered

in the following sections. Some of the quantities used in

the definition are not expected to change and are thus

held constant during an optimisation run, while others

which are identified as being more influential to the

design will be varied by the optimiser in the optimisation

loop and are known as design variables.

The first fir-tree geometry tackled is a simplified

version of an existing fir-tree model, which is composed

of straight lines and circular arcs only. The complete

geometry is described by approximately 20 quantities, as

shown in Table 1. The resultant blade/disc geometry is

illustrated in Fig. 3. Some of these quantities are iden-

tified as playing an important role in the stress distri-bution and thus will be optimised against known

constraints. Others will be kept at constant values based

on previous experience.

Every entity within the ICAD model can have addi-

tional non-geometric properties which will ease the use

of the geometry in other applications such as analysis

and manufacture. This object-oriented feature enables

various information related to a product design to be

integrated into a single model. For example, to apply

boundary conditions and loads to entities during the

analysis stage, it is desirable to name the entities with

unique labels which can then be referenced later. Usingunique tag names on each entity in the geometry enables

the boundary conditions, load properties and mesh

parameters to be specified in batch mode.

Geometric quantities such as the minimum thickness

of the blade root, the distance between the centre of the

contact face of the tooth on each side, etc., are calcu-

lated in the ICAD model based on a mathematical

representation of the geometry. Some of these are trea-

ted as constraints in the optimisation problem and some

are used to generate the FE model or to retrieve analysis

results. For example, point coordinates are normally

required to get the stress values at those points. The full

details of the geometric constraints used are covered in

Section 5.

4. Finite element analysis

The fir-tree joint used to hold a blade in place in a

turbine structure is usually identified as a critical com-

ponent which is subject to high mechanical loads. Most

often the attachment is a multi-lobe construction used to

transfer loads from blade to disk. It is generally assumed

that there are two forms of loading which act on theblade, the primary radial centrifugal tensile load result-

ing from the rotation of the disk, and bending of the

blade as a cantilever which is produced by the action of

the gas pressure on the airfoil and forces due to tilting of

the airfoil. The resulting stress distribution in the root

attachment area is a function of geometry, material and

loading conditions (which are of course related to the

speed of rotation). It is known that some critical ge-

ometry features exist for the stress distribution in the

bladedisk interface.

Many studies into the stress state of the blade root

attachment have been reported, originally using photo-

Fig. 2. Base tooth geometry parameters and an example of

unacceptable geometry with fixed end point.



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elastic methods [9,10], now mainly using finite element

analysis [1113]. Modern finite element codes already

have the capability of dealing with thermalmechanical

coupling and contact analysis between blade root and

disk head. It is now relatively easy to obtain the stressdistribution in the attachment area using commercial FE

codes. Also many in-house FE codes exist to handle

corporate-specific problems (these have some advanta-

ges over commercial tools among which the most no-

table is complete control over the source code).

Although there are many kinds of code available, the

general procedure of finite element analysis is almost

always as follows:

(1) create the geometry, or import the geometry from

another CAD system;

(2) apply the boundary conditions and loads;

(3) mesh the geometry;

(4) solve the problem and retrieve the results.

Most FE codes support batch running of the analysis

and this allows the analysis to be embedded into theoverall optimisation loop. To smoothly couple the

modelling process and analysis, however, is not an easy

task. It involves the transfer of the geometry itself and

related geometry dependent properties to the analysis

code, in this case, the finite element software.

It has been shown from previous preliminary opti-

misation work that it is possible to reduce the number of

teeth in a fir-tree joint. From aerodynamics, a constant

space/chord ratio ensures a constant lift coefficient, and

it is thus possible to trade blade number for blade chord.

Consequently a reduction in blade number will cause an

increase of centrifugal load transferring to the fir-tree

Table 1

Blade root/disk head geometry parameters

Variable Name Type

Skew (b) () Root skew angle Variable

Nteeth Number of teeth Variable

Rwa () Root wedge angle Variable

Alor (L) (mm) Axial length of root Variable

Snw (mm) Shank neck width Variable

Fsw (mm) Fir-tree shoulder width Variable

Rcrest (mm) Fir-tree tooth crest radius Variable

Rtrough (mm) Fir-tree tooth trough radius Variable

Bp 1, 2 (mm) Blade tooth pitch Variable

Btcr (mm) Bottom tooth crest radius Variable

Cpw (mm) Cooling passage width Variable

Bglr (R1) (mm) Bucket groove lower radius Variable

Bgur (R2) (mm) Bucket groove upper radius Variable

Dtp 1, 2 (mm) Disk tooth pitch Variable

Fcrest (mm) Disk tooth crest radius Variable

Ftrough (mm) Disk tooth trough radius Variable

Nblades Number of blades ParameterDrad (mm) Disk radius Parameter

Ninc Number of blades inclusive Parameter

Rtsn (mm) Radius to shank neck Parameter

Nblades Number of blades Parameter

Drad (mm) Disk radius Parameter

Snfr (R) (mm) Shank neck fillet radius Parameter

Tfa (/) () Top flank angle Parameter

Ufa (c) () Under flank angle Parameter

Ncfc (mm) Non-contact face clearance Parameter

Bac (Ca) (mm) Blade axial chord Parameter

Cpa (mm2) Cooling passage area Parameter

Tfa (/) () Top flank angle Parameter

Ufa (c) () Under flank angle Parameter

Fdcr (mm) First disk crest radius ParameterInr (mm) Inner radius Parameter

Dhnw (D) (mm) Disk head neck width Derived

Bga (mm2) Bucket groove area Derived

Bch (H) (mm) Bottom-to-contact height Derived

Bl (mm) Bedding length Derived

Fh (mm) Fir-tree height Derived

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root. The root will have to be redesigned to cope with

the larger load. At the same time, a reduced number of

blades leaves more disc material between adjacent fir-

tree root slots, and the wider disc post thus allows a

bigger tooth design and this makes a further reduction

of fir-tree teeth numbers viable. The reduced number of

teeth will bring further cost savings.

The loading on the root is mainly due to centrifugal

load which is dependent on the mass of the whole blade.

The design of the fir-tree root involves an iterative

process of controlling the blade mass, which incorpo-

rates the root mass. Also some key features, such as the

fillet radius, play very important roles in the stress dis-

tribution in notch regions. Thus a set of competitive

constraints ranging from geometrical, mechanical,

cooling requirements, etc., is established for use in ex-

ploration of various design candidates for the fir-tree

root. Finite element analysis is then utilised to obtain the

resulting stress distributions. This further complicatesthe situation. A traditional manual method is now

too slow for this process and thus automation is re-

quired. Four types of constraints are used to check the

design:

Crushing stress describes the direct tensile stress on

the teeth: bedding width is the main factor affecting

the stress.

Unzipping can occur after a blade release: the disc

post on either side of the released blade are then sub-

ject to high tensile and bending stresses. The disc post

must be able to withstand these stresses in order to

avoid a progressive unzipping and release of all

the blades.

Disc neck creep: the disc posts are subject to direct

tensile stress which causes material creep. Too much

creep, combined with low cycle fatigue, can dramat-

ically reduce the component life.

Peak stresses: peak stresses occur at the inner fillet ra-dii of both the blade and the disc. If the fillet radii are

too small and produce unacceptable peak stresses,

some bedding width has to be sacrificed to make

them bigger.

Apart from the above constraints, which are used to

check the candidate designs, some others are used to

check the optimised result. These include vibration

limits, neck stress, etc. From a preliminary blade num-

ber optimisation, these criteria are not deemed a sig-

nificant constraint here.

As the fir-tree geometry is constant along the rootcentre line, it is possible to think of the stresses as 2D.

However, the loading applied along the root centre line

is not uniform, so strictly speaking, the distribution of

stresses will be 3D. Nonetheless, it is still possible to

assume that each section behaves essentially as a 2D

problem with different loadings applied to it. The dif-

ference of loading on each section is affected by the ex-

istence of skew angle which will increase the peak

stresses in the obtuse corners of the blade root and the

acute corners of the disk head. From previous root

analysis research, it is feasible and convenient to use

a factor to estimate the peak stresses at each notch of

the blade and disc, and this factor takes different values

for different teeth.

Also, it is known from previous work using photo-

elastic and finite element methods, that the distribution

of centrifugal load between the teeth is very non-uni-

form and the top tooth may take a significant portion of

the load. This feature allows the possibility of using

different tooth sizes. The system implemented here also

allows designers to explore the effect of varying the

number of teeth, but this may cause difficulties when

gradient-based methods are used for optimisation.

Both a one sector model and a three sector model are

considered when estimating the mechanical constraints,the one sector model for the estimation of maximum

notch point stresses, crushing stresses and blade/disk

neck mean stresses and the three sector model for the

estimation of unzipping stresses. Typical FE results are

illustrated in Fig. 4a and b, for the one sector model and

three sector model, respectively. In general, finite ele-

ment analysis is computationally expensive, thus a

compromise between accuracy and computation cost

should always be made to obtain acceptable results as

quickly as possible when this is embedded in an opti-

misation run. This compromise is made by an appro-

priate choice of mesh density.

Fig. 3. Geometry definition of a (a) typical blade root and (b)

typical disk head.



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The whole process from the importing of geometry,

application of boundary conditions and loading, to re-

sults retrieval is implemented here as a SC03 Plugin,

which is a facility provided by the RR in-house FEA

code SC03 to extend the capability of its core func-

tionality. A command file is used by SC03 to carry out

jobs ranging from importing geometry from IGES files,applying boundary conditions and loads, to retrieving

stress results.

5. Optimisation

Here two different optimisation problems were

tackled using population based genetic algorithms (GAs)

and gradient-based methods. One was to minimise the

area outside of the last continuous radius of the turbine

disc for the 2D case, which is proportional to the rimload by virtue of the constant axial length. This quantity

is referred to as the fir-tree frontal area in the following

Fig. 4. Typical stress field for (a) single blade and (b) three blades with the middle out.



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effects of mesh density must be studied. Following the

set-up of the system, a series of systematic evaluations

has been carried out to establish appropriate mesh

density parameter values and to gain experience on the

effects of design variables changes. These were per-

formed by varying the mesh density control parameters

while holding all others constant. Two mesh control

parameters, the global and local edge node spacing were

chosen. Reducing the notch edge node spacing increases

the mesh density and therefore reduces the perturba-

tions in maximum notch stress. These studies resulted

in the use of 0.001 mm for both global and local edge

node spacing. The parameter study also revealed the

effect of different geometric features on the stress dis-tribution within the structure which are summarized

in Table 5. It can be also seen from the parameter

study results that the notch stress on the second tooth

takes the largest value, as already implied from previ-

ous work [10]: this aspect makes it desirable to design

each tooth using different values of tooth profile pa-

rameters.

5.2. Optimisation

The optimisation is performed here using the OP-

TIONS software package [2] which provides designers

with a flexible structure for incorporating problem spe-

cific code as well as more than 40 optimisation algo-

rithms. The critical parameters to be optimised, known

as design variables, are stored in a design database,

which also includes the objective, constraints and limits.

The design variables are transferred to ICAD by means

of a property list file which contains a series of pairs with

alternating names and values. This file is updated during

the process of optimisation and reflects the current

configuration. The geometry file produced by ICAD is

then passed to the FE code SC03 which is executed by a

command file. The analysis results are written out to

another file, which is read in by the optimisation code.

The design variables are then modified according to theoptimisation strategy in use until convergence or a

specified number of loops has been executed. The pro-

gram structure is illustrated in Fig. 5.

Due to the presence of a discrete design variable (the

number of teeth), most gradient based optimisa-

tion techniques will not work directly here. Therefore a

two-stage strategy of combining a GAs with gradient

search was used in this problem. A typical GA (as dis-

cussed by Goldberg [14]) is first employed in an attempt

to give a fairly even coverage on the search space, and

then gradient based search methods are applied on

promising individuals with the number of teeth fixed.

Table 4

Normalized constraint vector for the base design

Number Name of constraint Numeric values

Lower bound Value Upper bound

1 Ratio of R1 to R2R1=R2 )1.0 )0.8642

2 Ratio of Hto D H=D )

1.0 )

0.8955 3 Ratio of R1 to disk trough R1=Ftrough )1.0 )0.5268 4 Maximum ratio of tooth pitch to disk trough

DP=Ftroughmax 1.4779 1.0

5 Minimum ratio of tooth pitch to disk trough

DP=Ftroughmin)1.0 )0.5467

6 Ratio of axial length to blade axial chord [LCA] )1.0 )0.4961

7 Root stagger angle [RSA] 0.7499 1.0

8 Ratio of blade/disk serration pitch [PMIN] )1.0 )0.4540

9 Ration of blade bottom neck width to tooth pitch

[BNP]

)1.0 )1.1474

10 Minimum wall thickness of bottom blade notch

[BNMIN]

)1.0 )1.3038

11 Ratio of bucket groove region area to cooling passage

area [AR]

)1.0 )1.0900

1219 Maximum blade notch stress [NBL(R)(2a)]b )1.0 0.9270 1.0

2023 Maximum disk notch stress [NDL(R)(3)] )1.0 0.9948 1.0

2429 Maximum blade section stress [SB(1)] )1.0 0.6931 1.0

2935 Maximum disk section stress [SD(4)] )1.0 0.5623 1.0

3643 Maximum crushing stress [CS(1)] )1.0 0.6514 1.0

4445 Maximum bucket groove stress )1.0 0.9023 1.0

4653 Maximum unzipping stress [UZP(1)] )1.0 0.3688 1.0

a The numbers in bracket indicate the number of the tooth or the section where the maximum stress occurs.b For the purpose of compactness, only the maximum stresses are shown in the table.



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One of the considerations here is that GAs are capable

of dealing with discrete design variables. Another con-

sideration is that as the GA proceeds, the population

tends to saturate with designs close to all the likely op-

tima including sub-optimal and globally optimum de-

signs, while gradient based methods are more suited to

Table 5

Effect on stress distribution of design variables for the base design

Geometry features Unzip stress Notch stress Section stress Crushing

stressBlade Disk Blade Disk

Skew angle 1; rest

Shank neck width )

1; rest 1, 4)

; restBlade shoulder width ) 2,3; rest ) 1; rest) Bottom tooth crest radius 2; rest) ) 4; restCooling passage width 1,2; rest) 1); 2, 3 1, 2; 3, 4)Bucket groove lower radius

Bucket groove upper radius Tooth pitch )Blade crest radius 1); rest ) Blade trough radius ) ) 1, 2); 3, 4Disk crest radius 1); rest ) ) ) Disk trough radius ) ) 4); rest

indicates increase,) indicates decrease, means no significant effect when variable increases, number indicates the tooth number intop-down order.

Fig. 5. Optimisation program structure.



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gradient-based search will more likely get stuck in a to

local optimum.

Several steepest descent search methods have been

applied to the problem after the initial GA search: these

include the Hooke and Jeeves direct search method plus

various other methods discussed in Schwefels book [15].

The first method is very fast when the number of design

variables is small, as shown in Fig. 8, in which only 6

variables are chosen as design variables, while (the full

scale problem contains 14 design variables). Although

the complexity of this problem is only modest, the

computational cost in terms of the thousands of evalu-

ations required for some search techniques is still an

obstacle for a detailed search. It can be seen from the

contour maps that the objective function is rather

smooth, and this may justify the use of approximation

techniques alongside the accurate finite element model.

This will be the focus of further research.

A 20% reduction in the objective function was

achieved using the above methods while satisfying all the

Fig. 7. (a) Schwefel library method: repeated lagrangian interpolation (LAGR) on GA results and (b) comparison of search results

using different search strategies.



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same time improve the quality of the end product. Other

information such as a cost evaluation model or a man-ufacturing requirements model could be further included

without sacrificing the compatibility of the existing

model. A complete and consistent product model could

then be achieved to be set-up for evaluation in the design

optimisation process.

The system implemented has been used to carry out

fir-tree root optimisation using a two-stage hybrid

strategy combining gradient-based methods and GAs. It

is now clear that without a large number of evaluations,

it is not easy to explore the effects of different geometric

parameters and find the best possible solutions using

direct searches. Such large numbers of finite elementanalyses are computationally expensive even for a model

of modest complexity. Further research on the con-

struction of approximation models and strategies on

how to use them will be required to achieve further ef-

ficiency gains as well as and most benefit from the in-

tegration and automation offered by the use of ICAD

modelling.

Acknowledgements

This work was funded by the EPSRC under grant

reference GR/M53158 and the University Technology

Partnership for Design, which is a collaboration between

Rolls-Royce, BAE SYSTEMS and the Universities ofCambridge, Sheffield and Southampton.

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Turbine Blade Fir Tree FEA

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