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.

A Complex Variable Boundary Element-Free Method for Potential and Helmholtz Problems in Three Dimensions

2019 ◽  
Vol 17 (02) ◽  
pp. 1850129 ◽  
Author(s):  
Xiaolin Li ◽  
Shougui Zhang ◽  
Yan Wang ◽  
Hao Chen
Keyword(s):  
Boundary Element ◽  
Complex Variable ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Three Dimensions ◽  
Variable Boundary ◽  
Current Implementation ◽  
Element Free Method ◽  
Element Free ◽  
3D Surfaces

The complex variable boundary element-free method (CVBEFM) is a meshless method that takes the advantages of both boundary integral equations (BIEs) in dimension reduction and the complex variable moving least squares (CVMLS) approximation in element elimination. The CVBEFM is developed in this paper for solving 3D problems. This paper is an attempt in applying complex variable meshless methods to 3D problems. Formulations of the CVMLS approximation on 3D surfaces and the CVBEFM for 3D potential and Helmholtz problems are given. In the current implementation, the CVMLS shape function of 3D problems is formed with 1D basis functions, and the boundary conditions in the CVBEFM can be applied directly and easily. Some numerical examples are presented to demonstrate the method.

An adaptive orthogonal improved interpolating moving least-square method and a new boundary element-free method

2019 ◽  
Vol 353 ◽  
pp. 347-370 ◽  
Author(s):  
Qiao Wang ◽  
Wei Zhou ◽  
Y.T. Feng ◽  
Gang Ma ◽  
Yonggang Cheng ◽  
...  
Keyword(s):  
Boundary Element ◽  
Least Square Method ◽  
Least Square ◽  
Moving Least Square ◽  
Element Free Method ◽  
Element Free

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.

NUMERICAL ANALYSIS ON PROGRESSIVE FRACTURE BEHAVIOR BY USING ELEMENT-FREE METHOD BASED ON FINITE COVERS

2005 ◽  
Vol 02 (04) ◽  
pp. 543-553 ◽  
Author(s):  
MAOTIAN LUAN ◽  
XINHUI YANG ◽  
RONG TIAN ◽  
QING YANG
Keyword(s):  
Damage Model ◽  
Physical And Mechanical Properties ◽  
Least Square Method ◽  
Fracture Analysis ◽  
Least Square ◽  
Finite Cover ◽  
Moving Least Square ◽  
Progressive Fracture ◽  
Element Free Method ◽  
Element Free

The finite-cover element-free method FCEFM is applied to simulate the fracture and damage evolution process of geo-materials. This method is mathematically based on the finite-cover technique of manifold method and the multiple weighted moving least-square method to solve the continuous and discontinuous problems without meshing or re-meshing. The damage heterogeneity and evolutionary processes of rock mass with initial cracks are analyzed and numerically simulated by FCEFM. Using the method of probability to generate the parameters of materials randomly, the physical and mechanical properties of materials are randomly distributed in nodes or Gaussian points. And an alternating damage model together with numerical implementation which is adapted to microscopic elasto-brittle fracture analysis is proposed. Through analysis of several numerical examples, the validity and efficiency of progressive fracture analysis with use of the proposed FCEFM is demonstrated.

A Boundary Element-Free Method for Fracture Analysis of 2D Anisotropic Solids

2010 ◽  
Vol 139-141 ◽  
pp. 107-112
Author(s):  
Yu Zhou Sun ◽  
Dong Xia Li ◽  
Hui Wang
Keyword(s):  
Dislocation Density ◽  
Weight Function ◽  
Boundary Element ◽  
Fracture Analysis ◽  
Boundary Integral ◽  
Computational Method ◽  
Element Free Method ◽  
Mls Approximation ◽  
New Variables ◽  
Element Free

This paper presents a boundary element-free computational method for the fracture analysis of 2-D anisotropic bodies. The study starts from a derived traction boundary integral equation (BIE) in which the boundary conditions of both upper and lower crack surfaces are incorporated into and only the Cauchy singular kernal is involved. The boundary element-free method is achieved by combining this new BIE and the moving least-squares (MLS) approximation. The new BIE introduces two new variables: the displace density and The dislocation density. For each crack, the dislocation density is first expressed as the product of the characteristic term and unknown weight function, and the unknown weight function is approximated with the MLS approximation. The stress intensity factors (SIFs) can be calculated from the the weight function. The examples of the straight and circular-arc cracks are computed, and the convergence and efficiency are discussed.

The Element-Free Method for Steam Turbine Rotor

2013 ◽  
Vol 281 ◽  
pp. 343-346
Author(s):  
Su Ling Yuan
Keyword(s):  
Steam Turbine ◽  
Gaussian Quadrature ◽  
Turbine Rotor ◽  
Least Square ◽  
Gradient Field ◽  
Moving Least Square ◽  
Steam Turbine Rotor ◽  
Discrete Equations ◽  
Element Free Method ◽  
Element Free

The element-free method is a new numerical method, which requires only nodal data and whose shape functions are continual and differentiable. The element-free method employs moving least-square approximants to approximate original functions. In this paper, discrete equations of axial symmetry problem are obtained by variational principle and Gaussian quadrature. Several numerical examples indicate that the element-free method can obtain more accurate results about these problems, moreover, results and their gradients are continuous in the entire domain and post-processing to obtain a smooth gradient field is total unnecessary. Finaly, the element-free method is applied to heat conduction problems for steam turbine rotor.

Two dimensional element free Galerkin coupled flow function method and its application

2016 ◽  
Vol 33 (5) ◽  
pp. 1310-1326 ◽  
Author(s):  
Qingdong Zhang ◽  
Boyang Zhang ◽  
Xingfu Lu
Keyword(s):  
Function Method ◽  
Variational Equation ◽  
Least Square ◽  
Flow Function ◽  
Unknown Quantity ◽  
Content Type ◽  
Moving Least Square ◽  
Element Free Galerkin ◽  
Element Free Method ◽  
Element Free

Purpose – The purpose of this paper is to propose a hybridization numerical method to solve the plastic deformation of metal working based on the flow function method and meshless method. Design/methodology/approach – The proposed method is named as flow function-element free Galerkin (F-EFG) method. It uses the flow function as the basic unknown quantity to get the basic control equation, the compactly supported approximate function to establish a local approximate flow function by means of moving least square approximation, and the element free Galerkin (EFG) method to solve variational equation. The F-EFG method takes the upper limit method essence of flow function method, and the convergence, stability, and error characteristics of EFG method. Findings – The steady extrusion process of the axisymmetric extrusion problems as well as the extrusion deformation law and main field variables are subjects in the modeling and simulation analysis using F-EFG method. The results show that the F-EFG method has good computational efficiency and accuracy. Originality/value – The F-EFG method proposed in this paper has the advantages of high-solution precision of flow function method and large deformation solution of element free method. It overcomes the difficulties in global flow function establishment in flow function method and low-solution efficiency in element free method. The method is beneficial to the development of flow function method and element free method.

scholarly journals Complex Variable Meshless Manifold Method for Elastic Dynamic Problems

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Hongfen Gao ◽  
Gaofeng Wei
Keyword(s):  
Analytical Solution ◽  
Complex Variable ◽  
Meshless Method ◽  
Numerical Solutions ◽  
Least Square ◽  
Dynamic Problems ◽  
Finite Covering ◽  
Moving Least Square ◽  
Elastic Dynamics ◽  
Good Agreement

Combining the finite covering technical and complex variable moving least square, the complex variable meshless manifold method can handle the discontinuous problem effectively. In this paper, the complex variable meshless method is applied to solve the problem of elastic dynamics, the complex variable meshless manifold method for dynamics is established, and the corresponding formula is derived. The numerical example shows that the numerical solutions are in good agreement with the analytical solution. The CVMMM for elastic dynamics and the discrete forms are correct and feasible. Compared with the traditional meshless manifold method, the CVMMM has higher accuracy in the same distribution of nodes.

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