INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2012) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.3364 Construction of tetrahedral meshes of degree two P. L. George 1, * ,† and H. Borouchaki 2 1 INRIA, Équipe-projet Gamma 3, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France 2 UTT et INRIA, Équipe ICD-Gamma3, Université de Technologie de Troyes, BP 2060, 10010 Troyes Cedex, France SUMMARY There is a need for finite elements of degree two or more to solve various PDE problems. This paper dis- cusses a method to construct such meshes in the case of tetrahedral element of degree two. The first section of this paper returns to Bézier curves, Bézier triangles and then Bézier tetrahedra of degree two. The way in which a Bézier tetrahedron and a P2 finite element tetrahedron are related is introduced. A validity condition is then exhibited. Extension to arbitrary degree and dimension is given. A construction method is then pro- posed and demonstrated by means of various concrete application examples. Copyright © 2012 John Wiley & Sons, Ltd. Received 14 June 2011; Revised 13 September 2011; Accepted 24 October 2011 KEY WORDS: P2 tetrahedron; 10-node tetrahedron; P2 mesh; P2 finite element; Bézier curve; Bézier triangle; Bézier tetrahedron; high order simplex 1. INTRODUCTION High order (p-adaptation) finite elements are demanded to solve accurately and, with a good rate of convergence, a number of PDEs, see [1–5]. The order impacts two different aspects, one is con- cerned with the geometry, the other with the finite element approximation. These two aspects may be combined or not. For instance, a high order element in the case of a straight-sided geometry does not lead to any difficulty at the time the geometry is considered whereas even a not too high order element where the geometry is a curved geometry may lead to some tedious questions. In this paper we are not concerned with the finite element aspect but only with the geomet- ric aspect of the mesh. More precisely, we will discuss a construction method for P2 tet meshes (where the finite elements are P2 (or 10-node) tetrahedra, for example, the Lagrange tetrahedra of degree two). A precise look at the Jacobians of those elements reveals that it is unfeasible to decide about the element validity. Indeed we meet a (complete) polynomial of degree three and nothing can be said about its positiveness. This is why we consider the Bézier form of the P2 element and then, we demonstrate that the Jacobians read as a homogeneous polynomial of degree three. Because of this specific form, validity conditions can be now exhibited. Before going further in the mesh construction method, we give the general form of the Jacobians of arbitrary order simplices in arbitrary spatial dimension. Then, various algorithms are described and demonstrated by means of concrete applications examples. The proposed construction method consists constructing first a P1 (straight-sided) mesh irrespective of the geometry and then post-process this mesh so as to complete a P2 mesh. The construction method is not concerned with the P2 surface mesh (and therefore does not know the CAD), indeed the P2 surface is only a data, and these data cannot be modified in any way making the method constrained. *Correspondence to: P. L. George, INRIA, Équipe-projet Gamma 3, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, France. † E-mail: paul-louis.george@inria.fr Copyright © 2012 John Wiley & Sons, Ltd.