This paper presents a novel mesh generation scheme of all quadrilateral elements over a linear polygonal domain. We first decompose the linear polygon into simple sub regions in the shape of quadrilaterals. These simple regions are then quadrangulated to generate first into a fine mesh of four node quadrilateral elements using bilinear transformations.We propose then an automatic quadrilateral conversion scheme. Each four node quadrilateral is converted to an 8-node,9-node,12-node ,16-node,17-node and 25-noded quadrilaterals by inserting the midside nodes appropriately. Examples are presented to illustrate the simplicity and efficiency of the new mesh generation method for standard and arbitrary shaped domains. We have appended two important MATLAB programs which incorporate the mesh generation scheme for the 17-noded complete Lagrange elements developed in this paper.Other MATLAB programs can be coded on similar lines. These programs provide valuable output on the nodal coordinates ,element connectivity and graphic display of the all quadrilateral meshes for application to finite element analysis.
References
• R. Courant, Variational methods for the solution of problems of equilibrium and vibration, Bulletin of the American Mathematical Society, 49:1-23(1943).
• J.H. Argyris, Matrix displacement analysis of anisotropic shells by triangular elements, Journal of Royal Aeronautical Society 69:801 (1965).
• R.W. Clough, The finite element method in plane stress analysis, Proceedings of the 2nd ASCE Conference in Electronic Computation, Pittsburgh, PA 1960.
• Zienkiewicz. O. C, Philips. D. V, An automatic mesh generation scheme for plane and curved surface by isoparametric coordinates, Int. J.Numer. Meth.Eng, 3, 519-528 (1971) .
• Gordan. W. J and Hall. C. A, Construction of curvilinear coordinates systems and application to mesh generation, Int. J.Numer. Meth. Eng 3, 461-477 (1973).
• Cavendish. J. C, Automatic triangulation of arbitrary planar domains for the finite element method, Int. J. Numer. Meth. Eng 8, 679-696 (1974).
• Moscardini. A. O , Lewis. B. A and Gross. M A G T H M – automatic generation of triangular and higher order meshes, Int. J. Numer. Meth. Eng, Vol 19, 1331-1353(1983).
• Lewis. R. W, Zheng. Y, and Usmam. A. S, Aspects of adaptive mesh generation based on domain decomposition and Delaunay triangulation, Finite Elements in Analysis and Design 20, 47-70 (1995) .
• W. R. Buell and B. A. Bush, Mesh generation a survey, J. Eng. Industry. ASME Ser B. 95 332-338(1973).
• Rank. E, Schweingruber and Sommer. M, Adaptive mesh generation and transformation of triangular to quadrilateral meshes, Common. Appl. Numer. Methods 9, 11 121-129(1993)
• Ho-Le. K, Finite element mesh generation methods, a review and classification, Computer Aided Design Vol.20, 21-38(1988)
• Lo. S. H, A new mesh generation scheme for arbitrary planar domains, Int. J. Numer. Meth. Eng. 21, 1403-1426(1985)
• Peraire. J. Vahdati. M, Morgan. K and Zienkiewicz. O. C, Adaptive remeshing for compressible flow computations, J. Comp. Phys. 72, 449-466(1987).
• George. P.L, Automatic mesh generation, Application to finite elements, New York , Wiley (1991).
• George. P. L, Seveno. E, The advancing-front mesh generation method revisited. Int. J. Numer. Meth. Eng 37, 3605-3619(1994)
• Pepper. D. W, Heinrich. J. C, The finite element method, Basic concepts and applications, London, Taylor and Francis (1992)
• Zienkiewicz. O. C, Taylor. R. L and Zhu. J. Z, The finite element method, its basis and fundamentals, 6th Edn, Elsevier (2007)
• Masud. A, Khurram. R. A, A multiscale/stabilized finite element method for the advection- diffusion equation, Comput. Methods. Appl. Mech. Eng 193, 1997-2018(2004)
• Johnston. B.P, Sullivan. J. M and Kwasnik. A, Automatic conversion of triangular finite element meshes to quadrilateral elements, Int. J. Numer. Meth. Eng. 31, 67-84(1991)
• Lo. S. H, Generating quadrilateral elements on plane and over curved surfaces, Comput. Stuct. 31(3) 421-426(1989)
• Zhu. J, Zienkiewicz. O. C, Hinton. E, Wu. J, A new approach to the development of automatic quadrilateral mesh generation, Int. J. Numer. Meth. Eng 32(4), 849-866(1991)
• Lau. T. S, Lo. S. H and Lee. C. S, Generation of quadrilateral mesh over analytical curved surfaces, Finite Elements in Analysis and Design, 27, 251-272(1997)
• Park. C, Noh. J. S, Jang. I. S and Kang. J. M, A new automated scheme of quadrilateral generation for randomly distributed line constraints, Computer Aided Design 39, 258- 267(2007).
• Thompson.E.G, Introduction to the finite element method,John Wiley & Sons Inc.(2005).
• H.T.Rathod , Bharath Rathod , K.T.Shivaram , K. Sugantha Devi , 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,issue5(2014),pp5657-5713.
• H.T. Rathod, K. Sugantha devi, An All Quadrilateral Automesh Generation Technique and Explicit Integration Scheme for Finite Elements to Solve Some Elliptic Boundary Value Problems in Two Space, International Journal of Engineering and Computer Science, ISSN 2319-7242, Vol.5 issue 12(December 2016), pp 19674-19778.
• H.T.Rathod, 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.
• H. T. Rathod Md.Shafiqul Islam. H. Y. Shrivall , Bharath Rathod, 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:2319-7242Volume 6 Issue 11 November 2017, Page No. 22869-23058.
• H. T. Rathod, Md.Shafiqul.Islam, Bharath Rathod , 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 linear convex quadrilaterals of cubic order Serendipity and Lagrange families, International Journal Of Engineering And Computer Science ISSN:2319-7242,Volume 7 Issue 1 January 2018, Page No. 23329-23482.
• M.Okabe,Complete Lagrange family for the cube in Finite Element interpolations,Comput.Meth.Appl.Mech.Engng.29(1981)51-56.
• H.T,Rathod and Sridevi.Kilari,General complete Lagrange family for the cube in finite element interpolations,Comput.Meth.Appl.Mech and Engrg 181(2000)295-344.
