تحلیل و بهینه‌سازی شکل سازه های متقارن محوری با استفاده از روش ایزوژئومتریک

نوع مقاله : مقاله مستقل

نویسندگان

1 دانشیار دانشکده مهندسی عمران، دانشگاه صنعتی شاهرود

2 دانشجوی دکتری عمران، گرایش سازه، دانشگاه صنعتی شاهرود

3 دانش آموخته کارشناسی ارشد عمران، گرایش سازه، دانشگاه صنعتی شاهرود

چکیده

تحلیل ایزوژئومتریک یک روش عددی جدید در آنالیز مسائل مهندسی است. این روش بالقوه دارای ویژگی های منحصر به فرد و مناسبی است که شاید در آینده‌ای نه چندان دور بتواند جایگزین روشهای عددی متداول نظیر اجزای محدود و روش های بدون المان گردد. در این مقاله ضمن معرفی فرمول بندی روش ایزوژئومتریک در مسائل متقارن محوری به تحلیل و نیز کاربرد این روش در بهینه‌سازی شکل این نوع از سازه‌ها پرداخته شده است. در این راستا علاوه بر بررسی دقت حل با یک مثال نمونه، دو مثال بهینه‌سازی و حل آن به کمک الگوریتم برنامه ریزی درجه دوم پیاپی ارائه شده است. نتایج نشان می‌دهند که روش ایزوژئومتریک در مقایسه با روش‌های بهینه‌سازی مبتنی بر اجزای محدود با استفاده از تعداد بسیار کمی متغیر طراحی به جواب بهینه مسئله نایل شده است. همچنین به دلیل حذف فرآیند تولید شبکه در این روش، هزینه‌های محاسباتی به طور چشمگیری کاهش یافته است.

کلیدواژه‌ها


[1] Kagan P, Fischer A, Bar–Yoseph PZ (1998) New B–spline finite element approach for geometrical design and mechanical analysis. Int. J. numer.Methods Engrg. 41: 435–458.
[2] Hollig K, Reif U, Wipper J (2001) Weighted extended B–spline approximation of dirichlet problems. SIAM J Numer Anal. 39(2): 442–462.
[3] Kagan P, Fischer A, Bar–Yoseph PZ (2003) Mechanically based models: adaptive refinement for B-spline finite element. Int. J. numer.Methods Engrg. 57: 1145–1175.
[4] Hughes TGR, Cottrell JA, Bazilevs Y, (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput.Methods Appl. Mech. Engrg. 194: 4135–4195.
[5] Auricchio F, Beirao da Veiga L, Hughes TJR, Reali A, Sangalli G (2010) Isogeometric collocation methods. Math. Models Methods, Appl. Sci. 20(11): 2075–2107.
[6] Bazilevs Y, Beirao da Veiga L, Cottrell JA, Hughes TJR, Sangalli G (2006) Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16(7): 1031–1090.
[7] Cottrell JA, Hughes TJR, Reali A (2007) Studies of refinement and continuity in isogemetric analysis. Comput.Methods Appl.Mech. Engrg. 196: 4160–4183.
[8] Drfel M, Jüttler B, Simeon B (2010) Adaptive isogeometric analysis by local h-refinement with T–splines. Comput.Methods Appl. Mech. Engrg. 199(5–8): 264–275.
[9] Evans JA, Bazilevs Y, Babuška I, Hughes TJR (2009) n-width, sup–infs, and optimality ratios for the k-version of the isogeometic finite element method. Comput.Methods Appl. Mech. Engrg. 198: 1726–1741.
[10] Hughes TJR, Reali A, Sangalli G (2008) Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of p–method finite elements with k-method NURBS. Comput.Methods Appl. Mech. Engrg. 197(49–50): 4104–4124.
[11] Hughes TJR, Reali A, Sangalli G (2010) Efficient quadrature for NURBS–based isogeometric analysis. Comput.Methods Appl. Mech. Engrg. 199(5–8): 301–313.
[12] Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput.Methods Appl. Mech. Engrg. 197(1–4): 173–201.
[13] Bazilevs Y, Calo VM, Hughes TJR, Zhang Y (2008) Isogeometric fluid–structure interaction: theory, algorithms, and computations. Comput. Mech. 43(1): 3–37.
[14] 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.
[15] Bazilevs Y, Hughes TJR (2008) NURBS-based isogeometric analysis for the computation of flows about rotating components. Comput. Mech. 43: 143–150.
[16] Buffa A, deFalco C, Sangalli G (2010) Isogeometric analysis: new stable elements for the stokes equation. Int. J. Numer. Meth. Fluids 2000, 00:1–6.
[17] Gmez H, Calo V, Bazilevs Y, Hughes TJR (2008)  Isogeometric analysis of the Cahn–Hilliard phase–field model. Comput.Methods Appl. Mech. Engrg. 197(49–50): 4333–4352.
[18] Auricchio F, Beirao da Veiga L, Lovadina C, Reali A (2010) The importance of the exact satisfaction of the incompressibility constraint in nonlinear elasticity: mixed FEMs versus NURBS-based approximations. Comput.Methods Appl. Mech. Engrg. 199(5–8): 314–323.
[19] Auricchio F, Beirao da Veiga L, Buffa A, Lovadina C, Reali A, Sangalli G (2007) A fully‘‘locking-free’’ isogeometric approach for plane linear elasticity problems: astream function formulation. Comput.Methods Appl. Mech. Engrg. 197(1–4): 160–172.
[20] Benson DJ, Bazilevs Y, Hsu MC, Hughes TJR (2010) Isogeometric shell analysis: the Reissner–Mindlin shell. Comput.Methods Appl. Mech. Engrg. 199(5–8): 276–289.
[21] Cottrell JA, Reali A, Bazilevs Y, Hughes TJR (2006)  Isogeometric analysis of structural vibrations. Comput.Methods Appl. Mech. Engrg. 195(41–43): 5257–5296.
[22] Elguedj T, Bazilevs Y, Calo VM, Hughes TJR (2008) B and –F projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements. Comput.Methods Appl. Mech. Engrg. 197: 2732–2762.
[23] Lipton S, Evans JA, Bazilevs Y, Elguedj T, Hughes TJR (2010) Robustness of isogeometric structural discretizations under severe mesh distortion. Comput.Methods Appl. Mech. Engrg. 199(5–8): 357–373.
[24] Wall WA, Frenzel MA, Cyron C (2008) Isogeometric structural shape optimization. Comput.Methods Appl. Mech. Engrg. 197(33–40): 2976–2988.
[25] 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. Engrg. 196(29–30): 2943–2959.
[26] Hassani B, Khanzadi M, Tavakkoli SM, Moghadam NZ,(2009) Isogeometric shape optimization of three dimensional problems. 8th World Congress on Structural and Multidisciplinary Optimization June 1–5, Lisbon, Portugal.
[27] Buffa A, Rivas J, Sangalli G, Vazquez R (2010) Isogeometric analysis inelectromagnetics: theory and testing. Technical Report, Pubblicazione: 13PV10/13/0, Istituto di Matematica Applicata e Tecnologie Informatiche (I.M.A.T.I.)–C.N.R.
[28] Buffa A, Sangalli G, Vazquez R (2010) Isogeometric analysis in electromagnetics: Bsplines approximation. Comput.Methods Appl. Mech. Engrg. 199(17–20): 1143–1152.
[29] Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley.
[30] Rao SS (1995) Optimization–theory and applications, New Age International. New Delhi.
[31] Zienkiewicz OC, Campbell JS (1973) Shape optimization and sequential linear programming. In: Optimum structural design, theory and applications (eds. R.H. Gallagher & O.C.Zienkiewicz), Wiley and Sons, London, 109–126.
[32] Vanderplaats Research & Development, Inc. (2011) USERS MANUAL Version 5.0: DOT; DESIGN OPTIMIZATIONTOOLS. http://dakota.sandia.gov/licensing/release/Users5.0.pdf.
[33] Rogers DF (2001) Anintroduction to NURBS, Morgan Kaufmann Publishers.
[34] Piegl L, Tiller W (1997) The NURBS book. 2nd ed, Springer-Verlag, New York.
[35] Zienkiewicz OC, Taylor RL, Zhu JZ (2005) The finite element method. 6thed, Elsevier Butterworth-Heinemann.
[36] Afonso SMB (1995) Shape optimization of shells under static and free vibration conditions.PhD Thesis, Department of Civil Engineering, University of Wales, Swansea,UK.
[37] Ozaka M(1993) Analysis and optimal design of structures with adaptivity. Ph.D Thesis.Department of Civil Engineering, University of Wales, Swansea, UK.
[38] Wilson RB (1963) A simplicial algorithm for concave programming. Ph.D. Thesis,Harvard University, Graduate School of Business Administration.
[39] Han SP (1977) A globally convergent method for nonlinear programming. JOTA 22(3): 297–309.
[40] Powell MJD (1978) A fast algorithm for nonlinearly constrained optimization Calculations.  in: G.A. Waston (ed.), Numerical analysis, Springer, Berlin, 144–157.
[41] Hassani B, Hinton E (1999) Homogenization and Structural topology optimization. Springer.
[42] Sadd MH (2005)ELASTICITY theory, applications, and numerics. Elsevier Butterworth–Heinemann.