Keywords: elastodynamics, plane structures, large finite elements, modal analysis.

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.

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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