A construction of anisotropic meshes based on quasi-conformal mapping

We propose a novel method which is able to generate planar anisotropic meshes according to a given metric tensor. It is different from the classical metric-based or high dimensional embedding mesh adaptation methods. Our method resolves the anisotropy of a metric tensor field by finding a corresponding Euclidean metric in the plane. This is achieved via quasi-conformal mapping between two Riemannian surfaces. Given a planar source domain together with a metric tensor defined on it, and a target domain with a Euclidean metric, there exists a quasi-conformal mapping between them, such that the mapping is conformal with respect to the metric tensor on the source and the Euclidean metric on the target. A discrete quasi-conformal mapping can be constructed by solving the Beltrami equation on a Riemannian manifold. Our method first computes the Beltrami coefficient which is a complex-valued function from the given metric tensor. It then uses discrete Yamabe flow to construct this quasi-conformal mapping. We then construct an isotropic triangulation on the target domain. The constructed mesh is mapped back to the source domain by the inverse of the quasi-conformal mapping to obtain an anisotropic mesh of the original domain. This method has solid theoretical foundation. It guarantees the correctness for all symmetric positive definite metric tensors. We show experimental results on function interpolation problems to illustrate both of the features and limitations of this method.