等几何修正准凸无网格法

采用等几何B样条基函数的多项式再生条件对无网格形函数的多项式再生条件进行了修正,使得无网格形函数的负值部分明显减少,在域内趋于非负函数,即等几何修正准凸无网格形函数。该准凸无网格形函数仍然具有与传统再生核无网格形函数相似的构造形式,数值实现比较便捷,同时该准凸无网格形函数的多项式再生条件具有准确的修正系数,无需引入额外的人工节点松弛参数。更重要的是,等几何修正准凸无网格形函数可在确保形函数高阶光滑的前提下减小相对支持域,提高计算效率。最后,基于等几何修正准凸无网格形函数对杆梁和膜板结构进行了伽辽金无网格振动分析。结果表明,与标准再生核无网格法相比,等几何修正准凸无网格法具有更优的计算精度。

Isogeometric refined quasi-convex meshfree method

An isogeometric refined quasi-convex meshfree method is proposed.The present quasi-convexity of meshfree approximation is obtained through refining the consistency or reproducing conditions of meshfree shape functions by their counterparts in the convex isogeometric B-spline basis functions.The derived meshfree shape functions are thus called isogeometric refined quasi-convex meshfree shape functions which are almost positive.These shape functions still belong to the general reproducing kernel meshfree framework and their numerical implementation is quite straightforward.In contrast to the previous quasi-convex meshfree approximants that are related to artificial nodal relaxed parameters,the present quasi-convex meshfree shape functions are built upon the isogeometric refined reproducing conditions with analytical nodal relaxed coefficients,and consequently there is no need for any factitious adjustment.More importantly,compared with the standard meshfree shape functions,a unique feature of the present isogeometric refined quasi-convex meshfree shape functions is that they require much smaller support size to ensure smoothing shape functions,which is very desirable from the computational point of view.The accuracy of the proposed approach is demonstrated by performing Galerkin meshfree analysis of free vibration of rods,membranes and thin plates.The dispersion analysis results also consistently support the superiority of the proposed method.