Reconstructing high-order surfaces for meshing

We consider the problem of reconstructing a high-order surface from a given surface mesh. This problem is important for many meshing operations, such as generating high-order finite elements, mesh refinement, mesh smoothing, and mesh adaptation. We introduce two methods called Weighted Averaging of Local Fittings and Continuous Moving Frames. These methods are both based on weighted least squares polynomial fittings and guarantee C⁰ continuity. Unlike existing methods for reconstructing surfaces, our methods are applicable to surface meshes composed of triangles and/or quadrilaterals, can achieve third and even higher order accuracy, and have integrated treatments for sharp features. We present the theoretical framework of our methods, their accuracy, continuity, experimental comparisons against other methods, and applications in a number of meshing operations.