مهندسی مکانیک مدرس

مهندسی مکانیک مدرس

بهینه‌سازی توپولوژی سازه با در نظر گرفتن قیود تنش خوشه‌بندی شده

نویسندگان
1 گروه مکانیک، دانشگاه فردوسی مشهد، مشهد، ایران
2 هیئت علمی/گروه مکانیک، دانشگاه فردوسی مشهد، مشهد، ایران
3 گروه مکانیک، دانشکده مهندسی، دانشگاه فردوسی مشهد، مشهد، ایران
چکیده
در این مقاله روشی برای بهبود اعمال قید تنش در طراحی توپولوژی سازه‌ با جرم کمینه ارائه شده ‌است. برای فرمول‌بندی مسئله بهینه‌سازی توپولوژی از روش اجزای محدود و برای مدل‌سازی مصالح از تابع چگالی مصنوعی استفاده شده‌ است. برای محاسبه میزان تنش در اجزا از تنش فون‌میزز در نقاط انتگرال‌گیری گوس موسوم به نقاط فوق‌همگرا استفاده شده است. به منظور کاهش زمان و هزینه محاسبات از فن خوشه‌بندی قیود تنش مجتمع شده با روش P-norm برای کاستن از تعداد قیود مسئله بهینه‌سازی، که متناسب با تعداد المان‌های به کار رفته در مدل محاسباتی مسئله است، استفاده شده است. به این منظور تعداد زیادی از قیدهای تنش محلی با تعداد محدودی قید تنش سراسری جایگزین می‌شود. توصیف کاملی از فرمول‌بندی و تحلیل حساسیت قید تنش که با استفاده از روش الحاقی صورت پذیرفته، ارائه شده است. به علت پیچیدگی بهینه‌سازی توپولوژی با استفاده از قیدهای تنش، روش مجانب‌های متحرک برای حل مسئله بهینه‌سازی مورد استفاده قرار گرفته است. برای بررسی کارایی روش چند مثال تنش صفحه‌ای ارائه و با سایر پژوهش‌ها ارزیابی شده است. نتایج به‌دست آمده حاکی از مزیت روش محاسباتی ارائه شده در تولید توپولوژی‌های قابل قبول و کاربردی است.
کلیدواژه‌ها

عنوان مقاله English

Structural topology optimization considering clustered stress constraints

نویسندگان English

Habib Saffar Najib 1
Behrooz Hassani 2
Nima Yaghoobi 3
1 Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran
2 Mechanical Engineering Department,Ferdowsi University of Mashhad, Mashhad, Iran
3 Mechanical Engineering Department, Ferdowsi University of Mashhad, Mashhad, Iran
چکیده English

This paper presents an improved approach for handling stress constraints in minimum weight topological design. The Finite Element Method (FEM) and the material model of Solid Isotropic Material with Penalization (SIMP) is used to formulate the topology optimization problem. To evaluate the stress values in elements, the von Mises stresses are calculated at the so called super-convergent Gauss quadrature points. To reduce the time and computational cost, a clustering approach is here adopted and the P-norm integrated stress constraints are used. Doing this, a large number of local constraints are replaced with a few global ones and consequently the stress constraint sensitivities are calculated by using the adjoint method. The employed formulation as well as a complete explanation of the sensitivity analysis is provided. Due to the complexity of the topology optimization problem in the presence of stress constraints, the Method of Moving Asymptotes (MMA) is here employed. To demonstrate the performance and capability of the procedure, a couple of plane stress elasticity problems are taken into consideration. The resulted layouts indicate the superiority of the approach in generating acceptable and practical topological designs.

کلیدواژه‌ها English

Structural Topology Optimization
Stress Constraint
Clustering
Stress Penalization
SIMP
MMA
[1] B. Hassani, E. Hinton, Homogenization and Structural Topology Optimization: Theory, Practice and Software, pp. 104-136, London: Springer, 1999.
[2] F. Abbasi Parizad, B. Hassani, H. Ghasemnejad Moghari, Optimization of free form shells under stress constraint and using B-Spline functions, Modares Mechanical Engineering, Vol. 14, No. 16, pp. 190-200, 2015. (in (فارسی Persian
[3] S. M. Tavakkoli, Isogeometrical Analysis and topology Optimization of Continuum Strucutres Using NURBS Basis Functions, PhD Thesis Thesis, Department of Civil Engineering, Iran University of Science and Technology, Tehran, 2010. (in Persian فارسی (
[4] N. Yaghoobi, B. Hassani, Topological optimization of vibrating continuum structures for optimal natural eigenfrequency, International Journal of Optimization in Civil Engineering, Vol. 7, No. 1, pp. 1-12, 2017.
[5] C. Le, J. Norato, T. Bruns, C. Ha, D. Tortorelli, Stress-based topology optimization for continua, Structural and Multidisciplinary Optimization, Vol. 41, No. 4, pp. 605-620, 2010.
[6] E. Holmberg, B. Torstenfelt, A. Klarbring, Stress constrained topology optimization, Structural and Multidisciplinary Optimization, Vol. 48, No. 1, pp. 33-47, 2013.
[7] J. Roshanian, A. A. Bataleblu, M. H. Farghadani, B. Ebrahimi, Multiobjective multidisciplinary design optimization of a general aviation aircraft, Modares Mechanical Engineering, Vol. 17, No. 2, pp. 199-210, 2017. (in (فارسی Persian
[8] K. Svanberg, The method of moving asymptotes—a new method for structural optimization, International Journal for Numerical Methods in Engineering, Vol. 24, No. 2, pp. 359-373, 1987.
[9] T. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, pp. 109-120, Englewood cliffs: Prentice-hall, 1987.
[10] M. P. Bendsøe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, Vol. 71, No. 2, pp. 197-224, 1988.
[11] M. P. Bendsøe, Optimal shape design as a material distribution problem, Structural and Multidisciplinary Optimization, Vol. 1, No. 4, pp. 193-202, 1989.
[12] G. I. Rozvany, M. Zhou, T. Birker, Generalized shape optimization without homogenization, Structural and Multidisciplinary Optimization, Vol. 4, No. 3, pp. 250-252, 1992.
[13] U. Kirsch, On singular topologies in optimum structural design, Structural and Multidisciplinary Optimization, Vol. 2, No. 3, pp. 133-142, 1990.
[14] G. Rozvany, T. Birker, On singular topologies in exact layout optimization, Structural and Multidisciplinary Optimization, Vol. 8, No. 4, pp. 228-235, 1994.
[15] G. Rozvany, On design-dependent constraints and singular topologies, Structural and Multidisciplinary Optimization, Vol. 21, No. 2, pp. 164-172, 2001.
[16] G. Cheng, Z. Jiang, Study on topology optimization with stress constraints, Engineering Optimization, Vol. 20, No. 2, pp. 129-148, 1992.
[17] G. Cheng, X. Guo, ε-relaxed approach in structural topology optimization, Structural and Multidisciplinary Optimization, Vol. 13, No. 4, pp. 258-266, 1997.
[18] T. E. Bruns, D. A. Tortorelli, Topology optimization of non-linear elastic structures and compliant mechanisms, Computer Methods in Applied Mechanics and Engineering, Vol. 190, No. 26, pp. 3443-3459, 2001.
[19] M. Bruggi, On an alternative approach to stress constraints relaxation in topology optimization, Structural and Multidisciplinary Optimization, Vol. 36, No. 2, pp. 125-141, 2008.
[20] E. Holmberg, B. Torstenfelt, A. Klarbring, Fatigue constrained topology optimization, Structural and Multidisciplinary Optimization, Vol. 50, No. 2, pp. 207-219, 2014.
[21] J. D. Deaton, R. V. Grandhi, Stress-based design of thermal structures via topology optimization, Structural and Multidisciplinary Optimization, Vol. 53, No. 2, pp. 253-270, 2016.
[22] H. A. Jahangiry, A. Jahangiri, Topology optimization of heat conduction problem via level-set method and the finite elements analysis, Modares Mechanical Engineering, Vol. 16, No. 12, pp. 703-710, 2016. (in (فارسی Persian
[23] A. J. Ferreira, MATLAB Codes for Finite Element Analysis: Solids and Structures, pp.143-147, Netherlands: Springer, 2008.
[24] M. Hassanzadeh, Computation of shape design sensitivities for linear FEM using modified semi-analytical method, Modares Mechanical Engineering, Vol. 15, No. 11, pp. 73-80, 2015. (in Persian فارسی(
[25] O. C. Zienkiewicz, J. Z. Zhu, A simple error estimator and adaptive procedure for practical engineerng analysis, International Journal for Numerical Methods in Engineering, Vol. 24, No. 2, pp. 337-357, 1987.
[26] O. C. Zienkiewicz, J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique, International Journal for Numerical Methods in Engineering, Vol. 33, No. 7, pp. 1331- 1364, 1992.
[27] O. C. Zienkiewicz, J. Z. Zhu, The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity, International Journal for Numerical Methods in Engineering, Vol. 33, No. 7, pp. 1365-1382, 1992.
[28] K. S. Woo, J. S. Ahn, Adaptive refinement based on stress recovery technique considering ordinary kriging interpolation in L-shaped domain, Mathematical Problems in Engineering, Vol. 2017, No. 1, pp. 10, 2017.
[29] J. Barlow, Optimal stress locations in finite element models, International Journal for Numerical Methods in Engineering, Vol. 10, No. 2, pp. 243-251, 1976.
[30] E. Hinton, J. S. Campbell, Local and global smoothing of discontinuous finite element functions using a least squares method, International Journal for Numerical Methods in Engineering, Vol. 8, No. 3, pp. 461-480, 1974.
[31] U. Kirsch, Structural Optimization: Fundamentals and Applications, pp. 128-135, Berlin: Springer, 1993.
[32] C. Y. Kiyono, S. L. Vatanabe, E. C. N. Silva, J. N. Reddy, A new multi-pnorm formulation approach for stress-based topology optimization design, Composite Structures, Vol. 156, pp. 10-19, 2016.