Some computational aspects on generating numerical grids

Abstract

An elliptic method based on the application of both Beltrami and diffusion equations in monitor metrics is developed for generating adaptive and/or magnetic vector field-aligned grids. Some results of numerical experiments for constructing two-and three-dimensional grids are demonstrated.The elliptic systems of inverted Beltrami and diffusion equations in monitor metrics allow one to generate both structured and unstructured grids in domains or on surfaces in a unified manner, regardless of their dimension and geometry features. In particular, these systems can be applied to produce grids in spatial blocks by means of the successive generation of grids on curvilinear edges, faces, and parallelepipeds, using the solution at a step i n as the Dirichlet boundary condition for the following step i·1 n. Thus, both the interior and the boundary grid points of a domain or a surface can be calculated by the similar elliptic solver. In the same way the grid properties required in practical applications such as nonsingularity, adaptivity, vector-field alignment, and smooth block grid matching are readily realized by the monitor metrics. These features are indispensable for developing comprehensive automatic grid-generating codes. The paper is concerned with some aspects related to the development of grid generation techniques and codes based on the solution of the inverted Beltrami and diffusion equations.

Streamline integration as a method for two-dimensional elliptic grid generation

“…All in all, we show that we are able to construct various elliptic grids by the method of streamline integration combined with the solution of a suitably chosen elliptic equation. Compared to the TTM and related methods [4,6,7] we significantly simplify the construction and accuracy of the method. Compared to the flux-aligned grids suggested in the literature on magnetic fusion problems [16] we improve the distribution of cells across the domain and enable boundary orthogonality.…”

“…We follow Reference [3,6,7] and replace the canonical metric tensor g by a specifically tailored tensor G that takes the form…”

“…Both result in cells of similar size. The difference between these grids is the degree of flux alignment, which is clearly superior in the monitor grid in the region close to the X-point (Reference [6] provides a more quantitative investigation of the degree of alignment).…”

“…boundary integral element methods, however their implementation can be tricky. Among further improvements to the original TTM method are grid adaption methods and the monitor metric approach [6,7,3]. Both of these approaches modify the elliptic equation used to construct the coordinate system to a more suitable form.…”

Numerical simulation of four-field extended magnetohydrodynamics in dynamically adaptive curvilinear coordinates via Newton–Krylov–Schwarz

“…Various algebraic, differential and variational methods of constructing adaptive grids are described in [15][16][17][18][19]. There are three commonly identified primary types of mesh refinement: r-refinement, h-refinement, and p-refinement together with their variations and hybrids.…”