Indirect quadrangular mesh generation and validation of curved finite elements
Amaury Johnen · ORBi (University of Liège) · 2016
Among the different types of 3D finite element meshes, hexahedral meshes present properties that can be highly desirable, such as alignment with physical features or a lower computational cost. For this reason and despite the maturity of the tetrahedral mesh generators, hexahedral mesh generation has always been a prolific research domain. Yet, there exists currently no robust algorithm capable of generating conformal all-hexahedral meshes with prescribed input size field on any arbitrary geometry. One difficulty that remains is that there exists no method to robustly assert that a hexahedron is valid. Indeed, linear hexahedra can be folded (tangled) in the same way than curvilinear tetrahedra. This thesis addresses two subjects. First, two original quadrangular mesh generation techniques are investigated, with the aim to generalize them to 3D. Both are indirect methods and thus consider the problem of combining pairs of triangles of an initial input triangular mesh. The first technique, called Blossom-Quad, computes the optimal solution of this problem with respect to a given quality criterion. As for any indirect method, the quality of the solution strongly depends on the location of the nodes in the initial triangular mesh. The generalization to 3D is however unclear and a second technique is investigated. This one aims at computing a near-optimal solution by using a look-ahead tree technique. The corresponding algorithm allows tuning the quality of the final mesh by choosing the depth of the tree as a parameter. This technique gives a promising way forward, especially as it is directly applicable in 3D. The second subject concerns the development of a method that permits to compute, with respect to any prescribed tolerance, the extrema of Jacobian-based quantities defined on finite elements of any order and type. Applied to the Jacobian determinant, this method allows to assert the validity of any (curvi-)linear finite element. This method is also applied to a quality measure that quantifies the pointwise anisotropy of the elements. Besides being very attractive for hexahedral mesh generation, this method is especially useful for the analysis of curvilinear finite element meshes. It can moreover be an important component of optimization techniques for achieving robustness.