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Numerical method for shape optimization using T-spline based isogeometric method

Abstract

Numerical methods for shape design sensitivity analysis and optimization have been developed for several decades. However, the finite-element-based shape design sensitivity analysis and optimization have experienced some bottleneck problems such as design parameterization and design remodeling during optimization. In this paper, as a remedy for these problems, an isogeometric-based shape design sensitivity analysis and optimization methods are developed incorporating with T-spline basis. In the shape design sensitivity analysis and optimization procedure using a standard finite element approach, the design boundary should be parameterized for the smooth variation of the boundary using a separate geometric modeler, such as a CAD system. Otherwise, the optimal design usually tends to fall into an undesirable irregular shape. In an isogeometric approach, the NURBS basis function that is used in representing the geometric model in the CAD system is directly used in the response analysis, and the design boundary is expressed by the same NURBS function as used in the analysis. Moreover, the smoothness of the NURBS can allow the large perturbation of the design boundary without a severe mesh distortion. Thus, the isogeometric shape design sensitivity analysis is free from remeshing during the optimization process. In addition, the use of T-spline basis instead of NURBS can reduce the number of degrees of freedom, so that the optimal solution can be obtained more efficiently while yielding the same optimum design shape.

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References

  1. Annicchiarico W, Cerrolaza M (1999) Finite elements, genetic algorithms and b-splines: a combined technique for shape optimization. Finite Elem Anal Des 33:125–141

  2. Azegami H, Kaizu S, Shimoda M, Katamine E (1997) Irregularity of shape optimization problems and an improvement technique. In: Hernandez S, Brebbia CA (eds) Proceedings of the international conference on computer aided optimum design of structures, OPTI, pp 309–326

  3. Bazilevs Y, Hughes TJR (2007) Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput Fluids 36(1):12–26

  4. Bazilevs Y, Hughes TJR (2008) NURBS-based isogeometric analysis for the computation of flows about rotating components. Comput Mech 43:143–150

  5. Bazilevs Y, Calo VM, Zhang Y, Hughes TJR (2006) Isogeometric fluid-structure interaction analysis with applications to arterial blood flow. Comput Mech 38:310–322

  6. Bennett JA, Botkin ME (1985) Structural shape optimization with geometric description and adaptive mesh refinement. AIAA J 23(3):458–464

  7. Braibant V, Fluery C (1984) Shape optimal design using B-splines. Comput Methods Appl Mech Eng 44:247–267

  8. Cervera E, Trevelyan J (2005) Evolutionary structural optimisation based on boundary representation of NURBS. Part I: 2D algorithms. Comput Struct 83:1902–1916

  9. Chang KH, Choi KK, Tsai CS, Chen CJ, Choi BS, Yu X (1995) Design sensitivity analysis and optimization tool (DSO) for shape design applications. Comput Syst Eng 6(2):151–175

  10. Cho M, Roh HY (2003) Development of geometrically exact new shell elements based on general curvilinear coordinates. Int J Numer Methods Eng 56(1):81–115

  11. Cho S, Ha SH (2007) Shape design optimization of geometrically nonlinear structures using isogeometric analysis. In: 9th United States national congress on computational mechanics, San Francisco, California, USA, July 22–26

  12. Cho S, Ha SH (2009) Isogeometric shape design optimization: exact geometry and enhanced sensitivity. Struct Multidisc Optim 38:53–70

  13. Choi KK, Chang KH (1994) A study of design velocity field computation for shape optimal design. Finite Elem Anal Des 15:317–341

  14. Choi KK, Kim NH (2004) Structural sensitivity analysis and optimization. Linear systems, vol 1. Nonlinear systems and applications, vol 2. Springer, New York

  15. Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006) Isogeometric analysis of structural vibrations. Comput Methods Appl Mech Eng 195:5257–5296

  16. Cottrell JA, Hughes TJR, Reali A (2007) Studies of refinement and continuity in isogeometric structural analysis. Comput Methods Appl Mech Eng 196:4160–4183

  17. Dörfel MR, Jüttler B, Simeon B (2008) Adaptive isogeometric analysis by local h-refinement with T-splines. Comput Methods Appl Mech Eng 199:264–275

  18. Farin G (2002) Curves and surfaces for CAGD: a practical guide. Academic Press, New York

  19. Gómez H, Calo VM, Bazilevs Y, Hughes TJR (2008) Isogeometric analysis of the Cahn–Hilliard phase-field model. Comput Methods Appl Mech Eng 197:4333–4352

  20. Grindeanu I, Kim NH, Choi KK, Chen JS (2002) CAD-based shape optimization using a meshfree method. Concurr Eng Res Appl 10:55–66

  21. Gu Y, Cheng G (1990) Structural shape optimization integrated with CAD environment. Struct Optim 2:23–28

  22. Ha SH, Cho S (2007) Shape design optimization of structural problems based on isogeometric approach. In: 7th world congress on structural and multidisciplinary optimization. COEX, Seoul, Korea

  23. Hardee E, Chang KH, Tu J, Choi KK, Grindeanu I, Yu X (1999) A CAD-based design parameterization for shape optimization of elastic solids. Adv Eng Softw 30:185–199

  24. Haug EJ, Choi KK, Komkov V (1986) Design sensitivity analysis of structural systems. Academic, New York

  25. Hsu SY, Chang CL (2007) Mesh deformation based on fully stressed design: the method and 2-D examples. Int J Numer Methods Eng 72:606–629

  26. Hughes TJR (2000) The finite element method: linear static and dynamic finite element analysis. Dover Publications, Mineola

  27. Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195

  28. Kim NH, Choi KK, Botkin ME (2003) Numerical method for shape optimization using meshfree method. Struct Multidisc Optim 24:418–429

  29. Piegl L, Tiller W (1997) The NURBS book (monographs in visual communication), 2nd edn. Springer, New York

  30. Rogers DF (2001) An introduction to NURBS with historical perspective. Academic Press, San Diego

  31. Roh HY, Cho M (2004) The application of geometrically exact shell elements to B-spline surfaces. Comput Methods Appl Mech Eng 193:2261–2299

  32. Roh HY, Cho M (2005) Integration of geometric design and mechanical analysis using B-spline functions on surface. Int J Numer Methods Eng 62:1927–1949

  33. Sederberg TW, Zheng J, Bakenov A, Nasri A (2003) T-splines and T-NURCCSs. ACM Trans Graph 22(3):477–484

  34. Sederberg TW, Cardon DL, Finnigan GT, North NS, Zheng J, Lyche T (2004) T-spline simplification and local refinement. ACM Trans Graph 23(3):276–283

  35. Silva CAC, Bittencourt ML (2007) Velocity fields using NURBS with distortion control for structural shape optimization. Struct Multidisc Optim 33:147–159

  36. Uhm TK, Kim KS, Seo YD, Youn SK (2008) A locally refinable T-spline finite element method for CAD/CAE integration. Struct Eng Mech 30(2):225–245

  37. Wall WA, Frenzel MA, Cyron C (2008) Isogeometric structural shape optimization. Comput Methods Appl Mech Eng 197:2976–2988

  38. Yang H, Jüttler B (2007) 3D shape metamorphosis based on T-spline level sets. Vis Comput 23(12):1015–1025

  39. Yang H, Jüttler B (2008) Evolution of T-spline level sets for meshing non-uniformly sampled and incomplete data. Vis Comput 24(6):435–448

  40. Zhang Y, Bazilevs Y, Goswami S, Bajaj CL, Hughes TJR (2007) Patient-specific vascular NURBS modeling for isogeometric analysis of blood flow. Comput Methods Appl Mech Eng 196:2943–2959

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Acknowledgements

This work was supported by the Korea Research Foundation grant (KRF-2007-612-D00146) and National Research Foundation of Korea grant (R32-2008-000-10161-0) funded by the Korea government (MEST) in 2009. The supports are gratefully acknowledged.

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Correspondence to Seonho Cho.

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Ha, S., Choi, K.K. & Cho, S. Numerical method for shape optimization using T-spline based isogeometric method. Struct Multidisc Optim 42, 417–428 (2010). https://doi.org/10.1007/s00158-010-0503-0

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Keywords

  • Shape design sensitivity analysis
  • Shape design optimization
  • Isogeometric analysis
  • T-spline
  • NURBS
  • Design parameterization
  • Local refinement