A Study of the Elastodynamic Problem by Meshless Local Petrov-Galerkin Method Using the Laplace-Transform

The Meshless Local Petrov-Galerkin (MLPG) with Laplace transform is used for solving partial differential equation. Local weak form is developed using the weighted residual method locally from the dynamic partial differential equation and using the moving least square (MLS) method to construct shape function. This method is a more effective alternative than the finite element method for computer modelling and simulation of problems in engineering; however, the accuracy of the present method depends on a number of parameters deriving from local weak form and different subdomains. In this paper, the meshless local Petrov-Galerkin (MLPG) formulation is proposed for forced vibration analysis. First, the results are presented for different values of s α , and q α with regular distribution of nodes 55 t n = . After, the results are presented with fixed values of s α and q α for different time-step.


Introduction
The MLPG approach has been applied for solving 2D and 3D problems domain solutions by ; Atluri and al., [1,2], and for boundary integral equations, as the MLPG method by Sladek and al., [3,4].The MLPG method also applied to free and forced vibrations of a beam and a thin plate by [5] .These method is based on a local weak form and Moving Least Squares (MLS) approximation and proposed in this paper to extend the MLPG method to dynamic analysis and for solving the problem of a thin elastodynamic homogenous plate problem. In this study the Laplace transform is applied to eliminate the time variable, then, the local boundary integral equations are derived for Laplace transforms and the Stehfest inversion method is applied to obtain the time-dependent solutions. Moreover, both the contour and domain integrations can be easily carried out on rectangular sub-domains. The paper is organized as follows: In section 2 the Basic equations of elastodynamics and their Laplace transforms are proposed. in section 3 The MLPG formulation including the local weak form in Laplacetransformed domain using weighted residual method locally from the dynamic partial differential equation. The numerical results and discussions for 2D problem example are given in section V. Finally, the paper ends with a conclusion.

Elastodynamic basic equations
The strong form of the initial boundary value problem for small displacement elastodynamics is as follows: in (1) The Laplace-transform of a function f(x, t) is defined as: ( Then the Laplace-transforms of the basic Eq. (1) is given by :

The MLPG weak formulation in Laplace-transformed
Instead of writing the global weak-form for the governing equations, the MLPG methods construct the weak form over local subdomains such as . The local weak-form of the governing equation (1) and by considering: is the vector of MLS shape functions corresponding n nodes in the support domain of the point X, and can be written as With Laplace transform: Where is a test function. Applying the Gauss divergence theorem one can write: The constitutive equation gives the relationship between the stress and the strain: The traction vectors at a boundary point : (11) Substituting Eqs. (7), (10) and (11) into Eq. (9) we obtain the discretized LIEs: The time dependent values of the transformed variables can be obtained by an inverse transform.

Numerical results
In this section, we present a numerical study for elastodynamic 2-D problem of a cantilever rectangular homogeneous isotropic plate by using MLPG method, subjected to a dynamic force at the right end of the plate (See Fig. 2).

Figure 2 : Cantilever rectangular homogeneous isotropic plate subjected to a dynamic force at the right end of the plate.
We Consider a simple harmonic load where w is the frequency of the dynamic load, and w=27 is used in this example.  In the following curves, is employed.
Results for different time steps are plotted in Fig.5. It can be found that when the time step is less than 0.01, perfect results are obtained using the Laplace transform.
It also can be found that when a time step is larger than 0.01, the results are not reasonable any more.

Conclusion
In this study the equation formulation based on meshless MLPG method and in the Laplace transform and time domain has been successfully implemented to solve 2-D elastodynamic problems for an isotropic solid, subjected to a dynamic force at the right end of a cantilever plate. We found that the amplitude of the vibration decreases with time because of the damping effects. It also can be found that when the time step is less than 0.01, perfect results is obtained using the Laplace transform and when a time step is larger than 0.01, the results are not reasonable any more.