Comparison of 2D boundary curving methods with modal shape functions and a piecewise linear target mesh

It is well known that high-order simulation techniques demand an accurate geometric representation and a coarse mesh. To fulfill both requirements, curved meshes are generated. In most cases, curving methods assume that the exact geometry is known. But it can be useful to develop curving methods with only a limited knowledge of the target geometry. In this paper, three curving methods are described that take a piecewise fine linear mesh as input: a least squares approach, a direct optimisation in the H1-seminorm, and a H1-seminorm optimisation in a reference space. Hierarchic, modal shape functions are used as basis for the geometric approximation. The methods are compared on two test geometries, a unit circle and a distorted ellipse. Considering both test cases, the direct optimisation approach shows the most promising results. Finally, the main steps for the extension to 3D are outlined.