(A) study on the resolved p-adaptive hp-clouds method

Abstract

In the last decade, several meshfree methods have been developed. Those methods do not require mesh information in the formulation. Thus nodes can be easily added and removed. That is a prominent feature in h-adaptivity analysis. P-adaptive analysis can be easily achieved by using hp-clouds approximation. In hp-clouds method, higher order shape functions are obtained by the multiplication of PU meshfree shape function and monomial basis function. In general, the meshfree shape functions are highly continuous rational function. Thus the solution of meshfree method and its derivatives are also continuous, but the integration in the formulation cannot be exactly calculated by the complexity of the shape function. Another remarkable feature of meshfree shape functions is that they do not satisfy Kronecker delta condition. It makes the imposition of essential boundary condition cumbersome. In this study, effective p-adaptive scheme is investigated and the method to impose essential boundary condition and error detection scheme are presented. In the p-adaptive analysis of hp-clouds method, the generation of shape function corresponding to newly appended dof’s does not change the values of existing shape functions. The presented resolved hp-clouds method constructs equations for newly appended dof’s. And the equations of newly appended dof’s are solved to improve the solution. Consequently the existing dof’s do not contribute to the improvement of the solution in adaptive analysis. Thus the resolved adaptive scheme does not guarantee the convergence of solution but the solution can be improved at the cost of very small amount of additional computational efforts. A method to impose essential boundary condition has been presented. The method uses singular weight function constructed by introducing the auxiliary weight functions those have singular property. The resulting shape functions have the Kronecker delta property at the nodes on which essential boundary condit...