scholarly journals A New and Efficient Boundary Element-Free Method for 2-D Crack Problems

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Jinchao Yue ◽  
Liwu Chang ◽  
Yuzhou Sun
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Least Square ◽  
Moving Least Square ◽  
Crack Problems ◽  
Moving Least Square Approximation ◽  
New Variables ◽  
Element Free

An efficient boundary element-free method is established for 2-D crack problems by combining a pair of boundary integral equations and the moving-least square approximation. The displacement boundary integral equation is collated on the on-crack boundary, and a new traction boundary integral equation is applied on the crack surface without the separate consideration of the upper and lower sides. In virtue of integration by parts, only singularity in order 1/r is involved in the integral kernels of new traction boundary integral equation, which brings convenience to the numerical implementation. Meanwhile, the integration by parts produces the new variables, the displacement density, and displacement dislocation density, and they are the coexisting unknowns along with the displacement and displacement dislocation. With the high-order continuity of the moving-least square approximation, these new variables are directly approximated with the nodal displacement or displacement dislocation, and the final system of equations contains the unknowns of nodal displacements and displacement dislocations only. The boundary element-free computational scheme is established, and several examples show the efficiency and flexibility of the proposed method.

scholarly journals Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jufeng Wang ◽  
Fengxin Sun ◽  
Ying Xu
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Potential Problems ◽  
Element Free Method ◽  
Error Estimations ◽  
Element Free

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

scholarly journals The Interpolating Boundary Element-Free Method for Unilateral Problems Arising in Variational Inequalities

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Fen Li ◽  
Xiaolin Li
Keyword(s):  
Variational Inequalities ◽  
Boundary Element ◽  
Meshless Method ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Least Square ◽  
Moving Least Square ◽  
Unilateral Problems ◽  
Element Free Method ◽  
Element Free

The interpolating boundary element-free method (IBEFM) is developed in this paper for boundary-only analysis of unilateral problems which appear in variational inequalities. The IBEFM is a direct boundary only meshless method that combines an improved interpolating moving least-square scheme for constructing interpolation functions with boundary integral equations (BIEs) for representing governing equations. A projection operator is used to formulate the BIEs and then the formulae of the IBEFM are obtained for unilateral problems. The convergence of the developed meshless method is derived mathematically. The capability of the method is also illustrated and assessed through some numerical experiments.

AN INTERPOLATING BOUNDARY ELEMENT-FREE METHOD WITH NONSINGULAR WEIGHT FUNCTION FOR TWO-DIMENSIONAL POTENTIAL PROBLEMS

2013 ◽  
Vol 10 (06) ◽  
pp. 1350043 ◽  
Author(s):  
JUFENG WANG ◽  
JIANFEI WANG ◽  
FENGXIN SUN ◽  
YUMIN CHENG
Keyword(s):  
Integral Equation ◽  
Weight Function ◽  
Least Squares ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Moving Least Squares ◽  
Two Dimensional ◽  
Potential Problems ◽  
Element Free

In this paper, an improved interpolating moving least-squares (IIMLS) method with nonsingular weight function is presented. The shape function of the IIMLS method satisfies the property of Kronecker δ function. The IIMLS method can overcome the difficulties caused by the singularity of the weight function in the interpolating moving least-squares (IMLS) method presented by Lancaster and Salkauskas. By combining the boundary integral equation (BIE) method with the IIMLS method, an improved interpolating boundary element-free (IIBEF) method is presented for two-dimensional potential problems. The IIBEF method is a direct meshless boundary integral equation method in which the basic unknown quantities are the real solutions to the nodal variables, and the boundary conditions can be applied directly and easily. Thus, it gives greater computational precision. Some numerical examples are presented to demonstrate the IIMLS and IIBEF methods.

A Boundary Element Free Implementation Using NGF to Solve Fracture Mechanics Applications

2008 ◽  
Vol 383 ◽  
pp. 85-96 ◽  
Author(s):  
L.S. Miers ◽  
J.C.F. Telles
Keyword(s):  
Integral Equation ◽  
Fracture Mechanics ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Boundary Integral Equation Method ◽  
Equation Method ◽  
Local Boundary Integral Equation ◽  
Element Free

This work aims at introducing the concept of the numerical Green’s function (NGF) idea for elastostatic fracture mechanics using the boundary element-free method (BEFM). Unlike the local boundary integral equation method (LBIE), the BEFM only requires boundary interpolation. This method derives from the coupling of the boundary integral equation method and the orthogonal moving least-squares approximation scheme (OMLS). OMLS differs from standard MLS by using an orthogonal basis instead of only a linear independent one, which increases its accuracy and efficiency. Some illustrative examples are included in the end.

An Efficient Meshfree Method for Fracture Analysis of Cracks in Bi-Materials

2006 ◽  
Author(s):  
B. Nandulal ◽  
B. N. Rao ◽  
C. Lakshmana Rao
Keyword(s):  
Computational Efficiency ◽  
Basis Function ◽  
Meshfree Method ◽  
Fracture Analysis ◽  
Least Square ◽  
Moving Least Square ◽  
Linear Elastic ◽  
Moving Least Square Approximation ◽  
Material Fracture ◽  
Element Free

This paper presents an enriched meshless method based on an improved moving least-square approximation (IMLS) method for fracture analysis of cracks in homogeneous, isotropic, linear-elastic, two-dimensional bimaterial solids, subject to mixed-mode loading conditions. The method involves an element-free Galerkin formulation in conjunction with IMLS and a new enriched basis functions to capture the singularity field in linear-elastic bi-material fracture mechanics. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill-conditioned system of equations. The proposed enriched basis function can be viewed as a generalized enriched basis function, which degenerates to a linear-elastic basis function when the bimaterial constant is zero. Numerical examples are presented to illustrate the computational efficiency and accuracy of the proposed method.

A Novel Displacement Gradient Boundary Element Method for Elastic Stress Analysis With High Accuracy

1988 ◽  
Vol 55 (4) ◽  
pp. 786-794 ◽  
Author(s):  
H. Okada ◽  
H. Rajiyah ◽  
S. N. Atluri
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Numerical Differentiation ◽  
Boundary Integral ◽  
Displacement Gradient ◽  
Direct Integral ◽  
Displacement Boundary ◽  
Element Method

The boundary element method (BEM) in current usage, is based on the displacement boundary integral equation. The current practice of computing stresses in the BEM involves the use of a two-tier approach: (i) numerical differentiation of the displacement field at the boundary, and (ii) analytical differentiation of the displacement integral equation at the source point in the interior. A new direct integral equation for the displacement gradient is proposed here, to obviate this two-tier approach. The new direct boundary integral equation for displacement gradients has a lower order singularity than in the standard formulation, and is quite tractable from a numerical view point. Numerical results are presented to illustrate the advantages of the present approach.

Sign in / Sign up

Export Citation Format

Share Document