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Improved Meshless Local Boundary Integral Equation Method
Abstract
Combining the local boundary integral equation with the improved moving least-square method, an improved meshless local boundary integral equation method is presented. In the improved moving least-square method, the weighted orthogonal functions are used as basis ones so that the matrix inverse at each quadrature point is avoided and the algebra equations system is not ill-conditioned. In addition, the improved meshless local boundary integral equation method is applied to linear elasticity problems, the corresponding discrete equations are derived. Some numerical results to demonstrate the efficiency of the method are presented. Compared with the conventional local boundary integral equation method, the present method has higher computational efficiency and precision, and will not form ill-conditioned or singular equations.
In order to obtain a more effective and accurate method to study the fracture behavior of the piezoelectric materials, an interpolating element-free Galerkin scaled boundary method (IEFG-SBM) is proposed for two-dimensional fracture analysis of piezoelectric material based on the improved interpolating moving least-squares (IIMLS) method. This method allows the stress and electric displacement intensity factors to be calculated directly from their definitions. Only the boundary of the computational domain requires to be discretized by the element-free Galerkin (EFG) method and thus the spatial dimension is reduced by one. However, in contrast to the boundary element method, no fundamental solution is required. The solution in the radial direction is analytical, therefore the simulation precision of this method is relatively high. In the IIMLS method, the shape functions satisfy Kronecker delta property and the weight function involved is nonsingular. Moreover, the number of unknown coefficients in the trial function of the IIMLS method is less than that of the conventional moving least-squares (MLS) approximation. At last, numerical examples are presented to demonstrate the effectiveness and correctness of the proposed method for fracture analysis of piezoelectric material.
The aviation engine blade is the core component of high performance aero engine, the shape and the machining precision have great impact on its characteristics. Because of its working environment and functional requirements, the geometry shape of the blade was designed to be of variable cross-sections and high twist surfaces. One of the most important parameters to measure the blade geometry was the mean camber line. How to extract the mean camber line of blade accurately and efficiently was one of the problems to be solved urgently. Thus this paper combined MLS method and piecewise isometric curve principle to extract the mean line. This method first divided the engine blade section points into segments, and then the MLS method was applied to make the data of the blade much more intensive. Then, the piecewise isometric curve principal was used to extract the blade mean camber line. Besides, the algorithm accuracy was also evaluated, and the MLS algorithm was worked out when there were noises in the point cloud.
Combining the interpolation function, which has the delta function property and is constructed on the basis of radial basis functions and polynomial functions, using the local boundary integral equation. method (LBIE), the local boundary integral equation method based on radial basis functions is presented for potential problem in this paper. The corresponding discrete equations are obtained. Comparing with the other meshless boundary integral equation methods, the present method has simpler numerical procedures, lower computation cost and higher accuracy. In addition, the essential boundary conditions can be implemented directly. Some numerical results to show the efficiency of the present method are given.
In this paper, the local boundary integral equation (LBIE) for a Kirchhoff elastic plate is formulated. In order to get rid of the unknown functions Mn(x) and Vn(x) in the integrals along local boundaries the companion solution is introduced into the LBIE of the thin plate. In this work, first, the companion solution associated with the fundamental solution of the thin plate is given, then all the other known functions related to the companion solution and required in the meshless LBIE are presented.
The nonlinear integro-differential Berger equation is used for description of large deflections of thin plates. An iterative
solution of Berger equation by the local boundary integral equation method with meshless approximation of physical quantities
is proposed. In each iterative step the Berger equation can be considered as a partial differential equation of the fourth
order. The governing equation is decomposed into two coupled partial differential equations of the second order. One of them
is Poisson's equation whereas the other one is Helmholtz's equation. The local boundary integral equation method is applied
to both these equations. Numerical results for a square plate with simply supported and/or clamped edges as well as a circular
clamped plate are presented to prove the efficiency of the proposed formulation.