Eigenfunction Expansion Research Articles

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.

Resonance expansion for quantum walks and its applications to the long-time behavior

In this paper, resonances are introduced to a class of quantum walks on \mathbb{Z} . Resonances are defined as poles of the meromorphically extended resolvent of the unitary time evolution operator. In particular, they appear inside the unit circle. Some analogous properties to those of quantum resonances for Schrödinger operators are shown. Especially, the resonance expansion, an analogue of the eigenfunction expansion, indicates the long-time behavior of quantum walks. The decaying rate, the asymptotic probability distribution, and the weak limit of the probability density are described by resonances and associated (generalized) resonant states. The generic simplicity of resonances is also investigated.

A general analytical solution for axisymmetric electro‐osmotic consolidation of unsaturated soil with semi‐permeable boundary

AbstractThis study proposes a closed‐form solution for axisymmetric electro‐osmotic consolidation of unsaturated soil under semi‐permeable boundary conditions. The governing equations are formulated to allow for vertical and radial flows of liquid and air phases. The techniques of eigenfunction expansion and Laplace transformation are employed to develop the exact solution for excess pore‐air (EPAP) and pore‐water pressures (EPWP). The proposed solution is first validated by comparing it to an existing solution, followed by verification through finite element simulations. Both methods of validation confirm the accuracy of the analytical solution. Then, based on the obtained solution, the effects of vertical flow, semi‐permeable boundary conditions, electrical voltage, electro‐osmosis conductivity and spacing ratio re/rw on the consolidation profile have been further investigated. Parametric studies show that the EPWP at the steady state depends on the electro‐osmosis conductivity and applied electricity gradient. In addition, the dissipation rates of EPWP and EPAP in the axisymmetric electro‐osmotic consolidation would be underestimated if the vertical flows are neglected. The semi‐permeable boundary conditions have great influences on the dissipation rate and the steady‐state solution. The proposed solution could serve as a theoretical basis for axisymmetric electro‐osmotic consolidation of unsaturated soil.

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.

Interaction between a uniform current and a submerged cylinder in a marginal ice zone

The interaction between a uniform current with a circular cylinder submerged in a fluid covered by a semi-infinite ice sheet is considered analytically. The ice sheet is modelled as an elastic thin plate, and the fluid flow is described by the linearised velocity potential theory. The Green function or the velocity potential due to a source is first obtained. As the water surface is divided into two semi-infinite parts with different boundary conditions, the Wiener–Hopf method (WHM) offers significant advantages over alternative approaches and is consequently adopted. To do that, the distribution of the roots of the dispersion equation for fluid fully covered by an ice sheet in the complex plane is first analysed systematically, which does not seem to have been done before. The variations of these roots with the Froude number are investigated, especially their effects or factorisation and decomposition required in the WHM. The result is verified by comparing with that obtained from the matched eigenfunction expansion method. Through differentiating the Green function with respect to the source position, the potentials due to multipoles are obtained, which are employed to construct the velocity potential for the circular cylinder. Extensive results are provided for hydrodynamic forces on the cylinder and wave profiles, and some unique features are discussed. In particular, it is found that the forces can be highly oscillatory with the Froude number when the body is below the ice sheet, whereas such an oscillation does not exist when the body is below the free surface.

Unconditional convergence of eigenfunction expansions for abstract and elliptic operators

We study the most general class of eigenfunction expansions for abstract normal operators with pure point spectrum in a complex Hilbert space. We find sufficient conditions for such expansions to be unconditionally convergent in spaces with two norms and also estimate the degree of this convergence. Our result essentially generalizes and complements the known theorems of Krein and of Krasnosel'skiĭ and Pustyl'nik. We apply it to normal elliptic pseudodifferential operators on compact boundaryless $C^{\infty }$ -manifolds. We find generic conditions for eigenfunction expansions induced by such operators to converge unconditionally in the Sobolev spaces $W^{\ell }_{p}$ with $p>2$ or in the spaces $C^{\ell }$ (specifically, for the $p$ -th mean or uniform convergence on the manifold). These conditions are sufficient and necessary for the indicated convergence on Sobolev or Hörmander function classes and are given in terms of parameters characterizing these classes. We also find estimates for the degree of the convergence on such function classes. These results are new even for differential operators on the circle and for multiple Fourier series.

Integral Transform Solution for Natural Convection Within Horizontal Concentric Annular Cavities

Abstract Natural convection inside horizontal concentric annular cavities is dealt with through the generalized integral transform technique (GITT), offering a hybrid numerical-analytical solution of the continuity, Navier–Stokes, and energy equations in cylindrical coordinates. The flow is in steady-state, laminar regime, two-dimensional, buoyancy-induced, and the governing equations are written in the streamfunction-only formulation. Two strategies of integral transformation are adopted to verify the best computational performance, namely, the usual one with eigenvalue problems for both streamfunction and temperature defined in the radial variable, and a novel alternative with eigenvalue problems defined in the azimuthal angular coordinate. First, the eigenfunction expansions convergence behavior is analyzed to critically compare the two integral transform solution strategies. Then, test cases for different aspect ratios and Rayleigh numbers are validated with experimental data from the classical work of Kuehn and Goldstein. A maximum relative deviation of 5% is found comparing the GITT results for the average Nusselt number against the experimental data, while an 8% maximum relative deviation is found comparing against an empirical correlation by the same authors. It is concluded that the GITT solution with the eigenvalue problem in the angular coordinate yields better convergence rates than the more usual eigenfunction expansion in the radial variable. This is due to the originally homogeneous boundary conditions in the angular direction, which do not require filtering for convergence enhancement, as opposed to the required filter in the radial direction that introduces a source term in the filtered equation for the streamfunction.

Modal analysis of a submerged elastic disk: A hypersingular integral equation approach

A method based on the hypersingular integral equation approach and the modal analysis is presented to consider the effects of the motion of a submerged elastic disk on the incoming waves. Initially, the governing boundary value problem is reduced to a two-dimensional integral equation with a hypersingular kernel. This integral equation is further reduced to a one-dimensional Fredholm integral equation of the second kind with the help of Fourier series expansions and a newly defined function. As a part of modal analysis, eigenfunction expansion based on natural modes of structural motion is considered to describe the motion of a thin circular elastic disk. Physical quantities, such as hydrodynamic force, added mass, damping coefficient, and surface elevation, are numerically evaluated. The computed numerical results are verified by comparing them with those for the rigid disk horizontally submerged in deep water. Apart from this, as a part of the analytical verification of our present analysis, the reciprocity relation has been included. The effects of different parameters (disk's rigidity, radius, submergence depth, and mode of vibrations) on the aforementioned physical quantities have been studied. The maximum hydrodynamic force occurs around Ka = 0.5, while the maximum added mass and damping coefficient occur around the wavenumber Ka = 0.3 and Ka = 0.5, respectively. The peaks of the hydrodynamic force and free surface elevation become sharper with the increasing values of the disk's size. The numerical results emphasize that the wave focusing can be controlled by changing the submergence depth, size, and rigidity of the disk.

Phase change in a one-dimensional functionally graded material

Functionally graded materials (FGMs) are a class of materials with engineered microstructure to achieve spatial variation in properties so as to serve a desired function. While FGMs have been well-researched for their structural properties, there is a lack of investigation of FGMs in solid-liquid phase change heat transfer processes. A well-designed functionally graded phase change material (FG-PCM) may overcome some of the well-known limitations of the phase change process. In order to investigate this possibility, this work presents a theoretical model to predict phase change propagation in a one-dimensional FG-PCM containing an arbitrary spatial variation in thermal conductivity, thermal diffusivity and latent heat. The problem is solved using the method of eigenfunction expansion to determine the temperature distribution, wherein, spatial variation in thermophysical properties listed above is accounted for by deriving a first-order matrix ordinary differential equation in the coefficients of the series solution. Results are found to correctly reduce to the well-known Stefan solution under special conditions. The impact of variation in thermophysical properties is investigated in detail. It is found that while spatial variation in thermal conductivity and latent heat has a significant effect on the rate of phase change, the impact of the thermal diffusivity function is negligible for realistic values of the Stefan number. The effect of the Stefan number governing this problem on the rate of melting is investigated. The nature of phase change under linear (1+aξ), polynomial (1+aξn) and exponential (eaξ) functional forms of thermophysical properties is investigated. It is shown that an appropriate choice of these spatial functions may help overcome the well-known self-limiting nature of phase change and produce a nearly-linear phase change propagation curve. For example, results show 33% increase in phase change propagation for a linear (1+5ξ) variation in thermal conductivity compared to baseline. The theoretical tools developed here may help design and optimize functionally graded PCMs towards improved phase change heat transfer in applications of practical interest.

Impact of Two Vertical Porous Barriers in Reflection of Water Waves and Mitigation of Wave Forces on a Rigid Floating Structure with Consideration of Uniform Current Over a Porous Sea-Bed

This work examines an oblique water wave scattering by a rigid floating structure with the consideration of uniform current and no current, in the presence of two porous barriers with distinct height and porous properties placed in front of the structure. The objective is to study the associated reflection and to find means to mitigate the waveload on the floating structure. These barriers are considered at some distance apart from each other. It is assumed that the train of water waves interacts with the porous barriers and the floating structure due to which it undergoes partial reflection. The fluid domain is considered in terms of five distinct sub-regions in each of which the solution of the relevant boundary value problem in terms of the velocity potential is obtained through the employment of the techniques of separation of variables and eigenfunction expansion. The resulting system of equations is obtained by utilizing the matching conditions corresponding to the continuity of mass flux and pressure across each virtual vertical boundary. The roots of the dispersion relation are evaluated and utilized while solving the system of linear equations through a standard matrix technique which gives the unknown coefficients. For understanding the role of the porous barriers, the reflection coefficient and the hydrodynamic forces are evaluated and analyzed. In this context, the impact of the space between these two porous barriers, their height, and the values of the porous-effect parameters are thoroughly examined. It is observed that placing of these barriers at an appropriate distance from each other can substantially mitigate both horizontal force and vertical force acting on the floating structure. This study reveals a periodic nature of the reflection coefficient corresponding to an increase in the distance between the barriers. Further, another important observation is that the reflection coefficient results in an increment as the height of these porous barriers is increased. It, as a result, produces a reduction of the wave forces acting on the floating structure. The study of the oblique wave scattering by the floating structure is extended to the case of the presence of a uniform current. In this case, an enhancement in the reflection coefficient is observed, and subsequently, horizontal and vertical forces on the floating structure decrease. The impact of some other parameters on the reflection coefficient and the forces is also studied, and some observations are recorded. The results are validated against two established results which shows a very close match.

TOPOGRAPHICAL EFFECTS ON WAVE SCATTERING BY AN ELASTIC PLATE FLOATING ON TWO-LAYER FLUID

This article illustrates the hydroelastic interactions between surface gravity waves and a floating elastic plate in a two-layer liquid with variable bottom topography under the assumptions of small amplitude waves and potential flow theory. In this study, semi-infinite and finite-length plates are considered. The eigenfunction expansion method is applied in the fluid region with uniform bottom topography. A system of differential equations (mild-slope equations) is solved in the fluid region with variable bottom topography. From the matching and jump conditions, the solution is expressed as a linear algebraic system from which all the unknown constants are computed. We explored the effects of density ratio, depth ratio, and bottom topography on the bending moment, shear force, and the deflection of the elastic plate. Results show that when the density ratio becomes closer to one, the occurred bending moment and shear forces to the elastic plates tend to diminish. The bending moment and shear forces to the pates are higher and lower at a smaller depth ratiofor the incident surface wave and interfacial waves, respectively. The variations in the bending moment, shear force, and plate deflection, caused by surface and interfacial waves, are observed to be in opposite trends, respectively. Bottom profiles similarly affect semi-infinite and finite-length plates when they undergo free-edge conditions. These effects, however, are substantial when the plate is simply supported at the edges. Elastic plate with free edges experiences lower deflection for concave-up and plane-sloping bottoms for incident surface and interfacial waves, respectively.

Analytical Model of an E-core Driver-pickup Coils Probe Applied to Eddy Current Testing of Multilayer Conductor

An analytical model of an E-core driver-pickup coils probe located above a multilayer conductor containing a hidden cylindrical conductor is presented. The truncated region eigenfunction expansion (TREE) method is used to deal with the axial symmetry problem, and the closed-form final expression of the induced voltage in the pickup coil is derived. The changes of the induced voltage in the pickup coil due to the hidden cylindrical conductor are examined and calculated in Mathematica. Experiments and finite element simulations are performed and the results are compared with the analytical results, and they are in good agreement.

Radiation resistance matrix for baffled simply curved plates for sound power applications

Sound power, a standard metric used to quantify product noise, is determined through the vibration-based sound power (VBSP) method. This method involves measuring surface velocities and utilizing an acoustic radiation resistance matrix, R, dependent on the structure's geometry. While R matrix expressions have been established for baffled flat plates, fully closed cylinders, and fully closed spheres, this work presents the first analytical expression tailored for baffled simply curved plates with uniform curvature. This development, based on eigenfunction expansion and the uniform theory of diffraction, extends the VBSP method's capabilities for accurate sound power assessment from these structures. Experimental validation involved testing three plates of varying curvature in a reverberation chamber, comparing the VBSP method with the ISO 3741 pressure-based standard. One of the curved plates underwent additional testing in an anechoic chamber following the ISO 3745 standard, confirming the VBSP method's accurate sound power measurements down to the 160 Hz one-third octave band. The same plate was tested in uncontrolled acoustic environments — a busy hallway and an outdoor location. The VBSP results showed strong agreement with ISO 3741, affirming the method's robustness for measuring sound power from baffled simply curved plates in acoustically challenging real-world conditions. This underscores the practicality of the VBSP method, enabling accurate sound power measurements of baffled curved plates in the presence of substantial background noise and environmental variability.

A sectional method to estimate the transmission coefficients of a new type pile-supported permeable breakwater

This article investigates the transmission coefficient of a novel pile-supported permeable breakwater featuring inclined porous plates under regular waves. To analyze the transmission coefficient, we propose a sectional method that involves multiplying the transmission coefficients of three distinct sections: the upper retaining wall and porous plates, the horizontal platform, and the piles. The calculation employs the eigenfunction expansion method and empirical formula for pile-supported breakwaters. This method is applicable for water levels that fall between the upper and lower edges of the horizontal platform. The analytical results are compared with experimental tests and show good agreement. Further analysis reveals that the transmission coefficient is significantly influenced by the increasing length of the horizontal platform. Moreover, it is observed that the effect of porosity ε is not particularly noticeable when ε>20%.

Wave interaction with multiple adjacent floating solar panels with arbitrary constraints

The problem of wave interaction with multiple adjacent floating solar panels with arbitrary types and numbers of constraints is considered. All the solar panels are assumed to be homogeneous, with the same physical properties, as well as modeled by using the Kirchhoff-Love plate theory. The motion of the fluid is described by the linear velocity potential theory. The domain decomposition method is employed to obtain the solutions. In particular, the entire fluid domain is divided into two types, the one below the free surface, and the other below elastic plates. The velocity potential in the free surface domain is expressed into a series of eigenfunctions. By contrast, the boundary integral equation and the Green function are employed to construct the velocity potential of fluid beneath the entire elastic cover, with unknowns distributed along two interfaces and jumps of physical parameters of the plates. All these unknowns are solved from the system of linear equations, which is established from the matching conditions of velocity potentials and edge conditions. This approach is confirmed with much higher computational efficiency compared with the one only involving eigenfunction expansion for the fluid beneath each plate. Extensive results and discussions are provided for the reflection and transmission coefficients of water waves, maximum deflection, and principal strain of the elastic plates; especially, the influence of different types and numbers of edge constraints are investigated in detail.

Model of an E-cored probe over layered conductor containing corrosion for eddy current testing

PurposeIn eddy current nondestructive testing, ferrite-cored probes are usually used to detect and locate defects such as cracks and corrosion in conductive materials. However, the generic analytical model for evaluating corrosion in layered conductor using ferrite-cored probe has not yet been developed. The purpose of this paper is to propose and verify the analytical model of an E-cored probe for evaluating corrosion in layered conductive materials.Design/methodology/approachA cylindrical coordinate system is adopted and the solution domain is truncated in the radial direction. The magnetic vector potential of each region excited by a filamentary coil is derived first, and then the expansion coefficients of the solution are obtained by matching the boundary and interface conditions between the regions and the subregions. Finally the closed-form expression of the impedance of the multi-turn coil is derived by using the truncated region eigenfunction expansion (TREE) method, and the impedance calculation is carried out in Mathematica. In the frequency range of 100 Hz to 10 kHz, the impedance changes of the E-cored coil and air-cored coil due to the layered conductor containing corrosion are calculated, respectively, and the influences of corrosion on the coil impedance change are investigated.FindingsAn analytical model for the detection and evaluating of corrosion in layered conductive materials using E-cored probe is proposed. The model can quickly and accurately calculate the impedance change of E-cored coil due to corrosion in layered conductor. The correctness of the analytical model is verified by finite element method and experiments.Originality/valueAn accurate theoretical model of E-cored probe for evaluating corrosion of multilayer conductor is presented. The analytical model can be used to detect the inhomogeneity of layered conductor, design ferrite-cored probe or directly evaluate the corrosion defects of layered conductors.

Theory and modeling of molecular modes in the NMR relaxation of fluids

Traditional theories of the nuclear magnetic resonance (NMR) autocorrelation function for intra-molecular dipole pairs assume a single-exponential decay, yet the calculated autocorrelation of realistic systems displays a rich, multi-exponential behavior, resulting in anomalous NMR relaxation dispersion (i.e., frequency dependence). We develop an approach to model and interpret the multi-exponential intra-molecular autocorrelation using simple, physical models within a rigorous statistical mechanical development that encompasses both rotational diffusion and translational diffusion in the same framework. We recast the problem of evaluating the autocorrelation in terms of averaging over a diffusion propagator whose evolution is described by a Fokker–Planck equation. The time-independent part admits an eigenfunction expansion, allowing us to write the propagator as a sum over modes. Each mode has a spatial part that depends on the specified eigenfunction and a temporal part that depends on the corresponding eigenvalue (i.e., correlation time) with a simple, exponential decay. The spatial part is a probability distribution of the dipole pair, analogous to the stationary states of a quantum harmonic oscillator. Drawing inspiration from the idea of inherent structures in liquids, we interpret each of the spatial contributions as a specific molecular mode. These modes can be used to model and predict the NMR dipole–dipole relaxation dispersion of fluids by incorporating phenomena on the molecular level. We validate our statistical mechanical description of the distribution in molecular modes with molecular dynamics simulations interpreted without any relaxation models or adjustable parameters: the most important poles in the Padé–Laplace transform of the simulated autocorrelation agree with the eigenvalues predicted by the theory.

Two-Dimensional Wave Interaction with a Rigid Body Floating near the Marginal Ice Zone

The interaction problem of waves with a body floating near the marginal ice zone is studied, where the marginal ice zone is modeled as an array of multiple uniformly sized floating ice sheets. The linear velocity potential theory is applied for fluid flow, and the thin elastic plate mode is utilized to describe the ice sheet deflection. A hybrid method is used to solve the disturbed velocity potential; i.e., around the floating body, a boundary integral equation is established, while in the domain covered by ice sheets, the velocity potential is expanded into an eigenfunction series, and in the far-field with a free surface, a similar eigenfunction expansion is used to satisfy the radiation condition. The boundary integral equation and the coefficients of the eigenfunction expansions are solved together based on the continuous conditions of pressure and velocity on the interface between the sub-domains. Extensive results for the equivalent Young’s modulus of the ice sheet array and hydrodynamic force on the body are provided, and the effect of individual ice sheet length as well as wave parameters are investigated in detail.

Oblique Wave Scattering by a Combination of Two Asymmetric Trenches of Finite and Infinite Depths

Abstract The focal point of the current study lies in investigating oblique wave scattering within the framework of linear potential theory, with particular attention to scenarios involving asymmetric trenches of both finite and infinite depths. By employing the eigenfunction expansion method, the physical problem undergoes a transformation into an equivalent boundary value problem. This newly formulated problem is characterized by a system of four weakly singular integral equations, which pertain to the horizontal component of velocity across the gaps situated above the edges of the trenches. The solution to these integral equations is achieved through the utilization of a multi-term Galerkin approximation method. This approach involves expansions using ultraspherical Gegenbauer polynomials as basis functions, coupled with the appropriate weight functions tailored to address the one-third singularity. Graphical representations are employed to depict the numerical evaluations of reflection and transmission coefficients across various non-dimensional parameters. These visualizations offer insight into the behavior and dependencies of these coefficients under different conditions. To validate the accuracy of the current model, it is compared against previously published results available in the literature.

Wave Dynamics Around a Floating Circular Flexible Plate Over a Permeable Bed

Abstract In this paper, to determine the efficacy of the porous bed on damping far-field wave energy, the wave dynamics around a circular plate is studied. By combining the appropriate boundary conditions, the unknown potential is attained for the free surface and the plate-covered region. The Bessel series solution is attained further, by employing the matched eigenfunction expansion technique. Wave force excitation on the circular plate, deflection of the plate, and flow distribution is calculated and examined to comprehend the efficacy of the current investigation. Moreover, the motion of the plate is assessed in the time domain. The study reveals a substantial reduction in wave amplitude on the plate’s leeward side because of the energy dissipation by the porous bed. Also the study concludes that for intermediate values of porosity with larger wavenumbers, approximately 50% of wave power is dissipated with respect to incident wave power.