USE OF B-SPLINE FUNCTION IN ELEMENT FREE GALERKIN METHOD FOR NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATION

2009 ◽  
Vol 06 (03) ◽  
pp. 349-360
Author(s):  
K. SANDEEP ◽  
K. KAMAL KUMAR
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Spline Function ◽  
Least Square ◽  
Partial Differential ◽  
Order Continuity ◽  
B Spline ◽  
Moving Least Square ◽  
Element Free Galerkin ◽  
Element Free

The paper presents application of a B-spline function in place of a shape function obtained by using moving least square approximant with B-spline weight and linear basis. The piecewise polynomial function of B-spline with C2 and lower order continuity is efficiently used in element free Galerkin (EFG) method to generate a new fast algorithm for the solution of one-dimensional partial differential equation. The support of B-spline function is assumed as the domain of influence of EFG method. The results of bar deflection are compared with those obtained by other researchers.

The Improved Element-Free Galerkin Method for Diffusional Drug Release Problems

2020 ◽  
Vol 12 (08) ◽  
pp. 2050096
Author(s):  
Guodong Zheng ◽  
Yumin Cheng
Keyword(s):  
Drug Release ◽  
Weak Form ◽  
Equation System ◽  
Least Square ◽  
Moving Least Square ◽  
Scale Parameters ◽  
Element Free Galerkin ◽  
Penalty Factor ◽  
The Difference ◽  
Element Free

By using the improved moving least-square (IMLS) approximation to present the shape function, the improved element-free Galerkin (IEFG) method is investigated to solve diffusional drug release problems in this paper. In order to get the discretized equation system, Galerkin weak form of a diffusional drug release problem is used with applying essential boundary conditions using the penalty method. The difference method is applied for discretization of time domain. Then the formulae of IEFG method for solving diffusional drug release problems are presented. Three numerical example problems are given to study the convergence of solutions of IEFG method in this paper. The influences of scale parameters of influence domain, penalty factor and node distribution on the accuracy of the solutions of IEFG method are discussed. Compared with finite element method, the correctness of IEFG method in this paper is shown.

A MESHLESS ANALYSIS OF SHELLS BASED ON MOVING KRIGING INTERPOLATION

2007 ◽  
Vol 04 (04) ◽  
pp. 543-565 ◽  
Author(s):  
VILAYSAK SAYAKOUMMANE ◽  
WORSAK KANOK-NUKULCHAI
Keyword(s):  
Additional Treatment ◽  
Least Square ◽  
Shell Structures ◽  
Kriging Interpolation ◽  
Interpolation Function ◽  
Moving Least Square ◽  
Element Free Galerkin ◽  
Kronecker Delta ◽  
Mls Approximation ◽  
Element Free

An Element Free Galerkin Method (EFGM) for the analysis of degenerated shell structures is presented. The method is based on the Moving Kriging (MK) Interpolation function. The properties of the interpolation function possess the Kronecker delta property. With the MK Interpolation function no additional treatment required at the boundary conditions compared with that of using Moving Least Square (MLS) approximation. This deficiency of MLS at boundary condition has been definitely eradicated. The membrane and shear locking in the numerical analysis for degenerated shell problems has been alleviated by using higher order and removed by using quartic order of polynomials. Numerical benchmark examples for shell structures are presented to validate the proposed approach.

scholarly journals Behavior of beam on elastic foundation using element free Galerkin method

2021 ◽  
Vol 12 (4) ◽  
pp. 14-22
Author(s):  
H.T.T. Lan
Keyword(s):  
Galerkin Method ◽  
Elastic Foundation ◽  
Least Square ◽  
Element Free Galerkin Method ◽  
Moving Least Square ◽  
Beam On Elastic Foundation ◽  
Element Free Galerkin ◽  
Mesh Free ◽  
System Equations ◽  
Element Free

One of mesh free methods, element free Galerkin method, is presented to analyze the finite beam on elastic foundation. The shape functions are constructed by using the moving least square interpolation based on a set of nodes that are arbitrarily distributed in specified domain. Discrete system equations are derived from the variation form of system equations. Numerical examples of finite beam on elastic foundation are given by establishing Matlab code. The results of this paper demonstrate the effectiveness of the proposed method with small errors compared to analytical solutions. Keywords: mesh free method, element free Galerkin method, moving least square, finite beam, elastic foundation.

FURTHER INVESTIGATION OF ELEMENT-FREE GALERKIN METHOD USING MOVING KRIGING INTERPOLATION

2004 ◽  
Vol 01 (02) ◽  
pp. 345-365 ◽  
Author(s):  
P. TONGSUK ◽  
W. KANOK-NUKULCHAI
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Least Square ◽  
Kriging Interpolation ◽  
Shape Functions ◽  
Element Free Galerkin Method ◽  
Basic Premise ◽  
Moving Least Square ◽  
Element Free Galerkin ◽  
Element Free

Following its first introduction, this study further scrutinizes the new type of shape functions for Element-free Galerkin Method (EFGM) based on the Moving Kriging (MK) interpolation. Kriging is a geostatistical method of spatial interpolation. Its basic premise is that every unknown point can be interpolated from known scattered points in its specified neighborhood. This property is ideal for EFGM. Previously, a shortcoming of EFGM based on Moving Least Square (MLS) approximation is associated with its limitation to satisfy essential boundary conditions exactly. With MK interpolation functions, EFGM solution can satisfy essential boundary conditions automatically. Numerical tests on one and two-dimensional elasticity problems have confirmed the effectiveness of MK in addressing this specific shortcoming of EFGM. Furthermore, the study also finds the accuracy of EFGM to be greatly enhanced with the use of MK shape functions.

Study of Static Fracture Propagation by Element-Free Galerkin Method With Singular Weight Function at Crack Tip

2005 ◽  
Vol 21 (2) ◽  
pp. 125-129 ◽  
Author(s):  
K.-J. Shen ◽  
J. P. Sheng ◽  
C.-Y. Wang
Keyword(s):  
Galerkin Method ◽  
Crack Growth ◽  
Initial Crack ◽  
Least Square ◽  
Bending Test ◽  
Element Free Galerkin Method ◽  
Three Point Bending ◽  
Moving Least Square ◽  
Element Free Galerkin ◽  
Element Free

AbstractElement-free Galerkin method (EFGM) based on moving least-square curve fitting concept is presented and applied to elastic fracture problems. Because no element connectivity data are needed, EFGM is very convenient and effective numerical method for crack growth analysis. This paper is intended as an investigation of crack trajectory for different notch locations under three-point bending test. The initial crack growth angles obtained by element-free Galerkin method in comparison with those obtained by lab test reveal that both results are very close. However, numerical results also show that the location of an original notch can stronger affect the variation of crack path for different increment. The stress intensity factors (SIF) of cracks under three-point bending test with different increment are also investigated by EFGM.

The Element-Free Galerkin Method for the Unsteady Magnetohydrodynamic Flow in a Duct

2015 ◽  
Vol 723 ◽  
pp. 181-185
Author(s):  
Xing Hui Cai ◽  
Hong Fu Qiang ◽  
Jiang Ren Lu ◽  
Guo Liang Wang
Keyword(s):  
Galerkin Method ◽  
Numerical Solutions ◽  
Magnetohydrodynamic Flow ◽  
Least Square ◽  
Element Free Galerkin Method ◽  
Moving Least Square ◽  
Coupled Equations ◽  
Element Free Galerkin ◽  
Moving Least Square Approximation ◽  
Element Free

In this paper, a meshless global element-free Galerkin method is given to obtain the numerical solutions of the coupled equations in the velocity and magnetic field for the unsteady magnetohydrodynamic flow through a straight duct with arbitrary electrical conductivity, that is, from perfectly conducting to insulated duct walls. The moving least-square approximation scheme is employed to construct shape functions. A time stepping method is employed to deal with the time derivatives. Non-uniform background grids and nodes are applied for numerical simulations. Computations are performed for different Hartmann numbers and wall conductivities at different time.

A modified method to determine the radius of influence domain in element-free Galerkin method

2014 ◽  
Vol 229 (5) ◽  
pp. 795-805 ◽  
Author(s):  
Mao Sheng ◽  
Gensheng Li ◽  
Subhash Shah
Keyword(s):  
Least Square ◽  
Patch Tests ◽  
Moving Least Square ◽  
Modified Method ◽  
Point Of Interest ◽  
Element Free Galerkin ◽  
Radius Of Influence ◽  
Moving Least Square Approximation ◽  
Random Node ◽  
Element Free

The radius of the influence domain is an important parameter in the weight function and plays an essential role in the accuracy of approximations. A modified method was proposed for determining the radius of influence domain. The modification is that the radius of influence domain is prescribed by the desired amount of support nodes for any point of interest. The advantages of this strategy are the regularity of matrix can be ensured in moving least square approximation and the algorithm is simple and practicable. The modified method was validated by thousands of patch tests in the case of regular and irregular node collocations and Timoshenko beam problem with using random node collocation. Results show that the proposed method performs more capability to handle the arbitrary node collocation than the pre-existing methods. This paper provides an alternative way to determine the radius of influence domain.

Element Free Galerkin Formulation of Composite Beams with Longitudinal and Transverse Partial Interactions

2012 ◽  
Vol 166-169 ◽  
pp. 3237-3240
Author(s):  
Dzulkarnain Ahmad ◽  
Airil Yasreen Mohd Yassin ◽  
Ahmad Kueh
Keyword(s):  
Composite Beam ◽  
Weak Form ◽  
Least Square ◽  
Weight Functions ◽  
Moving Least Square ◽  
Element Free Galerkin ◽  
Interfacial Surface ◽  
Partial Interaction ◽  
Longitudinal Slip ◽  
Element Free

Partial interaction between two different materials of composite beam is consequent from longitudinal slip and transverse uplift effects at interfacial surface. The behaviour has yielded higher order differential equation as compared to beam that interacts fully. In this paper, a meshless approach for the analysis of composite beam with partial interaction by Element Free Galerkin (EFG) method is formulated and investigated. Discretized solely by nodes, the Moving Least Square (MLS) method is adopted for the EFG shape functions formulation and the variational approach is chosen in developing its Galerkin weak form. The weak form essential boundary conditions are enforced by Lagrange multiplier, where comparable results are obtained between developed EFG code and established analytical solutions. In addition, influence of various weight functions on shape function smoothness of EFG code is explored.

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