## 1 Introduction  
The structural instability that was induced by pressure pulsation and external force in nuclear power reactors, fuel pumps, air conditioning systems, and heat exchanger tubes has received increasing attention in recent years [1; 2; 3; 4; 5; 6]. Nuclear fuel rod bundles that were exposed to coolant flow should be protected from structural failure [7]. The vibrations are affected by the instability of the fluid flow through the pipe system. It is necessary to analyze the motion of the nuclear rod bundles induced by fluid flow and design the support system that has large damping and stiffness to protect the nuclear rod bundles subjected to instability [8]. To simulate the behavior of nuclear rod bundles induced by fluid flow, coupling of the fluid flow and structural vibration is required. The fluid-structure interaction analysis was performed while using the following procedures. Structural deformation occurs due to fluid force; as the structure deforms, the pressure magnitude of the surrounding fluid and its direction to the flow change, and this induces changes in the pressure fields. Consequently, the changed pressure fields act on the structure. A numerical analysis in the time domain was developed to analyze the interaction between the fluid and structure. The development of a numerical solution in the frequency domain is required for parametric studies to analyze the damping performance because the transient analysis requires a significant amount of computation time for accuracy. Fluid-structure interaction was solved using the finite element method [9; 10; 11]. The structural vibration of a liquid-filled pipe system was analyzed while considering the extended water hammer equations using a polynomial function [9; 10]. The shape function using non-uniform rational B-splines was used to increase the accuracy of the numerical results [11]. Because conventional finite element models are expressed by frequency-independent polynomial shape functions, the configuration of the model influences the accuracy in the high frequency modes of interest [12; 13; 14]. SEM using a shape function was developed to increase the accuracy of the numerical analysis [15; 16; 17; 18; 19]. The structural vibration of the pipelines conveying internal steady flow was simulated after considering the interaction with longitudinal and flexural vibrations. The momentum conservation for fluid flow in a pipe system and the governing equation of the vibration for pipelines conveying internal steady flow was analyzed using SEM [15]. Additionally, the flexural vibration of the pipeline considered the unsteady flow [16]. The behavior of blood flow, such as the viscoelastic nature of blood vessels, was analyzed using SEM [17]. The blood flow rate and pressure calculated by SEM was accurately more than those of the finite element method. In petroleum engineering applications, the effects on axial force, annulus fluid properties, and geometry for a braced well have been thoroughly investigated [18; 19]. Viscous coupling for each structure in the fluid medium was not considered when the structural vibration of nuclear rod bundles interacting with the surrounding fluid was simulated. There has been limited research regarding the numerical analysis of the structural behavior of nuclear rod bundles that were subjected to viscous fluid for the consideration of the supporting system. It is difficult to consider the viscous fluid coupling effects, because CFD analysis requires a large amount of computation time for accuracy and difficult numerical methods. Most studies focused on the simulation without the influence of hydrodynamic fluid coupling of each structure due to the computation time. The development of a new numerical method for calculating the vibration of nuclear rod bundles in viscous fluid for the design of support systems is necessary. This study simulated the flexural vibration of the nuclear rod bundles interacting with viscous fluid using SEM based on the wave approach.