1. INTRODUCTION High order (p-version) finite element methods, character- ized by the capability of exponential rate of convergence, are gaining popularity in industry. The basic functions of p- version finite elements, their convergence properties and aspects of their computer implementation have received extensive consideration in the literature (e.g. [1,3,10,16,17,18]). However, the critical technical issues of the appropriate geometric representation of p-version finite elements for solving partial differential equations over gen- eral three dimensional domains have not received adequate consideration. This paper first demonstrates that the accu- racy of finite element solutions is strongly influenced by how well the geometry is approximated. Consideration is then given to a set of procedures being developed for proper generation of curved elements for p-version analyses. Section 2 outlines the advantages of p-version finite ele- ments assuming the proper choice of meshing and mapping procedures as required to preserve the superior rates of con- vergence. Section 3 examines the role of the mesh geometric approximation on the accuracy of the results obtained in terms of a specific curved domain problem with a known exact solution. This simple example clearly dem- onstrates that the use of quadratic geometric approximations for p-version finite elements does not lead to satisfactory solution results, in the sense that using p-lev- els greater than 3 or 4 will produce results that are affected by the errors in mapping. The requirements of Section 2 and results of Section 3 demonstrate the need for new mesh generation technolo- gies to support p-version finite elements. Section 4 overviews current efforts on the development of such a mesh generation capability. Central to this new approach is the use of Bezier basis for the geometric representation of the element shapes. This basis allows one to effectively increase the order of geometric appropriation in an effi- cient manner to any order desired. In addition, this basis supports effective methods for the execution of key opera- tions such as determining the validity of curved finite elements and determining which mesh entities require shape change to make an invalid element valid. 2. p-VERSION FINITE ELEMENT MESHES A finite element mesh serves two purposes: First, to allow representation of an arbitrary body by a collection of ele- ments on which piecewise polynomial functions (occasionally augmented by other functions) are defined, and second, to control the error of approximation in terms of the data of interest. The error of approximation depends on the finite element mesh and the polynomial degree of elements. In conven- tional FEA codes, the polynomial degree of elements are p-Version Mesh Generation Issues Xiao-Juan Luo, Mark S. Shephard, Jean-François Remacle Scientific Computation Research Center, Rensselaer Polytechnic Institute, Troy, NY 12810, xluo@scorec.rpi.edu, shephard@scorec.rpi.edu, remacle@scorec.rpi.edu Robert M. O’Bara, Mark W. Beall Simmetrix, Inc., Clifton park, NY 12065 obara@simmetrix.com, mbeall@simmetrix.com Barna Szabó Washington University, St. Louis, MO 63130 szabo@me.wustl.edu Ricardo Actis Engineering Software Research and Development, Inc., St. Louis, Missouri ricardo@esrd.com ABSTRACT Higher order (p-version) finite element methods have been shown to be clearly superior to low order finite element methods when properly applied. However, realization of the full benefits of p-version finite elements for general 3-D geometries requires the careful construction and control of the mesh. A 2-D elasticity problem with curved boundary is used to clearly illustrate the influence of the mesh shape geometric approximation order and shape representation method on the accuracy of finite element solution in a p-version analysis. Consideration is then given to a new approach for the representation of mesh geometry for p-ver- sion meshes and to the automatic generation of p-version meshes. Keywords: geometric approximation, Bezier polynomials, curvilinear mesh