Element-free Method Research Articles

Hybrid element-free Galerkin-finite element method for electromagnetic field computations

A new hybrid computational procedure utilizing the Element-free Galerkin method (EFGM) and the finite element method (FEM) for electromagnetic field computations is proposed. With the proposed method, only a part of the analyzed domain is modeled using the EFGM, while the rest of the analysis domain is discretized using the ordinary FEM. The connection between both subdomains is enabled using an intermediate analysis subdomain where new hybrid shape functions are defined utilizing both EFGM and FEM shape functions as bases. The accuracy of the obtained results by the proposed method is of the same order with those obtained by the ordinary FEM, but with less computation effort.

Variational theory for Chandrasekharaiah thermopiezoelectricity

Via the semi-inverse method of establishing generalized variational principle for physical problems, a classical variational model (non Gurtin-type and not involving convolutions) for Chandrasekharaiah thermopiezoelectricity is established directly from the governing equations. The present theory aims at providing a more complete theoretical basis for the variational-based finite element applications and variational-based meshless method (element-free method).

A new approach for numerical modal analysis using the element‐free method

A novel approach in analysing the vibration of elastic body systems is presented and discussed in this paper. This method uses a meshless spatial approximation based only on nodes, which constitutes an advantage over the finite element method. As it is applicable to an elastic body with arbitrary shape, it also has advantages over the classical Rayleigh–Ritz method and its extensions. The paper is organized as follows: in Section 2, Hamilton's principle is used for obtaining the equations of motion for a three-dimensonal simply connected elastic body. In Section 3, the development of weighted base functions for the moving least- squares interpolant is described. In Section 4, the mass and stiffness matrices are derived using the previous results. In Section 5, a system synthesis description is given, in which we also develop the approach used for integrating the boundary conditions. Finally in Section 6, the theoretical results are validated by case studies chosen to highlight various features of the approach, including optimization of the shape function parameters. The particle method has already been used for structural dynamics (Liu et al., International Journal for Numerical Methods in Engineering 1995;38:1655–1679), but for free vibration of beams and plates, to the authors knowledge, this paper is the first application of the element-free method to modal analysis. Copyright © 1999 John Wiley & Sons, Ltd.

A new approach for numerical modal analysis using the element-free method

A novel approach in analysing the vibration of elastic body systems is presented and discussed in this paper. This method uses a meshless spatial approximation based only on nodes, which constitutes an advantage over the finite element method. As it is applicable to an elastic body with arbitrary shape, it also has advantages over the classical Rayleigh-Ritz method and its extensions. The paper is organized as follows: in Section 2, Hamilton's principle is used for obtaining the equations of motion for a three-dimensonal simply connected elastic body. In Section 3, the development of weighted base functions for the moving least-squares interpolant is described. In Section 4, the mass and stiffness matrices are derived using the previous results. In Section 5, a system synthesis description is given, in which we also develop the approach used for integrating the boundary conditions. Finally in Section 6, the theoretical results are validated by case studies chosen to highlight various features of the approach, including optimization of the shape function parameters. The particle method has already been used for structural dynamics (Liu et al., International Journal for Numerical Methods in Engineering 1995;38:1655-1679), but for free vibration of beams and plates, to the authors knowledge, this paper is the first application of the element-free method to modal analysis. Copyright (C) 1999 John Wiley & Sons, Ltd.

Development of Stress Analysis System Using a Meshless Method-Element-Free Galerkin Method.

Although finite element method (FEM) is effective in solving partial differential equations in numerical analysis, the creation of the mesh data has and will become a very burdensome process. Therefore, meshless or element free method, which require no element or grid, are believed to be effective and efficient. Since only the nodal data are necessary, it is belived that CAD/CAE seamless system can be created using these methods. Furthermore, because of the developments made in the field of multi-media, the processing of digital image data has become very simple tasks. With these method, it is believed that the implementation of these digital images to the numerical analysis is easily achievable. This paper will discribe a prototype of such system developed by the authors, based on the element-free Galerkin method (EFGM) 2D elasticity problems. Mainly the preprocessores, where the data is constructed from CAD and digital image data, will be the main focus of this paper.

Applications of Element-Free Trefftz Method to Two-Dimensional Elastic Problem and Its Shape Design Sensitivity Analysis.

This paper presents an application of the element free Trefftz method to the two-dimensional elastic problem and its shape design sensitivity analysis. The element-free Trefftz method is known as a boundary type solution procedure using the regular T-complete functions. Since the physical quantitles are represented by regular expressions, direct differentiation of the expressions with respect to shape parameters leads to the regular expressions of shape sensitivities. The present method is applied to the relatively simple examples in order to confirm its validity.

Application of Element-Free Trefftz Method to Second-Order Design Sensitivity Analysis of Two-dimensional Elastic Problem.

This paper presents a new scheme for the second-order shape design sensitivity analysis of the two-dimensional elastic problem by the element-free Trefftz method. In the Trefftz formulation, the physical quantities are approximated by the superposition of the regular T-complete functions. Therefore, direct differentiation of the approximate expressions with respect to shape parameters leads to the regular expressions of the sensitivities. The present schemes are applied to simple examples in order to confirm the validity.

Basic Study on Element-Free Galerkin Method(2nd Report, Application to Two-dimensional Potential Problem).

To introduce a numerical simulation technique for the design sites of industrial products, a CAD/CAE seamless system from modeling to computation is desired. A meshless or element-free method is expected to be an effective procedure for realizing such a system, because time-consuming mesh generation processes can be omitted. The element-free Galerkin method (EFGM) proposed by Belytschko et al. is one practical meshless method. This study describes an application of EFGM to solve two-dimensional potential problems. A new methodology of enforcing the essential boundary condition in the EFGM formulation is proposed. Numerical examples are shown and the solutions by EFGM are examined by comparing with exact solutions or those by the conventional FEM.

Basic Study on Element-Free Galerkin Method. 1st Report. Application to Ordinary Differential Equation.

To introduce the numerical simulation technique for the design sites of industrial products, a "CAD/CAE seamless" system from modeling to computation is desired. A gridless or element-free method is expected to be an effective procedure for realizing such system, because time-consuming mesh generation processes can be omitted. The Element-free Galerkin method (EFGM) proposed by Belytschko et al(6). is one of the practical gridless methods. This study describes an application of EFGM to solve boundary problems of a one-dimensional ordinary differential equation. A new methodology of enforcing the essential boundary condition in the EFGM formulation is proposed, and the accuracy of the present method is verified.

The technic of soldering auxiliary springs to lingual arches

The improved interpolating moving least-square (IIMLS) method has been widely used in data fitting and meshfree methods, and the obtained shape functions have the property of the delta function, compared with those obtained by the moving least-square (MLS) method. However, the moment matrix in IIMLS may be singular or ill-conditioned because of the ill quality of the point sets used. In this paper, the weighted orthogonal basis functions are applied in IIMLS to obtain a diagonal moment matrix, which can overcome the difficulty caused by directly inversing singular or ill-conditioned matrices. However, the weighted orthogonal basis functions cannot change the nature of the singular or ill-conditioned moment matrix, since the diagonal elements of the new moment matrix may be zero or close to zero. Thus, an adaptive scheme is further employed to resolve this problem by ignoring the contribution from the zero or very small diagonal elements in the diagonal moment matrix. Combined with shifted and scaled polynomial basis functions, a stabilized adaptive orthogonal IIMLS (SAO-IIMLS) approximation is obtained. Based on this approximation, a new boundary element-free method is proposed for solving elasticity problems. Numerical results for curve fitting, surface fitting and the new boundary element-free method have shown that the proposed SAO-IIMLS approximation is accurate, stable and performs well for ill quality point sets.