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Approximate Hessian for accelerated convergence of aerodynamic shape optimization problems in an adjoint-based framework. (English) Zbl 1390.74161
Summary: The current work explores the use of an approximate Hessian to accelerate the convergence of an adjoint-based aerodynamic shape optimization framework. Exact analytical formulations of the direct-direct, adjoint-direct, adjoint-adjoint, and direct-adjoint Hessian approaches are presented and the equivalence between the adjoint-adjoint and direct-adjoint formulations is demonstrated. An approximation of the Hessian is obtained from the analytical formulation by partially solving first-order sensitivities to reduce computational time, while neglecting second-order sensitivities to ease implementation. Error bounds on the resulting approximation are presented for the first-order sensitivities through perturbation analysis. The proposed method is first assessed using an inverse pressure problem for a quasi-one-dimensional Euler flow. Additionally, three-dimensional inviscid transonic test cases are used to demonstrate the effectiveness of the method.
74P15 Topological methods for optimization problems in solid mechanics
65K10 Numerical optimization and variational techniques
49Q10 Optimization of shapes other than minimal surfaces
76G25 General aerodynamics and subsonic flows
Full Text: DOI
[1] Pironneau, O., Optimal shape design for elliptic systems, System Modeling and Optimization, Lecture Notes in Control and Information Sciences, 38, 42-66, (1982), Springer Berlin Heidelberg · Zbl 0496.93029
[2] Jameson, A., Aerodynamic design via control theory, J Sci Comput, 3, 3, 233-260, (1988) · Zbl 0676.76055
[3] Jameson, A.; Martinelli, L.; Pierce, N., Optimum aerodynamic design using the Navier-Stokes equations, Theor Comput Fluid Dyn, 10, 1, 213-237, (1998) · Zbl 0912.76067
[4] Rumpfkeil, M. P.; Mavriplis, D. J., Efficient Hessian calculations using automatic differentiation and the adjoint method with applications, AIAA J, 48, 10, 2406-2417, (2010)
[5] Arian, E.; Taasan, S., Analysis of the Hessian for aerodynamic optimization: inviscid flow, Comput Fluids, 28, 7, 853-877, (1999) · Zbl 0969.76015
[6] Schmidt, S., Efficient large scale aerodynamic design based on shape calculus, (2010), University of Trier, Germany, Ph.D. thesis
[7] Sherman, L. L.; III, A. C.T.; Green, L. L.; Newman, P. A.; Hou, G. W.; Korivi, V. M., First- and second-order aerodynamic sensitivity derivatives via automatic differentiation with incremental iterative methods, J Comput Phys, 129, 2, 307-331, (1996) · Zbl 0933.76070
[8] Papadimitriou, D. I.; Giannakoglou, K. C., Direct, adjoint and mixed approaches for the computation of Hessian in airfoil design problems, Int J Numer Methods Fluids, 56, 10, 1929-1943, (2008) · Zbl 1141.76058
[9] Ghate, D.; Giles, M., Efficient Hessian calculation using automatic differentiation, Proceedings of the Twenty-Fifth AIAA Applied Aerodynamics Conference, 2406-2417, (2007)
[10] Papadimitriou, D. I.; Giannakoglou, K. C., Aerodynamic design using the truncated Newton algorithm and the continuous adjoint approach, Int J Numer Methods Fluids, 68, 6, 724-739, (2012) · Zbl 06059466
[11] Hicken, J. E., Inexact Hessian-vector products in reduced-space differential-equation constrained optimization, Optim. Eng., 15, 3, 575-608, (2014) · Zbl 1364.49037
[12] Gill, P. E.; Murray, W.; Saunders, M. A., SNOPT: an SQP algorithm for large-scale constrained optimization, SIAM Rev., 47, 1, 99-131, (2005) · Zbl 1210.90176
[13] Schittkowski, K., NLPQLP: A Fortran implementation of a sequential quadratic programming algorithm with distributed and non-monotone line search - user’s guide, version 3.0, Report, (2009), Department of Mathematics, University of Bayreuth
[14] Wächter, A.; Biegler, L. T., On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Math Program, 106, 1, 25-57, (2006) · Zbl 1134.90542
[15] Broyden, C. G., A new double-rank minimization algorithm, Not. Am Math Soc, 16, 670, (1969)
[16] Papadimitriou, D. I.; Giannakoglou, K. C., Computation of the Hessian matrix in aerodynamic inverse design using continuous adjoint formulations, Comput Fluids, 37, 8, 1029-1039, (2008) · Zbl 1237.76162
[17] Papadimitriou, D. I.; Giannakoglou, K. C., Aerodynamic shape optimization using first and second order adjoint and direct approaches, Arch Comput Methods Eng, 15, 4, 447-488, (2008) · Zbl 1170.76348
[18] Giannakoglou, K. C.; Papadimitriou, D. I., Adjoint methods for shape optimization, 79-108, (2008), Springer Berlin Heidelberg Berlin, Heidelberg · Zbl 1298.76145
[19] Papadimitriou, D. I.; Giannakoglou, K. C., The continuous direct-adjoint approach for second order sensitivities in viscous aerodynamic inverse design problems, Comput Fluids, 38, 8, 1539-1548, (2009) · Zbl 1242.76300
[20] Hascoët, L.; Pascual, V., The tapenade automatic differentiation tool: principles, model, and specification, ACM Trans Math Softw, 39, 3, (2013) · Zbl 1295.65026
[21] Poirier, V.; Nadarajah, S., Efficient reduced-radial basis function-based mesh deformation within an adjoint-based aerodynamic optimization framework, AIAA J. Aircr., 53, 6, 1905-1921, (2016)
[22] Balay S., Abhyankar S., Adams M.F., Brown J., Brune P., Buschelman K., et al. PETSc Web page. http://www.mcs.anl.gov/petsc; 2016a.
[23] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K., Petsc users manual, Tech. Rep., (2016), Argonne National Laboratory
[24] Balay, S.; Gropp, W. D.; McInnes, L. C.; Smith, B. F., Efficient management of parallelism in object oriented numerical software libraries, (1997), Birkhäuser Press · Zbl 0882.65154
[25] Petersen K.B., Pedersen M.S.. The matrix cookbook. 2012. Version 20121115, http://www2.imm.dtu.dk/pubdb/p.php?3274.
[26] Li, R.-C., Relative perturbation theory: I. eigenvalue and singular value variations, SIAM J Matrix Anal Appl, 19, 4, 956-982, (1998) · Zbl 0917.15009
[27] Ostrowski, A. M., A quantitative formulation of sylvester’s law of inertia, Proc. Natl. Acad Sci, 45, 740-744, (1959) · Zbl 0087.01802
[28] Tatsumi, S.; Martinelli, L.; Jameson, A., A new high resolution scheme for compressible viscous flows with shocks, Proceedimgs of the Thirty-Third AIAA Aerospace Sciences Meeting and Exhibit, (1995) · Zbl 0824.76058
[29] Martinelli, L.; Jameson, A., Validation of a multigrid method for the Reynolds averaged equations, Proceedings of the AIAA Twenty-Sixth Aerospace Sciences Meeting, (1988)
[30] Cagnone, J.; Sermeus, K.; Nadarajah, S.; Laurendeau, E., Implicit multigrid schemes for challenging aerodynamic simulations on block-structured grids, Comput Fluids, 44, 1, 314-327, (2011) · Zbl 1271.76179
[31] Sederberg, T. W.; Parry, S. R., Free-form deformation of solid geometric models, SIGGRAPH Comput Graph, 20, 4, 151-160, (1986)
[32] de Boer, A.; van der Schoot, M.; Bijl, H., Mesh deformation based on radial basis function interpolation, Fourth MIT Conference on Computational Fluid and Solid Mechanics, 85, 784-795, (2007)
[33] Jakobsson, S.; Amoignon, O., Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization, Comput Fluids, 36, 6, 1119-1136, (2007) · Zbl 1194.76253
[34] Skajaa, A., Limited memory BFGS for nonsmooth optimization, (2010), Master’s thesis, New York University
[35] Nocedal, J., Theory of algorithms for unconstrained optimization, Acta Numer, 1, 199-242, (1992) · Zbl 0766.65051
[36] Bisson, F.; Nadarajah, S., Adjoint-based aerodynamic optimization of benchmark problems, Proceedings of the Fifty-Third AIAA Aerospace Sciences Meeting, (2015), American Institute of Aeronautics and Astronautics
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