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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 83
PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 197

Transient Elastodynamic Analysis of Plane Structures Using Coons-Patch Macroelements and Modal Superposition

C.G. Provatidis

Department of Mechanical Engineering, National Technical University of Athens, Greece

Full Bibliographic Reference for this paper
C.G. Provatidis, "Transient Elastodynamic Analysis of Plane Structures Using Coons-Patch Macroelements and Modal Superposition", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 197, 2006. doi:10.4203/ccp.83.197
Keywords: elastodynamics, plane structures, large finite elements, modal analysis.

Summary
Early attempts to develop approximate methods by Ritz [1], Galerkin [2] and Trefftz [3] were based on global approximation of the displacement within the whole structure. Later, finite element methods (FEM) suggested several local approximation schemes as overviewed by Zienkiewicz [4] and Bathe [5]. However, due to high manual effort required for mesh generation as well as further needs for increased accuracy in calculations, a lot of attempts have been made in order to replace or improve conventional finite element methods. In this context, boundary element methods (e.g. Brebbia and Dominguez [6]) have been applied to the whole spectrum of mechanics as well as meshless and mesh-free techniques (Atluri [7], Liu [8]) have been applied for a over a decade. Concerning the particular area of dynamic analysis of structures, substructuring and dynamic condensation techniques aiming to reduce the order of the equations system have been applied by Guyan [9] and other researchers.

Within this context, the author has previously proposed a global approximation approach by constructing large finite elements with the nodal points mostly along the boundaries. The background of the relevant method is Coons' interpolation formula [10]. This method has been successfully applied to static and eigenvalue elasticity problems [11,12]. Concerning transient analysis, a preliminary study under uniform tensile loading using a Poisson's ratio of zero value in conjunction with a central-difference time-integration scheme has led to promising results [13]. Moreover, the proposed Coons-patch macroelements (CPM) have been successfully applied to potential time-dependent (hyperbolic and parabolic) problems [14,15].

This paper further investigates the applicability of two-dimensional Coons macroelements to solve transient dynamical problems using modal superposition and compare with conventional FEM for two typical test cases chosen from literature. The first example refers to a rectangular cantilever of dimensions (2x4 m2) under impulsive flexural load while the second to a dam-like structure subject to sinusoidal excitation. The proposed CPM methodology is successfully compared with conventional four-node FEM with the same boundary discretization. It was found that in the current formulation only consistent global mass matrices are applicable.

References
1
W. Ritz, "Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik", Zeitschrift für Angewandte Mathematik und Mechanik, 135(1), 1-61, 1908.
2
B.G. Galerkin, "Series solution of some problems of elastic equilibrium of rods and plates", Vestn. Inszh. Tech. 19, 897-908, 1915 (in Russian).
3
E. Trefftz, "Ein Gegenstück zum Ritz'schen Verfahren" in: Proceedings, 2nd International Congress in Applied Mechanics, Zurich, 1926.
4
O.C. Zienkiewicz, "The Finite Element Method", 3rd ed., McGraw-Hill, London, 1977.
5
K.J. Bathe, "Finite element procedures in engineering analysis", Prentice-Hall, New Jersey, 1982.
6
C.A. Brebbia, J. Dominguez, "Boundary Elements: An Introductory Course", 2nd edition, Computational Mechanics Publications, Southampton, 1992.
7
S.N. Atluri, "The Meshless Method (MLPG) for Domain and BIE Discretizations", Tech Science Press, Encino, 2004.
8
G.R. Liu, "Mesh Free Methods: Moving beyond the finite element method", CRC Press, Boca Raton, 2003.
9
R.J. Guyan, "Reduction of Stiffness and Mass Matrices", Journal AIAA, 3(2), 380, 1965. doi:10.2514/3.2874
10
G. Beer and J.O. Watson, "Introduction to Finite and Boundary Element Methods for Engineers", Wiley, Chichester, 1992, pp. 357-377 (Chapter 11).
11
C. Provatidis, "Analysis of axisymmetric structures using Coons' interpolation", Finite Elements in Analysis and Design, 39(5) 535-558, 2003. doi:10.1016/S0168-874X(02)00127-0
12
C. Provatidis, "Free vibration analysis of two-dimensional structures using Coons-patch macroelements", Finite Elements in Analysis and Design 42(6) 518-531, 2006. doi:10.1016/j.finel.2005.10.002
13
C. Provatidis, "Frequency analysis and transient response of two-dimensional structures using Coons-patch macroelements" in: M. Brennan et al. (Eds.), Proceedings VIII International Conference on Recent Advances in Structural Dynamics, ISVR, University of Southampton, 14-16 July 2003.
14
C.G. Provatidis, "Coons-patch macroelements in two-dimensional eigenvalue and scalar wave propagation problems", Computers & Structures 82(4-5) 383-395, 2004. doi:10.1016/j.compstruc.2003.10.012
15
C.G. Provatidis, "Coons-patch macroelements in two-dimensional parabolic problems", Applied Mathematical Modelling 30(4) 319-351, 2006. doi:10.1016/j.apm.2005.05.011

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