S. N. Atluri¹, Z. D. Han¹, A. M. Rajendran²

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.6, pp. 491-514, 2004, DOI:10.3970/cmes.2004.006.491

Abstract The Meshless Finite Volume Method (MFVM) is developed for solving elasto-static problems, through a new Meshless Local Petrov-Galerkin (MLPG) ``Mixed'' approach. In this MLPG mixed approach, both the strains as well as displacements are interpolated, at randomly distributed points in the domain, through local meshless interpolation schemes such as the moving least squares(MLS) or radial basis functions(RBF). The nodal values of strains are expressed in terms of the independently interpolated nodal values of displacements, by simply enforcing the strain-displacement relationships directly by collocation at the nodal points. The MLPG local weak form is then written for the equilibrium equations over… More >

Z. D. Han¹, S. N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.6, No.2, pp. 169-188, 2004, DOI:10.3970/cmes.2004.006.169

Abstract Three different truly Meshless Local Petrov-Galerkin (MLPG) methods are developed for solving 3D elasto-static problems. Using the general MLPG concept, these methods are derived through the local weak forms of the equilibrium equations, by using different test functions, namely, the Heaviside function, the Dirac delta function, and the fundamental solutions. The one with the use of the fundamental solutions is based on the local unsymmetric weak form (LUSWF), which is equivalent to the local boundary integral equations (LBIE) of the elasto-statics. Simple formulations are derived for the LBIEs in which only weakly-singular integrals are included for a simple numerical implementation.… More >

gping Shen¹, S. N. Atluri¹

CMES-Computer Modeling in Engineering & Sciences, Vol.5, No.3, pp. 235-256, 2004, DOI:10.3970/cmes.2004.005.235

Abstract A multiscale simulation technique based on the MLPG methods, and finite deformation mechanics, is developed, implemented, and tested. Several alternate time-dependent interfacial conditions, between the atomistic and continuum regions, are systematically studied, for the seamless multiscale simulation, by decomposing the displacement of atoms in the equivalent-continuum region into long and short wave-length components. All of these methods for enforcing the interface conditions can ensure the passage of information accurately between the atomistic and continuum regions, while they lead to different performances at short wavelengths. The presently proposed Solution Method 2 reduces the phonon reflections at the interface, without increasing the… More >

Z. D. Han¹, S. N. Atluri²

CMC-Computers, Materials & Continua, Vol.1, No.2, pp. 129-140, 2004, DOI:10.3970/cmc.2004.001.129

Abstract A Meshless Local Petrov-Galerkin (MLPG) method has been developed for solving 3D elasto-dynamic problems. It is derived from the local weak form of the equilibrium equations by using the general MLPG concept. By incorporating the moving least squares (MLS) approximations for trial and test functions, the local weak form is discretized, and is integrated over the local sub-domain for the transient structural analysis. The present numerical technique imposes a correction to the accelerations, to enforce the kinematic boundary conditions in the MLS approximation, while using an explicit time-integration algorithm. Numerical examples for solving the transient response of the elastic structures… More >

J. Sladek¹, V. Sladek¹, S. Krahulec¹, M. Wünsche², Ch. Zhang²

CMC-Computers, Materials & Continua, Vol.29, No.1, pp. 75-102, 2012, DOI:10.3970/cmc.2012.029.075

Abstract A meshless method based on the local Petrov-Galerkin approach is proposed, to solve static and dynamic problems of two-layered magnetoelectroelastic composites with specific properties. One layer has pure piezoelectric properties and the second one is a pure piezomagnetic material. It is shown that the electric potential in the piezoelectric layer is induced by the magnetic potential in the piezomagnetic layer. The magnetoelectric effect is dependent on the ratio of the layer thicknesses. Functionally graded material properties of the piezoelectric layer and homogeneous properties of the piezomagnetic layer are considered too. The magnetoelectric composites are analyzed under a pure magnetic or… More >

A. Rezaei Mojdehi^1,2, A. Darvizeh³, A. Basti²

CMC-Computers, Materials & Continua, Vol.29, No.1, pp. 15-40, 2012, DOI:10.3970/cmc.2012.029.015

Abstract The meshless local Petrov-Galerkin approach is proposed for the nonlinear dynamic analysis of three-dimensional (3D) elasto-plastic problems. Galerkin weak-form formulation is applied to derive the discrete governing equations. A weak formulation for the set of governing equations is transformed into local integral equations on local sub-domains by using a unit test function and local weak-form formulation in three dimensional continua for the general dynamic problems is derived. Three dimensional Moving Least-Square (MLS) approximation is considered as shape function to approximate the field variable of scattered nodes in the problem domain. Normality hypothesis of plasticity is adopted to define the stress-strain… More >

D. Soares Jr.¹, V. Sladek², J. Sladek², M. Zmindak³, S. Medvecky³

CMC-Computers, Materials & Continua, Vol.27, No.2, pp. 101-127, 2012, DOI:10.32604/cmc.2012.027.101

Abstract This work proposes a modified procedure, based on analytical integrations, to analyse poroelastic models discretized by time-domain Meshless Local Petrov-Galerkin formulations. In this context, Taylor series expansions of the incognita fields are considered, and the related integrals of the meshless formulations are solved analytically, rendering a so called modified methodology. The work is based on the u-p formulation and the incognita fields of the coupled analysis in focus are the solid skeleton displacements and the interstitial fluid pore pressures. Independent spatial discretization is considered for each phase of the model, rendering a more flexible and efficient methodology. The Moving Least… More >

Y. Sadeghi Ferezghi¹, M.R. Sohrabi¹, S.M Mosavi Nezhad ^{2, *}

CMES-Computer Modeling in Engineering & Sciences, Vol.113, No.4, pp. 497-520, 2017, DOI:10.3970/cmes.2017.113.497

Abstract In this paper, dynamic behavior of non-symmetric Functionally Graded (FG) cylindrical structure under shock loading is carried out. Dynamic equations in the polar coordinates are drawn out using Meshless Local Petrov-Galerkin (MLPG) method. Nonlinear volume fractions are used for radial direction to simulate the mechanical properties of Functionally Graded Material (FGM). To solve dynamic equations of non-symmetric FG cylindrical structure in the time domain, the MLPG method are combined with the Laplace transform method. For computing the inverse Laplace transform in the present paper, the Talbot algorithm for the numerical inversion is used. To verify the obtained results by the… More >

Annamaria Mazzia¹, Giorgio Pini¹, Flavio Sartoretto²

CMES-Computer Modeling in Engineering & Sciences, Vol.108, No.4, pp. 239-262, 2015, DOI:10.3970/cmes.2015.108.239

Abstract Meshless Local Petrov Galerkin (MLPG) methods are pure meshless techniques for solving Partial Differential Equations (PDE). MLPG techniques are nowadays used for solving a huge number of complex, real–life problems. While MLPG aims to approximate the solution of a given differential problem, its “dual” Direct MLPG (DMLPG) technique relies upon approximating linear functionals. Assume adaptive methods are to be implemented. When using a mesh–based method, inserting and/or deleting a node implies complex adjustment of connections. Meshless methods are more apt to implement adaptivity, since they does not require such adjustments. Nevertheless, ad–hoc insertion and/or deletion algorithms must be devised, in… More >

Mohammad Hossein Ghadiri Rad¹, Farzad Shahabian^1,2, Seyed Mahmoud Hosseini³

CMES-Computer Modeling in Engineering & Sciences, Vol.108, No.3, pp. 135-157, 2015, DOI:10.3970/cmes.2015.108.135

Abstract A meshless method based on the local Petrov-Galerkin approach is developed for elasto-dynamic analysis of geometrically nonlinear two dimensional (2D) problems in hyper-elastic functionally graded materials. The radial point interpolation method (RPIM) is utilized to build the shape functions and the Heaviside step function is used as the test function. The mechanical properties of functionally graded material are considered to continuously vary in a certain direction and are simulated using a nonlinear power function in volume fraction form. Considering the large deformations, it is assumed that the domain be made of large deformable neo-Hookean hyperelastic materials. Rayleigh damping is employed… More >