Improvement And Application Of The Basis Of Element-free Method

With the development of computer technology and rapid upgrade of calculation speed, FEA has become the most important achievement in numerical method field. Due to its dependence of background mesh, it has no advantage on the numerical simulations with repeated reconstruction mesh, such as solution of dynamic crack propagation, problems with large deformations, singularities in outer and inner boundaries, geometry distortions caused by high speed collisions, problems with high frequency shock waves and heavy gradients, and adaptive computations etc.Finite element mesh distortion may affect the accuracy of the calculation, resulting in the failure of the calculation.Therefore, the meshless method is proposed to meet the demands of the above-mentioned of FEA methods. The shape function of meshless method is an approximation of node, it needs no initial mesh construction or reconstruction. This method can assure the calculation accuracy even under structural distortions. Many researchers such as T.Belytschko, W.K.Liu, S.N.Atluri and Zhang xiong have investigated in this field.The meshless method needs no background grid or a little dependence on it. This method can solve those mesh reconstruction problems. The meshless method need only node information instead of grid information, and it can solve the numerical computation problem. It will enable engineers free from the heavy mesh work, and speed up the initial product verification. It has a prosperous future of application. From 1977 to date, dozens of different meshless methods have been developed.Some important theories as, Smoothed Particle Hydrodynamics, Diffuse approximation and Diffuse Element Method, the Element-free Glerkin Method, Reproducing Kernel Particle Method, Hp-Clouds, Finite Point Method, Partition of Unity Finite Element Method, Local Boundary Integral Equation Method, Hybrid Houndary Node Method, Meshless Weighted Least-Square Method, Stabilized Conforming Nodal Integration Method.All those meshless methods are not quite different, but have interrelations to some degree. As for those methods which need background integral grid, the background grid just be used in the integral calculation and has no relation to the approximation of function, so it is much simpler than the mesh construction. They are all based on node information, approximate the functions in the node support fields and obtain the approximate functions, which are substituted into the known boundary conditions and then can be solved.The mostly used approximation functions in meshless method are: kernel approximation, moving least-square approximation, partition of unit approximation, reproducing kernel approximation, radial basis function interpolation. The moving least-square method is one of the most widely used methods of those approximations. In this paper, the moving least–square method and meshless weighted function were introduced, and the treatment of discontinuity and elimination control program as well. At present, three methods are commonly used to deal with the discontinuity of the field functions, that is, visibility rule, diffraction rule and transmission rule. Three elimination control programs are adopted in the meshless method, they are: collocation method, Galerkin method and Local Petrov-Galerkin method. Being not satisfy the Kroneckerδconditions, the approximation function will not change through node variables, and the essential boundary conditions can not be loaded arbitrarily, it must be disposed properly. Till now, many essential boundary condition methods have been presented, such as Lagrange Multiplier Method, Penalty Function Multiplier Method, Coupling with Finite Element Method and Collocation Method.In the structure and mechanical engineering equipment, for a long time of fatigue, corrosion, wear and tear, the cracks would be produced and would lead to catastrophic accidents. Therefore, the analysis of the crack reliability on construction machinery would be of great theoretical and practical significance. The failure and crack of a multi-crack structure will show dispersed attributes due to the material properties, crack size and loads. It is need to establish a crack reliability model with multi-cracks from the probability and statistical point of view. In this paper, the reliability and relative theory are introduced, which include, the conception of reliability, the limit state function of a structure and the reliability index. Considering the difficulties of the calculation of failure probability on a structure with integral method, many approximate method are used in engineering project, hence the reliability index is introduced. Some of the typically used methods as, First-Order Quadric Moment Method, Higher-Order Quadric Moment Method, Response Method, Monte Carlo Method and Stochastic FEM, and those methods are all introduced in this paper.When the limit state function of structure is highly complex or with nonlinearity, it is time-consuming or cumbersome to carry out reliability analysis with traditional methods. To reduce the computational effort required for reliability analysis, response surface method combined with ANN could be helpful.RBF neural network is a forward network based on function approximation theory. It can approximate any non-linear function without local minimum. The combination of the RBF neural network response surface method and the proposed method has a broad application prospect in structural engineering reliability analysis. The model of RBF neural network and computation steps of the RBF response method are full described.The main work of this paper:Based on the least-square approximation, taken the Element-free Galerkin method as an example, the expressions of the shape function and its derivatives of meshless method were derived in inner product space, so did the expressions of the shape function and its derivatives with the orthogonal basis function being concerned, and proposed a method to improve the calculation. The expressions of the shape function and its derivatives of improved basis function were derived.With the results derived being compared, it is proved that the three kinds expressions are equivalent. Using orthogonal basis, this method can avoid the calculation of matrix inversion, it solved the difficulties brought by sick equations, and can improve the computation efficiency. The improved orthogonal basis provides the characteristics of orthogonal basis, and has the advantage on the settle of higher-order derivatives.The improved orthogonal basis and orthogonal basis were both applied in the Element-free Galerkin method and meshless weighted least-square method. The results were compared with the original basis ones, the computation time and calculating error of three basis were given, so did the diagram of the compared results.The Local Orthogonal Meshless Method is proposed in this paper. Keep the orthogonality of the shape function only in the field of nodes, which will avoid the ill-condition or singularity of a matrix, and can also simplify the calculation of the derivatives of the shape function. The proposed method was used to calculate the natural frequency of the structure, and the stress intensity factors of different structural cracks as well. Which include: the crack under unidirectional tension force, the crack between two round holes under unidirectional tension force, multi-cracks with multi-holes. The results were compared with the theoretical ones or FEM solution, and the relative errors were also listed. The feasibility of the proposed method was illustrated through the acceptable error results.In this paper, the material properties, crack size and load were taken as random variables. The radial basis function neural network was used to simulate the structural limit state function and its derivatives, together with meshless method, the reliability on the multiple crack structure was analyzed.