A meshless local Petrov-Galerkin (MLPG) method for free and forced vibration analyses for solids

Abstract

The meshless local Petrov-Galerkin (MLPG) method is an effective truly meshless method for solving partial differential equations using moving least squares (MLS) interpolants and local weak forms. In this paper, a MLPG formulation is proposed for free and forced vibration analyses. Local weak forms are developed using weighted residual method locally from the dynamic partial differential equation. In the free vibration analysis, the essential boundary conditions are implemented through the direct interpolation form and imposed using orthogonal transformation techniques. In the forced vibration analysis, the penalty method is used in implementation essential boundary conditions. Two different time integration methods are used and compared in the forced vibration analyses using the present MLPG method. The validity and ef®ciency of the present MLPG method are demonstrated through a number of examples of two-dimensional solids.

Analysis of 2-dimensional transient problems for linear elastic and piezoelectric structures using the consecutive-interpolation quadrilateral element (CQ4)

“…The bottom surface of the wall is clamped. The first six natural frequencies computed by using structured (476 elements) and unstructured (239 & 476 elements) meshes are compared in Table 7 with those from the numerical solutions based on the FEM (Gu and Liu, 2001), the MLPG (Gu and Liu, 2001), the MK (Bui et al, 2011a,b), the SFEM (Dai and Liu, 2007), and the BEM (Brebbia et al, 1984). The corresponding eigenmodes of a shear wall are depicted in Fig.…”

“…0.25(g þ 0.5) 2 , otherwise it is conditionally stable. Here we use both the unconditionally stable implicit method in which we take g ¼ 0.5 and b ¼ 0.25, and the conditionally stable explicit CDM for which g ¼ 0.5 and b ¼ 0 (Zienkiewicz et al, 2005;Gu and Liu, 2001). …”

“…BEM (Brebbia et al, 1984) FEM (Gu and Liu, 2001) MLPG (Gu and Liu, 2001) MK (Bui et al, 2011a,b) SFEM (Dai and Liu, 2007) The effect of the time step size has also been studied to check the stability of the integration technique. In the CDM Dt ¼ 1 Â 10 À4 s is used.…”

The Meshless Local Petrov-Galerkin Method for Large Deformation Analysis of Hyperelastic Materials

“…It offers a lot of flexibility to deal with large deformation problems. Remarkable successes of the MLPG method have been reported in solving the potential problems, the convection-diffusion problems and the nonlinear boundary value problems by Atluri et al [13][14][15]; the fracture mechanics problems by Kim and Atluri [16]; the NavierStokes flows by Lin and Atluri [17]; the elasticity problems and plate bending problems by Long [18,19]; static and free vibration analysis of thin plates by Gu and Liu [20]; Crack Tip Fields problems by Ching and Batra [21,22]; anisotropic elasticity problems and crack analysis in 3-D axisymmetric FGM bodies by Sladek et al [23,24]. However, there exist some inconveniences in the MLPG method when the shape functions are obtained from interpolation schemes, such as the Moving Least Squares method (MLS), Partition of Unity Method (PUM), Reproducing Kernel Particle Method (RKPM), or Shepard function.…”

A generalized plane strain meshless local Petrov–Galerkin method for the micromechanics of thermomechanical loading of composites

“…By contrast, MLPG requires no mesh either for the interpolation of the solution variable or for the integration of the weak form of equations. Some applications of this promising, efficient and flexible method include solving Poisson's equation ], elastostatic and elastodynamic problems [Atluri and Zhu 2000;Long et al 2006], plate bending [Gu and Liu 2001], fracture mechanics [Ching and Batra 2001], and Navier-Stokes flow [Lin and Atluri 2001;Atluri and Shen 2002].…”