Abstract




 
   

IJE TRANSACTIONS A: Basics Vol. 26, No. 10 (October 2013) 1191-1202   

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  THE PERFORMANCE OF AN HEXAHEDRON C* ELEMENT IN FINITE ELEMENT ANALYSIS
 
G. H. Majzoobi and B. Sharifi Hamadani
 
( Received: November 26, 2012 – Accepted: April 18, 2013 )
 
 

Abstract    The performance of an 8-noded hexahedron C1* element in elasticity is investigated. Three translational displacements and their derivatives as strain in each direction are considered as degrees of freedom (d.o.f.’s) at each node. The geometric mapping is enforced using a C0 element with no derivative as nodal d.o.f.’s . The stiffness matrix of the element is also computed using a transformation matrix obtained from an equivalent C0 element. The results obtained from elastic stress analysis of a cantilever show that: (i) the convergence rate of 8-noded C1* element is nearly equal to its equivalent C0 element, while it consumes less CPU time with respect to the C0 element; (ii) the element has successfully passed the patch and distortion tests; (iii) the condition number of the stiffness matrix for C1* element is less than the C0 element; (iv) the directly computation of strains as derivative degrees of freedom at the nodes along with excellent convergence makes the C1* element superior compared with its equivalent C0 element.

 

Keywords    Elasticity, Finite element method, elements, Convergence, elements.

 

چکیده    عملکرد یک المان مکعبی هشت گره ای نوع C1* در الاستیسیته بررسی گردیده است. سه تغییر مکان انتقالی و مشتقات آن ها به عنوان کرنش در هرجهت به عنوان درجات آزادی در هر گره در نظر گرفته شده است. نگاشت هندسی با استفاده از المان C0 که هیچ مشتقی به عنوان درجات آزادی گره ای ندارد اعمال می گردد. همچنین ماتریس سختی المان با استفاده از یک ماتریس تبدیل که از المان C0 معادل آن به دست می آید محاسبه گردیده است. نتایج به دست آمده از تحلیل تنش الاستیک یک تیر یک سرگیردار نشان می دهد که : (1) نرخ همگرایی المان هشت گره ای نوع C1* تقریباً معادل المان معادل نوع C0 آن است در حالی که زمان پردازش کمتری نسبت به المان C0 صرف می گردد. (2) المان به طور موفقیت آمیز تست های مسیر و اعوجاج را پشت سر گذاشته است. (3) عدد حالت مربوط به ماتریس سختی المان C1* کمتر از المان C0 است. (4) محاسبه کرنش ها به طور مستقیم و تحت عنوان مشتقات درجات آزادی گره ها همراه با همگرایی عالی، المان C1* را در مقایسه با المان C0 معادل آن برتر می سازد.

References   

1.     Zhu, J. and Zienkiewicz, O., "Adaptive techniques in the finite element method", Communications in Applied Numerical Methods,  Vol. 4, No. 2, (1988), 197-204.

2.     Zienkiewicz, O. C., Zhu, J. Z. and Gong, N. G., "Effective and practical h-p conversion adaptive analysis procedures for the finite element method", International Journal for Numerical Methods in Engineering,  Vol. 28, (1989), 879-891.

3.     Tocher, J. L. and Hartz, B. J., "Higher order finite element for plane stress", J. Eng. Mech. Div. Proc. ASCE,  Vol. 93, (1967), 149-172.

4.     William, K. J., Finite element analysis of cellular structures, in Department of Civil Engineering., University of California: Berkeley. (1969).

5.     YOSHIDA, Y., "A hybrid stress element for thin shell analysis", Finite Element Methods in Engineering, (1974), 271-284.

6.     Robinson, J., "Fournode quadrilateral stress membrane element with rotational stiffness", International Journal for Numerical Methods in Engineering,  Vol. 15, No. 10, (1980), 1567-1569.

7.     Macleod, I. A., "New rectangular finite element for shear wall analysis", J. Struct. Div., ASCE,  Vol. 95, No. 3, (1969), 399-409.

8.     Allman, D., "A compatible triangular element including vertex rotations for plane elasticity analysis", Computers & Structures,  Vol. 19, No. 1, (1984), 1-8.

9.     Bergan, P. and Felippa, C., "A triangular membrane element with rotational degrees of freedom", Computer Methods in Applied Mechanics and Engineering,  Vol. 50, No. 1, (1985), 25-69.

10.   Bergan, P. G. and Felippa, C. A., "“Efficient formulation of a triangular memberane element with drilling freedoms, infinite element methods for plate and shell structures, (eds) t.R. Hughes and e. Hinton, 1 (element technology)", pineridge Press International,  (1986).

11.   Allman, D., "Evaluation of the constant strain triangle with drilling rotations", International Journal for Numerical Methods in Engineering,  Vol. 26, No. 12, (1988), 2645-2655.

12.   Cook, R. D., "On the allman triangle and a related quadrilateral element", Computers & Structures,  Vol. 22, No. 6, (1986), 1065-1067.

13.   Macneal, R. H. and Harder, R. L., "A refined four-noded membrane element with rotational degrees of freedom", Computers & Structures,  Vol. 28, No. 1, (1988), 75-84.

14.   Kelly, D. W. and Kuruppu, M. D., Development of new four node plate and eight node hexahedron element with six nodal degrees of freedom, in Proceeding of the second Asian-pacific conference on computational mechanics.: Sydney, Australia, (1993). 145-149.

15.   Bigdeli, B., An investigation of  convergence in the finite element method, New South Wales University: Australia. (1996).

16.   Bigdeli, B. and Kelly, D. W.,  -convergence and nodal derivatives in the finite element method, in First Australian Congress on Applied Mechanics, The Institution on Engineers, Melbourne, Australia. (1996), 571-576.

17.   Bigdeli, B. and Kelly, D. W., "  -convergence in the finite element method", International Journal for Numerical Methods in Engineering,  Vol. 40, (1997), 4405-4425.

18.   Stassa, F. L., "Applied finite element method", Cbs international editions,  (1985).

19.   Sharifi Hamadani, B., The study of the convergence of  elements in 3-d elasticity (in persian), in Mechnaical Enginnering Department., Bu-Ali Sina University: Hamadan, Iran,. (2001)

20.   Irons, B. M. and Razzaque, A., "Experience with the patch test for convergence of finite element method, in mathematical foundation of the finite element method (eds) a.K. Aziz", Academic press,  (1972).

21.   Veubeke, D. and Fraeijs, B., "Variational principles and the patch test", International Journal for Numerical Methods in Engineering,  Vol. 8, No. 4, (1974), 783-801.

22.   de Arantes e Oliveira, E., "The patch test and the general convergence criteria of the finite element method", International Journal of Solids and Structures,  Vol. 13, No. 3, (1977), 159-178.

23.   Razzaque, A., "The patch test for elements", International Journal for Numerical Methods in Engineering,  Vol. 22, No. 1, (1986), 63-71.

24.   Rasanoff, R. A., Gloudeman, J. F. and Levy, S., Numerical conditioning of stiffness matrix formulations for frame structures, in Proceeding of the Second Conference on Matrix Method in Structural Mechanics., Wright-Patterson AFB: Ohio. (1968), 1029-1060.

25.   Smith, B. F., Domain decomposition algorithms for the partial differential equations of linear elasticity, in Mathematics Department., New York University. (1991)

26.   Babuška, I., Craig, A., Mandel, J. and Pitkäranta, J., "Efficient preconditioning for the p-version finite element method in two dimensions", SIAM Journal on Numerical Analysis,  Vol. 28, No. 3, (1991), 624-661.

27.   Dhatt, G. and Touzot, G., "Finite element method", John Wiley & Sons,  (2012).

28.           Di, S. and Ramm, E., "On alternative hybrid stress 2d and 3d elements", Engineering Computations,  Vol. 11, No. 1, (1994), 49-68.





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