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In numerical analyses based on unstructured meshes represented by the finite element method, quadrilateral and hexa-hedral meshes are used for a two-dimensional and a three-dimensional analyses, respectively, from the viewpoint of calculation accuracy and computation time. However, a method for automatically generating quadrilateral and hexahedral meshes has not been established thus far; in addition, a considerable amount of effort is required to perform manual mesh modifications. Mesh generation based on the frame field method is an effective technique to extract high quality quadrilateral and hexahedral meshes through the optimization of the posture of a frame with rotational symmetry. This method needs to solve a mixed-integer programming (MIP) problem while extracting the quadrilateral and hexahedral mesh using integer grid maps. When solving an MIP using a greedy method, solving a simultaneous equation is required approximately O(M) times, where M is the number of integer constraints. In this study, we propose a method to obtain a quadrilateral mesh by solving a simultaneous equation approximately O(1) times. In this method, the frame field is calculated using finite element discretization, and the finite element mesh is appropriately remeshed to be aligned with the integer grid resulting from the frame field optimization. The proposed method can reduce the amount of computation required to obtain integer grid maps for mesh extraction, without impairing the mesh quality. The proposed method is evaluated with respect to by mesh quality using the aspect ratio and the scaled Jacobian. Numerical examples demonstrate that the mesh quality is comparable to or better than that produced by conventional methods and manual procedures.