Abstract
Based on previous findings concerning the numerical solution of one-dimensional elastodynamical problems [Provatidis in Arch Appl Mech 78(4):241–250, 2008] this paper extends the methodology to the static analysis of two-dimensional problems in quadrilateral domains. This target is achieved by replacing the Galerkin/Ritz procedure involved in Lagrangian (or Gordon–Coons) type finite elements by a global collocation scheme. In brief, the boundary conditions are fulfilled at all boundary nodes, while the governing equation is fulfilled at internal points. The theory is supported by four test cases concerning rectangular and curvilinear structures under plane-stress or plane-strain conditions, where the convergence rate is successfully compared with that of conventional bilinear finite elements with the same mesh density.
Similar content being viewed by others
References
Bathe K.J.: Finite Element Procedures. Prentice-Hall/Upper Saddle River, New Jersey (1996)
Beissel S., Belytschko T.: Nodal integration of the element-free Galerkin method. Comput. Methods Appl. Mech. Eng. 139, 49–74 (1996)
Gordon W.J.: Blending functions methods of bivariate multivariate interpolation and approximation. SIAM J. Numer. Anal. 8, 158–177 (1971)
Gordon W.J., Hall C.A.: Transfinite element methods blending function interpolation over arbitrary curved element domains. Numer. Math. 21, 109–112 (1973)
Hughes T.J.R., Cottrell J.A., Bazilevs Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)
Inoue K., Kikuchi Y., Masuyama T.: A NURBS finite element method for product shape design. J. Eng. Des. 16(2), 157–174 (2005)
Kanarachos A., Provatidis C., Deriziotis D., Foteas N.: A new approach of the FEM analysis of two-dimensional elastic structures using global (Coons’s) interpolation functions. In: Wunderlich, J. (eds) CD Proceedings First European Conference on Computational Mechanics, München-Germany, August-September (1999)
Provatidis C.G.: Analysis of axisymmetric structures using Coons’ interpolation. Finite Elements Anal. Des. 39, 535–558 (2003)
Provatidis C.G.: Three-dimensional Coons macroelements in Laplace and acoustic problems. Comput. Struct. 83(19–20), 1572–1583 (2005)
Provatidis C.G.: Analysis of box-like structures using three-dimensional Coons’ interpolation. Commun. Numer. Methods Eng. 21(8), 443–456 (2005)
Provatidis C.G.: Three-dimensional Coons’ macroelements: application to eigenvalue and scalar wave propagation problems. Int. J. Numer. Methods Eng. 65(1), 111–134 (2006)
Provatidis C.G.: Coons-patch macroelements in two-dimensional parabolic problems. Appl. Math. Model. 30, 319–351 (2006)
Provatidis C.G.: Free vibration analysis of two-dimensional structures using Coons-patch macroelements. Finite Elements Anal. Des. 42(6), 518–531 (2006)
Provatidis C.G.: Free vibration analysis of elastic rods using global collocation. Arch. Appl. Mech. 78(4), 241–250 (2008)
Provatidis C.G.: Time- and frequency-domain analysis using lumped mass global collocation. Arch. Appl. Mech. 78(11), 909–920 (2008)
Provatidis C.G.: Global collocation method for two-dimensional rectangular domains. J. Mech. Mater. Struct. 3(1), 185–194 (2008)
Szabó B.A., Babuška I.: Finite Element Analysis. Wiley, New York (1991)
Timoshenko S.P., Goodier J.N.: Theory of Elasticity, 3rd edn. International Student Edition. McGraw-Hill Kogakusha, Tokyo (1970)
Zienkiewicz O.C.: The Finite Element Method, 3rd edn. McGraw-Hill, London (1977)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Provatidis, C.G., Ioannou, K.S. Static analysis of two-dimensional elastic structures using global collocation. Arch Appl Mech 80, 389–400 (2010). https://doi.org/10.1007/s00419-009-0317-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00419-009-0317-y