# Mapping Based Quality Metrics for Mesh Deformation Algorithms Using Radial Basis Functions
## 1 Introduction
Various RBFs are compared in Reference [22] and RBFs based on thin plate splines (TPS) shows good application prospects in mesh deformation, which is chosen as the mesh deformation method in this paper. Another aspect in research of the mesh deformation algorithm is how to evaluate the quality of deformed mesh. To compare the different meshes quality after deformation, mesh quality metrics are introduced. Traditional mesh quality metrics are based on a set of Jacobian matrices, which contain information on basic element qualities such as size, orientation, shape, and skewness [22]. We assume that the initial mesh is created with an optimal quality and the elements quality should be changed as little as possible after deformation. Both the angles and volume of the elements should be preserved. Several parametric evaluation criteria have been established in the past. The relative size metrics measuring the change in element size and skew metric measuring the skewness and distortion are often used in mesh quality evaluation. [PERSON] proposed other typical mesh quality metrics [31] in which both the area and the angles of the element are measured. In commercial software like Pointwise [32], area ratio, aspect ratio, equiaarea skewness, and equi-angle skewness are often used in mesh quality evaluation. The equiarea skewness are represented as a ratio of the mesh element area to the optimum cell area. It only applies to triangles and tetrahedral elements. The equi-angle skewness is represented as the maximum ratio of the element included angle-to-angle of an equilateral element. The angle skewness applies to all element types and is available for domains and blocks. Existing mesh quality metrics are based on specific coordinates and connectivity information of the mesh, which we called mesh-based metrics. It will lead to higher computational costs. On the other side, no attention has been paid to the quality of the mesh deformation method, which can reveal the characteristics of the deformation algorithm itself. Therefore, an efficient method that can give a criterion of the ability of the mesh deformation algorithm should be introduced. The objective of this paper is to develop a quality metric method for unstructured and structured mesh. The RBFs mesh deformation method based on generalized TPS is introduced, and is evaluated by this metric evaluating method. Different from traditional mesh-based metrics, this quality metric method is calculated based on the mapping information between the initial and deformed mesh, so it is named mapping-based metrics in this paper. The mapping-based metrics are based on the deformation principle of continuum mechanics [33; 34], which is especially suitable for nonlinear large deformation problems. Without the requirement of element connectivity information, this method possesses high efficiency. The rest of this paper is allocated as follows. Section 2 introduces the methodology of RBFs' mesh deformation method based on generalized TPS. Section 3 demonstrates the mapping-based metrics we proposed in detail. In Section 4, three 2-D mesh cases with structured and unstructured mesh are performed and the numerical results of the mapping-based metrics are presented. A comparison between mapping-based metrics, mesh-based metrics, and a geometric calculation directly are also been shown to illustrate the accuracy of the metrics evaluation method proposed in this paper. Finally, conclusion are described in Section 5.