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为了提高基于Galerkin弱积分形式的无网格方法求解瞬态热传导问题的计算效率,提出了两种方案:第一种方案在空间离散上采用基于任意凸多边形节点影响域的无网格形函数,并通过选取适当的节点影响半径因子,使背景网格内的积分点仅对该背景网格内的无网格节点有贡献,从而避免了节点搜索问题,减少了系统刚度矩阵的带宽,且当节点影响半径因子为1.01时,无网格方法的形函数近似具有插值特性;第二种方案在求解线性方程组时,引入质量矩阵集中技术,从而避免了系统方程组的求解。二维矩形区域、二维圆形区域的瞬态热传导数值算例结果表明:在保证计算精度的同时,采用任意多边形节点影响域的无网格方法比传统无网格方法的计算时间至少节省44.09%,采用质量矩阵集中技术的无网格方法比传统无网格方法的计算时间至少节省76.15%,且当节点影响半径因子为1.01时,其本质边界条件的施加和有限元方法一样简单;由于采用质量矩阵集中技术的无网格方法比采用任意多边形节点影响域的无网格方法精度较低,因此如仅从计算效率考虑,对精度要求不是很高(误差在5%以内),建议采用质量矩阵集中技术,如同时考虑计算精度和效率,建议采用多边形节点影响域的技术。
In order to improve the computational efficiency of the meshless method based on the Galerkin weak integral form to solve the transient heat conduction problem, two schemes are proposed: The first scheme adopts the meshless shape function based on the domain of arbitrary convex polygon nodes in the spatial discretization, And by choosing the appropriate node to influence the radius factor, the integral points in the background grid contribute only to the meshless nodes in the background grid, thus avoiding the node search problem and reducing the bandwidth of the system stiffness matrix, and when When the node influence radius factor is 1.01, the shape function approximation of meshless method has interpolation characteristic. The second one introduces the mass matrix concentration technique when solving linear equations, and avoids the solution of system equations. Two-dimensional rectangular region and two-dimensional circular region of the numerical results of transient heat transfer numerical results show that: in ensuring the accuracy of the calculation, the use of arbitrary polygon node domain-free mesh method than the traditional meshless method to save time at least 44.09 %, The meshless method using the mass matrix concentration technique saves at least 76.15% of the computation time of the traditional meshless method, and the application of the essential boundary conditions is as simple as that of the finite element method when the node influence radius factor is 1.01; The meshless method using the mass matrix concentration technique has a lower accuracy than the meshless method using any polygon node influence area. Therefore, if only the computational efficiency is not required, the precision is not very high (the error is within 5%), and it is recommended to use Quality matrix concentration technology, such as considering both the accuracy and efficiency, it is recommended to use the polygon node domain technology.