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# Meshless Local Petrov-Galerkin Method for Plane Piezoelectricity

Institute of Construction and Architecture, Slovak Academy of Sciences, 84503 Bratislava, Slovakia
Department of Civil Engineering, University of Siegen, D-57068 Siegen, Germany
Department of Civil Engineering, University of Malaga, 29031 Malaga, Spain

Computers, Materials & Continua 2006, 4(2), 109-118. https://doi.org/10.3970/cmc.2006.004.109

## Abstract

Piezoelectric materials have wide range engineering applications in smart structures and devices. They have usually anisotropic properties. Except this complication electric and mechanical fields are coupled each other and the governing equations are much more complex than that in the classical elasticity. Thus, efficient computational methods to solve the boundary or the initial-boundary value problems for piezoelectric solids are required. In this paper, the Meshless local Petrov-Galerkin (MLPG) method with a Heaviside step function as the test functions is applied to solve two-dimensional (2-D) piezoelectric problems. The mechanical fields are described by the equations of motion with an inertial term. To eliminate the time-dependence in the governing partial differential equations the Laplace-transform technique is applied to the governing equations, which are satisfied in the Laplace-transformed domain in a weak-form on small subdomains. Nodal points are spread on the analyzed domain and each node is surrounded by a small circle for simplicity. The spatial variation of the displacements and the electric potential are approximated by the Moving Least-Squares (MLS) scheme. After performing the spatial integrations, one obtains a system of linear algebraic equations for unknown nodal values. The boundary conditions on the global boundary are satisfied by the collocation of the MLS-approximation expressions for the displacements and the electric potential at the boundary nodal points. The Stehfest's inversion method is applied to obtain the final time-dependent solutions.

## Keywords

J. . Sladek, V. . Sladek, C. . Zhang, F. . Garcia-Sanche and M. W. . ¨unsche, "Meshless local petrov-galerkin method for plane piezoelectricity," Computers, Materials & Continua, vol. 4, no.2, pp. 109–118, 2006.

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