The main difficulty of the meshless methods is related to the support of shape functions. These methods become stable when sufficiently large support is used. Rather larger support size leads to higher calculation costs and greatly degraded quality. The continuous adjustment of the support size to approximate the shape functions during the simulation can avoid this problem, but the choice of the support size relative to the local density is not a trivial problem. In the present work, we deal with finding a reasonable size of influence domain by using a genetic algorithm coupled with high order mesh-free algorithms which the optimal value depends on the accuracy and stability of the results. The proposed strategy provides guarantees about the growth of approximation errors, monitor the level of error, and adapt the evaluation strategy to reach the required level of accuracy. This allows the adaptation of the proposed algorithm with problem complexity. This new strategy in meshless approaches are tested on some examples of structural analysis.
This paper presents a new method to solve a challenging problem and a topic of current research namely the selection of optimal shape parameters for the Radial Basis Function (RBF) collocation methods in both interpolation and nonlinear Partial Differential Equations (PDEs) problems. To this intent, a compromise must be made to achieve the conflict between accuracy and stability referred to as the trade-off or uncertainty principle. The use of genetic algorithm and path-following continuation allows us on the one hand to avoid the local optimum issue associated with RBF interpolation matrices, which are inherently ill-conditioned and on the other side, to map the original optimization problem of defining a shape parameter into a root-finding problem. Our computational experiments applied on nonlinear problems in structural calculations using our proposed adaptive algorithm based on genetic optimization with automatic selection of the shape parameter can yield more accuracy and a good precision compared to the same state of the art algorithm from literature with a fixed and given shape parameter and Finite Element Method (FEM).
This paper presents an identification method for continuous nonlinear systems whose the input-output functional is regular and homogeneous. The model is a Volterra series truncated in its firt terms. The Volterra kernels are expanded on multidimensional generalized orthonormal bases. A subset model selection is applied on the estimated model to provide a more parsimonious model.
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