- Volume 7 Issue 2
A previously proposed finite element formulation method is refined and modified to generate a new type of elements. The method is based on selecting a set of general solution modes for element formulation. The constant strain modes and higher order modes are selected and the formulation method is designed to ensure that the element will pass the basic single element test, which in turn ensures the passage of the basic patch test. If the element is to pass the higher order patch test also, the element stiffness matrix is in general asymmetric. The element stiffness matrix depends only on a nodal displacement matrix and a nodal force matrix. A symmetric stiffness matrix can be obtained by either modifying the nodal displacement matrix or the nodal force matrix. It is shown that both modifications lead to the same new element, which is demonstrated through numerical examples to be more robust than an assumed stress hybrid element in plane stress application. The method of formulation can also be used to arrive at the conforming displacement and hybrid stress formulations. The convergence of the latter two is explained from the point of view of the proposed method.
- Bazeley, G.P., Cheung, Y.K., Irons, B.M., and Zienkiewicz, O.C. (1966), "Triangular elements in plate bending. Conforming and nonconforming solutions", Proc. 1st Matrix Methods in Struct. Mech., AFFDL-TR-CC-80, Wright Patterson, A F. Base, Ohio, 547-576.
- Belytshko, T. and Lasly, S. (1988), "A fractal patch test" , Int. J. Numer. Methods Eng., 26, 2199-2210. https://doi.org/10.1002/nme.1620261005
- Bergan, P.G. and Hanssen, L. (1976), "A new approach for deriving 'good' element stiffness matrices" , Math. of Finite Elements and Appl. (Ed. Whiteman, J.R), Academic Press, London, 483-497.
- Bergan, P.G. (1980), "Finite elements based on energy orthogonal functions", Int. J. Numer. Methods Eng., 15, 1541-1555. https://doi.org/10.1002/nme.1620151009
- Bergan, P.G. (1981), "Correction and further discussion of 'finite element based on energy orthogonal functions'" , Int. J. Numer. Methods Eng., 17, 154-155. https://doi.org/10.1002/nme.1620170114
- Cook, R.D. (1974), "Improved two-dimensional finite element", J. Structural Div., ASCE, 100, 1851-1863.
- Cook, R.D., Malkus, D.S. and Plesha, M.E. (1989), Concepts and Applications of Finite Element Analysis, Third Ed., John Wiley & Sons, New York.
- Irons, B.M. and Razzaque, A. (1972), "Experience with the patch test for convergence of finite element methods", Math. Foundations of the Finite element Method (Ed. Aziz, AK.), Academic Press, 557-587.
- Mau, S.T. (1982), ''A direct vector combination procedure for finite element stiffness matrix formulation" , Int. J. Numer. Methods Eng., 18, 863-878. https://doi.org/10.1002/nme.1620180605
- Mau, S.T. and Dan H.C. (1986), "A restricted hybrid stress formulation based on a direct finite element method", Int. J. Numer. Methods Eng., 23, 1495-1507. https://doi.org/10.1002/nme.1620230807
- Park, Y.M. and Choi, C.K. (1997), "The patch test and convergence for nonconforming Mindlin plate bending elements" , Int. J. Struct. Eng. Mech., 5, 471-490. https://doi.org/10.12989/sem.19126.96.36.1991
- Pian, T.H.H. (1964), "Derivation of element stiffness matrices by assumed stress distributions" , AlAA J., 2, 1332-1336. https://doi.org/10.2514/3.2545
- Pian, T.H.H. and Mau, S.T. (1972), "Some recent studies in assuming stress hybrid models", Advances in Computational Methods in Structural Mechanics and Design (Ed. aden, J.T.), University of Alabama Press, Huntsville, Alabama, 87-106.
- Razzaque, A. (1986), "The patch test for elements", Int. J. Number. Methods Eng., 22, 63-71. https://doi.org/10.1002/nme.1620220106
- Stummel, F. (1980), "The limitations of the patch test", Int. J. Numer. Methods Eng., 15, 177-188. https://doi.org/10.1002/nme.1620150203
- Taylor, R.L., Beresford, P.J. and Wilson, E.L. (1976), "A nonconforming element for stress analysis" , Int. J. Numer. Methods Eng., 10, 1211-1219. https://doi.org/10.1002/nme.1620100602
- Taylor, R.L., Simo, J.C., Zienkiewicz, O.C. and Chan, A.C.H. (1986), "The patch test-A condition for assessing FEM convergence" , Int. J. Numer. Methods Eng., 22, 39-62. https://doi.org/10.1002/nme.1620220105
- Verma, A. and Melosh, R.J. (1987), "Numerical test for assessing finite element model convergence" , Int. J. Numer. Methods Eng., 24, 843-857. https://doi.org/10.1002/nme.1620240502