Ocean Engineering 217 (2020) 107870 Available online 21 October 2020 0029-8018/© 2020 Elsevier Ltd. All rights reserved. Numerical study on the second-order hydrodynamic force and response of an elastic body – In bichromatic waves Kyeonguk Heo a, b, * , Masashi Kashiwagi a a Osaka University, Japan b University of Ulsan, South Korea A R T I C L E INFO Keywords: Hydroelasticity Quadratic transfer function (QTF) Higher-order boundary element method (HOBEM) Generalized mode method Perturbation method Springing ABSTRACT The second-order hydrodynamic force and response of an elastic body in bichromatic waves are studied by using higher-order boundary element method (HOBEM). To solve the boundary value problem, the free-surface wave Green function is adopted in the boundary integral equation and the discretization is conducted by quadratic shape function. In the boundary condition, the normal vector variation on the body surface is re-defned to consider both rigid and elastic body motions. It is invoked especially for the derivation of second-order body boundary condition and several generalized forces in zero forward speed condition. In the second-order quan- tities, the second-order velocity potential force is obtained by using indirect formulation in the generalized mode and the contribution of each component is investigated. The validation for the second-order forces is frst con- ducted by comparing with published data. The excitation force on the rigid body motion of a hemisphere is calculated and showed a good agreement with other semi-analytic and numerical results. Using simplifed structural model, several vertical bending modes of an elastic hemisphere are considered as a numerical test. Second-order hydrodynamic force and response for the two-node vertical bending are obtained and properties of second-order quantities are confrmed. 1. Introduction The investigation on the hydroelastic response of large vessels has been conducted by many organizations. As the result, it has been confrmed that the effect of high frequency vibration should be considered in the design of the hull girder of ships (Kaminisk and Rigo, 2018). Slamming induced whipping and springing are representative phenomena among hydroelastic responses. Whipping is transient phe- nomenon induced by impact loads. On the other hands, the springing is a resonant vibration excited by harmonic incoming waves. Springing has been investigated due to the effect on the fatigue strength of a ship. It could not only happen at natural frequency of vi- bration but also 1/n of it by non-linear wave components. Several ex- periments have shown the existence of higher-order springing in waves and it has been confrmed at various types of a ship (e.g., Storhaug, 2007; Miyake et al., 2008; Hong and Kim, 2014). The non-linear fuid-- structure interaction should be considered for the analysis of non-linear springing but the numerical method has not been fully developed and relatively less conducted. Numerical study on the springing of a ship has been one of classical topics in naval architecture. Bishop and Price (1979) calculated linear hydrodynamic force and response of the elastic vessels by using strip theory and fnite element method. Jensen and Pedersen (1978) sug- gested quadratic strip theory which could consider several second-order hydrodynamic forces. Non-linear wave loads and responses are also calculated by using Timoshenko beam theory. Using time-domain simulation, the strip theory is also applied to the analysis of several non-linear phenomena (e.g., Wu and Moan, 1996; Xia and Wang, 1997; Wu and Hermundstad, 2002). In particular, nonlinear Froude-Krylov and hydrostatic restoring forces incorporated with the contribution from the instantaneous wetted surfaces have been considered. Recently, various kinds of three dimensional codes have been developed and used for the analysis of hydroelastic response of a ship (e.g., Malenica et al., 2003; Iijima et al., 2008; Kim et al., 2009; Kashiwagi et al., 2015). However, these numerical studies are focusing on the linear quantities and non-linear analysis has been carried out with some constraints and the hydrodynamic component generally has been ignored in the non-linear analysis. The non-linear wave-body interaction has been considered mainly for a stationary marine structure which has only rigid body motion * Corresponding author. University of Ulsan, Dept. of Naval Architecture & Ocean Engineering, South Korea. E-mail address: hhggoo@ulsan.ac.kr (K. Heo). Contents lists available at ScienceDirect Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng https://doi.org/10.1016/j.oceaneng.2020.107870 Received 8 January 2020; Received in revised form 15 June 2020; Accepted 26 July 2020