Anisotropic mesh adaptivity in CFD

MOX 29

Anisotropic mesh adaptivity in CFD

Tuesday 21st October 2003

Micheletti, Stefano; Perotto, Simona

Why anisotropy? The straightforward answer to this question could be: because anisotropy is everywhere! Actually, when numerically solving a problem in Computational Fluid Dynamics (CFD), or in some other areas, there are many instances where the solution shows directional features such as great variations along certain directions with less significant changes along other ones, e.g. boundary and internal layers, singularities or shocks. A correct orientation and deformation of the mesh elements (longest edges parallel to, e.g. the boundary layers) yields a great reduction of the number of triangles. Moreover, in the anisotropic case the layers are captured more sharply. The leitmotiv of an anisotropic analysis can be stated as: for a fixed solution accuracy, reduced the number of degrees of freedom involved in the approximation of the problem at hand by better orienting the mesh elements according to some suitable features of the solution, or vice versa, given a contraint on the number of elements, find the mesh maximizing the accuracy of the numerical solution. However, things may not be so straightforward. While anisotropy is proved to superior in terms of effectiveness for the most accurate computations in many cases, yet, there are some instances where a structured Adaptive Mesh Refinement (AMR) procedure turns out to be more simple to carry out, especially in view of an implementation in a parallel environment. Moreover, in the unstructured case, the main drawback of the anisotropic approach compared to the anisotropic one, is the more complex analysis required to fully describe the element dimensions and orientation. Though, this heavier burden is the strenght of the method. The outline of the article is the following. In the Sect. 2 we introduce the anisotropic framework by recalling some anisotropic interpolation error estimates, representing the main tool used in the a posteriori error analysis addressed in Sect. 3. This analysis is discussed in the case of a general differential operator, moving from the adjoint theory for goal-oriented error control, and it is then detailed for the advection-diffusion-reaction and the Stokes problems. Finally, in Sect. 4 the effectiveness of the anisotropic philosophy is assessed on some numerical test cases.

This report, or a modified version of it, has been also submitted to, or published on
Micheletti, S.; Perotto, S., Anisotropic mesh adaptivity in CFD, in Adaptive Mesh Refinement - Theory and Applications. Series: Lecture Notes in Computational Science and Engineering, Vol. 41, Springer-Verlag, T. Plewa, T. Linde, V.G. Weirs Eds. (2005), 1