Parallel Coordinates in Computational Engineering ...

54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference April 8-11, 2013, Boston, Massachusetts

AIAA 2013-1750

Parallel Coordinates in Computational Engineering Design Timoleon Kipouros∗ Department of Engineering, Cambridge University, Cambridge CB2 1PZ, United Kingdom

Alfred Inselberg† Computer Science and Applied Mathematics Departments, Downloaded by T. Kipouros on April 18, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-1750

Tel Aviv University, Israel

Geoffrey T. Parks‡ Department of Engineering, Cambridge University, Cambridge CB2 1PZ, United Kingdom

and A. Mark Savill§ Department of Power and Propulsion, Cranfield University, Cranfield MK43 0AL, United Kingdom

Modern Engineering Design involves the deployment of many computational tools. Research on challenging real-world design problems is focused on developing improvements for the engineering design process through the integration and application of advanced computational search/optimization and analysis tools. Successful application of these methods generates vast quantities of data on potential optimum designs. To gain maximum value from the optimization process, designers need to visualise and interpret this information leading to better understanding of the complex and multimodal relations between parameters, objectives and decision-making of multiple and strongly conflicting criteria. Initial work by the authors has identified that the Parallel Coordinates interactive visualisation method has considerable potential in this regard. This methodology involves significant levels of user-interaction, making the engineering designer central to the process, rather than the passive recipient of a deluge of pre-formatted information. In the present work we have applied and demonstrated this methodology in two different aerodynamic turbomachinery design cases; a detailed 3D shape design for compressor blades, and a preliminary mean-line design for the whole compressor core. The first case comprises 26 design parameters for the parameterisation of the blade geometry, and we analysed the data produced from a three-objective optimization study, thus describing a design space with 29 dimensions. The latter case comprises 45 design parameters and two objective functions, hence developing a design space with 47 dimensions. In both cases the dimensionality can be managed quite easily in Parallel Coordinates space, and most importantly, we are able to identify interesting and crucial aspects of the relationships between the design parameters and optimum level of the objective functions under consideration. These findings guide the human designer to find answers to questions that could not even be addressed before. In this way, understanding the design leads to more intelligent decision-making and design space exploration. ∗ Research

Associate, Engineering Design Centre, Member.

† Professor. ‡ Senior

Lecturer, Engineering Design Centre. Senior Member.

§ Professor,

1 of 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by By the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

I.

Introduction

omputational engineering design is a challenging field, which is widely recognized in the aerodynamic C design area, since highly performed innovative design products are achievable. The industrial and academic aeronautical design communities invest significant resources in the development of highly so1, 2

Downloaded by T. Kipouros on April 18, 2013 | http://arc.aiaa.org | DOI: 10.2514/6.2013-1750

3–6

phisticated automated integrated multi-disciplinary aerodynamic design optimization systems. As with most real-world problems, there are multiple (usually conflicting) performance metrics that an engineer might seek to improve in optimizing, for example, the design of turbomachinery blades, wings or other aerodynamic surfaces. This suggests that physical explanation of the optimum behavior is hard to extract. As a consequence, supplementary tools to the computational design process are necessary, with the intention of assisting the post-optimization analysis. It is evident that the visualization of multidimensional data is a challenging problem. The data is generally considered to be truly multidimensional if there is more than about five dimensions. Such data are not uncommon in optimization processes, and in particular when produced from the design parameters of an engineering design problem. Moreover, it is required to understand and study relationships between high numbers of attributes, such as those between design parameters and objectives under consideration for optimization. The suggested approach is based on parallel coordinates multidimensional geometries representation, introduced by Inselberg.7 The parallel coordinates technique is very popular in the field of information visualization.8 By this method, it is possible to visualize multivariate relations (i.e., subsets of RN ) by mapping them uniquely into indexed subsets of R2 .9, 10 This type of visualization directly reflects the conversion of the multidimensional design vector of an optimization design problem to a two-dimensional space. Hence, the complexity of analyzing the high-dimensional design space is reduced significantly when we perform the same analysis in two dimensions. In other words, we develop the ability of analyzing the optimum configurations lying in the high-dimensional design space. Moreover, new avenues are opened and explored in the field of post-optimization analysis and the potential for revealing the behavior of innovative optimum characteristics through the physical nature of the design problem is investigated. One of the advantages of parallel coordinates is the uniform treatment of multiple dimensions that is paramount in exploratory tasks. Another advantage is that the technique allows users to interact with the data in many ways. This is highly useful, as interaction is known to augment the knowledge acquisition process.8 Interacting with parallel coordinate plots in geographic visualization systems is presented by Edsall.11 He emphasizes the ability of the technique to facilitate geographic data exploration and understanding. The aerodynamic design optimization post-analysis is related to investigative visualization or knowledge mining. One of the most common tasks in information acquisition is cluster analysis, and parallel coordinate visualizations have been considered for this purpose in Chou et al.,12 as well as very recently in Woodruff et al.13

II.

Description and Analysis of the Aerodynamic Design Problems

In this paper we explore in depth the results of a three-objective design optimization problem for the detailed aerodynamic design of axial compressor blade rows as investigated by Kipouros et al.,14 in order to identify links between three-dimensional geometrical characteristics and optimum aerodynamic behaviour. Furthermore, we apply our post-optimization analysis technique to a bi-objective optimization for the preliminary design study of a generic 7-stage aero-engine compressor, as defined and described by Ghisu et al.,15 in order to explore any discontinuities in the multi-objective design space and relate them with the behaviour of the design parameter space. A.

Detailed Blade Shape Design for Axial Compressors

We consider the detailed aerodynamic design of axial compressor blade rows by reducing flow separation and the secondary losses that develop along the span. The three objective metrics are profile losses, endwall losses and blockage – detailed definitions and descriptions of the modelling of these metrics can be found in.14 These are the most critical and influential flow characteristics that express the overall efficiency of a compressor row. In addition, in order to maintain the practicality and applicability of the optimum design configurations, we have considered four constraints, both geometrical and aerodynamic. The mass flow rate is treated as an equality constraint, and the mass averaged flow turning, the minimum radius of the leading 2 of 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by By the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

edge, and the tip clearance of the blade are considered as inequality constraints. The objective functions, equation 1, are normalised and include penalty function terms to handle the aerodynamic and geometric constraints.

m ˙ 2 RLE Mi + 250 1 − + 0.4max2 0, 1 − Mi,0 m ˙0 RLE,0 C ∆θ + 0.5max2 0, 1 − + 500max2 0, 1 − ∆θ0 Clim

fi =

(1)


In equation 1, M represents any of the objective metrics, m ˙ is the mass flow rate, RLE is the minimum radius of the leading edge of the blade, ∆θ is the mass-averaged flow turning, and C measures the tip clearance of the blade. The zero subscripts identify the equivalent quantities for the datum blade geometry, the initial design in the optimization process.

Figure 1. The boundary conditions for the Partial Differential Equations blade parametrization technique.

It should be highlighted here that the design parameters of this test case do not reflect direct engineering characteristics, but express the boundary conditions to the 4th order elliptical differential equations that are solved in order to represent the 3D compressor blade geometry, based on the Partial Differential Equations parametrization technique as introduced and defined in.16 A schematic description of this technique is presented in Fig. 1. Table A describes in detail the significance of each design parameter, but it requires additional effort to translate the combination of design parameter values to actual blade geometrical characteristics, such as thickness, chord length, or camber. Table 1. Design parameters translation for the detailed shape optimization Parameter Number 1-10 11-15 16 17, 18 19, 20 21, 22 23, 24 25, 26

Description x-y coordinates of 5 B-Spline control points defining the hub profile 5 distances by which each control point is moved outward and normal to the hub surface, so as to define the tip profile The tip section is rotated by this angle Translate the tip trailing edge Scale the hub profile in x & y dimensions Shift the hub profile in x & y and impose the hub gradient Scale the tip profile in x & y dimensions Shift the tip profile in x & y and impose the tip gradient

In an earlier investigation by Kipouros et al.17 we considered a basic implementation of Parallel Coordinates for the post-optimization analysis of a bi-objective design problem of the same test case. We defined the objective functions following equation 1 and examined the trade-off between blockage (flow separation) and entropy generation rate (as an overall measure of losses). With this analysis we revealed that smooth combination of sweep and lean geometrical characteristics in compressor blades are the major features that 3 of 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by By the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


Figure 2. Full set of data for the three-objective design of compressor blades.

Figure 3. Full dataset with constant value variables eliminated.

control and balance the optimum behaviour of these objective metrics within a highly complex aerodynamic environment. With the new approach presented here using the ParallAX software18 we are able to explore the optimum designs in greater detail and also within the context of a higher number of objective flow metrics. This interactive approach is presented step-by-step through the following Figures of this Section. Initially, the whole dataset is presented in Fig. 2 comprising 54 optimum design configurations. It should be noted here that we express the 26 design parameters, as presented in Table A, with the notation x1 − x26, and the three objective functions as c1 − c3 (profile losses, endwall lossed and blockage respectively). We can directly identify the design parameters that express no variability between these data items, and hence eliminate them from the display (Fig. 3). In the context of an optimization problem, the fixed values of some of the design parameters among the Pareto designs mean that these are responsible for the satisfaction of the equality constraints that describe the objective functions (equation 1) within the optimum level achieved.



Figure 4. Highlighting the strong competition between the three objective functions.

Figure 5. Pinch query for lowest C2 (endwall losses) objective function values.

Figure 4 emphasizes the complexity and very strong competition between the three objective functions; there is not a single data item to satisfy the query for the simultaneous lower half of the objective functions of their value range. We then reveal the corresponding design configurations for the lower half of the value range of the second objective function, c2, as illustrated in Fig. 5. We can see that parameter x9 has a fixed value, which is not expressed through any other data items. Hence, an interval query for this dimension reveals exactly the same pattern (Fig. 6), which means that this particular value of design parameter x9 is responsible for the optimum behaviour of the endwall losses. With a closer look, there is a combination between dimensions x8 and x9, as it is also clear from the corresponding pattern presented in Fig. 7. Design parameter x9 translates into appropriate chord length and camber at the hub of the compressor blade, while design parameter x8 translates into thicker profile towards the trailing edge at the hub, and higher lean. These details are highlighted and presented in Fig. 8 among the extreme optimum blade designs for each of the three objective functions.



Figure 6. Interval query for design parameter X9.

Figure 7. The selected pattern.

However, the opposite behaviour of the extreme endwall losses design is not controlled from a particular parameter. This highlights the fact that the problem is truly multi-variate, but there are common geometrical aspects that describe the overall optimum behaviour, such as the lean at the tip, and the sweep at the midspan area of the blades, as was also explored in.17 B.

Preliminary Design of a 7-Stage Aero-engine Core Compressor

This optimization had the goal of identifying new design configurations that exhibit improvements in performance and operating margin without the requirement of large changes in the geometry of the other engine components. For these reasons, two objective functions were considered: isentropic efficiency and surge margin subject to a set of inequality constraints that secure the practicality and feasibility of the new optimum designs. Hence, a maximum allowed value for Koch factor, De Haller number and static pressure rise coefficient has been specified. In addition, the mass flow and the overall pressure ratio of the compressor were treated as equality constraints, in order to preserve compatibility with the rest of the components of 6 of 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by By the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


Figure 8. The geometrical details that distinguish and characterize the optimum behaviour of c2 (endwall losses) objective function.

the aero-engine. A proprietary mean-line multi-stage axial compressor performance prediction tool, similar to,19 was used for the evaluation of the performance metrics. The modeling of the design problem has been achieved with 45 design parameters as described in Table B. Table 2. Definition of the design parameters for the preliminary design optimization Parameter Number 1-6 7 8 9 10 11 12-18 19-25 26-31 32-38 39-45

Description Pressure ratio in stages First control point for mean line Exit to inlet area ratio Mid-length to inlet area ratio Area distribution inlet rate of change Area distribution exit rate of change Rotor blade length in each stage Stator blade length in each stage Stator exit flow angle in each stage Rotor number of blades in each stage Stator number of blades in each stage

For this design study we deploy the Parallel Coordinates analysis, in order to explore any discontinuities that develop along the trade-off surface. The full dataset, as presented in Fig. 9 is comprised by 724 data items, where c1 and c2 reflect the isentropic efficiency and surge margin objective functions respectively. We then visualize the Pareto front and highlight with a polygon the discontinuity area of the objective design space (Fig. 10). In this way, we can isolate the design configurations that are in the vicinity of the discontinuity (Fig. 11). It is very clear that there are two distinct patterns that express either side of the discontinuous trade-off surface, as highlighted in Fig. 12). Most importantly, there is an exact oposite trend to specific design parameters that reflects this behaviour of the design space. In particular, the exit flow angles of the stator stages 2 and 3. In constrast, the mid-range values for the exit flow angle of the stator stage 5 relate directly to the right hand side of the discontinuity (Fig. 13).

III.

Conclusion

The exploration of the suggested technique for the qualitative design optimization, post-analysis, demonstrates the ability to derive useful physical information through engineering optimization. We can identify patterns that relate design parameters with objective metrics, and constraints that can lead to the definition of various rules that relate geometrical characteristics with physical mechanisms. We can also identify patterns that can prompt the human designer to further perform such detailed exploration and investigation of particular areas of the design space. Furthermore, we can identify new



Figure 9. Full set of data for the preliminary design optimization of aero-engine compressor.

Figure 10. Simultaneous visualization of the Pareto front in Scatter Plot and the whole design space in Parallel Coordinates.

questions, and most importantly we can find the answers to these. During an optimization process the human designer could perform this analysis and identify such critical design characteristics. The automatic optimization process then can be manually redirected to emphasize and explore these particular regions that exhibit interesting features. Finally, we have demonstrated a capability for simultaneous visualization of the whole parametric space, objective functions space, and hard/soft constraints space if required; developing in this way a global picture of a complete hyper-space without any limitations to the dimensionality. Then, by letting the patterns prompt further analysis and investigation, we managed to reveal and analyze complexities of higherdimensional and non-conventional Pareto fronts. If we visualize all the candidate design vectors explored during the optimization process, we can identify the infeasible regions of the design problem. Hence, a better understanding of the complex morphology of the design landscape can be achieved. Moreover, in all the population-based optimization techniques we can assign the quality of the initial population by inspecting the diversity and the percentage covered of the 8 of 11 American Institute of Aeronautics and Astronautics Copyright © 2013 by By the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


Figure 11. The data items that describe the discontinuous area of the Pareto front.

Figure 12. The pattern that expresses the left hand side of the Pareto front.

design space. In this way, we use the human intelligence and judgment during the computational design process, in order to fully benefit from the available engineering design technology. In conclusion, the existing Parallel Coordinates analysis technique can also be extended and deployed directly to the computational design process in a pro-active way. In Kipouros et al.20 the implementation of such a technique and its application to a three-objective turbomachinery test case is described. It is demonstrated how reduction of the dimensionality of the design space can be achieved without compromising the overall efficiency of the optimization process and the revealed Pareto front surface. However, this type of automation should be used wisely and effectively, as there is the risk of losing information within the data - user interactivity is vital.



Figure 13. The regions of the design parameters space that express the discontinuity in the objective functions space.

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18 MDG,

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19 Wright,
