Meshless Method Based On Radial Basis Neural Networks And Its Applications

The meshless methods are primarily motivated to trace large gradient variation of the variables in local domain. It is distinctive in that approached field functions can be constructed independently of the grids, and that the numerical integral cells are not concerned with the manner the approached field functions are constructed by. As a class of newly developed methods with great appeal, the meshless methods have made significant successes in such fields as large deformations in elasticity and plasticity, dynamic crack tracing, crash problems, flow mechanics, and etc.However, it has been revealed that the derivative value may exhibit numerical oscillation in local domains and the peak values may be drifted even though the approached field functions are of high-level accuracy. Such local numerical oscillation and peak value drifts might have negative effects on tracing large gradient variation, making the calculation results less reliable. In addition, computation burden and low efficiency of the meshless methods is also an obstacle to solving large-scale numerical problems.Neural networks have many excellent properties such as self-adapted determinant, output insensitive to the initial weighted values and non-linear convergence. The procedure of the radial basis neural networks (RBNN) approaches agrees with the idea of the meshless methods. It is expected that a quasi-interpolated function can be obtained by introducing radial basis neural network algorithms based on the optimum ideas into the meshless methods for constructing approached field functions.The causes of local oscillations of the derivatives and drift of the peak values, as well as those factors that might have influence on the computation cost of the meshless methods, are analyzed in this dissertation. It is suggested that the local derivatives oscillations and the peak values drift should be eliminated by modifying the construction methods of the approached field functions with improved accuracy of the derivatives. On the other hand, in order to make less calculation, the fundamental solutions would be used for declining the dimensions by one, and the shape functions with Kroneckerδfunction properties be constructed so that the boundary conditions can be imposed easily.Two methods for eliminating local oscillations of the derivatives and drift of the peak values are proposed. One is to consider both approached field functions and their derivatives as independent variables respectively. Both are constructed with the same procedure to make the approached fields functions and their derivatives have the same accuracy. Applying two-dimensional prompted functions that are generated by tensor product method to the radial basis neural networks, a novel RBNN meshless method based on the tensor product is then developed, using the mixed variational principles to establish the governing equations of the system. In this meshless method, if the prompted functions are chosen properly, no more numerical integral is needed. Moreover, high accuracy results can be obtained in the normal region rapidly with only a few basis matrices required for completing numerical evaluation. A satisfactory result is also available if a moderate procedure is implemented in an irregular region.Another method proposed here for eliminating the local oscillations of the derivatives and drift of the peak values is to create new prompted functions with the integral operations by taking advantages of the error propagation laws. Generally, the error generated in the approached functions would get worse during the differential process. Inversely, little difference occurs while the integral calculi are executed. By constructing high order derivatives first and establishing indirect radial basis neural networks with prompted functions generated by indeterminate integral operations, the accuracy of the field functions and their derivatives are improved significantly. The local oscillations of the derivatives and drift of the peak values vanish evidently.So far as the calculation burden of meshless methods is concerned, two approaches are proposed to reduce the amount of computation.Firstly, meshless virtual boundary method is developed by taking the profits of the ideas of the indirect boundary numerical procedures. It might be able to decline the complexity and reduce the calculations by introducing the fundamental solutions into the boundary methods. In this proposed method, singular integral and boundary effect associated with other boundary type methods disappear because no intersection between the virtual boundaries and true boundaries exists. The shapes of the virtual boundaries are simple and can be chosen easily. The numerical results are kept stable and convergent when the intervals between the virtual boundaries and true boundaries vary in large range. In addition, the corner problem associated with the other boundary methods is nonexistent as the directional cosines of the outer normal along the virtual boundaries are no more required in the meshless virtual boundary method. The results of the two-dimensional elastic problems illustrate that stabilized solutions are available either in the simple regions or in the complex regions, and the variable values varied in large gradient can be obtained exactly.Secondly, a linear combination of the prompted functions neighboring the boundary neural units is made for generating new expanded prompted functions. Since the expanded prompted functions are subjected to the boundary conditions simply, the terms for imposing boundary conditions in system equations are eliminated. Consequently, the amount of calculation decreases.To meet the requirement of the numerical integral and approached field functions construction in the methods proposed here, the conditions that scattered nodes must be subjected to have also been discussed. A code for determining the nodes in the regions with different geometric characteristics is developed. The code is capable of determining the integral points in each integral cell according to the shape of each integral domain. As a result, accurate numerical integral results can be obtained conveniently.The numerical results have demonstrated the validity and effectiveness of the proposed RBNN meshless method. Such excellent performance the method exhibits as little calculation, high accuracy and rapid convergence, lays a foundation for the future progress.