### Downloads

## Abstract

This paper presents an explicit integration scheme to compute the stiffness matrix of twelve node and sixteen node linear convex quadrilateral finite elements of Serendipity and Lagrange families using an explicit integration scheme and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique, In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties matrices and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+) in the bivariates over a 2-square (-1 ) with the nodes on the boundary and in the interior of this simple domain. The finite elements up to cubic order have nodes only on the boundary for Serendipity family and the finite elements with boundary as well as some interior nodes belong to the Lagrange family. The first order element is the bilinear convex quadrilateral finite element which is an exception and it belongs to both the families. We have for the present ,the cubic order finite elements which havee 12 boundary nodes at the nodal coordinates {(-1,-1),(1,-1),(1,1),(-1,1),(-1/3,-1), (1/3,-1),(1,-1/3),(1,1/3),(1/3,1),(-1/3,1),(-1,1/3),(-1,-1/3)} and the four interoior nodal coordinates at the points (-1/3,-1/3),(1/3,-1/3),(1/3,1/3),(-1/3,1/3)} in the local parametric space ( In this paper, we have computed the integrals of local derivative products with linear denominator (4+) ** **in exact forms using

**the symbolic mathematics capabilities of MATLAB**. The proposed explicit finite element integration scheme can be then applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all 12-node and 16-node linear(straight edge) convex quadrilaterals for a polygonal domain which provides the nodal coordinates and the element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is an unknown physical variable and in with Dirichlet boundary conditions over the convex polygonal domain.

## References

2. Bathe K.J Finite Element Procedures, Prentice Hall, Englewood Cliffs, N J (1996)

3. Reddy J.N Finite Element Method, Third Edition, Tata Mc Graw-Hill (2005)

4. Burden R.L and Faires J.D Numerical Analysis, 9th Edition, Brooks/Cole, Cengage Learning (2011)

5. Stroud A.H and Secrest D , Gaussian quadrature formulas, Prentice Hall,Englewood Cliffs, N J, (1966)

6. Stoer J and Bulirsch R, Introduction to Numerical Analysis, Springer-Verlag, New York (1980)

7. Chung T.J Finite Element Analysis in Fluid Dynamics, pp. 191-199, Mc Graw Hill, Scarborough, C A , (1978)

8. Rathod H.T, Some analytical integration formulae for four node isoparametric element, Computer and structures 30(5), pp.1101-1109, (1988)

9. Babu D.K and Pinder G.F, Analytical integration formulae for linear isoparametric finite elements, Int. J. Numer. Methods Eng 20, pp.1153-1166

10. Mizukami A, Some integration formulas for four node isoparametric element Computer Methods in Applied Mechanics and Engineering. 59 pp. 111-21(1986)

11. Okabe M, Analytical integration formulas related to convex quadrilateral finite elements, Computer methods in Applied mechanics and Engineering. 29, pp.201-218 (1981)

12. Griffiths D.V Stiffness matrix of the four node quadrilateral element in closed form, International Journal for Numerical Methods in Engineering. 28, pp.687-703(1996)

13. Rathod H.T and Shafiqul Islam. Md , Integration of rational functions of bivariate polynomial numerators with linear denominators over a (-1,1) square in a local parametric two dimensional space, Computer Methods in Applied Mechanics and Engineering. 161 pp.195-213 (1998)

14. Rathod H.T and Sajedul Karim, Md An explicit integration scheme based on recursion and matrix multiplication for the linear convex quadrilateral elements, International Journal of Computational Engineering Science. 2(1) pp. 95- 135(2001)

15. Yagawa G, Ye G.W and Yoshimura S, A numerical integration scheme for finite element method based on symbolic manipulation, International Journal for Numerical Methods in Engineering. 29, pp.1539-1549(1990)

16. Rathod H.T and Shafiqul Islam Md, Some pre-computed numeric arrays for linear convex quadrilateral finite elements, Finite Elements in Analysis and Design 38, pp. 113-136 (2001)

17. Hanselman D and Littlefield B, Mastering MATLAB 7 , Prentice Hall, Happer Saddle River, N J . (2005)

18. Hunt B.H, Lipsman R.L and Rosenberg J.M, A Guide to MATLAB for beginners and experienced users, Cambridge University Press (2005)

19. Char B, Geddes K, Gonnet G, Leong B, Monagan M and Watt S, First Leaves; A tutorial Introduction to Maple ∨ , New York : Springer–Verlag (1992)

20. Eugene D, Mathematica , Schaums Outlines Theory and Problems, Tata Mc Graw Hill (2001)

21. Ruskeepaa H, Mathematica Navigator, Academic Press (2009)

22. Timoshenko S.P and Goodier J.N, Theory of Elasticity, 3rd Edition, Tata Mc graw Hill Edition (2010)

23. Budynas R.G, Applied Strength and Applied Stress Analysis, Second Edition Tata Mc Graw Hill Edition (2011)

24. Roark R.J, Formulas for stress and strain, Mc Graw Hill, New York (1965)

25. Nguyen S.H , An accurate finite element formulation for linear elasticcalculations, Computers and Structures. 42, pp.707-711 (1992)

26. Rathod H.T, Venkatesh.B, Shivaram. K.T,Mamatha.T.M, Numerical Integration over polygonal domains using convex quadrangulation and Gauss Legendre Quadrature Rules, International Journal of Engineering and Computer Science, Vol. 2,issue 8,pp2576-2610(2013)

27. Rathod H.T, Rathod Bharath, Shivaram.K.T,Sugantha Devi.K, A new approach to automatic generation of all quadrilateral mesh for finite analysis, International Journal of Engineering and Computer Science, Vol. 2,issue 12,pp3488-3530(2013)

28. Rathod.H.T, Bharath Rathod, Shivaram K.T , H. Y. Shrivalli , Tara Rathod , K. Sugantha Devi,An explicit finite element integration scheme using automatic mesh generation technique for linear convex quadrilaterals over plane regions , international Journal of Engineering and Computer Science, Vol. 3,issue 4,pp5400-5435 (2014)

29. Rathod.H.T, Bharath Rathod, K.T.Shivaram,Sugantha Devi.K,Tara Rathod , An explicit finite element integration scheme for linear eight node convex quadrilaterals using automatic mesh generation technique over plane regions, international Journal of Engineering and Computer Science, Vol. 3,issue 4,pp5657-5713 (2014)

30. Rathod.H.T, Sugantha Devi.K ,Finite element solution of Poisson equation over polygonal domains using an explicit integration scheme and a novel auto mesh generation technique,,International Journal of Engineering and Computer Science, ISSN:2319-7242,Volume(5),issue 8,August(2016), pp. 17397-17481

31. Rathod.H.T, Sugantha Devi.K,Nagabhushana.C.S , Chudamani.H.M ,Finite element analysis of linear elastic torsion for regular polygons, International Journal of Engineering and Computer Science, ISSN:2319-7242,Volume(5),issue 10,October(2016), pp. 18413-1842

32. Rathod.H.T, Bharath Rathod, Sugantha Devi.K,Hariprasad.A.S, Finite element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for eight node linear convex quadrilaterals, International Journal of Engineering and Computer Science, ISSN:2319-7242, Volume (6) Issue 10 October 2017, pp 22632-22787

33. H. T. Rathod,Md.Shafiqul. H. Y. Shrivall , Bharath Ratho K. Sugantha Devi, Finite element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for nine node linear convex quadrilateral of Lagrange family, International Journal of Engineering and Computer Science, ISSN:2319-7242, Volume 6 Issue 11 November 2017, Page No. 22869-23058