engleski

Hierarchic generation of the solutions of non-linear problems

This paper presents a new approach to the numerical modeling of non-linear engineering problems. Here, instead of a traditionally discretisation of the considered area into finite elements, the solution of arbitrary accuracy is attained by hierarchic increase of the number of basis functions on the area. The basis functions which are implementated in presented numerical models belong the class of finite functions with infinite differentiability named after they authors Rvachev's basis functions or, in short, Rbf. The possibility to calculate exactly and simply the derivation values of Rbf functions of a high degree enables an efficient application of the procedures of a strong formulation. The presented numerical models use the collocation method considering a unique criterion for the selection of collocation points. Hierarchic forming of numerical solution is conducted by an algorithm in which new functions, which are all images of the same mother basis function, are added to the base of an initial solution, but displaced and compressed or stretched in comparison with the initial base. The criterion of plastification is tested in the same points for which the values of solution function are calculated i.e. in collocation points. Described numerical procedure is illustrated on examples of elasto-plastic bending of a beam and elasto-plastic behavior of a prismatic bar subjected to torsion. The results of the analyses are compared to the existing exact solutions. It can be concluded that presented numerical models efficiently simulate real non-linear behavior of the structure. Hierarchic increase in number of basis functions in the model provides a simple way to increase the accuracy of an approximate solution in places where plastic yielding occurs and also accelerates the convergence of incremental-iterative procedure. This method provides excellent results for the elaborated problems and numerical procedure is stable until plastic failure occurs.

Numerical modeling; Basis functions; Universality; Fragment; Collocation method; Plastic failure

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