## 1 Introduction  
The dynamic behavior of slender bodies as aero-engine blades can be characterized simultaneously by large deformation and rubbing phenomena. A typical example is the fan blade of engines with ultra-high by-pass ratios [1]. On one side, large deformation activates distributed geometric nonlinearities [2; 3], while rubbing phenomena introduce localized nonlinearity to the system [4]. These nonlinearities can give rise to complex dynamic behavior, which must be accurately captured during the design process. The Finite Element (FE) method is typically used as an accurate and robust tool for simulating the structural response. However, the nonlinear dynamic analysis of a high-fidelity FE model with a large number of degrees of freedom can be prohibitively expensive. Model Order Reduction (MOR) techniques are employed to reduce this computing cost. In the literature, numerous reduction techniques have been developed to address either geometric or contact nonlinearity separately. Given the distinct nature of these two nonlinearities, the developed reduction techniques are essentially different. This paper focuses on a reduced model capable of capturing the nonlinear dynamics of a structure with both types of nonlinearity. Two categories of reduction method can be considered: linear methods and nonlinear methods, in the sense that in nonlinear reduction methods, new variables are nonlinearly related to the initial ones, while the linear methods are based on a linear change of basis [5]. Using linear methods, the Reduced Order Models (ROMs) are constructed by the Galerkin projection of the equation of motion on a low-dimensional basis. These techniques are widely used in the literature due to their ease of implementation and the fact that the linear projection preserves the general expression of the equation of motion, allowing the application of the same solution strategies as those used for the full-order model. The linear techniques face two challenges. The first is finding the best reduction basis and the second is computing the generalized nonlinear forces, which are the projection of the nonlinear forces onto the reduction basis. This work aims to address the first problem by providing the best basis in terms of the computing cost and its ability to present the nonlinear dynamics accurately. The approaches to address the second challenge in geometrically nonlinear structures and structures with friction contact are completely different. Geometric nonlinearity is distributed, which means that the degrees of freedom (DoFs) of a significant portion (or potentially the entirety) of the structure are subjected to nonlinear elastic forces and nonlinearly coupled [5]. As a result, the full displacement field is required to compute the elastic forces and a projection of the elastic forces has to be performed during the online stage at each iteration of the governing equations solution. This intrusive computation of nonlinear elastic forces, which involves nonlinear Finite Element formulation, is particularly time-consuming for a large system and results in a significant rise in the online computing cost. Several strategies are addressed to solve this issue [6; 7; 8]. However, these remedies will be successful if the reduced space basis is selected properly. This study uses the intrusive method of computing reduced nonlinear elastic forces and focuses on finding the best reduction basis. Due to the local nature of contact nonlinearity, a widely used method for their order reduction involves partitioning the structure into linear and nonlinear parts [9; 10; 11; 12], where the nonlinear part includes the DoFs of the contact region (contact DoFs). The resulting reduced model keeps the contact DoFs in reduced coordinates. It eliminates the need to project the nonlinear contact forces during the online stage.