高维线性插值方法及其在3维温度场计算中的应用

N-dimensional linear interpolation method and its application in 3D temperature field calculation

通过分析1维和2维线性插值可以推导出任意斜角直线坐标系下n维线性插值的一般计算公式以及有唯一解的条件,这一结论能够应用于三维温度场计算。可以将n维插值问题归结如下:已知n+1维空间中的n+1个点的坐标以及第n+2个点的n个坐标分量xn+2,1,x n+2,2,,xn+2,n,求解该点的第n+1个坐标分量xn+2,n+1。根据线性插值定义,第n+2个点位于前n+1个点所确定的n维超平面上。根据这一条件列写方程、求解方程可得到插值xn+2,n+1。n维插值问题有唯一解的条件是已知的n+1个点在n维空间中构成的多面体的体积不为0。推导过程在斜角直线坐标系中完成,因而结论具有较大普适性。

By analyzing 1- and 2-dimensional（2D） interpolation method, we derived anD linear interpolation formulae in oblique coordinate system and the condition that has a unique solution. The result can be applied in the calculation of 3D temperature field. The nD interpolation can be regarded as the following： Given the coordinates of n＋l points in （n＋1）D space and the coordinate x＋2,1,xn＋2,2,…,xn＋2,n of the （n＋2）th point, what is to solve is its （n＋l）th coordinate. According to the definition of linear interpolation, the （n＋2）th point locates on the nD superplane determined by the given n＋1 points. Based on this, we derived the equations, and the root Xn＋2,n＋1 is the interpolation if existing. The root uniquely exists if and only if the volume of the hyperpolyhedron constructed by the given n＋1 points is nonzero. The conclusion is of great universality due to derivating in oblique coordinate system.