High-order 2D mesh curving methods with a piecewise linear target and application to Helmholtz problems

High-order simulation techniques typically require high-quality curvilinear meshes. In most cases, mesh curving methods assume that the exact geometry is known. However, in some situations only a fine linear FEM mesh is available and the connection to the CAD geometry is lost. In other applications, the geometry may be represented as a set of scanned points. In this paper, two curving methods are described that take a piecewise fine linear mesh as input: a least squares approach and a continuous optimization in the H1-seminorm. Hierarchic, modal shape functions are used as basis for the geometric approximation. This approach allows to create very high-order curvilinear meshes efficiently (q>4) without having to optimize the location of non-vertex nodes. The methods are compared on two test geometries and then used to solve a Helmholtz problem at various input frequencies. Finally, the main steps for the extension to 3D are outlined.