# Time-Domain Aeroelasticity Analysis by a Tightly Coupled Fluid-Structure Interaction Methodology
## 1 Introduction  
Aeroelasticity phenomenon, which arises from the interaction of the aerodynamic, elastic and inertial forces, may have detrimental effect on the reliability, cost and safety of aircraft. However, the complicated interaction mechanism and multidisciplinary requirement of aeroelasticity problem make it difficult to be analyzed with efficiency and accuracy. With the development of high performance computers, coupled fluid-structure interaction (FSI) methodology is made available for aeroelasticity precondition and efforts have been made to develop the precondition capability of these methods. Generally, coupled FSI methodology can be divided into three types: the fluid and structural models are fully coupled, loosely coupled or tightly coupled. In the fully coupled method, the fluid and structural equations are integrated in both Eulerian and Lagrangian systems and solved in time simultaneously. The fully coupled method has a requirement in limitations on grid size. Moreover, this leads to the matrices being orders of magnitudes stiffer for structure system than fluid system, which makes it virtually impossible to solve the equations for large-scale problems [1]. For the loosely and tightly coupled methods, they use the existing solvers for computational fluid dynamics (CFD) and computational structure dynamics (CSD) [2, 3]. Because the aerodynamic and structure domains are discretized in a different manner, data communications at the fluid-structure interface are required. The difference between the loosely and tightly coupled methods in essence lies in the data communication times. The loosely coupled strategy transfers the aerodynamic forces and structural displacements between fluid and structure systems only once in a physical time step, so it is usually only first-order time accuracy. For tightly coupled methods, data communications are performed several times within each physical time step and second-order time accuracy can be obtained. The linear or nonlinear types of models can be coupled together in FSI methodology. Current industrial practice relies heavily on linear methods, which can lead to conservative design and flight envelope restrictions [4]. However, FSI methodology with linear aerodynamic model cannot reveal the nonlinear aeroelastic effects, such as the transonic flutter dip and the limit cycle oscillations. During the analysis of aeroelasticity problem, flexible structure deforms under the influence of air-loads, so it's necessary to update the fluid volume mesh according to the structural displacement. In the past decades, several mesh deformation methods have been investigated and successfully applied to aeroelastic analysis. Recently, extensive attention has been paid to the mesh deformation method based on radial basis functions (RBF), which can provide high-quality mesh deformation and perform reliable capability for arbitrary topology. The efficiency of RBF method is related to the number of control points on the wall surfaces of fluid mesh, and a reduced control points method proposed by [PERSON] [5] greatly improves the efficiency, which makes it possible to deal with large meshes. Unfortunately, the gain in deformation efficiency may cause the problem of deformation error on deformed surfaces and inverted boundary layer meshes near wall. Computational fluid dynamics (CFD) and computational structural dynamics (CSD) based FSI methodology become attractive for aeroelasticity analysis in the time domain. As CFD and computer technology progress, higher-order methods based on RANS coupled with CSD are preferable because they are able to model more accurately transonic, nonlinear, and viscous effects, compared with linear aerodynamic models. However, the traditional compressible RANS solver may lead to low precision and low efficiency solutions at low Mach number, on account of the overlarge numerical dissipation and condition number [6].