engleski

Numerical analysis of 2-D linear elastic problems by MLPG method

Meshless methods have attracted considerable attention of the scientific community over the last two decades due to their flexibility and high continuity of meshless approximation functions. However, high numerical costs in comparison to the Finite Element (FE) Method still impede their wide commercial use. In addition, primal meshless methods are in general plagued by various locking phenomena, similarly to the comparable FE formulations. Collocation meshless methods are computationally more efficient than those based on the integration of various weak forms, but they suffer from instability and lack of accuracy, caused primarily by the problematic imposition of Neumann boundary conditions (BCs). The mixed Meshless Local Petrov-Galerkin (MLPG) Method paradigm represents an efficient remedy for these deficiencies. So far it has been successfully applied for solving various engineering problems, such as the bending of thin beams, plates and shells, the topology-optimization, or electrodynamics. It is computationally superior to the primal meshless approaches because the differentiation of meshless functions in the entire domain is avoided, which increases numerical efficiency and stability. Furthermore, the mixed approach decreases the continuity requirements of the trail functions and allows the use of the shape functions of a lower order. This enables the use of smaller support domains of shape functions and further decreases computational costs. This contribution deals with the application of the MLPG approach in two-dimensional (2-D) linear elasticity, whereby special attention is dedicated to the mixed strategy. Various formulations based on either a weak or strong form of governing equations are presented. Discretization is performed by a set of nodes that are not connected into the mesh of elements. The local weak form (LWF) of the equilibrium equation is derived over the prismatic local sub-domains that surround the nodes. In the case of the collocation formulations, the Dirac Delta functions are applied at each nodal point as the test functions in LWF. The stress and displacements components are approximated independently by using the same trail shape function. Various approximation techniques are used, including the Moving Least Squares (MLS) functions, the Polynomial Point Interpolation Method (PPIM), the multi-quadrics Radial Basis Functions (MQ-RBF) and the B-splines functions. The approximated stresses are used to discretize the strong or weak forms of the equilibrium equations. The stress nodal values are then eliminated from the equations by enforcing the compatibility between the interpolated stresses and displacements at the nodes via the collocation, which yields a global system of equations with only nodal displacements as unknowns. In the collocation, special attention is dedicated to the imposition of the Force BCs, which is performed by the direct collocation and the penalty method. The performance of the presented algorithms is compared by few numerical examples.

meshless; MLPG method; mixed approach; weak form; strong form

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