01 March 2022
16:15
Louvain-la-Neuve
Auditorium BARB92
For the degree of Doctor of Engineering Sciences and Technology
This thesis aims at curvilinear mesh generation and adaptation, we start from summary ideally the research project as a simple yet fundamental question: assume a unit square $$\Omega = \{(x^1,x^2) \in [0,1] \times [0,1]\}$$ and a smooth function $f(x^1,x^2)$ defined on the square, and consider a mesh $\mathcal T$ made of $P^2$ triangles that exactly covers the square, how can we compute the mesh $\mathcal T$ that minimizes the interpolation error $\| f - \Pi f\|_\Omega$. Here, $\Pi$ is the nodal interpolation of $f$ on the mesh \cite{ern}.
We state the problem as to build a unit curvilinear mesh, i.e. build a mesh with unit edge lengths that are possibly curvilinear. We solve the problem in three stages of increasing complexity:
- input a unit square and a metric field $g(x,y)$, to build a unit curvilinear mesh that is possibly anisotropic;
- input a function $f(x,y)$, to build an anisotropic curvilinear mesh that minimizes the approximation error;
- input a high order finite element solution, to build an anisotropic curvilinear adapted mesh.
We propose \emph{a new framework} of curvilinear mesh generation and adaptation: metric field construction, generation of points (point sampling on the boundary and point sampling in the domain), straight-sided mesh generation and adaptation (triangulation and straight-sided edges swap), curvilinear mesh generation and adaptation (straight-sided edges curving, curvilinear edges swap, and Curvilinear Small Polygon Reconnection).
A unit curvilinear mesh containing only valid ``Geodesic Delaunay triangles'' is obtained this way.
In this approach, the curvature is not only used to match curved boundaries but also to capture features of the interpolated solutions, and it results in meshes that would not have been achievable by simply curving \emph{a posteriori} a straight-sided mesh. A number of application examples are presented in order to demonstrate the capabilities of the mesh adaptation procedure.
Jury members :
- Prof. Jean-François REMACLE (UCLouvain, Belgium), supervisor
- Prof. Renaud RONSSE (UCLouvai, Belgium), chairperson
- Dr. Jonathan Lambrecht, (UCLouvain, Belgium)
- Prof. Vincent Legat (UCLouvain, Belgium)
- Dr. Thomas TOULORGE (Cenaero, Belgium)
- Prof. Thierry COUPEZ (Mines-ParisTch, France)