engleski

Meshless numerical formulation for analysis of shell-like structures

Meshless computational methods for the analysis of plate and shell structures are proposed in this contribution. The developed algorithms are based on the local Petrov-Galerkin approach. A shell is considered as a three dimensional (3-D) solid continuum, and the solid-shell concept, which allows the implementation of complete 3-D material models, is adopted. It is assumed that the material fibres that are initially normal to the shell middle surface remain straight, but not necessarily normal to the mid-surface. However, the stretching of the fibre in the thickness direction is allowed, which enables the more realistic description of shell deformation response. The geometry of the shell is described by employing a parametric mapping technique, whereby the middle surface of the shell is defined mathematically exactly. Discretization is carried out by the couples of nodes located on the upper and lower surfaces of the structure. The nodes forming a couple lay on the same fibre in the direction of the normal vector to the shell middle surface. The governing equations are obtained by constructing the local weak forms (LWF) of the 3-D equilibrium equations, which are written over the local sub-domains surrounding the node couples. The approximation of all unknown field variables is carried out by using the Moving Least Squares (MLS) approximation scheme in the directions that are tangential to the structure middle surface, while simple polynomials are applied in the thickness direction. For simplicity, the test functions are assumed to be linear across the thickness, and the Heaviside step function is applied in the tangential directions. The causes of Poisson’s locking and transversal shear locking are identified and explained from the theoretical point of view for the classical displacement-based approach. Both the purely displacement-based (primal) and mixed formulations are proposed and special attention is given to the elimination of locking effects. Two different primal formulations are presented where only the displacement field is approximated. The Poisson’s thickness locking effect is circumvented by adopting the hierarchical quadratic interpolation over the thickness for the transversal displacement component. The transversal shear locking phenomenon is alleviated by applying a sufficiently high order of the MLS functions in the tangential directions. The proposed strategy performs well in the analysis of thick structures, but it is sensitive to the shear locking effects. The application of high-order approximation functions leads to prohibitively high computational costs. The mixed approaches, based on the mixed MLPG paradigm, are developed. Therein, the appropriate strain and stress components are approximated separately from the displacement field. A closed system of equations includes the LWF of the 3-D equilibrium equations, and suitable collocation relations enforcing the compatibility between approximated strains/stresses and approximated displacements at the nodes. The nodal strain and stress values are eliminated from the equations by the algebraic manipulation of the discretized equations, which is performed locally, and a global system of equations containing only the unknown nodal displacement variables is obtained. In the formulation for plates, Poisson’s thickness locking is eliminated by modifying the nodal values of the normal transversal strain component. In the curved shell structures, the transversal normal stress is approximated directly instead of the transversal normal strain, leading to a simpler and computationally less costly efficient algorithm. Transversal shear locking in the thin structural limit is efficiently suppressed by means of the separate strains approximation. It is theoretically proved that the developed mixed approach is numerically more efficient than the comparable primal meshless formulations. The numerical efficiency of the derived algorithms is demonstrated by numerical examples.

Meshless methods; Meshless Local Petrov-Galerkin method; Moving Least Squares approximation;

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