Open Access

ARTICLE

Dynamic Analysis by Meshless Local Petrov-Galerkin Formulations Considering a Time-Marching Scheme Based on Implicit Green's Functions

D. Soares Jr.¹, J. Sladek², V. Sladek²

1 Structural Engineering Department, Federal University of Juiz de Fora, CEP 36036-330 Juiz de Fora, MG, Brazil. Phone: +55 32 2102 3468. E-mail: delfim.soares@ufjf.edu.br
2 Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia. E-mail: usarslad@savba.sk

Computer Modeling in Engineering & Sciences 2009, 50(2), 115-140. https://doi.org/10.3970/cmes.2009.050.115

Abstract

In recent years the idea of using mesh-free or mesh-less methods for numerical solution of partial differential equations has received much attention, due to their potential advantage in eliminating the costly effort of mesh generation and re-meshing. A variety of meshless methods has been proposed so far. Many of them are derived from a weak-form formulation on global domain or a set of local subdomains. In the global formulation background cells are required for the integration of the weak form. In methods based on local weak-form formulation no cells are required and therefore they are often referred to as truly meshless methods. If for the geometry of subdomains a simple form is chosen, numerical integrations can be easily carried out over them. The meshless local Petrov-Galerkin (MLPG) method is a fundamental base for the derivation of many meshless formulations, since trial and test functions are chosen from different functional spaces. In the present paper, the MLPG is applied to solve linear and nonlinear dynamic 2-D problems. Both the Heaviside step function and the Gaussian weight function are used as the test functions in the local weak form to investigate which one is more convenient. The moving least square (MLS) method is used for approximation of physical quantities in the local integral equations. After spatial discretization, a system of ordinary differential equations of second order for unknown nodal quantities is obtained. A time-marching scheme which is based on implicit Green's functions is applied to solve this system of equations in the time domain. In this procedure, the time-domain Green's matrices of the elastodynamic problem are considered in order to generate a recursive relationship to evaluate displacements and velocities at each time-step. Taking into account the Newmark method, the Green's matrices of the problem are numerically and implicitly evaluated, establishing the Green-Newmark method. At the end of the work, numerical examples are presented, verifying the accuracy and potentialities of the new methodology.

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