In the last 15 years, automatic design optimisation has been the subject of ever ... system for a three-spool modern turbofan engine. Model. This study ... Optimiser: The multi-objective Tabu Search (TS) algorithm de- veloped by .... AIAA Multidisciplinary Design Optimization Specialist Conference, Honolulu,. Hawaii, USA ...
Accelerating Design Optimisation via Principal Components’ Analysis Tiziano Ghisu, Geoffrey T. Parks, Jerome P. Jarrett, P. John Clarkson Engineering Design Centre, Department of Engineering, University of Cambridge
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• Optimiser : The multi-objective Tabu Search (TS) algorithm developed by Jaeggi [4] was selected for this work. TS is a metaheuristic method designed to help a search negotiate difficult regions by imposing restrictions [3]. TS was found particularly effective in the solution of a number of aerodynamic problems, where the large number of variables and constraints generate multiple infeasible regions and a highly multi-modal and fragmented landscape that complicates notably the optimisation process. The TS approach is able to navigate this complex space more efficiently, while the larger changes applied by GAs generally lead to a considerable number of infeasible designs. The optimisation of the design-point performance of the system in Figure 1 was presented in [2]. The optimisation aimed to maximise system efficiency and IPC and HPC surge margin, subject to a number of aerodynamic constraints (summarised in Table 1(b)), for a total of 95 design variables and 3 objectives. Table 1: Design space and optimisation problem definition Variable Type Count mean-line 2m + 2 area distribution 3m + 2 stage pressure ratios n stator flow exit angles n blade axial chords 2n blade numbers 2n Total 5m + 6n + 4 ( a ) Design space (with n number of compressor stages and m number of modules)
maximise ηis,tot SMIP C SMHP C subject to DHmin ≥ DH SP Rmax ≤ SP R DFmax ≤ DF Kochmax ≤ Koch HmaxDU CT ≤ H
In this study, the choice of the parameterisation scheme has been mainly dictated by the evaluation software in use, but a decision has been taken not to reduce the number of variables arbitrarily, with the associated drawbacks of increased design space complexity (making the use of a stochastic optimiser essential) and computational time. Specific means of overcoming these drawbacks therefore need to be devised.
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( a ) Variances
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Figure 2: A different basis can reduce the effective dimensionality of the problem In more general terms, the objective is to find the best linear orthogonal transformation (or orthonormal for unicity) of the original set of coordinatess, so that the transformation a = ΦTx is optimal. Geometrically this is equivalent to rotating the system of coordinates (see Figure 2(a)) so that the new basis is optimal. The result is the well-known Karhunen-Loeve (KL) expansion, particularly important in Pattern Recognition and usually referred to as Proper Orthogonal Decomposition (POD), Principal Components’ Analysis (PCA) or Empirical Orthogonal Functions[5]. Mathematically, the best eigenvector φ(1) can be defined as the one that maximises the mean square projection of the data: φ(1)
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subject to (φ(1), φ(1)) = 1
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where (, ) represents the scalar product operator and hi represents summation over all the elements of the data set. This constrained optimisation problem can be solved via the Lagrange multipliers technique: it is equivalent to maximising g1(λ1, φ(1)) = h(φ(1), x)2i − λ1[(φ(1), φ(1)) − 1]
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The randomness associated with the use of a meta-heuristic optimiser for the solution of a multi-objective problem makes a direct quantification of the advantages of the new coordinate system complicated. To provide a simpler measure of the benefits, a singleobjective optimisation experiment has been conducted (aiming to maximise total system efficiency). The results are shown in Figure 4(b): not only is the search in the principal components’ system of coordinates much faster, but the results of the optimisation also improve because of the reduced complexity of the design space itself (Figure 4(a)). Considering only the most significant principal components leads to a further reduction in computational cost (less computational effort per time step) in comparison to the optimisation conducted in the full principal components’ space, with a small penalty on the value of the objective function. The computational time is reduced by a factor of 20.
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Figure 3: Principal Components’ Analysis
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Figure 4: Acceleration via PCA
Noting that h(φ(1), x)2i = (φ(1), xxTφ(1)) and defining the covariance matrix C = hxxTi, equation (3) can be rewritten as
The same approach has been used for the solution of a number of more complex multi-objective problems. Detailed discussion is presented in the companion paper.
g1(λ1, φ(1)) = h(φ(1), Cφ(1)) − λ1[(φ(1), φ(1)) − 1]
Conclusions
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The stationary points of this equation are found from ∂g1 ∂φ(1)
= 2Cφ(1) − 2λ1φ(1) = 0
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or Cφ(1) = λ1φ(1).
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The other components of this optimal basis can be found by solving a system equivalent to that in equation (3) with the further constraint of orthogonality with respect to the previously determined elements – this constraint is immediately satisfied because of the symmetry of matrix C [5]. The elements of the optimal basis come from solving an eigenvectors problem equivalent to 6. An important property of the KL expantion is that the variances along each coordinate direction are proportional to the associated eigenvalues of the matrix C. As the variance represents a measure of the “energy” associated with the specified direction, it is clear that this gives an immediate criterion for choosing the most significant components. The same approach used in Pattern Recognition to produce a more accurate model of the problem under consideration can be used in optimisation for determining both the most efficient system of coordinates and a reduced design space containing potentially the best designs.
( b ) Optimisation problem
Methodology
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• Geometry Modeler : The shape of each compressor annulus is specified through the definition of mean-line and area distributions: a fourth order polynomial is used in the mean-line definition, while a fifth order one is used to define the area distribution, in order to give greater flexibility in each stage flow function. Pressure ratios across each stage, stator exit angles, number of blades and axial chords are allowed to vary. The same parameterisation used for the compressor annulus is also used for the duct endwalls, with C1 continuity being imposed at the interfaces to ensure smooth hub and casing surfaces. The design space is summarised in Table 1(a).
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• Evaluation Tool: A proprietary code for mean-line performance prediction is used to evaluate the compressor. The duct is modeled using the Finite Volume axi-symmetric CFD solver developed by Ghisu et al. [1] in combination with a Reduced Order Model to minimise the number of CFD calls.
(7)
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where N is the design space size. It is evident (Figure 3(b)) that the new basis is much more efficient than the original one in describing the data set, which means that a smaller number of variables can be used to approximate the design space. The variance associated with each of the original variables is presented in Figure 3(a).
Model
Figure 1: Meridional view of a core compression system
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This study concentrates on the preliminary aerodynamic optimisation of the core compression system of a three-spool modern aeroengine, shown in Figure 1 and composed of an intermediate pressure compressor (IPC), an s-shaped duct and a high pressure compressor (HPC).
p X
V (fraction of total variance)
In the last 15 years, automatic design optimisation has been the subject of ever growing interest, thanks to the development of ever more reliable analysis software, efficient optimisation methods and powerful computers. The parameterisation represents the “critical enabling factor” for an efficient exploration of the design space: it is essential to ensure that the parameterisation scheme is able to cover all feasible designs, using the minimum possible number of parameters, as these affect the convergence time of the optimiser. The optimisation of complex products is likely to translate into a large design space and an a priori reduction is not advisable, as this could lead to the loss of potentially good designs. A method based on Principal Components’ Analysis is introduced: given a generic parameterisation, this allows an optimal representation to be derived that will facilitate a faster and more complete exploration of the design space. The advantages are demonstrated through two optimisation test cases from the design of a core compression system for a three-spool modern turbofan engine.
important coordinates and implementing an informed reduction of the design space dimensionality. The total fraction of the data set variance captured by p principal coordinates can be calculated as
ASSOCIATED VARIANCE
Introduction
In the same way that a physical problem that appears threedimensional in a Cartesian coordinate system can appear twodimensional in a cylindrical or spherical coordinate system, a different parameterisation of the design space can be more effective in describing the design space itself. For example, if the optimal designs in the three-dimensional space in Figure 2(a) are all located along a straight line, a rotation of the coordinate system can reveal the secondary importance of a variable and allow an effective reduction of the design space. The same is true in higher dimensionality problems, with potentially even more significant computational savings. The main advantages from an optimisation point of view are: • The search will be much faster than in the original coordinate system, as significantly fewer optimisation steps are needed to explore the same optimal designs. • The size of the design space can be reduced by considering only the most important components.
Results A possible approach for introducing the optimal basis in the optimisation process is the following: the optimisation problem is run for a pre-determined number of optimisation steps in the original design basis, and at this point the set of non-dominated solutions is used to generate the principal components’ space. The optimisation is then continued in the new coordinate system, which should offer a better parameterisation of the problem and thus make the search more efficient. The eigenvalues of the covariance matrix C are proportional to the variances calculated along the corresponding coordinates and thus offer a direct way for evaluating the most
The design of complex products is likely to translate into a large design space and, while an a priori reduction of this is not advisable, specific means of improving the search effectiveness are fundamental. In this study, a method by which the parameterisation can be reoriented and (temporarily) reduced in size based on the idea of Principal Components’ Analysis, from the field of Pattern Recognition, is introduced, implemented within a Tabu Search optimisation scheme and used in the solution of two optimisation problems of growing complexity from the design of the core compression system for a three-spool aeroengine. The capabilities of this approach in reducing optimisation times while improving the search effectiveness are clearly demonstrated. References [1] T. Ghisu, M. Molinari, G. T. Parks, W. N. Dawes, J. P. Jarrett, and P. J. Clarkson. Axial compressor intermediate duct design and optimisation. In 3rd AIAA Multidisciplinary Design Optimization Specialist Conference, Honolulu, Hawaii, USA, 2007. [2] T. Ghisu, G. T. Parks, J. P. Jarrett, and P. J. Clarkson. Design optimisation of gas turbine compression systems. In 4th AIAA Multidisciplinary Design Optimization Specialist Conference, Chicago, Illinois, USA, 2008. [3] F. Glover and M. Laguna. Tabu Search. Kluver Academic Publisher, USA, 1997. [4] D. M. Jaeggi, G. T. Parks, T. Kipouros, and P. J. Clarkson. The development of a multi-objective Tabu Search algorithm for continuous optimisation problems. European Journal of Operational Research, 185:1192–1212, 2008. [5] M. Kirby. Geometric Data Analysis. An Empirical Approach to Dimensionality Reduction and the Study of Patterns. John Wiley and Sons, 2002.
Acknowledgements This project has been funded by the Autonomous Region of Sardinia, the European Trust and the UK Engineering and Physical Sciencies Research Council (EPSRC) under grants GR/R64100/01 and EP/001777/1.
