Hydroelastic Problem Research Articles

Development of Reduced-Order-Discrete-Module method for hydroelastic analysis of floating flexible structures

The present study proposes a novel method for analyzing the hydroelastic response of floating flexible structures based on Reduced-Order-Discrete-Module (RODM) model. In this model, the floating flexible structure is discretized into a finite number of modules. The hydrodynamic problem is simplified as the interaction between waves and multiple modules. The hydroelastic response is approximated by solving the motion equation of the multibody system, in which the mass and stiffness of the structure are obtained from the reduced-order matrices by the finite element method with a system equivalent reduction expansion process. By using the transformation matrix, the detailed floating structure response can be reconstructed from the multibody dynamics. The validity of the proposed method was demonstrated by comparing the results with the experimental data and other existing methods. The results show that this study has developed an accurate hydroelastic model to analyze the hydroelastic response of floating flexible structures. A module number selection formula is proposed to select the appropriate number of modules based on the exciting force frequency. This model is relatively easy to implement for the hydroelastic problem of interconnection modules and take into account the spatial inhomogeneity of wind/wave field. The proposed model can offer a useful tool for analyzing the hydroelastic response of the offshore floating photovoltaic systems.

Nonlinear hydroelastic vibration of foamed concrete beams via peridynamic differential operator

Evaluating the hydroelastic responses of underwater cementitious structural elements is critical for ensuring the sustainability and durability of energy-saving marine infrastructures. Existing work on the hydroelastic analysis of porous structures has been mostly developed using the general elastic constitutive relation; however, it fails to capture the influence of saturation. To fill this knowledge gap, we for the first time propose a novel fluid-porous structure interactive model that incorporates the combined effects of hydrodynamic pressure and saturation-induced pore pressure. One more pioneering effort is to solve this nonlinear hydroelastic problem by introducing peridynamic differential operator (PDDO). It is worth noting that the introduction of PDDO removes the inherent drawback employing the local-theory based techniques, namely being prone to singularities arising from the presence of discontinuity. The accuracy and reliability of the proposed numerical framework are validated by comparing the results with the degraded model in the reported literature. Moreover, our results highlight that the angular frequencies are underestimated when ignoring the effect of saturation in foamed concrete beams. The presented method provides a profound understanding of the underwater structural dynamic monitoring that benefits the design of marine infrastructures.

Solitary deformation waves in two coaxial shells made of material with combined nonlinearity and forming the walls of annular and circular cross-section channels filled with viscous fluid

The aim of the paper is to obtain a system of nonlinear evolution equations for two coaxial cylindrical shells containing viscous fluid between them and in the inner shell, as well as numerical modeling of the propagation processes for nonlinear solitary longitudinal strain waves in these shells. The case when the stress-strain coupling law for the shell material has a hardening combined nonlinearity in the form of a function with fractional exponent and a quadratic function is considered. Methods. To formulate the problem of shell hydroelasticity, the Lagrangian–Eulerian approach for recording the equations of dynamics and boundary conditions is used. The multiscale perturbation method is applied to analyze the formulated problem. As a result of asymptotic analysis, a system of two evolution equations, which are generalized Schamel– Korteweg– de Vries equations, is obtained, and it is shown that, in general, the system requires numerical investigation. The new difference scheme obtained using the Grobner basis technique is proposed to discretize the system of evolution equations. Results. The exact solution of the system of evolution equations for the special case of no fluid in the inner shell is found. Numerical modeling has shown that in the absence of fluid in the inner shell, the solitary deformation waves have supersonic velocity. In addition, for the above case, it was found that the strain waves in the shells retain their velocity and amplitude after interaction, i.e., they are solitons. On the other hand, calculations have shown that in the presence of a viscous fluid in the inner shell, attenuation of strain solitons is observed, and their propagation velocity becomes subsonic.

Wave energy conversion by an array of oscillating water columns deployed along a long-flexible floating breakwater

Large-scale spatial configurations combining Wave Energy Converters (WECs) and coastal attenuating-wave facilities have the potential to exploit marine renewable energy sustainably. In this study, an integrated concept of multiple Oscillating Water Columns (OWCs) and a very long floating breakwater is introduced. Associated energy extraction, gap resonance and hydroelastic interaction problems are examined. A coupled numerical simulation methodology consisting of a Finite Volume Method (FVM)based solver and a Finite Element Method (FEM) solver, is developed to investigate the strong fluid and structure coupled problem. The fluid-structure information is matched in real-time and the flexible modes of the floating breakwater are obtained by imposing a restrained beam inside the pontoon. The accurate time-domain model is validated against both simulated and measured data. Extensive parametric studies indicate that the energy conversion has a conflict with the wave attenuation in terms of determining the along-shore number of OWCs. The highest energy conversion in medium-period and long-period waves are observed in the OWCs near the end and middle locations, respectively. Besides, the constructive resonant gap effect between OWCs and the breakwater can amplify the peaks of energy conversion efficiency, leads to a sudden collapse in transmission coefficient curves. With an increased sidewall draft, OWCs closer to oblique incident direction generate stronger piston-type and sloshing oscillations. Additionally, compared with a rigid breakwater, the elastic deformation of the breakwater plays a destructive role in wave energy conversion, which is attributed to the out-of-phase interference of multi-mode radiated waves.

Evolution of solitary hydroelastic strain waves in two coaxial cylindrical shells with the Schamel physical nonlinearity

The paper considers the formulation and solution of the hydroelasticity problem for studying wave processes in the system of two coaxial shells containing fluids in the annular gap between them and in the inner shell. We investigate the axisymmetric case for Kirchhoff–Lave type shells whose material obeys a physical law with a fractional exponent of the nonlinear term (Schamel nonlinearity). The dynamics of fluids in the shells is considered within the framework of the incompressible viscous Newtonian fluid model. The derivation of the Schamel nonlinear equations of shell dynamics makes it possible to develop a mathematical formulation of the problem, which includes the obtained equations, the dynamics equations of two shells, the fluid dynamics equations and the boundary conditions at the shell-fluid interfaces and at the flow symmetry axis. The asymptotic analysis of the problem is performed using perturbation techniques, and the system of two generalized Schamel equations is obtained. This system describes the evolution of nonlinear solitary hydroelastic strain waves in the coaxial shells filled with viscous fluids, taking into account the inertia of the fluid motion. In order to determine the fluid stress at the shell-fluid interfaces, we perform linearization of the fluid dynamics equations for fluids in the annular gap and in the inner shell. The linearized equations are solved by the iterative method. The inertial terms are excluded from the equations in the first iteration, while, in the second iteration, these are the values found in the first iteration. A numerical solution of the system of nonlinear evolution equations is obtained by applying a new difference scheme developed using the Gröbner basis technique. Computational experiments are performed to investigate the effect of fluid viscosity and the inertia of fluid motion in the shells on the wave process. In the absence of fluids in the inner shell, the results of calculations demonstrate that the strain waves in the shells during elastic interactions do not change their shape and amplitude, i.e., they are solitons. The presence of viscous fluid in the inner shell leads to attenuation of the wave process.

A Fully Explicit Lattice Spring Elastoplastic Solid Model for Fluid Structure Interaction Applications

The Fluid Structure Interaction (FSI) solver based on mesh-less methods developed at University of Nottingham Malaysia has been proven to be successful in solving hydroelastic problems involving elastic body. In this paper, the possibility of modelling elastocplastic material behaviour using mesh-less method is further explored. Elastoplastic material bahaviour is essential for modelling ductile fracture in a wide range of engineering problems. The use of a type of nonlocal lattice spring model (LSM) based on fully explicit integration scheme in modelling elastoplastic material behaviour is presented in this work. Specifically, the return mapping algorithm is incorporated in the solid solver for modelling J2 plasticity. The effect of time step size and particle spacing on the accuracy of the solver are investigated by checking the predicted strain against theoretical solution. It is found out that the strain results obtained by varying the time step size are somewhat consistent with the theoretical solution. The predicted strain also converges to the theoretical result as the particle spacing is refined.

An efficient method for estimating the structural stiffness of flexible floating structures

In the hydroelastic analysis of large floating structures, the structural and hydrodynamic analyses are coupled, and the structural stiffness plays an important role in the accurate prediction of the response. However, there is usually a large difference between the longitudinal and the cross-sectional scales of modern ships, and the sectional configurations are generally complex, making it difficult to obtain the exact structural stiffness. Using a full finite element model to calculate the structural stiffness is inevitably time-consuming. Since modern ship structures are usually nearly periodic in the longitudinal direction, we treat the hull as a periodic Euler–Bernoulli beam and use a novel implementation of asymptotic homogenization (NIAH) to calculate the effective stiffness. This can greatly improve the computational efficiency compared with a full finite element model. Based on a combination of finite element and finite difference methods, we develop an efficient analysis technique to solve the hydroelastic problem for nearly-periodic floating structures. The finite element method is used to efficiently calculate the structural stiffness, and the finite difference method is used to solve the hydrodynamic problem. This proposed technique is validated through several test cases with both solid and thin-walled sections. A range of representative mid-ship sections for a container ship are then considered to investigate the influence of both transverse and longitudinal stiffeners on the structural deformations. A simple method for including non-periodic end effects is also suggested.

On the solution of a complicated biharmonic equation in a hydroelasticity problem

On the solution of a complicated biharmonic equation in a hydroelasticity problem

An application of data-driven modeling for hydroelasticity of an elastically supported semi-circular pipe conveying fluid

PurposeThe paper aims to develop an efficient data-driven modeling approach for the hydroelastic analysis of a semi-circular pipe conveying fluid with elastic end supports. Besides the structural displacement-dependent unsteady fluid force, the steady one related to structural initial configuration and the variable structural parameters (i.e. the variable support stiffness) are considered in the modeling.Design/methodology/approachThe steady fluid force is treated as a pipe preload, and the elastically supported pipe-fluid model is dealt with as a prestressed hydroelastic system with variable parameters. To avoid repeated numerical simulations caused by parameter variation, structural and hydrodynamic reduced-order models (ROMs) instead of conventional computational structural dynamics (CSD) and computational fluid dynamics (CFD) solvers are utilized to produce data for the update of the structural, hydrodynamic and hydroelastic state-space equations. Radial basis function neural network (RBFNN), autoregressive with exogenous input (ARX) model as well as proper orthogonal decomposition (POD) algorithm are applied to modeling these two ROMs, and a hybrid framework is proposed to incorporate them.FindingsThe proposed approach is validated by comparing its predictions with theoretical solutions. When the steady fluid force is absent, the predictions agree well with the “inextensible theory”. The pipe always loses its stability via out-of-plane divergence first, regardless of the support stiffness. However, when steady fluid force is considered, the pipe remains stable throughout as flow speed increases, consistent with the “extensible theory”. These results not only verify the accuracy of the present modeling method but also indicate that the steady fluid force, rather than the extensibility of the pipe, is the leading factor for the differences between the in- and extensible theories.Originality/valueThe steady fluid force and the variable structural parameters are considered in the data-driven modeling of a hydroelastic system. Since there are no special restrictions on structural configuration, steady flow pattern and variable structural parameters, the proposed approach has strong portability and great potential application for other hydroelastic problems.

On the solution of a complicated biharmonic equation in a hydroelasticity problem

A hydroelastic problem of free vibrations of a thin plate that horizontally separates ideal incompressible liquids of different densities in a rigid cylindrical tank with an arbitrary cross-section has been considered in the linear formulation. To solve the corresponding complicated inhomogeneous biharmonic equation, the fundamental system of the solutions of biharmonic equation (FSS) and the eigenmodes of ideal liquid oscillations in a cylindrical cavity were used. The frequency equation was obtained for arbitrary fixation of the plate contour. On the example of a clamped plate, the frequency equation was simplified by decomposing the corresponding homogeneous biharmonic equation into two harmonic equations and using Green's formula for the Laplace operator. It was shown that in this case the frequency equation does not depend on the FSS and becomes greatly simplified because the FSS depends on the unknown frequency. The resulting equation has a single form for the cases of a right circular cylinder and a rectangular channel; in particular cases, it coincides with the previously obtained equations. Research of asymmetric vibration frequencies of a plate and a membrane, as well as axisymmetric vibration frequencies of a membrane in a circular cylinder, has been carried out. An approximation formula for high frequencies and approximate conditions for the stability of the plate and membrane vibrations were obtained.

The nonlinear wave interaction with a two-dimensional large-scale floating elastic plate

A two-dimensional fully nonlinear numerical wave tank is developed based on the higher order boundary element method and modal expansion to simulate the nonlinear interaction between waves and a finite horizontal floating elastic plate with draft. The large second harmonic displacement of a large-scale elastic plate near a second-order particular frequency is successfully calculated by this model. By the comparison with the results in a frequency model that involve perturbation expansion method, it is found that the large values of the second-order waves in a finite elastic plate near the particular frequency are not introduced by perturbation expansion. It is different from the case in infinite plate. The effect of the third-order particular triple frequency on the hydroelastic problem is small compared to the effect of the second-order particular frequency.

An improved M-SPEM for modeling complex hydroelastic fluid-structure interaction problems

Hydroelastic fluid-structure interaction (FSI) problems usually involve nonlinear free surface flows, large structure deformations and changing interfaces, which present great challenges for numerical modeling. To deal with violent FSIs, the authors proposed a multi-resolution smoothed particle element method (M-SPEM), which adaptively simulates local fluid regions with an improved smoothed particle hydrodynamics (SPH) method and models other solid/fluid regions with a smoothed finite element method (S-FEM). Both computational accuracy in local fluid domain and entire efficiency can be improved by M-SPEM compared with the conventional partitioned FSI solvers based on SPH-FEM coupling schemes. In this work, an improved M-SPEM is developed and extended to simulate the complex hydroelastic FSI problems with structural deformations considered in the numerical model. To solve the accuracy reduction problem in the multi-resolution simulations, we developed a velocity and position correction technique for the generated fluid and virtual particles. The computational results demonstrated that the new correction technique can effectively improve the computational accuracy of M-SPEM while the corrected schemes produce results very close to the reference ones. The positions of the generated particles should also be appropriately optimized during the element-particle conversion process. With both accuracy and efficiency improved, the improved M-SPEM is proven to be suitable to deal with some challenging hydroelastic FSI problems.

Slamming loads and responses on a non-prismatic stiffened aluminium wedge: Part II. Numerical simulations

An experimental study was carried out to examine the impact-induced loads and structural responses of a three-dimensional non-prismatic aluminium wedge. The findings of the experimental study were presented in the first part of this publication, entitled “Slamming loads and responses on a non-prismatic stiffened aluminium wedge: Part I. Experimental study.” This paper describes the second part of the study, which deals with numerical simulations of slamming loads acting on the wedge and its dynamic responses. Two different two-way coupling methods are assessed and compared to simulate the water entry problem. Initially, an explicit nonlinear finite element method with a Multi-Material Arbitrary Lagrangian-Eulerian (MMALE) solver is employed to evaluate the elastic response of the structure following a free-fall water impact. Subsequently, the hydroelastic slamming problem is modelled using a two-way coupled technique with a k-ε turbulence model and implicit unsteady solver for both the fluid and structural domains (Star-CCM+/ABAQUS). The effect of viscosity on slamming loads and responses was examined, and numerical results are compared with experimental data. To investigate the effect of rigidity on structural response and bottom deflection, the wedge was designed with bottom plates of varying thickness. The computed results from the two numerical models and the experimental data are in good agreement. The study concluded that the bottom plates deformation affects hydrodynamic loads during slamming. It is also observed that impact-induced loads depend on the water impact velocity and the flexibility of the bottom plates. This suggests that the slamming pressure increases with an increase in impact velocity and decreases when the structures become more flexible. According to this study, both numerical models are suitable for accurate and efficient computations of hydroelastic slamming; however, the MMALE method results in larger numerical fluctuations.

A discrete-module-finite-element hydroelasticity method in analyzing dynamic response of floating flexible structures

A discrete-module-finite-element (DMFE) based hydroelasticity method has been proposed and well developed. The DMFE solves the hydroelastic problem as follows. Firstly, a freely floating flexible structure is discretized into several macro-submodules in two horizontal directions to perform a multi-rigid-body hydrodynamic analysis. Each macro-submodule is then abstracted to a lumped mass at the center of gravity that bears the external forces acting on the macro-submodule. Apart from external forces, all lumped masses are also subjected to structural forces that reflect the structural deformation features of the original flexible structure, which is tackled following the finite element procedure. Finally, based on the d’Alembert principle, the DMFE method establishes the hydroelastic equation on all lumped masses with their displacements as unknown variables. Displacement and structural force responses are first calculated on the interfaces, after which at any given position of the flexible structure, the recovery of displacement is based on the structural stiffness matrix of the corresponding macro-submodule and that of structural force uses the spline interpolation scheme. The hydroelasticity of a narrow and a square pontoon-type VLFS is investigated. At last, a least square method to recover bending moment distribution of flexible-floating-structures with complicated shape is presented, including some unsolved problems.

О проблемах при моделировании плоских течений вязкой жидкости при повышенных значениях числа Рейнольдса вихревыми методами в программном комплексе VM2D

Vortex methods of computational fluid dynamics are an efficient tool in engineering practice for estimating hydrodynamic loads acting on bodies placed in a flow. Their usage allows for solving of coupled hydroelastic problems with relatively small computational cost. In many applications, the cross flow around structural elements with large elongation is considered, that allows one to use the flat cross-sections method providing the acceptable accuracy. Thus, flat flows simulation around airfoils is required. Modern modifications of vortex particle methods make it possible to simulate flows of a viscous incompressible medium. Based on the method of viscous vortex domains in 2017-2022 the VM2D code have been developed in Bauman University and Ivannikov Institute for System Programming. This code allows for flow simulating around airfoils with acceptable accuracy at low Reynolds numbers, while for higher Reynolds numbers, correct results are observed only for airfoils with sharp edges and corner points, and only in regimes where the most intensive flow separation takes place at these points. The reason for the error in the results for other regimes is seen in incorrect modeling of the flow separation on smooth airfoil surface line at high Reynolds numbers, which, in turn, is a consequence of incorrect modeling of vorticity evolution in the vicinity of separation points (zones). Some results of flow simulations around different airfoils at different values of the Reynolds number are presented and a hypothesis explaining the reason for the discrepancy between numerical results and experimental data is proposed. It is shown that the kinetic energy spectrum of turbulence corresponds to “two-dimensional turbulence”.

A numerical method to compute flexible vertical responses of containerships in regular waves

In this paper, a series of simulations of hydroelastic problems of containerships are conducted by a nonlinear 3D panel method which is based on the generalized Taylor Expansion Boundary Element Method (TEBEM). In the structure part, the Transfer Matrix Method is selected to perform a modal analysis which provides key parameters of hydroelastic calculation. In the hydrodynamic part, the nonlinearities arising from the variations of instantaneous wetted surface area are considered in the solutions of hydrostatic restoring forces as well as 1st order incident wave forces. Meanwhile, an extended Modified Logvinovich Model (MLM) is presented to predict the slamming forces due to the large-amplitude incident waves. The numerical predictions of the hydroelastic responses of a 6750-TEU containership and S175 in regular waves are presented. The effects of wave steepness and sailing speed as well as the variation characteristics of load components against with wave direction are studied. The numerical and experimental results are in acceptable agreement which proves the accuracy of the proposed method.

Modeling Nonlinear Hydroelastic Response for the Endwall of the Plane Channel Due to Its Upper-Wall Vibrations

A mathematical model for studying the nonlinear response of the endwall of a narrow channel filled with a viscous fluid to the vibration of the channel’s upper wall was formulated. The channel, formed by two parallel, rigid walls, was investigated. The right end-channel wall was supported by a nonlinear spring. At the end of the left channel, the fluid flowed into a cavity with constant pressure. The upper channel wall oscillated according to a given law. As a result of the interaction between the endwall and the upper wall via a viscous fluid, the forced, nonlinear oscillations of the channel endwall arose. The fluid motion was considered in terms of the hydrodynamic lubrication theory. The endwall was studied as a spring-mass system with a nonlinear cubic restoring force. The coupled hydroelasticity problem was formulated, and it was shown that the problem under consideration was reduced to a single equation in the form of the Duffing equation. The nonlinear hydroelastic response of the end wall was determined by means of the harmonic balance method. The results of numerical experiments on nonlinear hydroelastic response behavior and a comparison with the case when the support spring is linear were presented. The obtained results are of a fundamental nature and can be used in modeling various devices and systems that have narrow channels filled with viscous fluid and are subjected to vibrations on one side of the channel. For example, coolant pipes are subjected to vibrations from the engine. Of particular interest is the application of the presented solution to the mathematical modeling of nano- and micro-spacecraft systems with fluids since the proposed decision allows for the consideration of some boundary effects, which is important for nano- and micro-spacecraft due to their small size.

Convergence Analysis of Havelock-Type Eigenfunction Expansions for Hydroelastic Problems in Water having Infinite Depth

The present paper demonstrates the point-wise convergence of the Havelock-type eigenfunction expansion to the velocity potentials associated with the water waves interaction with flexible plate and membranes in water having infinite depth. To consider the higher-order boundary condition at the mean free surface of the water domain, flexible plate and membranes are assumed to float in the mean water level. In the convergence analysis procedure, firstly, the havelock-type eigenfunction expansion for the unknown velocity potentials associated with the physical problems are obtained. Hereafter, a suitable Green's function is developed for the associated physical problem. Using the developed Green's function and the associated properties, the vertical components of the Havelock-type eigenfunction expansion is expressed in terms of the Dirac delta function. Finally, using appropriate properties of the Dirac delta function, the point-wise convergence of the Havelock-type eigenfunction expansion is demonstrated.

A three-dimensional fluid-structure interaction model based on SPH and lattice-spring method for simulating complex hydroelastic problems

The present work revolves around the development of a 3D particle-based Fluid-Structure Interaction (FSI) solver to simulate hydroelastic problems that involve free surface. The three-dimensional Volume-Compensated Particle Method (VCPM) for modelling deformable solid bodies is developed within the open-source SPH software package DualSPHysics. Complex 3D FSI problems are readily simulated within a reasonable time frame thanks to the parallel scalability of DualSPHysics on both CPU and GPU. The Sequential Staggered (SS) scheme paired with a multiple time-stepping procedure is implemented in DualSPHysics for coupling the SPH and VCPM models. It is found that the SPH-VCPM method is computationally more efficient than the previously reported SPH-TLSPH method. Extensive validations have been performed based on some very recent 3D experimental setups that involve violent free surface and complex structural dynamics. Findings from this research highlight the capability of the 3D SPH-VCPM model to reproduce some of the physical observations that were not captured by previous 2D studies. Some preliminary 3D FSI results involving solid fracture are also demonstrated.

An Iterative Method for Interaction of Hydro-Elastic Waves with Several Vertical Cylinders of Circular Cross-Sections

The problem of ice loads acting on multiple vertical cylinders of circular cross-sections frozen in an ice cover of infinite extent is studied. The loads are caused by a flexural-gravity wave propagating in the ice cover towards the rigid bottom-mounted cylinders. This is a three-dimensional linearized problem of hydroelasticity with finite water depth. The flow under the ice is potential and incompressible. The problem is solved by the vertical mode method combined with an iterative method. The velocity potential is written with respect to each cylinder and is expanded into the Fourier series. The algorithm of the problem solving is reduced to calculations of the Fourier coefficients of the velocity potential. Numerical results for the forces acting on four circular cylinders are presented for different ice thicknesses, incident wave angles and cylinder spacing. The obtained wave forces are compared with the results by others. Good agreement is reported.