Meshless methods based on collocation with radial basis functions (RBFs) are investigated in detail in this paper. Both globally supported and compactly supported radial basis functions are used with collocation to solve partial differential equations (PDEs). Using RBFs as a meshless collocation method to solve PDEs possesses some advantages. It is a truly mesh-free method, and is space dimension independent. Furthermore, in the context of scattered data interpolation it is known that some radial basis functions have spectral convergence orders.This study shows that the accuracy of derivatives of interpolating functions are usually very poor on boundary of domain when a direct collocation method is used, therefore it will result in signi®cant error in solving a PDE with Neumann boundary conditions. Based on this fact, a Hermite type collocation method is proposed in this paper, in which both PDEs and prescribed traction boundary conditions are imposed on prescribed traction boundary. Numerical studies shows that the Hermite type collocation method improve the accuracy signi®cantly.
Because internal stresses in large steel workpieces are high during quenching, the coupling of stresses and phase transformations should be considered to predict the internal stresses. The influence of stresses on phase transformations was investigated by dilatometer experiments applied uniaxial loads, modified John-Mehl-Avrami equation and Koistinen-Marburger equation were suggested to model the stress-phase transformation coupling during quenching. The model suggested was used in an in-house FEM code NSHT developed by authors. The distribution of residual stresses of a hollow cylinder specimen was calculated to investigate the stress-induced phase transformation. The internal stresses of a generator rotor during different quenching processes and stresses at the groove bottom of a turbine rotor in quenching were analyzed.
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