Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems

Abstract

International audienceThe aim of this paper is to propose a method for dealing with the problem of mesh deformation (or mesh evolution) in the context of free and moving boundary problems, in three space dimensions. The method consists in combining two different numerical parameterizations of domains: on the one hand, domains are equipped with a computational tetrahedral mesh, and on the other hand, they are represented as the negative subdomains of 'level set' functions. We then consistently switch from one description to the other, depending on their respective convenience with respect to the operations to be performed. Among other things, doing so implies to be able to get a computational mesh from an implicitly-defined domain. This in turns relies on an algorithm for handling three-dimensional domain remeshing (that is, remeshing at the same time both surface and volume parts of a given tetrahedral mesh). Applications are considered in the fields of mesh generation, shape optimization, and computational fluid dynamics

Shape optimization with a level set based mesh evolution method

“…In a second step, this intermediate mesh T temp is remeshed into a high-quality mesh T Ω , which is a fine representative of Ω. For the many technical details the reader is refered to [28,29]. Remark that these meshing operations have many other applications, as described in [29].…”

“…For the many technical details the reader is refered to [28,29]. Remark that these meshing operations have many other applications, as described in [29].…”

“…To achieve this, we rely on the four usual local mesh modification operators (see [40], or [28,29] for more details around the actual implementation), which are briefly described hereafter (see Figure 3). Note that, in our application, each one of them exists under two different forms depending on whether it is applied to a surface configuration, or to a completely internal one.…”

“…However, it becomes much more combinatorial and tedious in 3d (and the designation 'edge swap' may seem a little be improper). In this case, the whole 'shell' of tetrahedra sharing the edge pq has to be reconnected, and there are many possible ways to do so, not all of them resulting in a valid configuration of suitable quality; see Figure 4 and [34,42], or [29] for further details. In either case, this operator too should be carefully controlled, as it may invalidate T temp , or degrade the geometric features of ∂Ω.…”

“…The proposed approach relies on meshing algorithms which may find other applications (like e.g. in the field of fluid-structure interaction) and have therefore been published in [29], independently from a particular physical or mechanical setting. The novelty of the present paper lies therefore in the interaction of these meshing devices with other level set type algorithms and shape optimization techniques.…”