حل مسائل غیرخطی الاستیک تراکم‌پذیر بااستفاده‌از روش تحلیل ایزوژئومتریک

نوع مقاله : مقاله پژوهشی

نویسندگان

• مهدی اردیانی ¹
• بهروز حسنی ¹
• سید مهدی توکلی ²

¹ دانشگاه فردوسی مشهد

² دانشگاه صنعتی شاهرود

10.22067/fum-mech.v30i2.65185

چکیده

  کاربرد روش تحلیلی ایزوژئومتریک (Isogeometric Analysis) برای حل مسائل غیرخطی الاستیک تراکم‌پذیر موسوم به مسائل هایپرالاستیسیته  (Hyperelasticity)موضوع این مقاله است. بدین‌منظور جهت استخراج ماتریس ضرایب با بهره­گیری از مفهوم ایزوژئومتریک، پس از خطی‌سازی روابط حاکم، معادلات تعادل در فرم گسسته‌شدۀ آن نوشته شده و در ادامه الگوریتمی برای تحلیل این دسته از مسائل ارائه می­گردد. برای بررسی کارایی روش و صحت نتایج به‌دست‌آمده در مسائل هایپرالاستیسیتۀ تراکم­پذیر، نتایج حاصل از روش‌های اجزای محدود و ایزوژئومتریک با یکدیگر مقایسه می­شود. استفاده از روش پیشنهادی علاوه بر امکان ایجاد هندسۀ مدل با دقت و انعطاف­پذیری بیشتر، باعث تشکیل دستگاه معادلات کوچکتر و در کل کاهش حجم محاسبات می­شود. به‌علاوه، علیرغم وجود تغییر­شکل­های بزرگ، وابستگی به نحوۀ گسسته‌سازی مسائل حداقل بوده و برخلاف روش اجزای محدود تا حد بسیار زیادی نیاز به تولید مجدد شبکه اجزای محدود وجود ندارد. همچنین، در این مقاله به بررسی تأثیر تعداد تقسیمات بار و نیز تعداد نقاط انتگرال­گیری گوسی در همگرایی جواب مسائل پرداخته شده است.

کلیدواژه‌ها

• تحلیل ایزوژئومتریک
• نربز
• هایپرالاستیسیته
• مصالح تراکم‌پذیر

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عنوان مقاله [English]

Solution of Nonlinear Compressible Hyperelastic Problems by Isogeometric Analysis Method

نویسندگان [English]

• Mehdi Ardiani ¹
• behrooz Hassani ¹
• S. Mehdi Tavakkoli ²

¹ Ferdowsi University of Mashhad

² Shahrood University of Technology

چکیده [English]

Employing the Isogeometric Analysis method for solution of nonlinear compressible elastic materials, generally known as hyperelasticity, is the subject of this article. For this purpose, the matrix of coefficients is derived and by the linearization of governing equations the discretized equilibrium equations are obtained and a solution algorithm is presented. To study the performance and accuracy of the method in compressible hyperelastic problems, the obtained results are compared with those of finite elements. The presented approach, besides providing a good flexibility in geometrical modeling, results in a smaller system of equations and consequently reducing the computational cost. Furthermore, despite having large deformations, the need for remeshings is alleviated. Also, the effects of the number of load increments, as well as, the number of Gauss integration points on the convergence of the solution are studied.

کلیدواژه‌ها [English]

• Isogeometric Analysis
• NURBS
• Hyperelasticity
• Compressible materials

مراجع

1. Tuner, M.J., Drill, E.H., Martin, H.C. and Melosh, R.J., "Large deflection of structures subject to heating and external load", Journal Aerospace Sciences., Vol. 27, pp. 97-106,) 1960(.

2. Kapur, W.W. and Hartz, B.J., "Stability of plates using the finite element method", Journal of the Engineering Mechanics Division, Vol. 92, pp. 177-195, )1966(.

3. Gallagher, R.J. and Padlog, J., "Discrete element approach to strucural stability", ", Journal of Aeronautics & Astronautics , Vol. 1, No. 6, pp. 1437-1439, (1963).

4. Gallagher, R.J., Gellatly, R.A., Padlog, J. and Mallet, R.H., "A discrete element procedure for thin shell instability analysis", Journal of Aeronautics & Astronautics, Vol. 5, No. 1, pp. 138-145, )1967(.

5. Holand, I. and Moan, T., "The finite element in plate buckling", Finite Element Methods in Stress Analysis, )1969(.

6. Argyris, J.H., "Recent Advance in Matrix Method of Structure Analysis, Progress in Aeronautical Sciences, )1964(.

7. Argyris, J.H.,"Countinua and discountinua", Proc, conf, Matrix Methods in Struct, mech, Air Force Institute of Technology, Wright Patterson Air Force Base, Ohio, Octobr )1965(.

8. Oden, J.T., "Numerical Formulation of non-linear elasticity problems", Journal of the Structural Division, Vol. 93, pp. 5290, )1967(.

9. Mallet, R.H. and Marcal, P.V., "finite element analysis of non-linear structures", Journal of the Structural Division, Vol. 94, pp. 2081-2105, )1968(.

10. Oden, J.T., "Finite elemnt application in non-linear structural analysis", Proc, Conf, on Finite elemnt Meth, Vanderbilt University Tennessee, 18 November )1969(.

11. Haisler, W.E., Stricklin, J.E. and Stebbins, F.J., "Development and evaluation of solution procedures for geometrically non-linear structural analysis by the discrete stiffnes method", AIAA/ASME, 12th structure, Structural Dynamics & Materials Conf, Anaheim, californa, 24 April )1971(.

12. Zinckiewicz, O.C.,"The Finite Element in Engeneering Science", Mc Graw-Hill, London, )1971(.

13. Brebbia, C. and Connor, J., "Geometrically non-linear finite element analysis", Journal of the Structural Division, pp. 6516, )1969(.

14. Crisfield, M.A., "Nonlinear finite element analysis of solids and structures", Vol. I & Vol. II, John Wiley & Sons, )1991(.

15. Belytschko, T., Liu, W.K. and Moran, B., "Nonlinera Finite Element for Countinua and Structures", John Wiley & Sons, )2000(.

16. Zinkiewicz, O.C. and Taylor, R.L., "The finite element method, Vol. II, 5nd edition", McGraw Hill, )2000(.

17. Bonet, J. and Wood, R.D., "Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd edition", Cambridge University Press, )2008(.

18. Wriggers, P.," Nonlinear finite element methods", Springer, )2008(.

19. Qiang, Z. and Qing-Sheng, Y., "Effects of large deformation and material nonlinearity on spherical indentation of hyperelastic soft materials", Mechanics Research Communications, Vol. 84, pp. 55-59, (2017).

20. Clayton J.D , "Geometry of nonlinear elastic solids with internal structure", Journal of Geometry and Physics, Vol. 112, pp. 118-146, (2017).

21. Piegel, L. and Wayne, T., "The Nurbs Book, 2nd edition", Springer,)1996(.

22. Rogers, D.F., "An Introduction to NURBS with Historical Perspective", Morgan Kaufmann Publishers, (2001(.

23. Hughes, T.J.R., Cottrell, J.A. and Bazilevs, Y., "Isogeometric analysis: Cad, finite elements, NURBS, exact geometry and mesh refinement", Computer Methods in Applied Mechanics and Engineering, Vol. 194, pp. 4135–4195, )2005(.

24. Penter, P.M., "Splines and Variational Methods", John Wiley & Sons, )1989(.

25. Hölling, K.," Finite elemnt methods with B-Splines", Society for industrial and applied mathematics Philadelphia, )2003(.

26. Kadapa, C., Dettmer, W.G. and Perić, D., "Subdivision based mixed methods for isogeometric analysis of linear and nonlinear nearly incompressible materials", Computer Methods in Applied Mechanics and Engineering, Vol. 305, pp. 241-270, (2016).

27. Nguyen, X., Atroschchenko, E. and nguyen, H., "Geometrically nonlinear isogeometric analysis of functionally graded microplates with the modified couple stress theory", Computers & Structures, Vol. 193, pp. 110-127, (2017).

28. Bauer, A.M., Breitenberger, M., Philipp, B., Wüchner, R. and Bletzinger, K.-U., "Nonlinear isogeometric spatial Bernoulli beam", Computer Methods in Applied Mechanics and Engineering, Vol. 303, pp. 101-127, (2016).

29. Antolin, P., Bressan, A., Buffa, A. and Sangalli, G., "An isogeometric method for linear nearly- incompressible elasticity with local stress projection", Computer Methods in Applied Mechanics and Engineering, Vol. 316, pp. 694-719, (2017).

30. Cottrell, J.A., Hughes, T.J.R. and Bazilves, Y., "Isogeometric Analysis: toward integration of CAD and FEA", John Wiley& Sons, )2009(.

31. Bazilevs, Y., Beirao, L., Cottrell, J., Hughes, T.J.R. and Sangalli, G., "Isogeometric analysis: approximation, stability and error estimates for h-refined meshes", Mathematical Models and Methods in Applied Sciences, Vol. 16, pp. 1031–1090, )2006(.

32. Bazilevs, Y., Calo, V., Cottrell, J., Hughes, T., Reali, A. and Scovazzi, G., "Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows", Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 173–201, )2007(.

33. Bazilevs, Y., Calo, V.M., Zhang, Y. and Hughes, T.J.R., "Isogeometric fluid structure interaction analysis with applications to arterial blood flow", Computer Methods in Applied Mechanics and Engineering, Vol. 38, No. 4, pp. 310–322, )2006(.

34. Cottrell, J.A., Reali, A., Bazilevs, Y. and Hughes, T.J.R., "Isogeometric analysis of structural vibrations", Computer Methods in Applied Mechanics and Engineering, Vol. 195, pp. 5257–5296, )2006(.

35. Elguedj, T., Bazilevs, Y., Calo, V.M. and Hughes, T.J.R., " and projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements", Computer Methods in Applied Mechanics and Engineering, Vol. 197, pp. 2732-2762, )2008(.

36. Hassani, B. and Moghadam, N.Z., "Development of a new numerical method for solution of ordinary differential equations by using spline basis functions", Technical Report, No. 1015, Shahrood University of Technology, Iran, )2009(, (In Farsi).

37. Hassani, B., Moghaddam, N.Z. and Tavakkoli, S.M.," Isogeometric solution of Laplace equation", Asian Journal of Civil Engineering, Vol. 10, No. 5, pp. 579-592, )2009(.