Turbine blade fir-tree root design optimisation usingintelligent 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, UK b 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 â ) 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 â 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. Ó 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 virtual environment 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) 1853–1867 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 Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S0045- 7949( 02) 00225- 0 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 flexible framework 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 in a 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 â 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 smooth transfer 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 distribution based 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 1854 W. Song et al. / Computers and Structures 80 (2002) 1853–1867 functional relationships between thousands of entities in a complex product model [6,7]. What is produced by this kind of CAD system is the final results of design, not the modelling and analysis process, which is often docu- mented separately. Whenever the designer would like to review the design or rebuild the modelling process with some even minor modifications this usually takes a long time. This kind of knowledge is documented or retained as the experience of the designers which is difficult to utilise during subsequent iterations by other designers. Expert system and knowledge-based approaches have made some progress in this regard. The intelligent CAD system (ICAD) from Knowledge Technologies Interna- tional is a combination of knowledge-based engineering (KBE) and CAD technology and its generative model- ling capability can be used to provide flexible and robust geometry for subsequent analysis. ICAD is a knowledge-based computer-aided design tool. Although geometry is the main object of manipu- lation, the most interesting feature is that it can be used to capture a corporate knowledge base along with best practice and performance, manufacturing and cost cri- teria into a complete product model, known as a ‘‘gen- erative model’’. The difference between using KBE tools and using conventional CAD lies in the following as- pects: KBE is knowledge oriented, while CAD is ge- ometry oriented; CAD works from the bottom up with detailed dimensions while KBE works on conceptual level and can harness all the design specifications in- cluding physical laws, material criteria, manufacturing constraints, and even non-technical aspects. KBE uses rule-based model generation methods in conjunction with a CAD modelling engine supporting solid model- ling techniques. The rules which are used to describe the geometric relationships can be expressed in various forms, such as equalities or inequalities using mathe- matical and logical relationships. It can further make some decisions about design on the basis of the rules supplied to it. This will give the model some kind of intelligence. It should be noted that there are other CAD/CAE tools such as Pro/Engineer [8], which also offers feature-based parametric solid modelling capa- bilities. Moreover, most CAD/CAE tools also provide their own ‘macro’ languages that form the basis for the extension of their functionality. The choice of ICAD in this study is based on the flexibility offered by incorpo- rating the LISP language into the IDL that is used in model construction. Analysis based optimisation has been identified as being able to play a significant part in the process of producing high quality designs against known con- straints. However the embedding of optimisation in the design process requires the parameterisation of the de- sign and automatic updating of the model used in the analyses. ICAD can be used to implement this capability because of its object-oriented, descriptive geometry modelling approach. This allows engineers to quickly and accurately explore and test multiple ‘‘what ifs’’ against all known constraints. This makes it possible to optimise the design in a reduced time scale. Here, the design of the fir-tree geometry is carried out using ICAD. The basic procedure of the geometry de- sign falls into two steps: first to identification of the features and rules used to define the geometry and sec- ondly the breaking down of the whole model into several modules, each of which becomes a building block in a hierarchical structure. In ICAD, each of these basic blocks is described using the ICAD design language (IDL) as a generic definition which can be implemented in the ICAD browser using a specific set of parameter values. Thus the model is defined parametrically: dif- ferent sets of parameter values will result in different Fig. 1. Overall system architecture. W. Song et al. / Computers and Structures 80 (2002) 1853–1867 1855 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 cancel the 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. Using unique 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 the blade, 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 blade–disk 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. 1856 W. Song et al. / Computers and Structures 80 (2002) 1853–1867 elastic methods [9,10], now mainly using finite element analysis [11–13]. Modern finite element codes already have the capability of dealing with thermal–mechanical coupling and contact analysis between blade root and disk head. It is now relatively easy to obtain the stress distribution 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 the overall 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 Parameter Drad (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 (mm 2 ) Cooling passage area Parameter Tfa (/) (°) Top flank angle Parameter Ufa (c) (°) Under flank angle Parameter Fdcr (mm) First disk crest radius Parameter Inr (mm) Inner radius Parameter Dhnw (D) (mm) Disk head neck width Derived Bga (mm 2 ) Bucket groove area Derived Bch (H) (mm) Bottom-to-contact height Derived Bl (mm) Bedding length Derived Fh (mm) Fir-tree height Derived W. Song et al. / Computers and Structures 80 (2002) 1853–1867 1857 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 complicates the 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 root centre 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. 1858 W. Song et al. / Computers and Structures 80 (2002) 1853–1867 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 rim load 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. W. Song et al. / Computers and Structures 80 (2002) 1853–1867 1859 sections. The number of teeth is treated as a design variable in this problem and the number of constraints is dependent on the number of teeth. The other was to find the optimum tooth profile to minimise the maximum notch stress. The design variables (Table 2) and con- straints (Table 3) used in the second problem are a subset of those defined in the first problem (which has 14 design variables and up to 53 constraints for a three- tooth design) although the goals are different. The constraints are divided into two categories, geometric and mechanical, which are summarized in Table 4 along with the normalized values for the initial design. For the meaning of symbols used in this section, please see Fig. 3a and b. The normalization adopted here is described as follows for upper and lower limits u and l, respectively: a. constraints with upper bounds only: y norm ¼ y=u; u 6¼ 0 y; u ¼ 0 ; b. constraints with lower bounds only: y norm ¼ Àl=y; y 6¼ 0; l > 0 Ày=l; l < 0 or y ¼ 0 y; l ¼ 0 ; 8 < : c. constraints with both upper and lower bounds: y norm ¼ 2 y Àl u Àl À1: These different formula will make it possible for all normalized constraints to have consistent behaviour when the design is moving from an infeasible region towards feasibility, and to have the values of )1 or þ1 at the boundary of the constraints. 5.1. Mesh density parameters As mentioned earlier, it is necessary to establish the appropriate values for mesh control parameters. The purpose is to find a compromise between the high computational costs that are incurred for very fine me- shes and the accuracy required to capture the maximum stresses in the notch area. Therefore, the local and global Table 2 Definition of optimisation problems Variables Number of vari- ables 6 14 Tooth profile parameters Root wedge angle (°) 20–40 20–40 Tooth pitch (mm) 2.0–4.0 2.0–4.0 Blade crest radius (mm) 0.2–1.0 0.2–1.0 Blade trough radius (mm) 0.2–1.0 0.2–1.0 Disk crest radius (mm) 0.2–1.0 0.2–1.0 Disk trough radius (mm) 0.2–1.0 0.2–1.0 Fir-tree root/disk head parameters Skew angle (°) [15] a 10–20 Axial length of root (mm) [20] 15–25 Shank neck width (mm) [6.7615] 6.5–7.5 Fir-tree shoulder width (mm) [9.8943] 8–12 Bottom tooth crest radius (mm) [1.0668] 0.8–1.2 Cooling passage width (mm) [1.3455] 1.2–1.5 Bucket groove lower radius (mm) [3.5] 3.0–4.0 Bucket groove upper radius (mm) [2.2] 1.5–2.5 a [ ] indicates value when not used in optimisation. Table 3 Constraints and objective function applied in the optimisation problems Number Constraints Number of variables 6 14 1 R1=R2   2 H=D   3 DP=F   4–5 min < DP < max   6 Lca   7 RSA   8 Minimum serration pitch   9 Bottom neck width > pitch   10 Minimum wall thickness   11 Bucket groove area > cooling passage area   12–23 Notch stresses  24–35 Section stresses   36–43 Crushing stresses   44–45 Bucket groove stresses   46–53 Unzipping stresses   Objective function Maximum notch stress Fir-tree frontal area 1860 W. Song et al. / Computers and Structures 80 (2002) 1853–1867 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 the optimisation 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 R2 ½R1=R2 )1.0 )0.8642 – 2 Ratio of H to D ½H=D )1.0 )0.8955 – 3 Ratio of R1 to disk trough ½R1=Ftrough )1.0 )0.5268 – 4 Maximum ratio of tooth pitch to disk trough ½DP=FtroughðmaxÞ – 1.4779 1.0 5 Minimum ratio of tooth pitch to disk trough ½DP=FtroughðminÞ )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 12–19 Maximum blade notch stress [NBL(R)(2 a )] b )1.0 0.9270 1.0 20–23 Maximum disk notch stress [NDL(R)(3)] )1.0 0.9948 1.0 24–29 Maximum blade section stress [SB(1)] )1.0 0.6931 1.0 29–35 Maximum disk section stress [SD(4)] )1.0 0.5623 1.0 36–43 Maximum crushing stress [CS(1)] )1.0 0.6514 1.0 44–45 Maximum bucket groove stress )1.0 0.9023 1.0 46–53 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. W. Song et al. / Computers and Structures 80 (2002) 1853–1867 1861 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 stress Blade Disk Blade Disk Skew angle  þ þ þ  1þ; rest  Shank neck width )  1þ; rest  þ  1, 4); rest  Blade shoulder width þ ) 2,3þ; rest ) 1þ; rest ) þ þ Bottom tooth crest radius  2þ; rest) )   4þ; rest  Cooling passage width  1,2þ; rest ) 1); 2, 3þ þ  1, 2þ; 3, 4) Bucket groove lower radius       Bucket groove upper radius       Tooth pitch þ þ þ  þ ) Blade crest radius 1); rest þ ) þ   þ Blade trough radius ) þ )   1, 2); 3, 4þ Disk crest radius 1); rest þ ) ) þ ) þ Disk trough radius )  )   4); rest þ þ indicates increase, ) indicates decrease,  means no significant effect when variable increases, number indicates the tooth number in top-down order. Fig. 5. Optimisation program structure. 1862 W. Song et al. / Computers and Structures 80 (2002) 1853–1867 locating the exact position of individual optimum given suitable starting points. Here the GA is used to give good starting points for the gradient search meth- ods. An initial analysis on the base design revealed that several geometric and mechanic constraints were vio- lated. These include geometric constraints 4, 9, 10 and 11, and the disk notch stress constraints. Table 4 shows the resulting normalized constraint values for the base design. Note that this means that the GA must first locate feasible designs before it can begin optimi- zation. 5.3. Optimisation results 5.3.1. Results for minimising the fir-tree frontal area GAs often require large number of evaluations of the objective function and constraints. The computational cost involved can soon become prohibitively high when computationally intensive finite element analysis is used to calculate the stresses in the structure. In this problem each evaluation takes about 5–6 min to finish, and not surprisingly, most of this time is engaged in finite ele- ment analysis. This means that it takes about 80 h to finish a 10 generation GA search with a population size of 100 using serial processing. (Note that for some spe- cific sets of parameters, there is no viable geometry that can be constructed, and when this occurs SC03 is simply signalled to cancel the analysis.) Because of the large number of design variables, the optimisation trace may only be simply plotted on con- tour maps of any two variables if these maps are as these maps are necessarily produced while holding all the other variables constant. Furthermore, if only a small number of quantities are chosen as design variables, there may be no feasible designs at all. For an infeasible starting design, it is easier for the optimiser to find a feasible region if a large number of quantities are left as design variables and broad exploratory searches meth- ods are used. A contour map for two design variables has been generated using results from the GA search (Fig. 6). Infeasible geometries and possible analysis failures are also illustrated in the figure. It is noted that identifica- tion of this type of failure is useful for identifying any problems with the implementation of the system (sometimes calculations fail simply because of delays due to overloading of the network), but this is not a concern for optimisation as long as appropriate mea- sures are employed to avoid misleading the optimiser. This is important especially when approximations such as response surfaces are introduced to improve run speeds. A gradient based search is illustrated in Fig. 7a. It can be seen that better starting points do not always converge to better results, depending on the location in the design space. This justifies the use of the whole final population of the GA results instead of just the best one as starting points for gradient search. A com- parison between this simple two-stage strategy and a direct gradient-based search is provided in Fig. 7b, which shows that this two-stage strategy, although simple, works better than a direct gradient search as the initial GA search offers a better chance of steer- ing the optimiser towards global optima, while a direct Fig. 6. Contour map of fir-tree frontal area for root wedge angle and tooth-pitch based on GA results. È: Geometry or analysis failure þinfeasible points; Ã: minimum as shown. W. Song et al. / Computers and Structures 80 (2002) 1853–1867 1863 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 Schwefel’s 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. 1864 W. Song et al. / Computers and Structures 80 (2002) 1853–1867 geometric and mechanic constraints. By looking at the trace data of the search process, it can be seen that the three geometric constraints and the disk notch stresses (identified earlier) remain the major factors af- fecting the optimisation results. Note that the primary changes to geometry occurred during the GA search and are as illustrated in Fig. 9. These are a decrease in tooth pitch, an increase in shank neck width and a decrease in three of the four fillet radii. In addition, the root wedge angle is slightly increased. 5.3.2. Results for minimising the maximum notch stress–– local shape tuning Although minimising the frontal area, and thus the rim-load will reduce overall weight, the life of blade/disc is highly dependent on the notch stresses, and therefore the notch stress may be ultimately minimised for the to achieve required life targets. Therefore, following the search on the full scale problem, a second optimisation problem to minimise the maximum notch stress has been carried out, starting from the best design found in the by previous search. It is expected that this search will drive the geometry in a different direction given the changed goal. The result is shown in Fig. 10. In this case only the six tooth profile parameters were chosen as design variables. It can be seen that although a 25% reduction in the maximum notch stress can be achieved, the fir-tree area is now increased by approximately 11%. Also note that the root wedge angle has dropped significantly while the pitch and all the radii have risen. Clearly, the choice of objective function has significant impact on the final design. 6. Concluding remarks The generative modelling facility provided by the ICAD system enables the rapid evaluation of different design alternatives in an engineering environment. In- corporating such capabilities into a FEA-based struc- tural optimisation process has been shown to be an effective way to reduce design time scales and at the Fig. 8. Contour map of fir-tree frontal area and Hooke and Jeeves search terrace. Fig. 9. Comparison of geometry between the base design and the best of the GA search results. W. Song et al. / Computers and Structures 80 (2002) 1853–1867 1865 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 element analyses 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 of Cambridge, Sheffield and Southampton. References [1] Biedron RT, Mehrotra T, Nelson ML, Preston FS, Rehder JJ, et al. Compute as fast as the engineers can think! NASA/TM-1999-209715. [2] Keane AJ. The OPTIONS design exploration system, Reference Manual and User Guide Version B3.0, Febru- ary, 2000. [3] Samareh JA. Status and future of geometry modelling and grid generation for design and optimisation. J Aircraft 1999;36(1). [4] Knowledge Technologies International, http://www. ktiworld.com. [5] Theory and design. 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