Abstract: To solve the problem of shape distortion when generating the finite element mesh of parametric surfaces with mapping method,a new algorithm for parametric surface mesh generation based on advancing front technique is proposed.The algorithm selects the direction along which a new node is located according to an elastic vector.The topological compatibility of the new element is then checked against the surrounding mesh in the parameter domain.Moreover,a method for adjusting the boundary of closed surface is suggested by introducing new edges related to its real boundary and the virtual boundary in the parametric domain.We also present a method for initializing the poles of closed surface and set appropriate conditions to check if one element edge being equal to another,enabling our algorithm applicable to the mesh generation of various closed surfaces.Experimental examples demonstrate that the proposed method is suitable for the high quality finite element mesh generation of the parametric surface and the combo surface.