Eigenfunction Expansion Approach Research Articles

Wave interaction with multiple thin flexible porous barriers in water of uniform finite depth

The interaction of water waves with multiple flexible porous barriers partially immersed in a uniform finite-depth fluid is investigated in the present study. Using linear theory, the physical problem is mathematically modeled as a boundary value problem. The eigenfunction expansion approach is used to generate the velocity potential function in terms of orthogonal eigenfunctions. The Havelock’s inversion formula is then used to convert the boundary value problem into a set of Fredholm-type integral equations. By choosing a suitable weight function, the multi-term Galerkin’s approximation technique is used to obtain the mathematical solutions of these integral equations in terms of the Chebyshev polynomial. Numerical solutions regarding the reflection and transmission coefficients, hydrodynamic forces, and dissipated wave energy at the barriers are obtained and graphically represented in several figures for various non-dimensional parameters. It is observed from the graphical representations that the permeability and flexibility of the barriers play a crucial role in the modeling of efficient breakwaters.

Time-dependent water wave scattering by a marine structure consisting of an array of compound porous cylinders

Under the assumption of linear wave theory, a semi-analytical model is developed to address the time-dependent water wave scattering problem involving a marine structure consisting of several circular rigid vertical cylinders, each of which is surrounded by a thin cylindrical porous wall with a water region between the inner cylinder and the wall. The problem is tackled by applying the eigenfunction expansion approach. The energy dissipation relation is also derived for the system of compound cylinders. The principal focus of this work lies in locating the optimum geometrical configurations for which the wave forces acting on the structure are minimal along a given wave direction in the frequency domain. Subsequently, the time-dependent response of the structure under the impact of a Gaussian wave pulse is examined. The wavenumber at which the force acting on the inner cylinder is at its maximum is obtained for a given value of the porous-effect parameter. Moreover, the value of that corresponding wavenumber decreases as the magnitude of the porous-effect parameter increases. The model is also carefully validated numerically against results available in the existing literature.

A mathematical study of a two-layer fluid flow system in the presence of a floating breakwater in front of VLFS

The present study deals with the usefulness of a surface-piercing porous structure of definite width in a two-layer fluid placed at a finite distance from a floating elastic plate to mitigate the hydrodynamic response of the elastic plate. We use linear water wave theory with the eigenfunction expansion approach to develop analytical solutions to the model problem in finite water depth. Here, we identify and explore the complete parameter space associated with the problem by examining the hydrodynamic coefficients for waves in surface and internal modes, plate elevation, shear force, bending moment, wave loads on the elastic plate, flow amplitude in the plate-covered region and wave attenuation. We investigate the effectiveness of structures of various configurations and geometries on wave scattering. This study demonstrates an oscillatory pattern of wave reflection in the confined region between the plate and the breakwater. Moreover, optima in the oscillatory pattern can be observed due to the resonating interaction of waves and the breakwater-plate system. Furthermore, the research shows that an ideal width and distance between the breakwater and floating plate can be calculated while assuming an appropriate configuration of the porous structure to create a breakwater-plate system with a reasonable efficiency that possesses both reflection and dissipation properties. The design of various types of coastal structures employed in the marine environment for the reflection and dissipation of wave energy at continental shelves dominated by stratified fluid, described here as a two-layer fluid, will significantly benefit from the findings of the current study.

Wave interaction with a concentric bottom-mounted cylinder system with dual porous ring plates

The interaction between linear surface waves and a concentric bottom-mounted cylinder system has been investigated using the eigenfunction expansion approach. This system contains an outer porous cylinder and an inner impermeable cylinder, which are connected by dual porous ring plates. Both cylinders are surface-piercing and rigidly installed on the flat bottom of the ocean, while the porous ring plates are fixed below the free water surface. The analytical solution of the velocity potentials can be obtained by matching the boundary conditions. After obtaining the velocity potentials, the wave force and free surface elevation are computed. The numerical results obtained for limiting cases agreed well with the published results. The results of the study show that reducing the draft, spacing, and permeability of the dual plates all contribute to reduce the horizontal force of the inner cylinder. However, the significance of the lower plate is mainly to cope with the condition where the water surface is lower than the upper plate.

Theoretical solution for thermo-mechanical crack-tip stress field for transversely graded materials

The present work deals with the structure of the thermo-mechanical stress field near the crack-tip in a graded material, where properties vary along the crack front in an exponential manner. With assumptions of steady-state conditions and insulated crack faces, expansion of the temperature field around the crack-tip is obtained. Results indicate that the heat flux in the radial and angular direction has inverse square root singularity near the crack-tip. The heat flux in the direction of gradation does not have any singularity. Navier’s equation for a thermo-mechanical system is solved using the Eigenfunction expansion approach to obtain the near crack-tip stress field. The explicit form of the first four terms in the expansion of the displacement field is obtained. It is observed that terms corresponding to in the stress field are directly affected by the temperature or heat flux. Effect of temperature is observed in out-of-plane normal stress only for Considering the first two terms in the asymptotic expansion, a parametric analysis, shows that thermal stresses help relax the constraint ahead of the crack-tip when the structure is constrained in the thickness direction. It helps in the suppression of crack growth mechanisms, which delays the failure of structures.

An eigenfunction expansion approach for the derivation of asymptotic expansions in financial valuation problems

An eigenfunction expansion approach for the derivation of asymptotic expansions in financial valuation problems

Analytical modelling of water wave diffraction by a bottom-mounted surface-piercing concentric dual-arc porous thin wall

The interaction of Airy waves with a bottom-mounted surface-piercing concentric dual-arc porous wall was theoretically investigated in this study via the eigenfunction expansion approach. The outer and inner arc-shaped walls were considered to be thin in thickness and rigidly fixed on a flat seabed. As critical elements, two virtual concentric cylinders extending the dual-arc wall structure were introduced to divide the entire fluid domain into three major sub-domains. Subsequently, a set of linear equations of unknown coefficients in the velocity potential was derived and solved by matching the boundary conditions between different regions. The current results of the three cases were consistent with the literature reports. Moreover, the dual-arc wall delivered a better shielding performance than the single-arc wall. In addition, parametric studies on porosity, annular spacing, opening angle, and incident wave angle were given and discussed. Furthermore, the hydrodynamic loads and wave elevations for a concentric structure with an outer arc-shaped wall and a concentric two-cylinder structure were compared using the same parameters. The hydrodynamic performance of the two structures were found to be similar, which reveals that former structure can replace the latter in certain water areas.

Solitary wave diffraction around a concentric porous cylindrical structure in front of a vertical wall

This paper investigates solitary wave diffraction around concentric porous cylindrical structure in front of a vertical wall, where the exterior cylinder is permeable and the interior cylinder is impermeable. This problem is transformed into an issue of bi-directional incident waves diffraction around two concentric cylindrical systems based on the image theory. An analytical solution of the problem is obtained by applying the eigenfunction expansion approach and Graf's addition theorem. Unlike previous studies using Airy wave, this paper uses solitary wave as incident wave. The hydrodynamic loads and wave elevations on the concentric cylindrical system are calculated and compared with existing work. An excellent agreement is obtained between the model and data. Parametric studies on porosity, annular spacing, incident wave angle, distance between the concentric structure and wall were also explored. The results indicated that wave loads on the exterior cylinder near a vertical wall could reach twice as large as the force on it in open water, while the magnitudes of wave loads on the interior cylinder are similar for the two cases. In addition, we compared the modelling results of wave surface elevations using solitary wave and Airy waves. It shows that the amplitude of the wave surface elevations caused by the solitary wave is significantly higher than that of the Airy wave, which means the wave effects on offshore structures will be underestimated when using the Airy wave model in the same shallow water conditions.

On the effects of chemical reaction on controlled heat and mass transfer in magnetized non-Newtonian biofluid through a long rectangular tunnel

A comprehensive analysis of heat transfer and chemical reaction of flow driven by the complex peristaltic wave in non-uniform long tunnel is presented in this article. Wall properties and the effects of chemical reaction in MHD flow are also observed in the current study. The flow problem is modeled using the law of conservations for mass, momentum, heat and chemical reaction. The nonlinear problem is simplified via long-wavelength/creeping flow assumptions and certain transformations. The resulting coupled linear partial differential equations are evaluated using the eigenfunction expansion approach; analytic solution is obtained. Simultaneous effects of all the physical parameters are discussed through graphs. Trapping phenomenon is also presented by drawing streamlines. In addition, a comparative analysis of fluid with Newtonian properties is performed with non-Newtonian ones. It is observed that the flow tends to reduce in non-Newtonian fluid than in Newtonian fluids.

Eigenfunction approach to generalized thermo-viscoelasticity with memory dependent derivative due to three-phase-lag heat transfer

The present article investigates a new mathematical model of generalized thermo-viscoelasticity based on memory-dependent derivative in the context of three-phase-lag heat transfer subjected to an instantaneous point heat source. The governing coupled equations associated with kernel function and time delay are constituted for a two-dimensional problem of semi-infinite space in an isotropic medium. The bounding surface of the half-space is assumed to be subjected to a time-dependent thermal shock and rigidly fixed. The Eigenfunction expansion approach is proposed to analyze the problem and, consequently, the analytical solutions of the thermodynamic field variables viz. temperature, displacement and stresses are obtained in the Laplace-Fourier transformed domain. The effects of the kernel function and time-delay factor on the distributions of the non-dimensional thermodynamic field variables are presented graphically and compared with the several well-known generalized thermoelasticity theories present in the literature. Finally, some remarkable points are mentioned in conclusion.

Hydrodynamic interactions between cnoidal waves and a concentric cylindrical structure with arc-shaped outer cylinder

The interaction between cnoidal waves and a concentric cylindrical structure with an arc-shaped outer wall has been investigated theoretically using eigenfunction expansion approach. The arc-shaped outer cylinder is permeable and thin, protecting the porous dual-cylinder structure. Three cylinders are surface-piercing and rigidly installed on the flat bottom of an ocean. Analytical solutions were obtained based on the first approximation to cnoidal waves. A series of linear equations pertaining to unknown coefficients were solved using matching boundary conditions. Hydrodynamic forces on the cylinders and the diffracted wave faces around the cylinders were evaluated, and the effects of water depth, annular spacing, angle of incidence and structure configuration including the porosity parameter of the cylinders and the opening angle are illustrated via numerical results. The numerical results obtained for limiting cases agreed well with published results for a solid cylinder. The results indicated that the wave forces on the inner structure did not change significantly when the opening angle was greater than 2π∕3. The hydrodynamic loads on the inner cylinder of the concentric cylindrical structure were significantly lower than the wave force on a single cylinder. Moreover, the value of wave force on the inner cylinder of a concentric dual-cylinder structure with an arc-shaped outer wall was similar to that on the inner cylinder of a concentric two-cylinder system.

On applicability and uniqueness of the correspondence principle to pie-shaped wedge (“wedge paradox”) with various boundary conditions

A new paradigm for re-examination of the concept of the correspondence principle (originally due to Dundurs) in association with the much-studied issue of “wedge paradox” is presented. This study connects for the first time two parallel streams of research: (i) Carothers-Sternberg-Koiter’s wedge paradox study and (ii) Williams' eigenfunction approach. First, a three-dimensional eigenfunction expansion approach for determination of the asymptotic (singular) stress fields in the vicinity of the fronts of infinite wedges of finite thickness, subjected to various wedge-flank boundary conditions, is presented. This analysis is then exploited to establish rigorous rules or conditions, which would depend on the prescribed wedge-flank boundary conditions, the related state of deformation and the elastic property (Poisson’s ratio), for validity (or lack thereof) of the assumptions implicit in the heuristic enunciation of the correspondence principle. More important, a hitherto unavailable analysis for uniqueness (or lack thereof) of the prevalence of such a heuristically postulated correspondence is also developed. Finally, the present study offers both mathematical as well as (heretofore missing) physical explanations behind the working of the correspondence principle.

The Combined Method of Ray Tracing and Diffraction and Its Application to Ultra-wideband Pulse Propagation

In this study the scattered-filed computation using the combined method of ray tracing and diffraction (CMRD) is revisited but with an extension to the backscattering computation. The concept of the equivalent phase object is considered as the key part in the developed CMRD method, and is analyzed mathematically with accurately derived expressions for its amplitude and phase function. A formulated CMRD method for the backscattering computation is developed in this work, which is then used in the forward modeling and numerical computations for ultra-wideband pulse propagation and backscattering from a perfectly conducting circular cylinder. The numerical simulation indicates that reasonable and good agreements can be achieved for comparisons between our CMRD method and exact eigenfunction expansion approach. It is expected that the theoretical model and method of backscattering calculation using CMRD can be applied to the image processing and target identification with measurements of backward-scattered electromagnetic and acoustic waves.

The Hamiltonian System Method for a Cylinder under the Action of Gravity

The classical problem of a viscoelastic circular cylinder under its own weight is analysed in the Hamiltonian system. On the basis of the theory of state space for axisymmetric problems and the application of dual variables of stresses, the technology of variable separation can be used. Hence the fundamental solutions, zero eigenfunctions and non-zero eigenfunctions, are derived. Because of the integrity of the solution space, various boundary conditions can be described by certain combinations general eigenfunctions. By using the adjoint symplectic relations, the eigenfunction expansion approach is applied to satisfying the boundary conditions.

Stochastic finite element response analysis using random eigenfunction expansion

A mathematical form for the response of the stochastic finite element analysis of elliptical partial differential equations has been established through summing products of random scalars and random vectors. The method is based upon the eigendecomposition of a system’s stiffness matrix. The computational reduction is achieved by only summing the dominant terms and by approximating the random eigenvalues and the random eigenvectors. An error analysis has been conducted to investigate the effect of the truncation and the approximations. Consequently, a novel error minimisation technique has been applied through the Galerkin error minimisation approach. This has been implemented by utilising the orthogonal nature of the random eigenvectors. The proposed method is used to solve three numerical examples: the bending of a stochastic beam, the flow through a porous media with stochastic permeability and the bending of a stochastic plate. The results obtained through the proposed random eigenfunction expansion approach are compared with those obtained by using direct Monte Carlo Simulations and by using polynomial chaos.

Analysis of magnetohydrodynamic natural convection in closed cavities through integral transforms

A hybrid numerical-analytical solution is proposed to analyze MHD (magnetohydrodynamic) natural convection of an electrically-conducting fluid within a square cavity, differentially heated at the sidewalls and subjected to an inclined external magnetic field. The first goal is to expand the spectrum of application of the so called Generalized Integral Transform Technique (GITT), dealing with a multiphysics formulation, while further demonstrating the relative merits of the proposed eigenfunction expansion approach in handling highly nonlinear and coupled systems of partial differential equations. The second goal is to provide a set of benchmark results in this important application for quantities of practical interest in determining the heat transfer rates, such as the average Nusselt number. The two-dimensional steady state equations are written in dimensionless form using the streamfunction-only formulation and are subsequently solved with the GITT approach, under automatic relative error control. Critical comparisons are performed against previous work reported in the literature, both computational and experimental, together with the corresponding physical interpretations, for different values of the governing parameters, such as Grashof number, Hartmann number, Prandtl number, and magnetic field inclination angle.

Dynamic analysis of a cable-stayed bridge subjected to a continuous sequence of moving forces

In this work, an eigenfunction expansion approach is used to study the dynamic response of a cable-stayed bridge excited by a continuous sequence of identical, equally spaced moving forces. The nonlinear dynamic response of the cable-stayed bridge is obtained by simultaneously solving nonlinear and linear partial differential equations that govern transverse and longitudinal vibrations of stay cables and transverse vibrations of segments of the deck beam, respectively, along with their boundary and matching conditions. Orthogonality conditions of exact mode shapes of the linearized cable-stayed bridge model are employed to convert the coupled nonlinear partial differential equations of the original nonlinear model to a set of ordinary differential equations by using the Galerkin method. The dynamic response of the cable-stayed bridge is numerically solved. Convergence of the dynamic response from the Galerkin method is investigated. Effects of close natural frequencies, mode localization, the distance between any two neighboring forces, and geometric nonlinearities of stay cables on the forced dynamic response of the cable-stayed bridge are captured using a convergent modal truncation.

Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach

This paper develops an eigenfunction expansion approach to solve discretely monitored first passage time problems for a rich class of Markov processes, including diffusions and subordinate diffusions with jumps, whose transition or Feynman–Kac semigroups possess eigenfunction expansions in $L^{2}$ -spaces. Many processes important in finance are in this class, including OU, CIR, (JD)CEV diffusions and their subordinate versions with jumps. The method represents the solution to a discretely monitored first passage problem in the form of an eigenfunction expansion with expansion coefficients satisfying an explicitly given recursion. A range of financial applications is given, drawn from across equity, credit, commodity, and interest rate markets. Numerical examples demonstrate that even in the case of frequent barrier monitoring, such as daily, approximating discrete first passage time problems with continuous solutions may result in unacceptably large errors in financial applications. This highlights the relevance of the method to financial applications.

Wave Interaction with Porous Vertical Breakwater in Two-layer Fluid

In this paper, the two-layer fluid model is used to investigate the wave interaction with porous vertical breakwater resting on permeable seabed in two-layer fluid. Based on the Biot seepage consolidation theory and linear wave theory, the analytical solutions to the wave potentials and the seepage pressures inside permeable seabed induced by surface wave and inner wave are derived by applying eigenfunction expansion approach. And accordingly, the analytical expressions of wave-induced uplift force and overturning moment acting on the bottom of porous breakwater are obtained. Numerical results are presented to illustrate the seepage loads caused by inner wave may have same order of magnitude as that induced by surface wave, and the wave-induced seepage loads may have the same order of magnitude as direct wave loads. It is found that the porosity of the breakwater surface may lead an obvious reduction in direct wave loads on breakwater and a certain reduction in wave-induced seepage loads on the bottom of breakwater, especially, lead to a significant reduction in the inner wave-induced overturn moment.

Effects of Cnoidal Wave-Induced Seepage on Vertical Breakwater Resting on Permeable Elastic Seabed

Theoretical investigations on cnoidal waves interacting with breakwater resting on permeable elastic seabed are presented in this paper. Based on the shallow water reflected wave theory and Biot consolidation theory on wave-induced seepage pressure, the analytical solutions to first order cnoidal wave reflection and wave-induced seepage pressure are obtained by the eigenfunction expansion approach. Numerical results are presented to show the effects of depth of water, breakwater geometry on cnoidal wave-induced seepage uplift force and overturning moment. Compared with Airy wave theory, in certain shallow water conditions, the shallow water wave theory can more effectively illustrate wave nonlinearity effect in wave load prediction.