Upper Bound Limit Analysis By The Cell-based Smoothed Radial Point Interpolation Method

As a recently-developed meshless method,the cell-based smoothed radial point interpolation method possesses some excellent properties which is widely used in many fields.With this method,the problem domain was discretized using triangular background cells,and each cell was regarded as a smooth domain.The strain smoothing technique is utilized and therefore area integration over each smoothing cell can be recast into line integration along its edges.The cellbased smoothed radial point interpolation can deal with the mesh distortion effectively,which leads to the fact that it can offer very accurate solutions even if the nodal arrangement is very irregular.Furthermore,the shape functions in the proposed method possess the Kronecker delta properties and as a result the essential boundary conditions can be imposed directly.In order to take full advantages of the cell-based smoothed radial point interpolation and expand the numerical methods of limit analysis,the cell-based smoothed radial point interpolation is developed for limit analysis of plane and thin plate structures.Based on the upper bound theorem of limit analysis,the cell-based smoothed radial point interpolation method is developed for the upper bound limit analysis of plane structures in the present paper.The radial point interpolation method is utilized to construct the displacement velocity field and the plastic incompressibility condition of plane stress and plane strain problems are dealt with by two different methods respectively.The non-smooth optimization problems are solved by two different methods.In the first method,the direct iterative algorithm is employed which distinguish the rigid zone from the plastic zone gradually.In the second way,the non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms,which can be solved by an efficient second-order cone programming algorithm.Classical numerical examples demonstrate that the proposed method is effective and feasible.The cell-based smoothed radial point interpolation method is also developed for the limit analysis of perfectly rigid-plastic thin plate structures.The plastic incompressibility condition is directly fulfilled through its introduction in the goal function.Only one displacement variable is required for each node,therefore the total number of variables in the resulting optimization is kept to a minimum.Through the introduction of strain smoothing technique,calculation of the second order derivatives of shape functions is avoided.In order to overcome the difficulties caused by non-differentiable objective function,the direct iterative algorithm and the second order cone programming method are introduced in the present paper.Numerical examples show that the proposed method has good numerical stability and high accuracy.