无网格法中边界畸变的控制与计算效率的提高

Improvement of boundary aberration and computation efficiency in meshless method

分析比较了常用的2种无网格法的形函数,即采用光滑粒子流体动力学(SPH)法与移动最小二乘(MLS)法构造的形函数,指出SPH形函数在特定情况下易在边界处产生畸变的原因,并提出了在边界外围设置虚节点以改善边界畸变的方法.分别通过配点法和无网格Galerkin(EFG)法计算了一维和二维算例,讨论了不同的边界条件处理方式对计算精度的影响,结果表明Lagrange乘子法处理边界条件的精度比点插值法高.在EFG法的一维悬臂梁算例分析中,讨论了节点支撑域半径和高斯积分阶次对计算量和计算精度的影响.分析表明,当使用单点高斯积分时,节点支撑域的变化易导致计算结果不稳定,提高高斯积分阶次能够降低计算结果对节点支撑域大小变化的敏感性并提高计算精度,但同时增加了计算量.

Two usual methods for constructing the shape functions of meshless methods, smooth particles hydrodynamics （SPH） and moving least square （MLS）, were analyzed and compared. The reason of boundary aberration in SPH was indicated, and the virtual points method which assigned extra points outside the boundary was presented to control the boundary aberration. Allocation method and element-free Galerkin method were used to resolve the one-dimensional and two-dimensional differential equations. Computation results showed that Lagrange multiplier method has better accuracy than point interpolation method in dealing with boundary conditions. Cantilever beam example illustrated the influence of support radius and Gauss integral point on accuracy and cost. The result indicates that single Gauss integral point is affected by the variation of support radius, however higher order Gauss integral points reduce the sensitivity to the variation of support radius and improve the computational accuracy, while the computation cost increases.