AN ASSESSMENT OF THE MESHLESS WEIGHTED LEAST-SQUARE METHOD

The meshless weighted least-square (MWLS) method was developed based on the weighted least-square method. The method possesses several advantages, such as high accuracy, high stability and high e?ciency. Moreover, the coe?cient matrix obtained is symmetric and semi- positive de?nite. In this paper, the method is further examined critically. The e?ects of several parameters on the results of MWLS are investigated systematically by using a cantilever beam and an in?nite plate with a central circular hole. The numerical results are compared with those obtained by using the collocation-based meshless method (CBMM) and Galerkin-based meshless method (GBMM). The investigated parameters include the type of approximations, the type of weight functions, the number of neighbors of an evaluation point, as well as the manner in which the neighbors of an evaluation point are determined. This study shows that the displacement accuracy and convergence rate obtained by MWLS is comparable to that of the GBMM while the stress accuracy and convergence rate yielded by MWLS is even higher than that of GBMM. Furthermore, MWLS is much more e?cient than GBMM. This study also shows that the instability of CBMM is mainly due to the neglect of the equi- librium residuals at boundary nodes. In MWLS, the residuals of all the governing equations are minimized in a weighted least-square sense.

A WEIGHTED RESIDUAL METHOD FOR ELASTIC-PLASTIC ANALYSIS NEAR A CRACK TIP AND THE CALCULATION OF THE PLASTIC STRESS INTENSITY FACTORS

In this paper, a weighted residual method for the elastic-plastic analysis near a crack tip is systematically given by taking the model of power-law hardening under plane strain condition as a sample. The elastic-plastic solutions of the crack lip field and an approach based on the superposition of the nonlinear finite element method on the complete solution in the whole crack body field, to calculate the plastic stress intensity factors, are also developed. Therefore, a complete analvsis based on the calculation both for the crack tip field and for the whole crack body field is provided.

A Unified Variational Principle of Fluid Mechanics and Application on Solitary Subdomain or Point

-According to basic equations of fluid mechanics, this paper presents a unified variational principle of fluid mechanics (UVPFM) by using the optimization method of weighted residuals (OMWR). The advantages are as follows, the establishment of the functional and the variational principle is easy, it can change various problems of fluid mechanics derived by basic equations into a unified optimization problem, and the solution is the optimum one in some sense. According to the OMWR for the solitary subdomain, this paper uses UVPFM onto any solitary subdomain and gives the solution of the hydrodynamics equation which is suitable only for that solitary subdomain. According to the OMWR for solitary point, this paper uses UVPFM to any solitary point and gives the solution of the hydrodynamics equation (point solution) which is suitable only for that solitary point. As the solution for the solitary subdomain or solitary point is developed independently, the compatibility with other subdomain or other points, does not need to be considered, but all the boundary conditions and the supplementary derived residual equations obtained by running the derivative operations to the differential equation should be taken into account.

AN APPROXIMATE METHOD FOR DETERMING THE VELOCITY PROFILE IN A LAMINAR BOUNDARY-LAYER ON FLAT PLATE

In this paper, using the integration method, it is sought to solve the problem for the laminar boundary_layer on a flat plate. At first, a trial function of the velocity profile which satisfies the basical boundary conditions is selected. The coefficients in the trial function awaiting decision are decided by using some numerical results of the boundary_layer differential equations. It is similar to the method proposed by Peng Yichuan, but the former is simpler. According to the method proposed by Peng, when the awaiting decision coefficients of the trial function are decided, it is sought to solve a third power algebraic equation. On the other hand, in this paper, there is only need for solving a linear algebraic equation. Moreover, the accuracy of the results of this paper is higher than that of Peng.

A GENERAL SOLUTION OF AXISYMMETRIC PROBLEM OF ARBITRARY THICK SPHERICAL SHELL AND SOLID SPHERE

In this paper, the axisymmetric problems of arbitrary thick spherical shell and solid sphere are studied directly from equilibrium equations of three-dimensional problem, and the general solutions informs of Legendre serifs for thick spherical shell and solid sphere are given by using the method of weighted residuals.

APPLICATION OF SUBREGION FUNCTION METHOD FOR SOLVING BEAM-BOARD STRUCTURE

In this paper, the solution to the structure consisting of a bead and a board is given as a result of the application of the subregion function method which was suggested in ref. [1]. The same problem is also computed with finite element method. The comparison between the two results shows that the application of the subregion function in the method of weighted residuals is practical and effective, especially for solving compound structures.

AN EFFICIENT PREDICTION METHOD FOR THE TWO-DIMENSIONAL COUPLED STRUCTURAL-ACOUSTIC ANALYSIS

A wave number method （WNM） is proposed to deal with the two-dimensional coupled structural-acoustic problem. Based on an indirect Trefftz approach, the displacement and the pressure response are approximated respectively by a set of wave functions, which exactly satisfy the governing equations and are independent of the size of the coupled system. The wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions, which arise from the external excitation. The weighting coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions and it is performed by applying the weighted residual formulation. The example is computed by the WNM and the BEM. The results show that, the WNM can attain the same accuracy and convergence as the BEM with less degrees of freedom.

Determination of Optimal Work Roll Profile for Cold Strip Mill

The model of predicting distribution of strip thickness at the exit and tension stress is established by using weighted residual method and variational method with consideration of rolled metal lateral flow.A New method of calculating loaded roll gap in relation to target tension stress is proposed.The minimum square summation of loaded roll gap deviation is used as an objective function for the first time to optimize initial work roll profile.

An analysis of the incompressible viscous flows problem by meshless method

Generally the incompressible viscous flow problem is described by the Navier-Stokes equation. Based on the weighted residual method the discrete formulation of element-free Galerkin is inferred in this paper. By the step-bystep computation in the field of time, and adopting the least-square estimation of the-same-order shift, this paper has calculated both velocity and pressure from the decoupling independent equations. Each time fraction Newton-Raphson iterative method is applied for the velocity and pressure. Finally, this paper puts the method into practice of the shear-drive cavity flow, verifying the validity, high accuracy and stability.

AN ITERATIVE PARALLEL ALGORITHM OF FINITE ELEMENT METHOD

In this paper, a parallel algorithm with iterative form for solving finite element equation is presented. Based on the iterative solution of linear algebra equations, the parallel computational steps are introduced in this method. Also by using the weighted residual method and choosing the appropriate weighting functions, the finite element basic form of parallel algorithm is deduced. The program of this algorithm has been realized on the ELXSI-6400 parallel computer of Xi'an Jiaotong University. The computational results show the operational speed will be raised and the CPU time will be cut down effectively. So this method is one kind of effective parallel algorithm for solving the finite element equations of large-scale structures.

Wave Number Method for Three-Dimensional Steady-State Acoustic Problems

Based on the indirect Trefftz approach, a wave number method （WNM） is proposed to deal with three-dimensional steady-state acoustic problems. In the WNM, the dynamic pressure response variable is approximated by a set of wave functions, which exactly satisfy the Helmholtz equation. The set of wave functions comprise the exact solutions of the homogeneous part of the governing equations and some particular solution functions. The unknown coefficients of the wave functions can be obtained by enforcing the pressure approximation to satisfy the boundary conditions. Compared with the boundary element method （BEM）, the WNM have a smaller system matrix, and is applicable to the radiation problems since the wave functions are independent of the domain size. A 3D acoustic cavity is exemplified to show the properties of the method. The results show that the wave number method is more efficient than the BEM, and it is fairly accurate.