## 3 Treatment of Curved Boundaries
### High-Order Re-Mesh  
It is recognized that the order of the DG method is limited by the order of the boundary representation [25]. Thus, it is necessary to have a proper treatment of the curved boundary. Here, a practical approach is proposed through high-order re-meshing in view of easy implementation and curved-mesh quality. The approach is NURBS surface-based, and a NURBS surface is enhanced with a RBF interpolation in the high-order re-mesh procedure to have an appropriate treatment of curved boundaries. Specifically, NURBS-based surface representations are firstly generated based on the initial surface elements of the curved boundaries (often available in AutoCAD shapes) (see Figure 2a), and then used to create high-order-required control points on the curved surfaces or edges (see Figure 2b-d). A key to form an appropriate set of high-order-required control points lies in adding new control points, which can be easily operated from the initial surface elements. Take the second order approximation as an example; the midpoints of the vertexes of the initial surface elements are usually selected to be control points, as illustrated by the red points in Figure 2c. Such newly added control points are certainly required to be mapped to the curved NURBS surface generated, which can be done by solving the problem associated with the rule of shortest displacement [11], which reads:  
\[\underset{a,b}{min}\|\mathbf{x}_{new}-\mathbf{S}(a,b)\|^{2},\hskip 14.226378 pts.t\ 0\leq a,b\leq 1,\]  
where \(\mathbf{S}(a,b)\) denotes a point on NURBS surface and \(\mathbf{x}_{new}\) is a newly added control point. Keep in mind that with the NURBS surface generated, the gradient at the point \(\mathbf{S}(a,b)\) can be directly computed. Thus, the above problem can be efficiently solved by gradient-based  
Figure 1: Tetrahedron element in reference space (**left**) and physical space (**right**). The bold integer numbers are the indexes of the corresponding mapping points. optimization, such as the Gauss-Newton algorithm, to obtain the required new control points (see red points in Figure 2d). However, as pointed out in References [9; 31], such kind mapping may cause low quality or even invalid elements in the vicinity of the curved boundary with high curvature (see intersected surfaces existed at the vertex B of the element in Figure 3b). Such a problem is reported to be ameliorated by improving the quality of the mesh near the boundary through adopting optimization processes [32]. An alternative approach used widely is to propagate the boundary movement to the interior through mesh-deforming techniques, including the RBF interpolation method [31], the linear elasticity method [9; 33], etc. Among them, the RBF interpolation method has the ability to achieve high-quality meshes at a relatively low computational costs [31], and hence, whether the invalid elements exist or not, the RBF interpolation is applied as the last step of the present approach to retain the element quality (see Figure 3c). In order to assure the quality of the resulting mesh, an additional step of quality check is usually required. But it is learned from the test cases presented (see Section 5) that it is not necessary after having the RBF interpolation. Figure 2: Illustration of the high-order surface re-mesh. (**a**) Surface of curved boundary; (**b**) initial boundary mesh; (**c**) midpoints introduced; and (**d**) high order curved surface. The black points are the vertexes of the mesh, and the red points are the newly added control points.