Study Of Meshless Method Theory And Applications

As a new-style numerical method, meshless method has started yet. All its theory and methods have much room for improvement. Based on the experience summed up by our predecessors, this paper has made some beneficial probe about meshless method's theory and application. And the main work can be summarized as follows.1. Based on the weighted residual method with compactly supported trial function, the fundamentals of meshless method have been explained explicitly. And then we have focused our discussion on the principle of Galerkin-based meshless method (GBMM) and meshless weighted least square method (MWLS) and the application of two methods to elastic mechanics.2. Meshless method has been applied to reservoir simulations in this paper. Its principles and meshless numerical forms have been elucidated explicitly. And the meshless method has been applied to the solution of oil-water two-phase flow through porous media for the first time. Some preliminary simulation results, which will be beneficial for us to further investigate reservoir simulations by meshless method, have been obtained.3. In the meshless numerical calculation, nodes distribution directly affects the precision of meshless method. So nodes distribution and characters of nodes are the emphasis and difficulty of meshless method. On the basis of nodal sensitivities in finite element method and shape optimization, the sensitivity analysis was used to study nodal arranging scheme in meshless method. By choosing potential energy as response quantities, calculating nodal sensitivities modulus, the relation between sensitivity modulus and nodal distribution scheme was gotten. The results of the case showed that the meshless method could guide and improve the nodal distribution scheme effectively. The precision of calculations was improved.4. How to confirm the radius of the influence domain reasonably and exactly? Which type of weight function is satisfied to numerical calculation? They are the two difficult problems in meshless method research. This paper has advanced a new weight function, named Normal weight function, which dated from probability theory and error theory. Its feasibility has been proved in terms of theory. And then two simple numerical simulation results have shown that Normal weight function possess good potential to meshless calculation, such as high accuracy and good stability. At last, some rules to make sure the radius of influence domain for Normal weight function were listed.5. A new mehsless method, named meshless method based on weighted least square method and ridge basis function interpolation (WLSR), has been established. In WLSR, the ridge basis function was the trial function, and the weighted least square method was used to disperse the control equations. The WLSR equation of elastic mechanics was established, and the results of cases showed that WLSR was better in accuracy and enforcing essential boundary condition than EFGM and MWLS.