Shape optimization frequently works with geometries involving several dozen design variables. The high dimensionality itself can be an impediment to efficient optimization. Moreover, a possibly high number of explicit/implicit constraints restrict the design space. Traditional CAD geometric parameterization methods present serious difficulties in expressing these constraints leading to a high failure rate of generating admissible shapes. In this paper, we discuss shape interpolation between admissible instances of finite element/CFD meshes. We present an original approach to automatically generate a hyper-surface locally tangent to the manifold of admissible shapes in a properly chosen linearized space. This permits us to reduce the size of the optimization problem while allowing us to morph exclusively between feasible shapes. To this end, we present a two-level a posteriori mesh parameterization approach for the design domain geometry. We use Principal Component Analysis and Diffuse Approximation to replace the geometry-based variables with the smallest set of variables needed to represent an admissible shape for a chosen precision. We demonstrate this approach in two typical shape optimization problems.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Allaire G, Jouve F, Toader A-M (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 194:363–393
Berkooz G, Holmes P, Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Ann Rev Fluid Mech 25:539–575
Bregler C, Omohundro SM (1995) Nonlinear image interpolation using manifold learning. In: Tesauro G, Touretzky DS, Leen TK (eds) Advances in neural information processing systems 7. MIT Press, Cambridge, pp. 973–980
Breitkopf P (1998) An algorithm for construction of iso-valued surfaces for finite elements. Eng Comput 14:146–149
Breitkopf P, Rassineux A, Touzot G, Villon P (2000) Explicit form and efficient computation of mls shape functions and their derivatives. Int J Numer Methods Eng 48:451–466
Breitkopf P, Naceur H, Rassineux A, Villon P (2005) Moving least squares response surface approximation: formulation and metal forming applications. Comput Struct 83:1411–1428
Bui-Thanh T, Willcox K, Ghattas O, van Bloemen Waanders B (2007) Goal-oriented, model-constrained optimization for reduction of large-scale systems. J Comput Phys 224:880–896
Canny J (1986) A computational approach to edge detection. IEEE Trans Pattern Anal Mach Intell 8:679–698
Carlberg K, Farhat C (2008) A compact proper orthogonal decomposition basis for optimization-oriented reduced-order models. In: 12th AIAA/ISSMO multidisciplinary analysis and optimization conference, Victoria
Carlberg K, Farhat C (2010) A low-cost, goal-oriented compact proper orthogonal decomposition basis for model reduction of static systems. Int J Numer Methods Eng 86:381–402
Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Curr Sci Spec Sect Comput Sci 78:808–817
Chinesta F, Ammar A, Cueto E (2010) Proper generalized decomposition of multiscale models. Int J Numer Methods Eng 83:1114–1132
Coelho RF, Breitkopf P, Knopf-Lenoir C (2009) Bi-level model reduction for coupled problems. Int J Struc Multidisc Optim 39:401–418
Cordier L, El Majd BA, Favier J (2010) Calibration of pod reduced order models using tikhonov regularization. Int J Numer Methods Fluids 63:269–296
Couplet M, Basdevant C, Sagaut P (2005) Calibrated reduced-order pod-galerkin system for fluid flow modeling. J Comput Phys 207:192–220
Dulong J-L, Druesne F, Villon P (2007) A model reduction approach for real-time part deformation with nonlinear mechanical behavior. Int J Interact Des Manuf 1:229–238
Duvigneau R (2006) Adaptive parameterization using free-form deformation. INRIA Research Report RR-5949
Fukunaga K, Olsen D (1971) An algorithm for finding intrinsic dimensionality of data. IEEE Trans Comput 20:176–183
Jan S, Zolesio J-P (1992) Introduction to shape optimization: shape sensitivity analysis. Springer, Berlin
Kaufman A, Cohen D, Yagel R (1993) Volume graphics. IEEE Comput 26:51–64
LeGresley P, Alonso J (2000) Airfoil design optimization using reduced order models based on proper orthogonal decomposition. In: Fluids 2000 conference and exhibit, Denver
Murat F, Simon J (1976) Sur le controle par un domaine geometrique. Pre-publication du Laboratoire d’Analyse Numerique, no 76015. Universite de Paris 6
Nayroles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10:307–318
OMD2-project home-page (2009) http://omd2.scilab.org/. Accessed 22 Feb 2011
Ravindran SS (2000) A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. Int J Numer Methods Fluids 34:425–448
Sahan RA, Gunes H, Liakopoulos A (1998) A modeling approach to transitional channel flow. Comput Fluids 27:121–136
Schulz V (2012) A riemannian view on shape optimization. Found Comput Math
Sofia AYN, Meguid SA, Tan KT (2010) Shape morphing of aircraft wing: status and challenges. Mater Des 31:1284–1292
Willcox K, Peraire J (2002) Balanced model reduction via the proper orthogonal decomposition. AIAA J 40:2323–2330
This work has been supported by the French National Research Agency (ANR), through the COSINUS program (project OMD2 no. ANR-08-COSI-007). The authors acknowledge the Projet Pluri-Formations PILCAM2 at the Universite de Technologie de Compiegne (URL: http://pilcam2.wikispaces.com) for providing HPC resources that have contributed to the results reported.
About this article
Cite this article
Raghavan, B., Breitkopf, P., Tourbier, Y. et al. Towards a space reduction approach for efficient structural shape optimization. Struct Multidisc Optim 48, 987–1000 (2013). https://doi.org/10.1007/s00158-013-0942-5
- Model reduction
- Diffuse approximation
- Space reduction