×

Full analytical sensitivities in NURBS based isogeometric shape optimization. (English) Zbl 1231.74352

Summary: Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and its popularity among CAD systems. The advent of NURBS based isogeometric analysis has made it even more advantageous to use NURBS in shape optimization since it can potentially avoid the inaccuracy and labor-tediousness in geometric model conversion from the design model to the analysis model.Although both positions and weights of NURBS control points affect the shape, until very recently, usually only control point positions are used as design variables in shape optimization, thus restricting the design space and limiting the shape representation flexibility.This paper presents an approach for analytically computing the full sensitivities of both the positions and weights of NURBS control points in structural shape optimization. Such analytical formulation allows accurate calculation of sensitivity and has been successfully used in gradient-based shape optimization.The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids and spheres in 3D that are otherwise not possible without the weights as design variables.

MSC:

74P10 Optimization of other properties in solid mechanics
65D17 Computer-aided design (modeling of curves and surfaces)
65D07 Numerical computation using splines
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Braibant, V.; Fleury, C., Shape optimal design using b-splines, Computer Methods in Applied Mechanics and Engineering, 44, 247-267 (1984) · Zbl 0525.73104
[2] Samareh, J. A., A Survey Of Shape Parameterization Techniques. NASA Langley Technical Report (1999)
[3] Hughes, T. J.R.; Cottrell, J. A.; Bazilevs, Y., Isogeometric analysis: Cad, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, 4135-4195 (2005) · Zbl 1151.74419
[4] Qian, X.; Dutta, D., Physics-based modeling for heterogeneous objects, ASME Transactions Journal of Mechanical Design, 125, 416-427 (2003)
[5] Yang, P.; Qian, X., A b-spline based approach to heterogeneous object design and analysis, Computer-Aided Design, 34, 2, 95-111 (2007)
[6] Schramm, U.; Pilkey, W. W., The coupling of geometric descriptions and finite elements using nurbs — a study in shape optimization, Finite Elements in Analysis and Design, 15, 11-34 (1993) · Zbl 0801.73074
[7] Choi, J. H., Shape design sensitivity analysis and optimization of general plane arch structures, Finite Elements in Analysis and Design, 32, 119-136 (2002) · Zbl 1213.74245
[8] Nadir, W.; Kim, I. Y.; de Weck, O. L., Structural shape optimization considering both performance and manfuacturing cost, (10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference (2004))
[9] Zhang, X.; Rayasam, M.; Subbarayan, G., A meshless, compositional approach to shape optimal design, Computer Methods in Applied Mechanics and Engineering, 196, 2130-2146 (2007) · Zbl 1173.74371
[10] Silva, C. A.C.; Bittencourt, M. L., Velocity fields using nurbs with distortion control for structural shape optimization, Structural and Multidisciplinary Optimization, 33, 147-159 (2007)
[11] Wall, W. A.; Frenzel, M. A.; Cyron, C., Isogeometric structural shape optimization, Computer Methods in Applied Mechanics and Engineering, 197, 2976-2988 (2008) · Zbl 1194.74263
[12] Cho, S.; Ha, S. H., Isogeometric shape design optimization: exact geometry and enhanced sensitivity, Structural and Multidisciplinary Optimization, 38, 53-70 (2009) · Zbl 1274.74221
[13] Poueymirou, D.; Tribes, C.; Trepanier, J. Y., A nurbs-based shape optimization method for hydraulic turbine stay vane, (In Proceedings of the Third International Conference on Computational Fluid Dynamics, ICCFD3, Toronto, 12-16 July 2004 (2004)), 415-421
[14] Nagy, A. P.; Abdalla, M. M.; Gurdal, Z., Isogeometric sizing and shape optimization of beam structures, Computer Methods in Applied Mechanics and Engineering, 199, 1216-1230 (2010) · Zbl 1227.74047
[15] Haftka, R. T.; Gurdal, Z., Elements of Structural Optimization (1992), Kluwer Academic Publishers · Zbl 0782.73004
[16] Brockman, R. A., Geometric sensitivity analysis with isoparametric finite elements, Computer in Applied Numerical Methods, 3, 495-499 (1987) · Zbl 0623.73081
[17] Haslinger, J.; Makinen, R. A.E., Introduction to Shape Optimization: Theory, Approximation, and Computation (2003), SIAM: SIAM Philadelphia · Zbl 1020.74001
[18] Christensen, P. W.; Klarbring, A., An Introduction to Structural Optimization (2009), Springer · Zbl 1180.74001
[19] Piegl, L.; Tiller, W., The NURBS Book (1997), Springer-Verlag: Springer-Verlag New York · Zbl 0868.68106
[20] Fish, J.; Belytschko, T., A First Course in Finite Elements (2007), John Wilsey and Sons: John Wilsey and Sons Ltd · Zbl 1135.74001
[21] Svanberg, K., The method of moving asymptotes: a new method for structural optimization, International Journal of Numerical Methods in Engineering, 24, 359-373 (1987) · Zbl 0602.73091
[22] Herskovits, J.; Dias, G.; Santos, G.; Soares, C. M., Shape structural optimization with an interior point nonlinear programming algorithm, Structural and Multidisciplinary Optimization, 20, 107-115 (2000)
[23] Wilke, D.; Kok, S.; Groenwold, A., A quadratically convergent unstructured remeshing strategy for shape optimization, International Journal for Numerical Methods in Engineering, 65, 1-17 (2006) · Zbl 1122.74478
[24] Norato, J.; Haber, R.; Tortorelli, D.; Bendsoe, M. P., A geometry projection method for shape optimization, International Journal for Numerical Methods in Engineering, 60, 2289-2312 (2004) · Zbl 1075.74702
[25] Pedersen, P., On optimal shapes in materials and structures, Structural and Multidisciplinary Optimization, 19, 169-182 (2000)
[26] Bletzinger, K. U.; Firl, M.; Linhard, J.; Wüchner, R., Optimal shapes of mechanically motivated surfaces, Computer Methods in Applied Mechanics and Engineering, 199, 324-333 (2010) · Zbl 1227.74043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.