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An adaptive T-spline finite cell method for structural shape optimization

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Abstract

In this work, an adaptive T-spline finite cell method is developed for structural shape optimization based on the finite cell method (FCM). This method has the advantage of using local T-mesh refinement on the uniform B-mesh to ensure a good computing accuracy in the shape optimization process. To do this, we first carry out a pre-analysis of the structure using a uniform coarse mesh with relatively low computational effort. Then, cells for local refinements are identified by adopting the stress gradient criterion. Subsequently, an adaptive quadtree-based refinement scheme is employed to generate the adaptive T-mesh which preserves the linear independence of the corresponding T-spline shape functions. Finally, the shape optimization framework is correspondingly established, and the proposed adaptive refinement scheme is validated by numerical examples. Its efficiency is fully demonstrated by shape optimization procedures of mechanical structures.

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References

  • Ainsworth M, Senior B (1998) An adaptive refinement strategy for hp-finite element computations. Applied Numerical Mathematics 26(1-2):165–178

    MathSciNet  MATH  Google Scholar 

  • Bazilevs Y, Calo VM, Cottrell JA, Evans JA, Hughes TJR, Lipton S, Scott MA, Sederberg TW (2010) Isogeometric analysis using t-splines. Computer Methods in Applied Mechanics and Engineering 199 (5-8):229–263

    MathSciNet  MATH  Google Scholar 

  • Belytschko T, Krongauz Y, Organ D, Fleming M, Krysl P (1996) Meshless methods: an overview and recent developments. Computer Methods in Applied Mechanics and Engineering 139(1-4):3–47

    MATH  Google Scholar 

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

    Google Scholar 

  • Buffa A, Cho D, Sangalli G (2010) Linear independence of the t-spline blending functions associated with some particular t-meshes. Computer Methods in Applied Mechanics and Engineering 199(23-24):1437–1445

    MathSciNet  MATH  Google Scholar 

  • Cai S, Zhang W, Zhu J, Gao T (2014) Stress constrained shape and topology optimization with fixed mesh: a b-spline finite cell method combined with level set function. Computer Methods in Applied Mechanics and Engineering 278:361–387

    MathSciNet  MATH  Google Scholar 

  • Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of cad and fea. John Wiley & Sons

  • da Veiga LB, Buffa A, Cho D, Sangalli G (2011) Isogeometric analysis using t-splines on two-patch geometries. Computer Methods in Applied Mechanics and Engineering 200(21-22):1787–1803

    MathSciNet  MATH  Google Scholar 

  • Dörfel MR, Jüttler B, Simeon B (2010) Adaptive isogeometric analysis by local h-refinement with t-splines. Computer Methods in Applied Mechanics and Engineering 199(5-8):264–275

    MathSciNet  MATH  Google Scholar 

  • Farin GE (1993) Curves and surfaces for computer aided design: a practical guide. Academic Press, Cambridge, MA

  • Forsey DR, Bartels RH (1988) Hierarchical b-spline refinement. In: Proceedings of the 15th annual conference on Computer graphics and interactive techniques, pp 205–212

  • Garau EM, Vázquez R (2018) Algorithms for the implementation of adaptive isogeometric methods using hierarchical b-splines. Applied Numerical Mathematics 123:58–87

    MathSciNet  MATH  Google Scholar 

  • Ghasemi H, Park HS, Rabczuk T (2017) A level-set based iga formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering 313:239–258

    MathSciNet  MATH  Google Scholar 

  • Giannelli C, Jüttler B, Speleers H (2012) Thb-splines: the truncated basis for hierarchical splines. Computer Aided Geometric Design 29(7):485–498

    MathSciNet  MATH  Google Scholar 

  • Greville TNE (1969) Spline functions and applications., WISCONSIN UNIV MADISON MATHEMATICS RESEARCH CENTER

  • Grindeanu I, Choi KK, Chen JS, Chang KH (1999) Shape design optimization of hyperelastic structures using a meshless method. AIAA Journal 37(8):990–997

    Google Scholar 

  • Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194(39-41):4135–4195

    MathSciNet  MATH  Google Scholar 

  • Idelsohn SR, Onate E, Calvo N, Del Pin F (2003) The meshless finite element method. International Journal for Numerical Methods in Engineering 58(6):893–912

    MathSciNet  MATH  Google Scholar 

  • Joulaian M, Düster A (2013) Local enrichment of the finite cell method for problems with material interfaces. Computational Mechanics 52(4):741–762

    MATH  Google Scholar 

  • Kuru G, Verhoosel CV, Van der Zee KG, van Brummelen EH (2014) Goal-adaptive isogeometric analysis with hierarchical splines. Computer Methods in Applied Mechanics and Engineering 270:270–292

    MathSciNet  MATH  Google Scholar 

  • Li K, Qian X (2011) Isogeometric analysis and shape optimization via boundary integral. Computer-Aided Design 43(11):1427–1437

    Google Scholar 

  • Li X, Scott MA (2014) Analysis-suitable t-splines: characterization, refineability, and approximation. Mathematical Models and Methods in Applied Sciences 24(06):1141–1164

    MathSciNet  MATH  Google Scholar 

  • Li X, Zheng J, Sederberg TW, Hughes TJR, Scott MA (2012) On linear independence of t-spline blending functions. Computer Aided Geometric Design 29(1):63–76

    MathSciNet  MATH  Google Scholar 

  • McKinley S, Levine M (1998) Cubic spline interpolation. College of the Redwoods 45(1):1049–1060

    Google Scholar 

  • Meng L, Zhang W-H, Zhu J-H, Xia L (2014) A biarc-based shape optimization approach to reduce stress concentration effects. Acta Mechanica Sinica 30(3):370–382

    MathSciNet  MATH  Google Scholar 

  • Meng L, Zhang W, Zhu J, Xu Z, Cai S (2016) Shape optimization of axisymmetric solids with the finite cell method using a fixed grid. Acta Mechanica Sinica 32(3):510–524

    MathSciNet  MATH  Google Scholar 

  • Meng L, Zhang W, Quan D, Shi G, Tang L, Hou Y, Breitkopf P, Zhu J, Gao T (2020) From topology optimization design to additive manufacturing: today’s success and tomorrow’s roadmap. Archives of Computational Methods in Engineering, vol 27

  • Morgenstern P, Peterseim D (2015) Analysis-suitable adaptive t-mesh refinement with linear complexity. Computer Aided Geometric Design 34:50–66

    MathSciNet  MATH  Google Scholar 

  • Nguyen-Thanh N, Nguyen-Xuan H, Bordas SPA, Rabczuk T (2011) Isogeometric analysis using polynomial splines over hierarchical t-meshes for two-dimensional elastic solids. Computer Methods in Applied Mechanics and Engineering 200(21-22):1892–1908

    MathSciNet  MATH  Google Scholar 

  • Parvizian J, Düster A, Rank E (2007) Finite cell method. Computational Mechanics 41(1):121–133

    MathSciNet  MATH  Google Scholar 

  • Piegl L, Tiller W (2012) The nurbs book. Springer Science & Business Media

  • Rassineux A, Breitkopf P, Villon P (2003) Simultaneous surface and tetrahedron mesh adaptation using mesh-free techniques. International Journal for Numerical Methods in Engineering 57(3):371–389

    MATH  Google Scholar 

  • Reddy JN (1993) An introduction to the finite element method. New York

  • Rhino (2019) Autodesk t-splines plug-in for rhino software. https://www.rhino3d.com/

  • Schillinger D, Ruess M (2015) The finite cell method: A review in the context of higher-order structural analysis of cad and image-based geometric models. Archives of Computational Methods in Engineering 22(3):391–455

    MathSciNet  MATH  Google Scholar 

  • Schillinger D, Kollmannsberger S, Mundani RP, Rank E (2010) The hierarchical b-spline version of the finite cell method for geometrically nonlinear problems of solid mechanics. In: The European Conference on Computational Mechanics

  • Schillinger D, Dede L, Scott MA, Evans JA, Borden MJ, Rank E, Hughes TJR (2012) An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of nurbs, immersed boundary methods, and t-spline cad surfaces. Computer Methods in Applied Mechanics and Engineering 249:116–150

    MathSciNet  MATH  Google Scholar 

  • Schillinger D, Cai Q, Mundani R-P, Rank E (2013) Nonlinear structural analysis of complex cad and image based models with the finite cell method. pp 1–23

  • Scott MA, Li X, Sederberg TW, Hughes TJR (2012) Local refinement of analysis-suitable t-splines. Computer Methods in Applied Mechanics and Engineering 213:206–222

    MathSciNet  MATH  Google Scholar 

  • Sederberg TW, Zheng J, Bakenov A, Nasri A (2003) T-splines and t-nurccs. ACM Transactions on Graphics (TOG) 22(3):477– 484

    Google Scholar 

  • Sederberg TW, Cardon DL, Finnigan GT, North NS, Zheng J, Lyche T (2004) T-spline simplification and local refinement. In: ACM Transactions on Graphics (TOG), vol 23. ACM, pp 276–283

  • Shapiro V (1991) Theory of r-functions and applications: a primer, Cornell University

  • Tang L, Gao T, Song L, Meng L, Zhang C, Zhang W (2019) Topology optimization of nonlinear heat conduction problems involving large temperature gradient. Computer Methods in Applied Mechanics and Engineering 357:112600

    MathSciNet  MATH  Google Scholar 

  • Tie Y, Hou Y, Li C, Meng L, Sapanathan T, Rachik M (2020) Optimization for maximizing the impact-resistance of patch repaired cfrp laminates using a surrogate-based model. International Journal of Mechanical Sciences 172:105407

    Google Scholar 

  • Wang JG, Liu GRs (2002) A point interpolation meshless method based on radial basis functions. International Journal for Numerical Methods in Engineering 54(11):1623–1648

    MATH  Google Scholar 

  • Wang D, Zhang W, Wang Z, Zhu J (2010) Shape optimization of 3d curved slots and its application to the squirrel-cage elastic support design. Science China Physics, Mechanics and Astronomy 53(10):1895–1900

    Google Scholar 

  • Wang D, Zhang W, Zhu J (2016) A moving bounds strategy for the parameterization of geometric design variables in the simultaneous shape optimization of curved shell structures and openings. Finite Elements in Analysis and Design 120:80–91

    MathSciNet  Google Scholar 

  • Wang Z-P, Poh LH, Dirrenberger J, Zhu Y, Forest S (2017) Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Computer Methods in Applied Mechanics and Engineering 323:250–271

    MathSciNet  MATH  Google Scholar 

  • Wang Y, Wang Z, Xia Z, Poh LH (2018) Structural design optimization using isogeometric analysis: a comprehensive review. Computer Modeling in Engineering & Sciences 117(3):455–507

    Google Scholar 

  • Xia Q, Shi T, Liu S, Wang MY (2012) A level set solution to the stress-based structural shape and topology optimization. Computers & Structures 90:55–64

    Google Scholar 

  • Xu M, Xia L, Wang S, Liu L, Xie X (2019) An isogeometric approach to topology optimization of spatially graded hierarchical structures. Composite Structures 225:111–171

    Google Scholar 

  • Yagawa G, Yamada T (1996) Free mesh method: a new meshless finite element method. Computational Mechanics 18(5):383–386

    MATH  Google Scholar 

  • Yang TY (1986) Finite element structural analysis. Prentice-Hall Englewood Cliffs, NJ, vol 2

  • Zhang W, Huang Q (2017) Unification of parametric and implicit methods for shape sensitivity analysis and optimization with fixed mesh. International Journal for Numerical Methods in Engineering 109(3):326–344

    MathSciNet  Google Scholar 

  • Zhang W, Zhao L, Cai S (2015) Shape optimization of dirichlet boundaries based on weighted b-spline finite cell method and level-set function. Computer Methods in Applied Mechanics and Engineering 294:359–383

    MathSciNet  MATH  Google Scholar 

  • Zhang W, Zhao L, Gao T, Cai S (2017) Topology optimization with closed b-splines and boolean operations. Computer Methods in Applied Mechanics and Engineering 315:652–670

    MathSciNet  MATH  Google Scholar 

  • Zhou JX, Zou W (2008) Meshless approximation combined with implicit topology description for optimization of continua. Structural and Multidisciplinary Optimization 36(4):347–353

    Google Scholar 

  • Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method: its basis and fundamentals. Elsevier

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Funding

This work is supported by the Research and Development Program of China (2017YFB 1102800), National Natural Science Foundation of China (11620101002, 11902251), and Fundamental Research Funds for the Central Universities (G2018KY0304).

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Authors and Affiliations

  1. State IJR Center of Aerospace Design and Additive Manufacturing, School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

    Liang Chen, Weihong Zhang, Liang Meng, Lipeng Jiu & Shengqi Feng

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  1. Liang Chen
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  2. Weihong Zhang
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Chen, L., Zhang, W., Meng, L. et al. An adaptive T-spline finite cell method for structural shape optimization. Struct Multidisc Optim 61, 1857–1876 (2020). https://doi.org/10.1007/s00158-020-02645-w

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