Elsevier

Thin-Walled Structures

Volume 49, Issue 1, January 2011, Pages 185-196
Thin-Walled Structures

A natural neighbour meshless method with a 3D shell-like approach in the dynamic analysis of thin 3D structures

https://doi.org/10.1016/j.tws.2010.09.023Get rights and content

Abstract

This work presents the dynamic analysis of three-dimensional plate and shell structures based on an improved meshless method, the Natural Neighbour Radial Point Interpolation Method (NNRPIM) using a shell-like formulation. In the NNRPIM, the nodal connectivity is imposed using the natural neighbours concept. An integration background mesh is constructed, totally node-dependent, and used in the numerical integration of the NNRPIM interpolation functions, which possess the delta Kronecker property. Several dynamic plate and shell problems are studied to demonstrate the effectiveness of the method.

Introduction

Several engineering structures, such as aeroplane fuselages, boat hulls, roof structures, among many others, are constructed assembling spatial thin shells structures. In our days shell structures are design to be light, being the shells themselves the main load supporting structure, reducing the number of structure stiffeners. On the other hand this structural material optimization leads to lower fundamental frequencies, increasing the risk of collapse by resonance. Thus, the dynamic analysis became an important part in shell structures design.

The modulation of such complex structures is made using numerical methods. For many years the numerical method used was the Finite Element Method (FEM) [1]. However in the last 15 years meshless methods [2] enlarge their application field, and are today a competitive and alternative numerical method in structural analysis. Numerous shell structures present elaborated curvatures and several holes or discontinuous essential boundaries, and for these conditions meshless methods are efficient. Since it was in the beginning with the FEM, in this work the analysed thin structures are solved as three-dimensional (3D) problems, with some awareness in the integration along the smallest dimension in order to obtain the most reliable results. The nodes discretizing the problem domain can be randomly distributed in meshless methods, since the field functions are approximated within a flexible influence domain rather an element. In meshless methods the influence domains may and must overlap each other, in opposition to the no-overlap rule between elements in the FEM.

Meshless methods that use the weak form solution can be divided in two categories, the ones that use approximation functions [3], [4], [5], [6], [7], [8] and others that use interpolation functions. Meshless methods based in approximation functions have been successfully applied in computational mechanics and even its difficulty on imposing the essential and natural boundary conditions, due to the lack of the delta Kronecker property φi(xj)≠δij, has been overcome with the use of efficient numerical methods. At the time, to solve the mentioned difficulty of the approximation functions, several meshless methods, using interpolation functions, were developed [9], [10], [11], [12], [13], [14], [15].

In this work an interpolation meshless method is used—the Natural Neighbour Radial Point Interpolation Method (NNRPIM) [16], [17]. In the NNRPIM, the nodal connectivity and the background integration mesh, totally dependent on the nodal mesh, are achieved using mathematic concepts, such as Voronoï diagrams [18] and the Delaunay tessellation [19]. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed with the Radial Point Interpolators (RPI) [15] and possess the delta Kronecker property.

Within the 3D shell-like formulation a bi-dimensional nodal mesh, coincident with the middle surface of the plate or of the shell, is firstly created. Based on this bi-dimensional mesh the background integration mesh is constructed. Therafter the middle surface nodal mesh is projected to the top surface and bottom surface of the plate or shell, and the bi-dimensional integration mesh is distributed along the plate or shell thickness respecting a Gauss quadrature scheme.

The outline of this paper is as follows: In Section 2 the meshless 3D shell-like approach is presented, the creation of the influence-cells and the used integration scheme are summarized, as well as the construction of the interpolation functions. In Section 3, the dynamic discrete system of equations is presented and developed. In Section 4, benchmark dynamic examples of plates and shells in free and force vibrations are solved, and finally the paper ends with the conclusions and remarks in Section 5.

Section snippets

Natural neighbours

The natural neighbours [20] determination of each node belonging to the global nodal set N={n1n2nN}R3 is achieved in the NNRPIM using the Voronoï diagrams and the Delaunay triangulation. This theory is applicable to an N-dimensional space. The Voronoï diagram of N is the partition of the domain defined by N in sub-regions VI, closed and convex. Each sub-region VI is associated with the node I, nI, in a way that any point in the interior of the VI is closer to nI than any other node nJ, where n

Dynamic discrete system equations

Consider the solid with a domain Ω bounded by Γ. In the absence damping effects, the dynamic equilibrium based on the principle of virtual work can be written asΩδεTσdΩ+ΩδuTρüdΩΩδuTbdΩΓtδuTtdΓ=0xu and ü are, respectively, the displacement and the acceleration field, b is the body force vector and t the traction on the natural boundary. The strain vector ε is defined asε=Luwhere L is the differential operator defined in Eq. (12):L=[x00y0z0y0xz000z0yx]T

The linear

Dynamic examples

In this section, in order to show the accuracy of the NNRPIM 3D shell-like (3DSL) approach, in the context of dynamic analysis, several examples are presented and the numerical results are compared with analytical solutions and FEM solutions, which are available in the literature. Firstly, free vibration plates and shells examples are shown and afterwards are presented examples of the forced vibration analysis of plates and shells. All results are obtained using the consistent mass matrix.

Conclusions

In this work, a three-dimensional shell-like (3DSL) approach for the dynamic analysis of thin plates and shells using the Natural Neighbour Radial Point Interpolation Method (NNRPIM) was proposed. Several well-known plate and shells benchmark examples were also solved. The obtained results, and the experience acquired along the development of this work, permit to conclude the following:

  • (a)

    The NNRPIM is a stable and accurate interpolator meshless method.

  • (b)

    In general, the convergence rate is high and

Acknowledgments

The authors truly acknowledge the funding provided by Ministério da Ciência, Tecnologia e Ensino Superior—Fundação para a Ciência e a Tecnologia (Portugal), under Grant SFRH/BD/31121/2006, and by FEDER/FSE, under Grant PTDC/EME-PME/81229/2006.

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