# Mapping Based Quality Metrics for Mesh Deformation Algorithms Using Radial Basis Functions
## 1 Introduction  
For computational fluid dynamics (CFD) computations and fluid-structure interaction (FSI) computations, a flows simulation with a moving boundary are often encountered at various important problems such as stability analysis of bridges, aircraft optimizations, flutter calculations, and bioengineering simulations [1, 2, 3, 4, 5]. The mesh deformation algorithm is required to adapt the CFD computational mesh with the motion of the boundary. With regards to the problems with large deformation of boundary, performance of mesh deformation methods often becomes a limiting factor to CFD simulations accuracy [6]. Meanwhile, the computational mesh needs to be updated repeatedly during an unsteady simulation, bringing large computational cost. Accuracy and computational efficiency of mesh deformation algorithm is crucial for the previously mentioned problem [7, 8]. Variety of mesh deformation methods have been developed in literature in terms of accuracy, efficiency, and robustness [9]. These methods can be classified into two main categories: connectivity-based methods and node-based method. The spring analogy method [10, 11] is a typical one of connectivity methods. The idea of the spring analogy method is to create a network of springs connecting all nodes in the mesh and the stiffness of springs is inversely proportional to the edge length [12]. The whole mesh can be viewed as a spring system. It has been successfully applied to many unsteady and optimization problems. However, this method is relatively expensive in computational cost due to the necessity of total mesh connectivity information, and it may lead to an invalid mesh for large deformation [13, 14]. Although several strategies have been introduced to improve the ability of the spring analogy method [15, 16, 17], it is still difficult to guarantee both highefficiency and applicability. In the other classification of the node-based method, each node can be modified independently to its adjacent nodes, and then the deformation of mesh can be adjusted indiscriminately [12]. This method allows for large deformation and does not require connectivity information between mesh nodes. The node-based method is useful for unstructured mesh due to those characteristics. Interpolation techniques are often introduced in a node-based method. [PERSON] proposed the Inverse Distance Weighed (IDW) interpolation method [18] in which the deformation of internal nodes depends on the weighted-averaging of the displacement of boundary nodes. Transfinite interpolation (TFI) developed by [PERSON] [19] is a fast scheme used in a node-based method. Despite its efficiency, the deformed mesh quality is not guaranteed, especially the orthogonality of the mesh near the boundary. [PERSON] proposed a coarse Delaunay graph with barycentric interpolation to propagate deformation from boundary nodes to internal ones [20]. In addition, [PERSON] et al. decreased computational cost of the previously mentioned method via tree-code optimization [21]. Although the common node-based methods increase the efficiency of generation of adapted mesh, it does not perform well with complex mesh and is not capable of preserving the mesh orthogonality. A well-established method based on radial basis functions (RBFs) proposed by [PERSON] [22] is a desirable approach for both structured and unstructured meshes. It can generally preserve the mesh quality with reasonable orthogonality near the boundary in mesh deformation [23]. The research shows that different RBFs introduced could result in different results [22; 24; 25]. [PERSON] et al. describe suggestions for choosing RBFs, and demonstrate that the computational cost will still grow as the mesh scale increases, even choosing the appropriate functions [1]. Considering the high computational cost, several approaches are proposed with greedy algorithm [21; 26; 27] or reducing the interpolated nodes [28; 29; 30]. The improved RBFs method has better computational efficiency and can also preserve the mesh quality well.