弹性薄板的广义多维微分求积分析方法

多维微分求积DC(differential cubature)法是近年迅速发展起来的一种处理多维微分方程的高精度数值方法。本文围绕多元函数插值适定性这一基本问题，探讨了实现DC法时插值结点组和插值空间应满足的条件，研究了离散结点组选择及其构造适定试函数的方法。提出了一种沿各坐标方向可以布置不同数量结点的广义交错网格DC方法，并给出了确定相应的插值空间及其选择试函数的具体方案。通过弹性薄板变形分析研究了广义交错网格DC法解决实际结构力学问题的可靠性及其潜力。研究结果表明，广义交错网格DC法较传统的交错网格DC格式具有更强的适用性和灵活性，采用数量少得多的离散结点即可达到传统交错网格DC法相同的数值精度，表现出精度高和计算工作量小的优点。

Differential cubature solution of Kirchhoff plates with generalized staggered mesh pattern

The differential cubature(DC) method was developed for solving numerically multi-dimensional differential equations.We focus on the well-posedness of multivariate polynomial interpolation and research the fundamental conditions that the appropriate node group of interpolation polynomial and the interpolation space should meet when applying the DC method.Furthermore,the specific operation for selecting the grid points in the solution domain and constructing the well-posed test functions is explored carefully,and a generalized staggered-grid scheme in which different numbers of grid points may be permitted along each coordinate direction is proposed.The interpolation space as well as test functions corresponding to this generalized staggered-grid DC scheme are also researched and presented.Finally,the reliability and potential of the generalized staggered grid-based DC method are examined in solving actual problems of structural mechanics by the deformation analysis of the rectangular elastic thin plate.The results show that the proposed DC method has stronger applicability and flexibility than the traditional staggered grid DC method,and similar accuracy could be achieved by means of a much smaller number of grid points.The proposed method appears characteristics of high precision solution without more computational efforts.