Eigenfunction Expansion Method Research Articles

Mitigation of hydroelastic responses of a very large floating structure by flexible porous barriers

The role of a vertical flexible porous barrier of two different configurations, namely (i) surface-piercing barrier (SPB) and (ii) bottom-standing barrier (BSB) in the mitigation of hyroelastic responses of a very large floating structure (VLFS) is analysed. Both the flexible barrier and the floating plate are modelled using Euler-Bernoulli beam equation under the assumption of small amplitude structural response. The solution of the associated mathematical problem is obtained using the eigenfunction expansion method. Various results illustrating the effect of the barrier length, porous-effect parameter, and the barrier-VLFS spacing on various hydroelastic responses of the VLFS are presented. Moreover, results on barrier deflection and overturning moments of the flexible porous barrier for various structural parameters are presented. Energy identity is derived using Green’s identity and used to check the accuracy of the computational results. The study reveals that for suitable values of the structural flexibility, porosity and barrier configurations, wave-induced structural responses on the VLFS and barriers can be reduced significantly. Moreover, the study depicts that the presence of flexible porous barriers can significantly reduce the wave reflection and wave forces exerted on the floating elastic plate.

Eigenfunction expansion solution for the radiation problem of a uniform cylinder in front of a partially reflective wall

In this paper, the wave radiation problem of a vertical cylinder in front of a partially reflective vertical wall is solved by combining the eigenfunction expansion method and the mirror image method. To satisfy the partial reflection condition at the vertical wall, a scattering incident wave is introduced, which corresponds to the reduction scattering potential from the mirror image cylinder symmetrically distributed about the partially reflective wall and under symmetrical motion about the vertical wall with respect to the original cylinder. Subsequently, the scattering potentials generated by each of the two cylinders are derived using the eigenfunction expansion method on cylindrical coordinate systems with origins at the centers of the two cylinders. By applying Graf's addition theorem, the wave radiation problem is finally solved by transforming the incident scattering potential in the imaginary cylindrical coordinate system back to the original one of the cylinder itself and satisfying the body surface boundary condition. Using the numerical model developed in this paper, the effects of the parameters of the modulus and phase of the partial reflection coefficients as well as the distance between the partially reflective wall and the center of the cylinder on the added mass and radiation damping are investigated. The numerical results show that there are oscillations present in both the curves of added mass and radiation damping versus nondimensional wavenumber ka, the amplitude of the oscillations increases with the increase in the modulus of the reflection coefficient, and the peaks and troughs of the oscillations move to lower wavenumber ranges with increasing phase shift of the reflection coefficient. In addition, the distance between the cylinder and the vertical wall significantly affects the wavenumber range in which the peaks and troughs of the hydrodynamic coefficient curves appear, and the period of the oscillations decreases as the distance increases.

Water wave radiation by an array of bottom-mounted surface-piercing concentric cylinders

This paper develops a wave radiation analysis model in horizontal oscillation mode for concentric cylindrical arrays with arbitrary numbers and arrangements within the framework of linear potential flow theory. The concentric cylinders have solid inner cylinders and porous outer cylinders, and they are both bottom-mounted and surface-piercing. The porous boundary conditions of the outer cylinders are treated using Darcy's law, and the flow field is divided into one outer region and N (number of cylinders) inner regions with the outer cylinders as the boundary. Employing eigenfunction expansion and region-matching methods, expressions for the radiation velocity potential of each region are obtained, and then expressions of the added mass and damping coefficient of each cylinder are obtained. Following a rigorous verification of the results against published literature, this paper proceeds to analyze the impact of various parameters, including oscillation modes, outer cylinder permeability, radius ratio, spacing, and the number of cylinders, on the wave radiation characteristics of tandem-arranged cylindrical arrays. The research reveals that when the oscillation direction of the structure is parallel with its arrangement direction, the hydrodynamic coefficient curves exhibit significant spikes, attributed to the influence of the Dirichlet trapped mode. Outside the peak regions, the hydrodynamic coefficient curves display regular fluctuations, which are attributed to the alternating occurrence of constructive and destructive interference effects within the wave field.

Generalised eigenfunction expansion and singularity expansion methods for canonical time-domain wave scattering problems

The generalised eigenfunction expansion method (GEM) and the singularity expansion method (SEM) are applied to solve the canonical problem of wave scattering on an infinite stretched string in the time domain. The GEM, which is shown to be equivalent to d’Alembert’s formula when no scatterer is present, is also derived in the case of a point-mass scatterer coupled to a spring. The discrete GEM, which generalises the discrete Fourier transform, is shown to reduce to matrix multiplication. The SEM, which is derived from the Fourier transform and the residue theorem, is also applied to solve the problem of scattering by the mass–spring system. The GEM and SEM are also used to solve the problem of wave scattering by a mass positioned a fixed distance from an anchor point, which supports more complicated resonant behaviour.

Effect of step bottom and waterway on flexural gravity wave scattering

Flexural gravity wave scattering by two semi-infinite non-identical ice sheets, which are separated by a clean water surface, is investigated in the presence of a step bottom topography. The problem is investigated in the cases of (i) intermediate water depth and (ii) shallow water. The effect of two edge conditions, (i) simply supported edge and (ii) free edge, on wave scattering is also analyzed. Employing linear velocity potential theory, the problem is studied in the frequency domain. The physical phenomenon is modeled as a boundary value problem having the Laplace equation as the governing equation, and it is solved using the eigenfunction expansion method. Considering the bottom topography and upper boundary, the fluid domain is divided into three regions, and in each region, velocity potential is expressed in terms of infinite Fourier series. Velocity and pressure are matched at the intermediate surface of two regions, and a system of algebraic equations with unknown coefficients is obtained. The complete solution of the present problem is recognized by solving the system of equations numerically. The energy relation is derived using Green's theorem. The reflection and transmission coefficients are computed and compared with the energy relation to check the accuracy of the present method. For different parameters (depth ratio, clean waterway, and different ice properties), reflection coefficient and transmission coefficient are computed and presented as a function of angular frequency. The value of the reflection coefficient ranges from zero to unity, whereas at certain frequencies, the transmission coefficient attends value more than unity.

Wave excited motion of a body floating on water confined between a semi-infinite ice sheet and a side wall

In this paper, we investigate the diffraction and radiation coupling problem of a floating body confined between a fixed wall and a semi-infinite ice sheet under the action of incident waves. Based on the linear water wave theory and Kirchhoff Love elastic thin plate assumption, the ice sheet is considered as an elastic beam, and the boundary conditions of fluid boundaries covered by the ice sheet are obtained. Dividing flow domains with different upper surfaces, the eigenfunction expansion method was used to obtain the velocity potential expansion equations for each sub-domain. Then, through the matching conditions at the interface of sub-domains, a system of equations was constructed to solve the unknown coefficients of the expansion equations. The influence of the existence of the ice sheet on the diffraction and radiation motion of the floating body is analyzed. The effects of changes in the draft and open water width of the floating body on the wave excitation force and hydrodynamic coefficient of the floating body are discussed. The influence of the reflection effect of the ice sheet on the added mass and damping coefficient of the floating body has been discovered, especially the changes and oscillation rules of their extreme points. This research achievement provides theoretical support for the safety design of floating structures in frozen harbors.

Hydrodynamic analysis of an oscillating water column array in presence of variable bathymetry

In this study, hydrodynamic analysis of an oscillating water column (OWC) array in the presence of the variable bathymetry (seafloor depth) is performed by using a semi-analytical potential-flow solver. Within the framework of linear theory, the boundary-value problem associated with wave diffraction and radiation by an OWC array with variable bathymetry is formulated and the semi-analytical hydrodynamic model is developed using the matched eigenfunction expansion method. The fluid domain above variable bathymetry is mathematically discretized into multiple subdomains using the boundary approximation method. The Haskind relation and the energy flux conservation law are used to verify the semi-analytical model. The hydrodynamic performance of the OWC array in the presence of the ripple and coral reef bathymetries are investigated in further. Pronounced oscillations are observed for the curve of reflection coefficient Cr and hydrodynamic efficiency η for cases with ripple and coral reef bathymetries. Bragg resonance is identified as the triggering of the strong oscillations in Cr and η for case of ripple bathymetry. The primary frequency of Bragg resonance and the peaked value of Cr converge as the ripple number increases (i.e., Ns > 3). Particularly, for cases with coral reef bathymetry, the wave resonance modal in open-ended basins results in a significant wave amplification at the weather side of the OWC array.

Viscous effects on the hydrodynamic performance of a two-body wave energy converter with a damping plate

In this study, the hydrodynamic forces and power absorption performance of an autonomous underwater vehicle (AUV)-based two-body wave energy converter (2BWEC) are investigated. A theoretical model is developed within the framework of linear potential flow to solve for added mass, radiation damping, and wave excitation force using the matched eigenfunction expansion method (MEEM). A computational fluid dynamics (CFD) model is employed to account for vortex-shedding effects of the floater and inner cylinder with a damping plate under various excitation conditions. Empirical formulas for supplementary added mass and drag coefficients caused by flow separation are proposed based on curve-fitting the differences between CFD results and MEEM calculations. These formulas are integrated into motion equations to enhance accuracy in evaluating the power absorption of the 2BWEC. It has been found that in the context of viscous flow, both the added mass and damping coefficients are increased, particularly for the inner cylinder with a damping plate. In addition, the viscous hydrodynamic coefficients exhibit strong dependence on the Keulegan–Carpenter number, while showing insensitivity to changes in the frequency parameter β. The supplementary (viscous) added mass provides additional inertia for the AUV with a limited mass itself, which is advantageous for the power absorption of the AUV-based 2BWEC. Conversely, the presence of viscous damping from the damping plate impedes wave energy capture.

Wave Pressure and Wave Height Distribution around Seawall Structure Constructed by an Array of TSP Circular Piles

An analytic solution for the interaction between an array of circular piles made by joining trapezoid steel pipes (TSP) and waves was obtained using an eigenfunction expansion method. First, an analytic model for the wave scattering of multiple piles fixed at arbitrary positions was derived, and then a simplified model was obtained assuming that an infinite array of identical piles were deployed perpendicular to the propagating direction of incident waves. A regular wave experiment was conducted using an experimental model with a scale ratio of 1/100 in a two-dimensional wave tank to verify the analytic solutions. The analytic results and experimental results were qualitatively consistent with each other. Using a developed analytic model, we examined the wave force on the multiple piles and the wave deformation in front of the arrayed piles. The period for the installation is greatly reduced as the TSP pile can be prefabricated in a factory. In particular, it is possible to install at the soft seabed. A seawall structure using arrayed TSP piles will be an ideal complement for a concrete seawall in future.

Unified theory of alpha, mu, and tau rhythms via eigenmodes of brain activity.

A compact description of the frequency structure and topography of human alpha-band rhythms is obtained by use of the first four brain activity eigenmodes previously derived from corticothalamic neural field theory. Just two eigenmodes that overlap in frequency are found to reproduce the observed topography of the classical alpha rhythm for subjects with a single, occipitally concentrated alpha peak in their electroencephalograms. Alpha frequency splitting and relative amplitudes of double alpha peaks are explored analytically and numerically within this four-mode framework using eigenfunction expansion and perturbation methods. These effects are found to result primarily from the different eigenvalues and corticothalamic gains corresponding to the eigenmodes. Three modes with two non-overlapping frequencies suffice to reproduce the observed topography for subjects with a double alpha peak, where the appearance of a distinct second alpha peak requires an increase of the corticothalamic gain of higher eigenmodes relative to the first. Conversely, alpha blocking is inferred to be linked to a relatively small attention-dependent reduction of the gain of the relevant eigenmodes, whose effect is enhanced by the near-critical state of the brain and whose sign is consistent with inferences from neural field theory. The topographies and blocking of the mu and tau rhythms within the alpha-band are explained analogously via eigenmodes. Moreover, the observation of three rhythms in the alpha band is due to there being exactly three members of the first family of spatially nonuniform modes. These results thus provide a simple, unified description of alpha band rhythms and enable experimental observations of spectral structure and topography to be linked directly to theory and underlying physiology.

Non-standard interface conditions in flexure of mixture unified gradient Nanobeams

Structural schemes of applicative interests in Engineering Science frequently encounter the intricate phenomenon of discontinuity. The present study intends to address the discontinuity in the flexure of elastic nanobeam by adopting an abstract variational scheme. The mixture unified gradient theory of elasticity is invoked to realize the size-effects at the ultra-small scale. The consistent form of the interface conditions, stemming from the established stationary variational principle, is meticulously set forth. The boundary-value problem of equilibrium is properly closed and the analytical solution of the transverse displacement field of the elastic nanobeam is addressed. As an alternative approach, the eigenfunction expansion method is also utilized to scrutinize the efficacy of the presented variational formulation in tackling the flexure of elastic nanobeams with discontinuity. The flexural characteristic of mixture unified gradient beams with diverse kinematic constraints is numerically illustrated and thoroughly discussed. The anticipated nanoscopic features of the characteristic length-scale parameters are confirmed. The demonstrated numerical results can advantageously serve as a benchmark for the analysis and design of pioneering ultra-sensitive nano-sensors. The established variationally consistent size-dependent framework paves the way ahead in nanomechanics and inspires further research contributing to fracture mechanics of ultra-small scale elastic beams.

Green’s functions in the presence of a bubble wall

Field theoretical tools are developed so that one can analyze quantum phenomena such as transition radiation that must have occurred during the Higgs condensate bubble expansion through plasma in the early universe. Integral representations of Bosonic and Fermionic propagators are presented for the case that particle masses are varied continuously during the passage through the bubble wall interface between symmetry-restored and symmetry-broken regions. The construction of propagators is based on the so-called eigenfunction expansion method associated with self-adjoint differential operators, developed by Weyl, Stone, Titchmarsh, Kodaira and several others. A novel method of field quantization in the presence of the bubble wall is proposed by using the spectral functions introduced in constructing the two-point Green’s functions.

Analytical model of an I-core coil inside a metal tube of finite length

An analytical solution is presented for the interaction of an I-core coil with a metal tube of finite length. Using the Truncated Region Eigenfunction Expansion (TREE) method, the impedance variation of the I-core coil is expressed in a closed form. Notably, the 1D finite element method (1D FEM) is adopted to calculate the complex eigenvalues and eigenfunctions of the steel tube, by which the overflow issue of complex eigenfunctions has been surmounted. And the 1D FEM also simplifies the calculation in the ferrite core region immensely. A performant algorithm is obtained due to the matrix formalization and the high speed of the sparse eigenvalue solver. The tubes made of aluminum alloy and stainless steel are investigated by the proposed method and tested in the experiment. The high reliability and efficiency of the proposed method are proved, and the results are also validated by the comparison with simulation and measurement data.

Analyzing Vertical Earthquake Vibrations and Moving Vehicle Loads for Structural Health Monitoring and Vibration Suppression in Bridges

The design of bridges often overlooks the vertical component of earthquakes or considers it of secondary importance, despite compelling evidence indicating specific structural damage caused by primary earthquake waves. Conversely, during the operational phase, the combined influence of ground motion and moving loads from vehicles can significantly impact the structural health monitoring (SHM) of bridges. This study aims to evaluate the simultaneous effect of vertical earthquake vibrations and moving vehicle loads on simply supported bridges. The research employs a practical methodology based on the eigenfunction expansion method to analyze change of deflection due to the effect of these concurrent forces under seven different earthquake records. It is shown that within a realistic range of vehicle mass and velocity, the average of changing the maximum deflection at the mid-span of the main beam (denoted as M_n) reaches up to 163% under various scenarios. Subsequently, the seismic parameters influencing this phenomenon are identified through a statistical analysis of set of 100 different earthquake records with unique features. A linear regression equation is presented to predict the M_n based on the earthquake specific properties. Additionally, to control the vertical vibration of bridge systems, a novel vibration suppression system utilizing steel pipe dampers is introduced, and its reliability is examined across a broad spectrum of bridge flexural rigidity. The results indicate that the system's efficiency depends on M_n and the soil type of the bridge construction, enabling a reduction in structural sections (up to 27%) while achieving the same maximum target deflection in the initial state. This efficiency leads to a more economical design solution, emphasizing the potential benefits of the proposed system for practical application.

Two-dimensional diffraction and radiation problems of a floating body in varying bathymetry

In this study, a coupling method of the higher order boundary element method and the eigenfunction expansion method is applied to investigate the diffraction and radiation problems of a floating body over an uneven seabed under the assumption of linear small-amplitude wave theory. The wave exciting forces and hydrodynamic coefficients are calculated for a floating rectangular box. It is found that the curves of the wave exciting forces and hydrodynamic coefficients versus frequency become fluctuating as the horizontal length L and vertical height D of the sloping seabed gradually increase. The period of the fluctuations decreases with the increase of L, and the magnitude of the fluctuations increases with the increase of D. The mechanism of the fluctuation phenomenon is then examined, and found that it is associated with the wave reflection from the slope seabed. With increasing horizontal scale L and vertical scale D of the sloping seabed, the amplitude and phase of the reflection wave change, which leads to the combined action of the incident, scattering and reflection waves fluctuating with the wave frequency.

Gravity wave interaction with a heaving membrane above a thick porous bed

The present study analyzes diffraction and radiation phenomena of oblique waves interacting with a heaving floating membrane in the presence of a thick porous bed. Following the linear water wave theory, the physical problem is framed mathematically. The significance of the article resides in the following: (1) progressive wave analysis (water and membrane-covered region), (2) solving the boundary value problem (BVP) using the matched eigenfunction expansion method for diffraction and radiation problems, and (3) numerical illustration of various hydrodynamic coefficients for different membrane and porous bed parameters. Bragg scattering with varying frequency is observed for smaller values of membrane tension. Also, the present study demonstrates that the number of oscillations experienced by the reflection coefficient increases proportionally with the length of the membrane. Furthermore, cut-off membrane properties exist at a given frequency for which the zero minimum of wave force is obtained. Also, the porous bed's thickness impacts wave reflection and membrane deflection significantly. Thus, we found that the maximum reflection is observed for a fully permeable bed; however, it decreases with a decrease in the porosity of the porous medium because of its dissipative nature. Conversely, the added mass and damping coefficient increases with increased membrane length. The collective numerical observations for both diffraction and radiation provide insight into resonance phenomena, the role of membrane properties, and the intricate relationship between wave characteristics and membrane properties. The findings from this study could assist geologists and marine engineers in designing and managing ports and harbor infrastructure.

Theoretical and numerical study on transient acoustic wave propagation across ice layers in the Arctic Ocean.

The study of transient acoustic wave propagation across the Arctic Ocean ice layer provides theoretical guidance for the design of trans-ice acoustic communication systems. In this study, the Arctic Ocean was modeled as an ice-water composite structure, where the ice and water are regarded as an elastic solid and liquid, respectively. An analytical transient solution for acoustic wave propagation in this structure was derived using the eigenfunction expansion method. Further, the numerical procedures were presented and used to analyze the acoustic wave propagation characteristics across the ice layer. The results show that waveforms corresponding to the radial displacements are more severely distorted than the axial displacements. The amplitudes of the radial and axial displacements decreased rapidly with increasing propagation distance. The ice thickness had a greater impact on the radial displacement than axial displacement; the thicker the ice, the greater the distortion for both radial and axial displacements.

Analysis of Hydrodynamic Loads by a Vertical Hollow Cylindrical Structure in A Two-Layer Fluid of Finite Depth

This paper aims to study the effect of surge radiation by a vertically hollow cylinder floating in a two-layer fluid of finite depth. Since most of the important characteristics of the oscillating water column are preserved by the hollow cylindrical structure, so the proposed device can be considered as an oscillating water column (OWC) which is one of the important wave energy device. In this two-layer fluid system, the upper layer is bounded above by a free surface and the lower layer is bounded below by a solid impermeable bottom, and the submerged hollow cylindrical structure allows it to oscillate in a surge mode of motion in the upper layer. On the assumption that wave amplitudes are small in comparison to wavelengths, hence the linear water wave theory is applied to formulate boundary value problems by dividing whole domain into 2 sub-domains. The formulated boundary value problems for surge radiated potentials in each sub-domain; we execute to obtain solutions in each sub-domain by using variable separation and matched eigenfunction expansion methods. With the help of obtained surge radiated potentials, we derive the radiation forces by the device in terms of added mass and damping coefficients. A set of influences of different parameters of the device on the added mass and damping coefficients are demonstrated in both surface and internal wave modes of motion. Hydrodynamic coefficients oscillate highly near a particular frequency, which leads to the resonance phenomena. In addition to that, the 3D plots of the free surface elevation in surface and internal wave modes are presented. A density ratio γ = 0.97 has a higher oscillation than a density ratio γ = 0.93, 0.95. High oscillations occur around a particular frequency ω = 3.83 due to the resonance phenomenon in which the waves incoming match with the natural motion of the object. HIGHLIGHTS A floating hollow vertical cylindrical structure is considered in a two-layer fluid system having finite depth The separation of variables and matched eigenfunction expansion methods are utilized to obtain the solutions to the boundary-value problems The diffracted velocity potentials under the surge mode of motion are evaluated in the identified regions The added mass, damping coefficients, free surface and interface elevations in surface and internal wave modes are evaluated with different set of parameters of the device A number of results are presented regarding the effect of parameters of the device on the added mass, damping coefficients, free surface and interface elevations It is observed that the density ratio of the two-layer fluid has significant effect on the added mass, damping coefficient and the elevations GRAPHICAL ABSTRACT

Control co-design optimization of nonlinear wave energy converters

This paper presents a study in which both the control and the shape of a Wave Energy Converter (WEC) are optimized simultaneously. A heaving point absorber WEC is assumed. To optimize the shape of the WEC’s buoy, nonlinear hydrodynamics need to be evaluated. One main contribution of this paper is the integration of nonlinear hydrodynamics and nonlinear control during the optimization of the WEC’s buoy shape. This approach is referred to as Control Co-Design (CCD). In this work, we present a control co-designed nonlinear heaving point absorber WEC that leverages the nonlinear dynamic, static Froude–Krylov (FK) forces to maximize power extraction. The nonlinear FK forces are approximated using a variation of the algebraic solution; the hydrodynamic forces of the body are computed using an analytic formulation leveraging the methods of eigenfunction expansion and separation of variables. The nonlinear geometry of the buoy is modeled as a series of inclined panels; the inclination angles are optimized to arrive at the optimal shape. The performance of the optimized shape is compared to that of a nonlinear spherical WEC. It is found that an average of 20% improvement is achieved by the optimized geometry over the spherical device.

Wave trapping by porous breakwater near a wall under the influence of ocean current

In this paper, oblique water wave interaction with a surface piercing box-type porous structure placed in front of a vertical rigid wall is investigated in the presence of current, which flows parallel to vertical wall. The current speed is considered distinct in different subdomains. However, in a particular subdomain current speed is uniform throughout the water depth. In this analysis, Sollitt and Cross model is used for the study of wave motion inside the porous structure. Employing small amplitude linear water wave theory, the present model of our interest is developed into a boundary value problem and solved analytically using the eigenfunction expansion method. Using the developed solution, the reflection coefficient and hydrodynamic wave forces on porous structure are computed for different values of parameters: the dimension of porous structure, the friction coefficient of porous structure, the current speed of following current and opposing current. This research paper suggests that suitable arrangement of rigid walls and surface-piercing partially extended porous structures can be an effective and economical approach for guiding ocean currents and protecting marine facilities against wave attacks.