Eigenfunction Expansion Research Articles

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.

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.

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.

Analysis of Wave-Induced Forces on a Floating Rectangular Box with Analytical and Numerical Approaches

A three-dimensional mathematical hydrodynamic model associated with surface wave radiation by a floating rectangular box-type structure due to heave, sway, and roll motions in finite water depth is investigated based on small amplitude water wave theory and linear structural response. The analytical expressions for the radiation potentials, wave forces, and hydrodynamic coefficients are presented based on matched eigenfunction expansion method (MEFEM). The correctness of the analytical results of wave forces is compared with the construction of a numerical model using the open-source boundary element method code NEMOH. In addition, the present result is compared with the existing published experimental results available in the literature. The effects of the different design parameters on the floating box-type rectangular structure are studied by analyzing the vertical wave force, horizontal wave force, torque, added mass, and damping coefficients due to the heave, sway, and roll motions, and the comparison analysis between the forces is also analyzed in detail. Further, the effect of reflection and transmission coefficients by varying the structural width and drafts are analyzed.

Dual-sided melting of a one-dimensional composite wall comprising two distinct phase change material (PCM) layers

Past literature on theoretical modeling of solid-liquid phase change heat transfer mostly addresses a single phase change material (PCM), whereas, a series combination of two or more PCMs is often used in practical applications. This work presents an approximate analytical model for the melting/freezing of a one-dimensional composite of two PCMs being heated/cooled from both ends. It is shown that up to three phase change fronts may simultaneously exist in this problem. In addition to direct melting/freezing from the respective walls, it is shown that each PCM may also melt/freeze due to heat transfer through the other PCM. Using a quasi-stationary assumption, temperature field during the multi-stage phase change process is approximated by solving a multilayer transient thermal conduction problem using eigenfunction expansion. Good agreement with numerical simulations, and with the Stefan solution for a special case is demonstrated. The impact of key non-dimensional parameters on performance metrics including total time for phase change is determined. A quantitative prediction of conditions leading to the thermodynamically favorable situation of simultaneous completion of phase change of two commonly used PCMs is carried out. This work improves the theoretical understanding of phase change in composite PCMs, leading to optimization of practical thermal systems.

Nonlinear vibrations of a slightly curved Maxwell viscoelastic pipe conveying two phase flow under various boundary conditions

Two phase flows are common in pipeline engineering, and most times, the assumption of single phase flow is usually assumed. In this paper, the dynamics of a slightly curved Maxwell viscoelastic pipe conveying two phase flow under four boundary conditions; cantilever, simple supported, clamped simply supported and clamped clamped supported is investigated. Eigenfunction expansion was used to discretize the PDE into four modes. The linear analysis results show that the Maxwell viscoelastic parameter has no significant effect on the first and the second modes of the vibration, but on the third and fourth modes for the three boundaries conditions considered. The void fraction has effects on the frequency and critical velocity only if the densities of the components of the two phase fluid are significantly large. The nonlinear dynamics shows that Maxwell viscoelastic parameter reduces the bifurcation position and changes the bifurcation point from being sharp to a gentle curve at the bifurcation point. Further results were presented for the Maxwell parameter and velocity of fluid and temperature loading on the pipe, which are very important to understanding the behavior of Maxwell viscoelastic pipe conveying two phase flow.

Oblique wave trapping by periodic array of two-layer pile rock breakwaters placed on uneven seabed

ABSTRACT This research work evaluates the wave trapping efficiency of the periodic array of two-layer pile-rock breakwaters (TPRBs) placed far away from vertical rigid wall. The analytical model is developed to study the effect of TPRBs on the variation of wave reflection and reduction of fluid force experienced by wall based on the classical potential flow and porous media flow theory. The effect of the uneven seabed on wave trapping by TPRBs is added to show its significance, and the porous structure consists of a pair of layers with individual porosities. The seaside and leeside protection walls are introduced to secure the two-layer rock from vertical and horizontal failures. Thus, the continuity of velocity and pressure is applied in the vertical and horizontal directions. The eigenfunction expansion method based on orthogonal mode-coupling relation is used to solve the unknown coefficients. The effect of incident wave properties and TPRBs physical properties on wave trapping is reported as a function of breakwater width, gap region, and trapping chamber length. The correctness of the present study results is verified with the available results. The study strongly suggests that the variation of wave reflection by TPRBs is a trivariate function, which would depend upon the width, spacing, and porosity of the TPRBs. Around 30–40% reduction of wave reflection is obtained after introducing the TPRBs.

INTEGRAL TRANSFORM SOLUTION OF HEAT CONDUCTION IN ANISOTROPIC HETEROGENEOUS MEDIA

The generalized integral transform technique (GITT) is advanced to deal with conduction heat transfer in anisotropic heterogeneous media. A formal solution for exact integral transformation of conduction in anisotropic media is extended to account for heterogeneities expressed as space variable equation coefficients and source terms. The proposed eigenfunction expansion is based on biorthogonal eigenvalue problems, which results in an exact integral transformation for a class of linear problems and in a coupled transformed ordinary differential system for nonlinear situations. An algorithm is proposed for the associated eigenvalue problems, also handled through the GITT, by considering simpler biorthogonal eigenvalue problems of known analytical solution, leading to transformed algebraic eigenvalue problems. A single domain reformulation strategy is adopted to merge the information from multiple regions and materials, either isotropic or anisotropic, into one single diffusion equation. A two-dimensional transient test case is considered that presents an abrupt transition between isotropic and anisotropic materials yielding a marked change in thermal behavior in a defined region of interest formed by the anisotropic inclusion. Convergence behavior of the integral transform solution is illustrated, and the fully converged results are employed as a benchmark to inspect the accuracy of a commercial finite element code for automatically defined mesh refinement levels.

Flexural-gravity wave scattering by an array of bottom-standing partial porous barriers in the framework of Bragg resonance and blocking dynamics

Flexural-gravity wave scattering by an array of vertical porous barriers of various configurations is investigated in finite water depth from the viewpoint of blocking dynamics. A scattering matrix is introduced for the velocity potentials using the canonical eigenfunction expansion method developed for a single propagating wave mode to account for the multiple propagating wave modes. Subsequently, appropriate matching conditions are applied at the interface boundaries and edges to solve the physical problem. Apart from multiple barriers of equal length, the efficiency of four different barrier configurations of unequal lengths is investigated. This study shows that out of these four barrier configurations, the convex and increasing order of the barrier arrangements are more effective as wave-dissipating systems than the concave and decreasing order of the barriers. Bragg reflection occurs in the case of two or more barriers for a specific value of porosity and suitable barrier configuration, whose amplitude decreases with an increase in the number of barriers due to the dissipation of wave energy. The presence of three propagating wave modes in the blocking paradigm leads to mode conversion within a certain range of the frequency space. Both the scattering and dissipation coefficients are influenced by the wave energy transfer rates and the amplitudes of incident, reflected, and transmitted wave modes. This investigation exhibits the presence of discontinuities in the scattering coefficients at frequencies where blocking and mode conversion occur. The frequency domain results are used to simulate the plate displacement in the time domain by applying the Fourier transform.

Analyzing monopole sources modeling with structural variations and material contrast: An analytical perspective

This study presents a comprehensive analysis of modeling acoustic radiation from monopole sources within waveguides, considering variations in structural parameters, including backward passage and the use of sound-absorbent materials. The investigation employs analytical techniques, including mode-matching methods and Fourier transforms, to address the intricate boundary value problem governing acoustic behavior in waveguides. Eigenfunction expansions are utilized to describe acoustic responses in various segments, leading to systems of linear algebraic equations that are numerically solved after truncation. The research examines the influence of critical factors such as frequency, thickness of absorbent materials, and chamber length on power components, including reflection, absorption, transmission, and propagation, across different regions of the waveguide. Notably, the choice of sound-absorbent material emerges as a significant factor affecting scattering characteristics, with A-Glass outperforming E-Glass and Steel Wool in various scenarios. These insights offer valuable knowledge for optimizing acoustic systems in engineering applications, facilitating the design and enhancement of sound control mechanisms within waveguides.