Abstract: The primary task of trimmed isogeometric analysis is searching trimmed elements of trimmed surfaces, so a numerical algorithm searching trimmed elements of a rectangular structured grid is proposed in this paper. Firstly, the boundary of every element in the grid was segmented into 12 parts. Some parametric points on the trimming curve were selected, and then the grid was trimmed by the obtained discrete trimming curve rather than the original trimming curve. Therefore, all the trimmed elements fell into 156 categories according to the intersections of the discrete trimming curves and boundaries of elements. Besides, specific points on the trimming curve can also be obtained depending on situations using the algorithm. When the parametric grid of a trimmed NURBS surface was trimmed by a closed anticlockwise rectangular curve, all the trimmed elements can be detected using the algorithm. The point at which the curvature changes greatly and the images in physical space of trimmed elements can also be obtained. The optimal topology structure of a short cantilever beam demonstrates that the algorithm can searching trimmed elements of any rectangular structured grid effectively and efficiently.