Section: New Results

Metric field interpolation

Participants : Patrick Laug [correspondant] , Houman Borouchaki.

To solve a physical problem formulated in terms of partial differential equations, the finite element method is generally used, based on a spatial discretization, or mesh, of the domain studied. Local adaptations of meshes to the behavior of the physical phenomena can improve the accuracy to the computed solutions, and in particular it is possible to capture high variations of the solution in specific areas while maintaining a reasonable number of degrees of freedom. In an initial phase, a mesh of the domain is built by using any particular method, then a first calculation of the solution of the problem is made. After choosing an appropriate criterion (Hessian and/or gradient of the solution, error estimate in general), areas that must be adapted by refinement or coarsening are detected in the initial mesh, and a new mesh is generated which is better adapted to the problem. This process is iterated until obtaining a mesh which satisfies the specified criterion (for which the finite element error is bounded by a specified threshold).

In practice, via an a posteriori analysis of the finite element error, a discrete map of sizes or metrics is set to the mesh vertices. This discrete size or metric field is made continuous by interpolating on the mesh, and the new mesh is generated according to this new field. In general, for a given point of the domain, a mesh element containing this point is found, and the interpolation of the size or metric field at this point is made from the sizes or metrics associated with the vertices of the containing element. For a scalar size field, the interpolation is straightforward by considering any interpolation scheme (for instance linear or geometric). On the other hand, the same scheme cannot be applied in the case of metrics representing a tensor field. However, several approaches have been proposed based on the link between a size and the corresponding metric and, in most cases, the interpolation scheme for sizes is applied to a power or the logarithm of the metrics. In particular, as a size $h$ is represented by the isotropic metric $ℳ=\frac{1}{h^2}ℐ$, where $ℐ$ is the identity matrix, a possible link consists in approximating the size by ${ℳ}^{-\frac{1}{2}}$, then applying the size interpolation scheme to this new metric and finally recovering the interpolated metric. These schemes are still an approximation and require the calculation of the eigenvalues of $ℳ$ which is generally costly.

In this work, a new method for interpolating discrete metric fields is proposed. It is based on the “natural decomposition” of metrics using the LU factorization. With this decomposition, for each metric, the natural sizes along particular (or natural) directions can be retrieved, thus the size interpolation scheme can be applied to both natural directions and sizes, and the interpolation on the metrics is obtained. The proposed method is faster than those mentioned above and provides a continuous metric field with low variations. Some numerical examples illustrate our methodology.