Hongfen Gao¹, Gaofeng Wei^2,3,*

CMES-Computer Modeling in Engineering & Sciences, Vol.131, No.3, pp. 1793-1814, 2022, DOI:10.32604/cmes.2022.019687

Abstract By introducing the radial basis functions (RBFs) into the reproducing kernel particle method (RKPM), the calculating accuracy and stability of the RKPM can be improved, and a novel meshfree method of the radial basis RKPM (meshfree RRKPM) is proposed. Meanwhile, the meshfree RRKPM is applied to transient heat conduction problems (THCP), and the corresponding equations of the meshfree RRKPM for the THCP are derived. The two-point time difference scheme is selected to discretize the time of the THCP. Finally, the numerical results illustrate the effectiveness of the meshfree RRKPM for the THCP. More >

Hossein M. Shodja^{1, 2, 3}, Alireza Hashemian^{2, 4}

The International Conference on Computational & Experimental Engineering and Sciences, Vol.13, No.3, pp. 57-58, 2009, DOI:10.3970/icces.2009.013.057

Abstract Gradient reproducing kernel particle method (GRKPM) is a meshless technique which incorporates the first gradients of the function into the reproducing equation of RKPM. Therefore, in two-dimensional space GRKPM introduces three types of shape functions rather than one. The robustness of GRKPM's shape functions is established by reconstruction of a third-order polynomial. To enforce the essential boundary conditions (EBCs), GRKPM's shape functions are modified by transformation technique. By utilizing the modified shape functions, the weak form of the nonlinear evolutionary Buckley-Leverett (BL) equation is discretized in space, rendering a system of nonlinear ordinary differential equations (ODEs). Subsequently, Gear's method is… More >

Mani Khezri¹, Alireza Hashemian¹, Hossein M. Shodja^1,2,3

The International Conference on Computational & Experimental Engineering and Sciences, Vol.11, No.4, pp. 99-108, 2009, DOI:10.3970/icces.2009.011.099

Abstract Meshless methods using kernel approximation like reproducing kernel particle method (RKPM) and gradient RKPM (GRKPM) generally use a set of particles to discretize the subjected domain. One of the major steps in discretization procedure is determination of associated volumes particles. In a non-uniform or irregular configuration of particles, determination of these volumes comprises some difficulties. This paper presents a straightforward numerical method for determination of related volumes and conducts a survey on influence of different assumption about computing the volume for each particle. Stress intensity factor (SIF) as a major representing parameter in fracture of solids is calculated by employing… More >

Mohammad Mashayekhi¹, Hossein M. Shodja^1,2, Reza Namakian¹

CMES-Computer Modeling in Engineering & Sciences, Vol.76, No.1, pp. 19-60, 2011, DOI:10.3970/cmes.2011.076.019

Abstract A meshless method called reproducing kernel particle method (RKPM) is exploited to cope with elastic-plastic fracture mechanics (EPFM) problems. The idea of arithmetic progression is assumed to place particles within the refinement zone in the vicinity of the crack tip. A comparison between two conventional treatments, visibility and diffraction, to crack discontinuity is conducted. Also, a tracking to find the appropriate diffraction parameter is performed. To assess the suggestions made, two mode I numerical simulations, pure tension and pure bending tests, are executed. Results including J integral, crack mouth opening displacement (CMOD), and plastic zone size and shape are compared… More >

H.M. Shodja^1,2,3, M. Khezri⁴, A. Hashemian¹, A. Behzadan¹

CMES-Computer Modeling in Engineering & Sciences, Vol.62, No.2, pp. 171-204, 2010, DOI:10.3970/cmes.2010.062.171

Abstract An accurate numerical methodology for capturing the field quantities across the interfaces between material discontinuities, in the context of reproducing kernel particle method (RKPM), is of particular interest. For this purpose the innovative numerical technique, so-called augmented corrected collocation method is introduced; this technique is an extension of the corrected collocation method used for imposing essential boundary conditions (EBCs). The robustness of this methodology is shown by utilizing it to solve two benchmark problems of material discontinuities, namely the problem of circular inhomogeneity with uniform radial eigenstrain, and the problem of interaction between a crack and a circular inhomogeneity. Moreover,… More >

D. Asprone¹, F. Auricchio², G. Manfredi¹, A. Prota¹, A. Reali², G. Sangalli³

CMES-Computer Modeling in Engineering & Sciences, Vol.62, No.1, pp. 1-22, 2010, DOI:10.3970/cmes.2010.062.001

Abstract Particle methods represent some of the most investigated meshless approaches, applied to numerical problems, ranging from solid mechanics to fluid-dynamics and thermo-dynamics. The objective of the present paper is to analyze some of the proposed particle formulations in one dimension, investigating in particular how the different approaches address second derivative approximation. With respect to this issue, a rigorous analysis of the error is conducted and a novel second-order accurate formulation is proposed. Hence, as a benchmark, three numerical experiments are carried out on the investigated formulations, dealing respectively with the approximation of the second derivative of given functions, as well… More >

Hua Li¹, Shantanu S. Mulay¹, Simon See²

CMES-Computer Modeling in Engineering & Sciences, Vol.48, No.1, pp. 43-82, 2009, DOI:10.3970/cmes.2009.048.043

Abstract Differential Quadrature (DQ) is one of the efficient derivative approximation techniques but it requires a regular domain with all the points distributed only along straight lines. This severely restricts the DQ while solving the irregular domain problems discretized by the random field nodes. This limitation of the DQ method is overcome in a proposed novel strong-form meshless method, called the random differential quadrature (RDQ) method. The RDQ method extends the applicability of the DQ technique over the irregular or regular domains discretized using the random field nodes by approximating a function value with the fixed reproducing kernel particle method (fixed… More >

Vikas Namdeo^1,2, C S Manohar^1,3

CMES-Computer Modeling in Engineering & Sciences, Vol.32, No.3, pp. 123-160, 2008, DOI:10.3970/cmes.2008.032.123

Abstract The problem of identification of nonlinear system parameters from measured time histories of response under known excitations is considered. Solutions to this problem are obtained by using the force state mapping technique with two alternative functional representation schemes. These schemes are based on the application of reproducing kernel particle method (RKPM) and kriging techniques to fit the force state map. The RKPM has the capability to reproduce exactly polynomials of specified order at any point in a given domain. The kriging based methods represent the function under study as a random field and the parameters describing this field are optimally… More >

Hossein M. Shodja^1,2,3, Alireza Hashemian^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.32, No.1, pp. 17-34, 2008, DOI:10.3970/cmes.2008.032.017

Abstract Gradient reproducing kernel particle method (GRKPM) is a meshless technique which incorporates the first gradients of the function into the reproducing equation of RKPM. Therefore, in two-dimensional space GRKPM introduces three types of shape functions rather than one. The robustness of GRKPM's shape functions is established by reconstruction of a third-order polynomial. To enforce the essential boundary conditions (EBCs), GRKPM's shape functions are modified by transformation technique. By utilizing the modified shape functions, the weak form of the nonlinear evolutionary Buckley-Leverett (BL) equation is discretized in space, rendering a system of nonlinear ordinary differential equations (ODEs). Subsequently, Gear's method is… More >

Chih-Ping Wu¹, Kuan-Hao Chiu, Yun-Ming Wang

CMES-Computer Modeling in Engineering & Sciences, Vol.27, No.3, pp. 163-186, 2008, DOI:10.3970/cmes.2008.027.163

Abstract A differential reproducing kernel particle (DRKP) method is proposed and developed for the analysis of simply supported, multilayered elastic and piezoelectric plates by following up the consistent concepts of reproducing kernel particle (RKP) method. Unlike the RKP method in which the shape functions for derivatives of the reproducing kernel (RK) approximants are obtained by directly taking the differentiation with respect to the shape functions of the RK approximants, we construct a set of differential reproducing conditions to determine the shape functions for the derivatives of RK approximants. On the basis of the extended Hellinger-Reissner principle, the Euler-Lagrange equations of three-dimensional… More >