Keywords: mesh generation, hexahedral mesh, mesh smoothing, mesh untangling, mesh optimization.

The accuracy and performance of a finite element analysis depends on the quality of the mesh on which the spatial domain has been discretized. Most three-dimensional meshes are composed of tetrahedral elements as they can be automatically generated to discretize an arbitrary domain by rapid and mature algorithms. While more challenging to mesh, hexahedral elements have several advantages over tetrahedral meshes both in structural and in fluid mechanics. Unfortunately, ready access to these advantages is hampered by the significant challenges regarding the generation of the hexahedral mesh. At this stage no automated technique exists which which is comparable to those used for for tetrahedral generation such as the Delaunay or advancing front techniques. Moreover, hexahedral meshes generated automatically can include poorly formed or even inverted elements that can affect the accuracy and invalidate any subsequent analysis. Therefore, it is of the major importance to develop an algorithm that can improve (smooth and untangle) the quality of a given hexahedral mesh.

Several quality measures have been developed to smooth tetrahedral meshes. For instance, algebraic quality measures [1] have been widely used to improve the quality of these meshes. Moreover, smoothing techniques have been developed to improve the quality of tetrahedral meshes based on a continuous optimization of algebraic qualities. In particular, in [2] an optimization technique is presented that is able to simultaneously smooth and untangle tetrahedral meshes. In this work, we extend this technique to quadrilateral and hexahedral meshes [3]. Specifically, we present a technique that iteratively untangles and smoothes a given quadrilateral or hexahedral mesh by minimizing an objective function defined in terms of an algebraic quality measure [4]. It is important to point out that the proposed method optimizes the quality of quadrilateral and hexahedral meshes by a local node relocation process. That is, without modifying the mesh connectivity.

Finally, we present several examples to show that the proposed technique can be used to obtain valid meshes (untangled) composed by high-quality (smoothed) quadrilaterals and hexahedra.

1

P.M. Knupp, "Algebraic mesh quality metrics", Society for Industrial and Applied Mathematics, Journal for Scientific Computing, 21(1), 193-218, 2001. doi:10.1137/S1064827500371499

2

J.M. Escobar, E. Rodríguez, R. Montenegro, G. Montero, J.M. González-Yuste, "Simultaneous untangling and smoothing of tetrahedral meshes", Computer Methods in Applied Mechanics and Engineering, 192(5), 2775-2787, 2003. doi:10.1016/S0045-7825(03)00299-8

3

T.J. Wilson, "Simultaneous untangling and smoothing of hexahedral meshes", MSc Thesis, Universitat Politecnica de Catalunya, 2011.

4

P.M. Knupp, "A method for hexahedral mesh shape optimization", International Journal for Numerical Methods in Engineering, 58, 319-332, 2003. doi:10.1002/nme.768

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