ORIGINAL ARTICLE Xiao-Juan Luo Æ Mark S. Shephard Æ Robert M. O’Bara Rocco Nastasia Æ Mark W. Beall Automatic p-version mesh generation for curved domains Received: 28 July 2003 / Accepted: 12 April 2004 / Published online: 28 July 2004 Ó Springer-Verlag London Limited 2004 Abstract To achieve the exponential rates of conver- gence possible with the p-version finite element method requires properly constructed meshes. In the case of piecewise smooth domains, these meshes are character- ized by having large curved elements over smooth por- tions of the domain and geometrically graded curved elements to isolate the edge and vertex singularities that are of interest. This paper presents a procedure under development for the automatic generation of such me- shes for general three-dimensional domains defined in solid modeling systems. Two key steps in the procedure are the determination of the singular model edges and vertices, and the creation of geometrically graded ele- ments around those entities. The other key step is the use of general curved element mesh modification procedures to correct any invalid elements created by the curving of mesh entities on the model boundary, which is required to ensure a properly geometric approximation of the domain. Example meshes are included to demonstrate the features of the procedure. Keywords p-version method Æ Curved meshes Æ Graded meshes 1 Introduction The performance of the p-version finite element method is characterized by its ability to attain exponential rates of convergence, thus, allowing the desired level of accuracy to be obtained with many fewer degrees of freedom than other methods, such as the standard h- version finite element method. However, to attain the exponential rates of convergence, the meshes must sat- isfy specific requirements in terms of mesh entity geo- metric approximation and mesh gradation that are not satisfied by current mesh generation methods developed for h-version meshes of linear or quadratic elements. As the order of the finite element basis functions are increased, so must the level of geometric approximation of those finite element faces and edges that lie on curved domain boundaries. Otherwise, the error due to geometric approximation will become the dominate error, thus, yielding any further increase in finite element order meaningless [6, 13]. To attain the optimal convergence, p-version finite elements require strongly graded coarse meshes. In the neighborhood of singularities, the meshes must be geometrically graded in the directions of the singular gradients [1–4, 15]. In the smooth portions of the domain away from the singularities, the meshes must be very coarse with satisfactory geometric approximation. Approaches to generate high-order meshes for the p-version method usually start from linear mesh gener- ation developed for the h-version finite element method, followed by assigning high-order geometric shapes to the mesh entities on the curved model boundaries [5, 14]. Optimization of the surface mesh generation, hybrid meshing with prismatic elements near the domain boundaries, and curvature-driven surface mesh adapta- tion are the three strategies employed in [14], and the procedure presented in [5] incrementally corrects the invalid high-order elements by applying available local mesh modification operations. The ideal approach to accomplish p-version mesh generation is to create curved elements of the size and shape desired one at a time. However, the lack of a known algorithm and the high level of computational effort required by such an approach make it unfeasible. A compromise approach is to combine operations to curve a mesh initially constructed based on linear geometry mesh X.-J. Luo (&) Æ M. S. Shephard Scientific Computation Research Center, Rensselaer Polytechnic Institute, Troy, NY 12180, USA E-mail: xluo@scorec.rpi.edu E-mail: shephard@scorec.rpi.edu R. M. O’Bara Æ R. Nastasia Æ M. W. Beall Simmetrix Inc., Clifton Park, NY 12065, USA E-mail: obara@simmetrix.com E-mail: nastar@simmetrix.com E-mail: mbeall@simmetrix.com Engineering with Computers (2004) 20: 273–285 DOI 10.1007/s00366-004-0295-1