## 4 Concluding Remarks  
A new and comprehensive analytical framework for the coupled flight dynamics and nonlinear aeroelasticity of highly flexible aircrafts is presented. The theoretical methodology is based on the statics and dynamics of a 3D co-rotational geometrically nonlinear beam model for elastic deformations and rigid-body motions, with the original equations for nodal inertial force vector of element modified to enhance its versatility. The coupling of axial, bending and torsional effects is considered by adding the coupling terms into the element stiffness and mass matrices of traditional Euler-Bernoulli beam in order to capture the most relevant characteristics of real wing structure. The Peters' finite-state unsteady aerodynamic model is coupled with the structural model to simulate the unsteady aerodynamics in flight. The flight dynamics are coupled with nonlinear aeroelasticity in a more intuitive and straightforward way by re-defining the ground, body and airflow axes analogously to the global and elemental reference frames for structural dynamics at the local element level, followed by the definitions of local Euler angles and airflow angles of each element. The scheme of mixed end-point and mid-point time-marching algorithm is proposed and applied into the implicit predictor-corrector integration for the first time, by which the computation efficiency, accuracy and stability are improved.  
Two representative numerical benchmarks, especially a highly flexible flying-wing aircraft, are presented and analyzed with the trimmed states calculated and dynamic responses after flap perturbations computed numerically. The results show that with the increase of the concentrated payload, large deformations occur in the highly flexible wings, resulting in significant changes in trim state and the reduction in phugoid mode stability. The effects of spanwise mass distribution on trim and longitudinal stability are further studied, indicating that, for highly flexible aircrafts, the longitudinal performances will be affected by the spanwise mass distribution; span-loaded aircrafts with distributed payloads  
Figure 13: Altitude at mid-span with the same weight but different payload distributions.  
have much smaller deformations and higher longitudinal stabilities than the center-loaded ones with concentrated payloads.  
The present framework accounts for realistic design elements including arbitrary structural shapes and properties, concentrated masses, multiple thrusts and control surfaces and unsteady aerodynamics. It is adaptive and easy to accommodate with other models, and can be used for complete aircraft analyses in the conceptual and preliminary design phases, including the quick linear and nonlinear trim, aerodynamic load estimation, stability assessment, as well as linear and nonlinear time-domain response simulations and further control evaluations. Comparisons with published literature show the validity and reliability of the present framework. The investigations on lateral and directional characteristics and symmetric and unsymmetrical gust responses with dynamic stall effects of large-scale flexible aircrafts based on the present framework will be the focus of the future work.  
Conceptualization, Z.Z. and C.Z.; methodology, C.Z. and L.Q.; software, C.Z. and L.Q.; validation, C.Z. and L.Q.; formal analysis, C.Z.; resources, Z.Z. and X.Z.; writing--original draft preparation, C.Z.; writing--review and editing, Z.Z.; supervision, Z.Z.; project administration, Z.Z. and X.Z.; and funding acquisition, Z.Z. and X.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Equipment Advanced Research Project grant number 41411020401, the Special Project of Civil Aircraft grant number MJ-2015-F-009, the Key Project of Research and Development of Shaanxi Province grant number 2018 ZDCXL-GY-03-04, and the Innovation Project of Key Institutes grant number TC2018 DYDS24.  
The authors declare no conflict of interest.