Body Boundary Condition Research Articles

Transient Electromagnetic Field Response Analysis Considering Dam Boundary Conditions

Abstract The transient electromagnetic (TEM) method has outstanding advantages in detecting depth and response information and can be used to detect anomalies hidden deep inside dams. However, the current data analysis method based on geological exploration is inapplicable to the interpretation of dam detection data, the main reason being the lack of consideration of the influence of dam boundary conditions on the diffusion of the TEM field. In this work, by establishing a numerical model of a dam for TEM detection, the effects of the dam slope ratio, upstream water body, overburden, and other dam boundary conditions on the TEM field were studied. The results showed that the slope ratio made the propagating TEM field to exhibit a constraining behavior, and a negative deviation could be seen in the attenuation curve, indicating a false high-resistance anomaly. The upstream water body distorted the diffusion of the electromagnetic field, and the low-resistance covering layer caused a low-retardation retention phenomenon in the magnetic field, leading to a positive deviation in the attenuation curve, indicating a false low-resistance anomaly. The influence of the boundary conditions was present throughout the entire observation window in which the detection was performed, and the sensitivity of the magnetic field to these boundary conditions was significantly higher than that of the change rate of the magnetic field. With the increase in the observation time, the main influencing factors were the slope ratio, overburden layer, and upstream water body.

Forced surge motion of a floating bridge pontoon to evaluate the hydrodynamic damping properties

Dedicated model tests were carried out to investigate the hydrodynamic force coefficients in surge for a pontoon that is designed to support a floating bridge. The experiments were carried out by oscillating the pontoon at various Keulegan–Carpenter (KC) numbers and oscillation periods in forced motion. A pontoon designed for the floating bridge in Bjørnarfjorden, Norway, is investigated herein. Several numerical simulations are performed to supplement the experiments. The estimated damping coefficients in surge are of significant importance at the first horizontal resonant mode of the floating bridge, which may be excited due to swell sea in the fjord. The first horizontal resonant mode of the floating bridge is around 15–16 s. The model tests were conducted with and without turbulence triggers. A finding from the model tests were that turbulence triggers did not prove to be necessary to eliminate model scale effects in oscillatory flow.The KC-dependent added mass and damping coefficient in oscillatory flow is investigated. The added mass and damping coefficients are clearly KC-dependent. The wave radiation damping in surge is small at the tested oscillation periods, and the damping in surge is dominated by the damping due to flow separation.Numerical simulations using a hybrid potential flow-CFD solver with linear free-surface conditions, PVC3D, are carried out with both linear and nonlinear body-boundary conditions. The simulations with nonlinear body-boundary conditions predict the hydrodynamic force coefficients with reasonable accuracy, whilst the simulations with linear body-boundary conditions fail to predict the hydrodynamic coefficients with the desired accuracy — both used to investigate the viscous damping which arises due to flow separation along the pontoon body.

On the Hinch–Kim dualism between singularity and Faxén operators in the hydromechanics of arbitrary bodies in Stokes flows

We generalize the multipole expansion and the structure of the Faxén operator in Stokes flows obtained for bodies with no-slip to generic boundary conditions, addressing the assumptions under which this generalization is conceivable. We show that a disturbance field generated by a body immersed in an ambient flow can be expressed, independently on the boundary conditions, as a multipole expansion, the coefficients of which are the moments of the volume forces. We find that the dualism between the operator giving the disturbance field of an nth order ambient flow and the nth order Faxén operator, referred to as the Hinch–Kim dualism, holds only if the boundary conditions satisfy a property that we call Boundary-Condition reciprocity (BC-reciprocity). If this property is fulfilled, the Faxén operators can be expressed in terms of the (m, n)th order geometrical moments of the volume forces (defined in the article). In addition, it is shown that in these cases, the hydromechanics of the fluid-body system is completely determined by the entire set of the Faxén operators. Finally, classical boundary conditions of hydrodynamic applications are investigated in light of this property: boundary conditions for rigid bodies, Newtonian drops at the mechanical equilibrium, porous bodies modeled by the Brinkman equations are BC-reciprocal, while deforming linear elastic bodies, deforming Newtonian drops, non-Newtonian drops, and porous bodies modeled by the Darcy equations do not have this property. For Navier-slip boundary conditions on a rigid body, we find the analytical expression for low order Faxén operators. By using these operators, the closed form expressions for the flow past a sphere with arbitrary slip length immersed in shear and quadratic flows are obtained.

Numerical method for whipping response of ultra large container ships under asymmetric slamming in regular waves

In this paper, a 3-D nonlinear time-domain hydroelastic analysis method for ship wave loads considering asymmetric slamming is proposed, and a 3-D Rankine panel method is used to solve the seakeeping problem. To accurately predict the hydroelastic response of the hull girder, the interaction between the hull structure and the flow field is considered in the body boundary condition. The motions and sectional loads of the ship are expressed by the superposition of the modal, which is obtained by modal analysis on a 3-D finite element model of the hull. After calculating asymmetric slamming loads using the Modified Logvinovich Model (MLM), the hydroelastic response of a 21,000-TEU container ship is analyzed, which incorporates the horizontal bending-torsional coupling vibration. Furthermore, the hull motion, slamming force, and section load are analyzed in detail to properly identify the mechanism of the slamming-whipping phenomenon. The results show that asymmetric slamming significantly affects the horizontal-torsional whipping response of the hull, especially when the wave encounter frequency matches the natural frequency.

Effect of Torso Boundary Conditions on Spine Kinematic and Injury Responses in Head-First Impact Assessed with a 50th Percentile Male Human Body Model

<div>Computational and experimental studies have been undertaken to investigate injurious head-first impacts (HFI), which can occur during automotive rollovers. Recent studies assume a torso surrogate mass (TSM) boundary condition, wherein the first or first two thoracic vertebrae are potted and constrained to only move in the vertical loading direction. The TSM boundary condition has not been compared with a full body (FB) model computationally or experimentally for HFI. In this study, the Global Human Body Models Consortium 50th percentile male detailed human body model (M50-O, Version 6.0) was applied to compare the kinematic, kinetic, and injury response of an HFI with a TSM boundary condition (M50-TSM), and a full body boundary condition (M50-FB). Impacts (to M50-TSM and M50-FB) were simulated between the head and a rigid plate using a commercial FE code (LS-DYNA). The impact velocity of 3.1 m/s corresponded to the onset of spinal injury in diving reconstructions, and the impact velocity reported in experiments. The TSM boundary condition was simulated by applying a mass of 16 kg to the first thoracic vertebra (T1), and constraining motion to only the vertical direction. A quantitative comparison of the head and spine impact forces, spine kinematics, and prediction of hard tissue fracture was reported. The M50-TSM model demonstrated a 53.4% lower (straighter) spinal curvature 10 ms after impact, compared to the M50-FB. The lower curvature of the M50-TSM resulted in higher neck loads during that timeframe (2.26 kN M50-TSM, 1.44 kN M50-FB). The resulting hard tissue fracture in M50-TSM was attributed to direct compression at an early time (<5 ms) in the impact, while M50-FB demonstrated compression-extension fractures later (>16 ms) in the simulation. It was concluded that kinematics, kinetics, and injury response differed for the TSM and FB boundary conditions, and therefore these conditions are critical to consider when investigating HFI.</div>

ON THE INFLUENCE OF THE MECHANICAL CHARACTERISTICS OF A THIN ADHESION LAYER ON THE COMPOSITE STRENGTH. PART 1. ELASTIC DEFORMATION

The problem of deformation of a DCB sample, which is a composition of bodies bound by an adhesive layer of finite thickness, is considered. Based on the variational equilibrium equation containing the layer thickness as a linear parameter, a finite element solution of the problem of loading the layer with a normal discontinuity in the plane strain state is constructed. The stresses averaged over the layer thickness are related to the stresses along the layer boundary by the equilibrium equations. The boundary stresses of the layer form the boundary conditions for the mating bodies. In the layer, along with shear stresses, stresses orthogonal to shear are also taken into account. The constitutive relations in the layer are represented in terms of average stresses. With a significant difference in the Young's moduli of the adhesive and mating bodies, the convergence of the value of the J-integral with a decrease in the layer thickness is shown. To find the J-integral, its representation is used as a product of the specific free energy at the end of the layer and its thickness. It has been established that the Poisson's ratio of the bodies affects the value of the J-integral, and the Poisson's ratio of the adhesive layer has almost no effect on the value of the J-integral. Using the theory of plates Mindlin - Reisner at zero Poisson's ratio of the adhesive, an analytical representation of the J-integral is obtained. The representation includes energy terms related to the pull-off stress and the axial stress in the layer. In this case, the term associated with the axial stress in the layer is proportional to the square of the ratio of the Young's moduli of the adhesive layer and the bodies mating with it. From the solution obtained, it follows that the mechanical properties of the adhesive layer with a small thickness compared to bodies do not affect the value of the J-integral if the elastic modulus of the adhesive layer is significantly less than the elastic modulus of the mating bodies. Thus, the use of replacing the adhesive layer with a layer of zero thickness is correct under these restrictions.

An In-Silico Study on the Therapeutic Effect of Low Back Belts: Biomechanical Correlation between Belt Design and Patient Morphology

A comparative study of eight different lumbar belts, which are representative of the French market, was carried out on four typical morphologies of patients to assess their therapeutic effects and identify the correlation between the therapeutic parameters and mechanical ones. Four typical morphologies were chosen among 15 patients that had been chosen for the clinical study: tall-large, small-large, tall-thin, and small-thin. Simplified 3D finite elements (FE) models of the trunk according to each patient’s morphology were used for numerical analyses using Abaqus SimuliaTM. The same material properties of the body structures and boundary conditions were taken for all models to only focus on morphological variations. The material properties of eight lumbar belts were obtained by mechanical testing. The pressure applied by the belt to the trunk was modelled by Laplace’s law. The influences of belt types on typical morphologies were analyzed and synthetized to show which parameters are significant for biomechanical efficacy and attendance to the therapeutic effects. Finally, we found the following belt effects: (i) the lumbar belt is more efficient on the thin morphology than the large one, (ii) all mechanical values checked on the vertebral disks and vertebrae have a strong correlation with the correction of lordosis angle, and (iii) the belt’s global stiffness is an important parameter for generating the pressure applied to the trunk.

Numerical investigation on the water entry of bow-flare section with different bulbous bow shapes

Water entry is one of the important topics in ship and offshore structures. Among the structures, water entry of bow-flare section with bulbous bow could be one of the most difficult problems due to its complex geometry. To investigate the problem, a numerical model which can simulate water entry of complex objects is developed in this paper. In the framework of Cartesian grid, Volume of Fluid method combined with Piecewise Linear Interface Construction (VOF-PLIC) are used to capture free interface. Besides, Volume of Solid based Immersed Boundary (VOS-IB) method are adopted to enforce the body boundary condition. To validate the model, water entry of a wedge and a ship hull section with bulbous bow in calm water are simulated. Time histories of pressure and local force are compared with previous experimental results. Then, a series of simulations are conducted for the water entry of bow-flare sections with different bulbous bows. Three bulbous bow shapes namely ‘Delta’, ‘Oval’ and ‘Nabla’ are considered. The influence of bulbous bow shape on the hydrodynamic forces are analyzed and three peaks are observed. A detailed discussion on the three peaks is performed with special attention paid to the second peak. It is observed that the shape of bulbous bow shape has non-negligible influence on the pressure distribution during the secondary flare impact. Based on the impact characteristics, recommendations for which shape of bulbous bow to use are given.

A Fully Coupled Time-Domain BEM-FEM Method for the Prediction of Symmetric Hydroelastic Responses of Ships with Forward Speed

This paper presents a direct time-domain method for the prediction of symmetric hydroelastic responses of ships progressing with forward speed in small amplitude waves. A transient time-domain free surface Green function is used for the idealisation of the seakeeping problem using an Earth fixed coordinate system. Free surface ship hydrodynamics are idealised in the time domain by a Green function, and forward speed effects are idealised by a space-state model. Modal actions are accounted for by Timoshenko beam structural dynamics. Flexible fluid structure interaction (FFSI) coupling is enabled by a body boundary condition, and a direct integration Newmark- β scheme is used to obtain symmetric dynamic responses. The method is validated against available published numerical and experimental results. A parametric study for different container ship hull forms confirms that (i) forward speed effects should be taken under consideration as far as practically possible and (ii) hull flexibility effects accounting for hull shear deformation and rotary inertia are more notable for slender hull forms.

Numerical modeling of nonstationary hydrodynamic forces and induced motions of a coupled offshore floating installation system

Offshore installations always involve multi-body systems in which the hydrodynamic and mechanical interactions are complicated for the properties keep changing under certain operation procedures. Float-over deck (FOD) installation is a typical example, where the draft of barge continuously varies during the load transfer operation. Numerical analysis of such continuous operation is crucial to identify the extreme responses, but it's difficult to model the nonstationary properties involved in the hydrodynamic loads and motions and contact loads of the system. In the study, a numerical modeling methodology is developed to calculate these nonstationary hydrodynamic forces and induced responses, considering an active update of the hydrodynamic properties with the time varying body boundary conditions. The effects of time varying body boundary conditions on hydrodynamic forces and induced nonstationary motions and contact loads of a FOD installation system is studied. The results indicate that the dynamic analysis of the continuous load transfer operation could be effectively carried out with one update of hydrodynamic forces at the end of the operation. The barge roll motions and its natural period are the most affected. The proposed method could be used in intensive simulation of offshore multi-body systems with obvious draft variation during operations.

RESEARCH ON ANALYTICAL METHOD OF CIRCULAR CYLINDRICAL SCATTERED WAVE PRESSURE SUBJECTED TO UNDERWATER EXPLOSION

In order to improve the anti-blast and protection design of offshore structures such as bridges and wharves, this paper proposes an efficient analytical method for calculating the circular cylindrical scattered wave pressure subjected to underwater explosion, aiming at the common cylindrical structures with constant cross-sections in offshore structures. Based on the theory of diffracted waves, this paper derives the time-domain solution of the scattered wave pressure of the cylindrical structure by the method of separation of variables in the cylindrical coordinate system according to the water body state equation and boundary conditions; The calculation accuracy of the scattered wave pressure time-domain solution proposed in this paper is verified by numerical simulation comparison. The numerical example shows that the calculation errors of the two methods are within the engineering allowable range. Based on the time-domain substructure analysis method that considers the compressibility of water, by comparing the analytical methods proposed in this paper without considering the compressibility of water, the influence of water compressibility on scattered wave pressure is analyzed.

ДИНАМИКА ТЕПЛОВЫХ ПОТЕРЬ КОНВЕКЦИЕЙ И ИЗЛУЧЕНИЕМ СФЕРИЧЕСКОГО АККУМУЛЯТОРА ТЕПЛА СОЛНЕЧНЫХ УСТАНОВОК

There are needed energy (heat) accumulators to increase the efficiency of solar installations, including solar collectors (water heaters, air heaters, dryers). One of the tasks of designing heat accumulators is to ensure its minimal heat loss. The article considers the problem of determining the distribution of temperatures and heat losses by convection and radiation of the heat insulation-accumulating body (water) system for a ball heat accumulator under symmetric boundary conditions. The problem is solved numerically according to the program developed on the basis of the proposed «gap method». (Research purpose) The research purpose is in determining heat losses by convection and radiation of a two-layer ball heat accumulator with symmetric boundary conditions. (Materials and methods) Authors used the Fourier heat equation for spherical bodies. The article presents the determined boundary and initial conditions for bodies and their surfaces. (Results and discussion) The thickness of the insulation and the volume of the heat accumulator affect the dynamics and values of heat loss. The effect of increasing the thickness of the thermal insulation decreases with increasing its thickness, starting with a certain volume of the heat accumulator or with R > 0.3 meters, the heat losses change almost linearly over time. The dynamics of heat loss decreases with increasing shelf life, but the losses remain large. (Conclusions) Authors have developed a method and program for numerical calculation of heat loss and temperature over time in a spherical two-layer heat accumulator with symmetric boundary conditions, taking into account both falling and intrinsic radiation. The proposed method allows to unify the boundary conditions between contacting bodies.

Numerical study on the second-order hydrodynamic force and response of an elastic body – In bichromatic waves

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-defined 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 quantities, 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 first conducted 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 simplified 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 confirmed.

Multi-Person Pose Estimation Using an Orientation and Occlusion Aware Deep Learning Network.

Image based human behavior and activity understanding has been a hot topic in the field of computer vision and multimedia. As an important part, skeleton estimation, which is also called pose estimation, has attracted lots of interests. For pose estimation, most of the deep learning approaches mainly focus on the joint feature. However, the joint feature is not sufficient, especially when the image includes multi-person and the pose is occluded or not fully visible. This paper proposes a novel multi-task framework for the multi-person pose estimation. The proposed framework is developed based on Mask Region-based Convolutional Neural Networks (R-CNN) and extended to integrate the joint feature, body boundary, body orientation and occlusion condition together. In order to further improve the performance of the multi-person pose estimation, this paper proposes to organize the different information in serial multi-task models instead of the widely used parallel multi-task network. The proposed models are trained on the public dataset Common Objects in Context (COCO), which is further augmented by ground truths of body orientation and mutual-occlusion mask. Experiments demonstrate the performance of the proposed method for multi-person pose estimation and body orientation estimation. The proposed method can detect 84.6% of the Percentage of Correct Keypoints (PCK) and has an 83.7% Correct Detection Rate (CDR). Comparisons further illustrate the proposed model can reduce the over-detection compared with other methods.

Numerical Study on Hydrodynamics of Ships with Forward Speed Based on Nonlinear Steady Wave

In this paper, an improved potential flow model is proposed for the hydrodynamic analysis of ships advancing in waves. A desingularized Rankine panel method, which has been improved with the added effect of nonlinear steady wave-making (NSWM) flow in frequency domain, is employed for 3D diffraction and radiation problems. Non-uniform rational B-splines (NURBS) are used to describe the body and free surfaces. The NSWM potential is computed by linear superposition of the first-order and second-order steady wave-making potentials which are determined by solving the corresponding boundary value problems (BVPs). The so-called mj terms in the body boundary condition of the radiation problem are evaluated with nonlinear steady flow. The free surface boundary conditions in the diffraction and radiation problems are also derived by considering nonlinear steady flow. To verify the improved model and the numerical method adopted in the present study, the nonlinear wave-making problem of a submerged moving sphere is first studied, and the computed results are compared with the analytical results of linear steady flow. Subsequently, the diffraction and radiation problems of a submerged moving sphere and a modified Wigley hull are solved. The numerical results of the wave exciting forces, added masses, and damping coefficients are compared with those obtained by using Neumann–Kelvin (NK) flow and double-body (DB) flow. A comparison of the results indicates that the improved model using the NSWM flow can generally give results in better agreement with the test data and other published results than those by using NK and DB flows, especially for the hydrodynamic coefficients in relatively low frequency ranges.

Comparative study on ship motions in waves based on two time domain boundary element methods

This paper presents a comparative study on the motions of a ship advancing in waves using two different three-dimensional time domain boundary element methods: a Transient Free surface Green's Function (TFGF) method and a Rankine Higher Order Boundary Element Method (HOBEM). Three models based on the HOBEM are also considered to identify the dominant factors affecting the accuracy of solutions: (i) the N–K model using Neumann–Kelvin (N–K) linearization, (ii) the N–K+DB model using the N–K linearization for the free surface boundary conditions and the double-body (DB) m-terms in the body boundary condition, (iii) the Rankine HOBEM based on DB linearization. The Wigley I and the Series 60 (CB = =0.7) are taken as study objects. The comparisons show that the Rankine HOBEM based on DB linearization is generally more accurate than the Rankine HOBEM with other two models and the TFGF method. The hydrodynamic coefficients are mainly affected by the m-terms, especially at low frequencies; while the influences of different linearizations of the free surface boundary conditions are negligible. Both the m-terms and the leading terms of steady velocity potential reserved in the free surface boundary conditions are important for motion responses, while the influence of the former is stronger than the latter.

A numerical study of second-order springing of an elastic body using higher-order boundary element method (HOBEM)

Second-order springing on an elastic body with forward speed is analyzed by numerical simulations. The boundary-value problem for the velocity potential is solved by means of the direct time-domain higher-order boundary element method (HOBEM). The free-surface boundary condition in the boundary-value problem is approximated on the mean surface up to second order by use of perturbation and Taylor-series expansion methods. The body boundary condition for an elastic body is derived with various quantities which are redefined in the generalized mode. These variables such as mode shape, normal vector, etc. are obtained by using directional derivative and continuum mechanics, and the same mathematical expressions are used to obtain several second-order generalized forces. To validate the numerical results, the second-order hydrodynamic force on the bottom-mounted rigid/elastic cylinders without forward speed is compared with other semi-analytic results. The property of second-order forces on an elastic ship is studied by changing the flexural rigidity and forward speed with elastic response. It is confirmed that the second-order velocity potential is important for a body with forward speed and investigation should be made more on numerical methods for accurate computation of the second-order velocity-potential force with forward speed.

Stability analysis of high-order finite-difference discretizations of the linearized forward-speed seakeeping problem

A high-order finite-difference method solution of the linearized, potential flow, seakeeping problem for a ship at steady forward speed was recently presented by Amini-Afshar et al. [1,2]. In this paper, we provide a detailed matrix-based eigenvalue stability analysis of this model, highlighting the sources of instability and the effects of possible remedies. In particular, we illustrate how both boundary treatment and grid stretching are important factors which are not typically captured by a von Neumann-type analysis. The new analysis shows that when grid stretching is used together with centered finite difference schemes, the method is generally unstable. The source of the instability can in some cases be traced to an effective downwinding of the convective terms. Stable solutions can be obtained either by introducing upwind-biased schemes for computing the convective derivatives on the free-surface, or by application of a mild filter at each time-step. A second source of instability is associated with the treatment of the convective derivatives of the free-surface elevation at points close to the domain boundaries. Here it is necessary to consider whether the surrounding fluid points lie in an upwind or a downwind direction. For upwinded points, ordinary one-sided differencing can be used, but for downwinded points we instead impose a Neumann-type boundary condition derived from the body and free-surface boundary conditions. As an example application to complement those already given in [1,2], the method is applied to solve the steady wave resistance problem and comparison is made to reference solutions for a two-dimensional floating cylinder and a submerged sphere. Estimates of the wave resistance of the Wigley hull are also compared with experimental measurements.

Characteristics of rolling with drift in regular beam sea for small hard-chine craft at rest

The characteristics of rolling with drift in regular beam sea for a small hard-chine craft at rest is investigated experimentally, and modification of a strip method with Ikeda's roll damping prediction method for hard-chine boat (Ikeda et al., 1990) is proposed. From the comparison between calculated and measured RAO of rolling, it is confirmed that the calculation overestimates roll amplitude at the condition where wave encounter period is shorter than its roll natural period, especially. The reasons of overestimation are investigated and it is confirmed that the coupling viscous roll moment from swaying cannot be ignored for small hard-chine craft for shorter wave period than its roll natural period. Because the roll natural period is shorter than that of a conventional displacement type vessel and its resonant wavelength is short comparing with its breadth, therefore its swaying can't follow orbital motion of wave, then coupling viscous roll moment from swaying becomes significant. Moreover, as another reason of the overestimation, the estimation accuracy of wave exciting roll moment is investigated and it is recommended that the diffraction potential should be solved under strict satisfying the body boundary condition for short wave length (λ/B < 7.0).

A two-dimensional numerical and experimental study of piston and sloshing resonance in moonpools with recess

The piston and first sloshing modes of two-dimensional moonpools with recess are investigated. Dedicated forced heave experiments are carried out. Different recess lengths are tested from $1/4$ to $1/2$ of the length of the moonpool at the mean waterline. A theoretical model to calculate the natural frequencies is developed based on linearized potential flow theory and eigenfunction expansion. Two numerical methods are implemented: a boundary element method (BEM) and a Navier–Stokes solver (CFD). Both the BEM and CFD have linearized free-surface and body-boundary conditions. As expected, the BEM over-predicts the moonpool response significantly, in particular at the first sloshing mode. The CFD is in general able to predict the maximum moonpool response adequately, both at the piston and first sloshing modes. Both numerical methods fail to predict the Duffing-type behaviour at the first sloshing mode, due to the linearized free-surface conditions. The Duffing behaviour is more pronounced for the largest recess. The main source of damping in the proximity of the first sloshing mode is discussed.