In the present paper, we propose to define the Shepard interpolation by using the geodesic distance. The geodesic distance between a pair of points is the length of the shortest geodesic line, and geodesic line is the generalization of straight line in the Euclidean geometry to general spaces, for instance 3D surfaces. When the shape of structure is non-convex, the geodesic distance is more reasonable than the Euclidean distance to define influence domain for interpolation points. In view of the fact that the length of a geodesic line can be considered as the time it takes to go with a certain velocity from one point to another, the fast marching method is used to compute the geodesic distance. In the design optimization of fiber reinforced composite structures, the proposed Shepard interpolation based on geodesic distance is used to construct a continuous global function that represents the fiber angle arrangement. The design variables to be optimized are the angles at scattered design points. Several examples with in-plane load are investigated. In the simple representative numerical examples, the proposed method shows good performance.

在本文中，我们建议使用测地线距离定义Shepard插值。一对点之间的测地距离是最短测地线的长度，而测地线是欧几里得几何中直线到一般空间（例如3D曲面）的推广。当结构形状为非凸时，测地距离比欧几里得距离更合理，以定义插值点的影响域。考虑到测地线的长度可以视为以一定速度从一个点到另一点所花费的时间，因此使用快速行进法来计算测地距离。在纤维增强复合材料结构的设计优化中，提出的基于测地距离的Shepard插值用于构造表示纤维角度排列的连续全局函数。要优化的设计变量是分散的设计点处的角度。研究了平面载荷的几个例子。在简单的代表性数值示例中，所提出的方法表现出良好的性能。