Abstract
Multi-objective optimization has been rising in popularity, especially within an industrial environment, where several cost functions often need to be considered during the design phase. Traditional gradient-free approaches, such as evolutionary algorithms, can be employed to compute a front of equally suitable solutions a designer can choose from, with a high computational cost though, particularly in high-dimensional design spaces. In this paper, a gradient-based algorithm is developed for efficiently tracing the Pareto front in bi-objective aerodynamic shape optimization problems, where an adjoint method is used for the computation of the objective functions’ gradients with respect to the design variables. After obtaining a starting point on the front, a prediction–correction approach is employed to compute new Pareto points. Satisfying the Karush–Kuhn–Tucker conditions provides a prediction for the next point, which is, then, corrected by solving a minimum distance problem. The prediction step, though, requires the costly computation of the Hessian matrix. This is avoided here using the BFGS (Nocedal and Wright in Numerical optimization, Springer, New York, 2006) technique. The proposed method is first demonstrated in a less expensive lift/drag optimization of an isolated airfoil and then applied to the bi-objective optimization of a 3D compressor stator. The extension of the proposed method to cases with more than two objectives is straightforward, on condition that an algorithm is found to coordinate the way the Pareto front is swept.
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References
Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, New York
Marler RT, Arora JS (2004) Survey of multi-objective optimization methods for engineering. Struct Multidiscip Optim 26(6):369–395
Zingg DW, Nemec M, Pulliam TH (2008) A comparative evaluation of generic and gradient-based algorithms applied to aerodynamic optimization. Eur J Comput Mech 17(1–2):103–126
Kyriacou SA, Asouti VG, Giannakoglou KC (2014) Efficient PCA-driven EAs and metamodel-assisted EAs, with applications in turbomachinery. Eng Optim 46(7):895–911
Das I, Dennis JE (1998) Normal-boundary intersection: a new method for generating the pareto surface in nonlinear multicriteria optimization problems. SIAM J Optim 8(3):631–657
Kim IY, de Weck OL (2005) Adaptive weighted-sum method for bi-objective optimization: Pareto front generation. Struct Multidiscip Optim 29(2):149–158
Mueller-Gritschneder D, Graeb H, Schlichtmann U (2009) A successive approach to compute the bounded pareto front of practical multiobjective optimization problems. SIAM J Optim 20(2):915–934
Gebken B, Peitz S, Dellnitz M (2019) On the hierarchical structure of Pareto critical sets. J Glob Optim 73(4):891–913
Shankaran S, Barr B (2011) Efficient gradient-based algorithms for the construction of Pareto fronts. In: ASME Turbo Expo, (2011) ASME paper GT2011-45069. Vancouver, Canada
Fike JA (2013) Multi-objective optimization using hyper-dual numbers. PhD thesis, Stanford University, USA
Peitz S, Ober-Blöbaum S, Dellnitz M (2018) Multiobjective optimal control methods for the navier-stokes equations using reduced order modeling. Acta Applicandae Mathematicae. Springer, The Netherlands, pp 1–29
Banholzer S, Beermann D, Volkwein S (2017) POD-based error control for reduced-order bicriterial PDE-constrained optimization. Annu Rev Control 44:226–237
Schmidt S, Schulz VH (2008) Pareto-curve continuation in multi-objective optimization. Pac J Optim 4(2):243–257
Gkaragkounis K, Papoutsis-Kiachagias E, Asouti V, Giannakoglou K (2018) Adjoint-based pareto front tracing in aerodynamic shape optimization. In: 10th international conference on computational fluid dynamics (ICCFD10), Barcelona, Spain
Papadimitriou DI, Giannakoglou KC (2012) Aerodynamic design using the truncated newton algorithm and the continuous adjoint approach. Int J Num Methods Fluids 68(6):724–739
Tsiakas KT, Trompoukis XS, Asouti VG, Giannakoglou KC (2018) Shape optimization of wind turbine blades using the continuous adjoint method and volumetric NURBS on a GPU cluster. In: Advances in evolutionary and deterministic methods for design, optimization and control in engineering and sciences. Springer, pp 131–144
Giles MB (2000) On the use of Runge-Kutta time-marching and multigrid for the solution of steady adjoint equations. In: AD2000 Conference, Nice, France
Ehrgott M (2005) Multicriteria optimization. Springer, Berlin
Nocedal J, Wright S (2006) Numerical optimization. Springer, New York
Piegl L, Tiller W (1997) The NURBS book. Springer, Berlin
Gagliardi F, Tsiakas KT, Giannakoglou KC (2017) A two-step mesh adaptation tool based on RBF with application to turbomachinery optimization loops. In: EUROGEN international conference 2017, Madrid, Spain
Asouti VG, Trompoukis XS, Kampolis IC, Giannakoglou KC (2011) Unsteady CFD computations using vertex-centered finite volumes for unstructured grids on Graphics Processing Units. Int J Num Methods Fluids 67(2):232–246
Spalart PR, Allmaras SR (1992) A one-equation turbulence model for aerodynamic flows. In: AIAA 30th aerospace sciences meeting and exhibit, Reno, USA
EASY: The Evolutionary Algorithms SYstem. http://velos0.ltt.mech.ntua.gr/EASY/. Accessed 24 Apr 2019
Vasilopoulos I, Flassig P, Meyer M (2017) CAD-based aerodynamic optimization of a compressor stator using conventional and adjoint-driven approaches. In: ASME Turbo Expo, (2017) ASME Paper GT2017-63199. Charlotte, NC, USA
Müller JD (2016) TurboLab Stator. http://aboutflow.sems.qmul.ac.uk/events/munich2016/benchmark/testcase3/. Accessed 26 Apr 2019
Gräsel J, Keskin A, Swoboda M, Przewozny H, Saxer A (2004) A full parametric model for turbomachinery blade design and optimisation. In: ASME 2004 international design engineering technical conferences and computers and information in engineering conference. ASME Paper DETC2004-57467, Salt Lake City, Utah
Dutta AK (2011) An automated multi-objective optimization approach for aerodynamic 3D compressor blade design. Dissertation, Brandenburg University of Technology Cottbus-Senftenberg, Germany
Abbott IH, Doenhoff AE (1960) Theory of wing sections. Including a summary of airfoil data. Dover, New York
Shahpar S, Lapworth L, (2003) PADRAM: Parametric design and rapid meshing system for turbomachinery optimisation. In: ASME Turbo Expo, (2003) ASME Paper GT2003-38698. Atlanta, Georgia
Launder BE, Spalding DB (1974) The numerical computation of turbulent flows. Comput Methods Appl Mech Eng 3(2):269–289
Lapworth BL (2004) HYDRA-CFD: a framework for collaborative CFD development. In: International conference on scientific and engineering computation (IC-SEC) 2004, Singapore
Shahpar S, Caloni S (2013) Aerodynamic optimization of high-pressure turbines for lean-burn combustion system. J Eng Gas Turb Power 135(5):055001
Moinier P (1999) Algorithm developments for an unstructured viscous flow solver. PhD thesis, University of Oxford, UK
Moinier P, Müller JD, Giles MB (2002) Edge-based multigrid and preconditioning for hybrid grids. AIAA J 40(10):1954–1960
Mihalyovics J, Brück C, Vasilopoulos I, Meyer M (2018) Numerical and experimental investigations on optimized 3D compressor airfoils. In: ASME Turbo Expo 2018. ASME Paper GT2018-76826, Lillestrøm (Oslo), Norway
Vasilopoulos I, Agarwal D, Meyer M, Robinson TT, Armstrong CG (2016) Linking parametric CAD with adjoint surface sensitivities. In: ECCOMAS congress 2016, Crete Island, Greece
Xu S, Radford D, Meyer M, Müller JD (2015) CAD-based adjoint shape optimisation of a one-stage turbine with geometric constraints. In: ASME Turbo Expo, (2015) ASME Paper GT2015-42237. Montreal, Canada
Initial Training Network (2015) Industrial optimal design using adjoint CFD. http://ioda.sems.qmul.ac.uk/. Accessed 26 Apr 2019
Acknowledgements
The first author was an ESR of the IODA project [39], funded by the European Union HORIZON 2020 Framework Programme for Research and Innovation under Grant Agreement No. 642959, during the period 2015-2018. In addition, the authors would like to thank Rolls-Royce Deutschland for the permission to publish the results regarding the turbomachinery application.
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Vasilopoulos, I., Asouti, V.G., Giannakoglou, K.C. et al. Gradient-based Pareto front approximation applied to turbomachinery shape optimization. Engineering with Computers 37, 449–459 (2021). https://doi.org/10.1007/s00366-019-00832-y
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DOI: https://doi.org/10.1007/s00366-019-00832-y