Eigenfunction Expansion Technique Research Articles

Compressible ocean waves generated by sudden seabed rise near a step-type topography

A typical tsunami generation occurs through submarine earthquakes leading to large volume displacement. The corresponding mathematical problem involves modeling surface water waves generated by an arbitrary temporal motion of the ocean floor. The propagation of tsunami wave and the subsequent scattering from a sudden drop in bathymetry away from the ground motion is studied following linearized water wave theory and a weakly compressible ocean, including static oceanic background compression. The Fourier transformation and eigenfunction expansion techniques are employed to find the surface displacement and pressure profiles by leveraging appropriate matching conditions between regions of different depths. A novel energy balance relationship is derived by considering both the pure-gravity and acoustic-gravity modes. The model is validated in the limit that the depth difference approaches zero, showing a vanishing reflection contribution from the depth change. An efficient numerical code is developed that accurately captures the contribution of the cutoff frequencies of acoustic-gravity modes. Apart from the time-domain propagation of tsunami waves away from the origin, standing wave formations are observed within the shallow region, supported by significantly large pressure fluctuations in time. These standing waves or, equivalently, the pressure fluctuations sustain longer for larger ocean depth. The increase in tsunami speed in the deeper region is readily visible in the time-domain simulations. A three-dimensional axisymmetric solution is also developed, and results show a more gradual sloping tsunami wavefront compared to the equivalent two-dimensional solution for shallower depths. Animation movies corresponding to the two- and three-dimensional surface profiles are provided for better visualization.

Scattering of Oblique Water Waves by Two Vertical Barriers Over Trench-Type Bottom

The problem of scattering of oblique water waves by two surface-piercing vertical thin rigid barriers over a trench-like bottom is investigated within the framework of linear water wave theory. The eigenfunction expansion technique is used to obtain a system of algebraic equations, which is further solved using the algebraic Least squares method. The reflection and transmission coefficients are computed and contrasted with the prior results. Physical characteristics such as surface elevation and horizontal force on both barriers are also determined and depicted graphically against various parameters to see their behavior. It is possible to strategically construct barriers and trench-like bottoms to regulate how waves reflect and propagate. This might mitigate the wave impact on ships, improving the safety, effectiveness and longevity of ships navigating through the channels. The highlights of the given problem include comparing the impacts of different depth ratios and barrier placements, followed by a study of the modulation parameter to evaluate standing wave patterns in the trench region. The specific circumstances for the formation of the standing wave are identified. Hence, understanding this study is critical for safe navigation because standing waves may produce hazardous conditions for boats, making navigation difficult and causing substantial damage.

Interaction of short-crested waves with cylinder and concentric segmented arc breakwaters

Using the outer breakwater of Palm Jumeirah in Dubai as a physical geometric model, the diffraction problem of short-crested waves interacting with concentric segmented arc structures was theoretically investigated using linear water wave theory. In this study, the watershed was partitioned into two regions, and a unique coordinate system was established to describe the velocity potential function within each region. The interface between the two regions connects the velocity potentials across them using the matching eigenfunction expansion technique. By accounting for the boundary conditions, the integral equation was streamlined into a series of algebraic equations, which were subsequently resolved to determine an unidentified function. The precision of the model was validated through a comparison with findings from the literature. Numerical investigations have elucidated the correlation between changes in wave force, cylinder run-up, and parameters such as the spacing between adjacent arcs, opening angle, and porous coefficient. The numerical results indicated that the impermeable segmented arcs structure provides better protection against short-crested and plane waves, whereas the permeable segmented arc structure provides better protection against standing waves. Interestingly, for a segmented three-arc structure, when the segments are moderately spaced, the protection of the interior region is superior to that of an unsegmented arc structure.

Three-Dimensional Singular Stress Fields and Interfacial Crack Path Instability in Bicrystalline Superlattices of Orthorhombic/Tetragonal Symmetries

First, a recently developed eigenfunction expansion technique, based in part on the separation of the thickness variable and partly utilizing a modified Frobenius-type series expansion technique in conjunction with the Eshelby–Stroh formalism, is employed to derive three-dimensional singular stress fields in the vicinity of the front of an interfacial crack weakening an infinite bicrystalline superlattice plate, made of orthorhombic (cubic, hexagonal, and tetragonal serving as special cases) phases of finite thickness and subjected to the far-field extension/bending, in-plane shear/twisting, and anti-plane shear loadings, distributed through the thickness. Crack-face boundary and interface contact conditions as well as those that are prescribed on the top and bottom surfaces of the bicrystalline superlattice plate are exactly satisfied. It also extends a recently developed concept of the lattice crack deflection (LCD) barrier to a superlattice, christened superlattice crack deflection (SCD) energy barrier for studying interfacial crack path instability, which can explain crack deflection from a difficult interface to an easier neighboring cleavage system. Additionally, the relationships of the nature (easy/easy, easy/difficult, or difficult/difficult) interfacial cleavage systems based on the present solutions with the structural chemistry aspects of the component phases (such as orthorhombic, tetragonal, hexagonal, as well as FCC (face-centered cubic) transition metals and perovskites) of the superlattice are also investigated. Finally, results pertaining to the through-thickness variations in mode I/II/III stress intensity factors and energy release rates for symmetric hyperbolic sine-distributed loads and their skew-symmetric counterparts that also satisfy the boundary conditions on the top and bottom surfaces of the bicrystalline superlattice plate under investigation also form an important part of the present investigation.

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.

Simplified analytical solution for time dependent deformation of soft soil improved with pervious column considering load transfer between column and soil

This paper presents a simplified analytical solution to analyse the coupled excess pore water pressure dissipation and deformation response of a composite stone column – soft soil foundation. The mathematical formulation is derived considering the combined radial – vertical flows of excess pore water pressure in stone column and soft soil areas (orthotropic permeability for each area), and adopting the settlement pattern for the composite ground suggested by an existing study in the literature. The homogeneous consolidation formulations are solved first to develop Green’s function for final excess pore water pressure solutions, employing the method of separation of variables and eigenfunction expansion technique. Then, the final solutions for excess pore water pressure corresponding to the nonhomogeneous consolidation formulations are derived in terms of Green’s formula, which can be expanded readily into a double series solution. The obtained analytical solutions can be used to examine the variation of excess pore water pressure against time at any point in the composite foundation. Thus, the consolidation settlements of stone column and soft soil areas and the differential settlement between them can be captured. The average differential settlement between the soil and column areas is used to compute the shear strains and shear stresses in the composite ground during the consolidation process. The proposed analytical solution is validated via a worked example in conjunction with a verification exercise against finite element simulation. The analytical predictions are presented in terms of the total vertical stress variations and excess pore water pressure dissipations against depth and time, the settlements and average degrees of consolidation for stone column and surrounding soft soil, and the shear stress distribution in the soil region. The capability of proposed analytical solution is also verified against field measurements of a full-scale test of soil - column composite ground (pervious column). The verifications show that the obtained analytical solution can predict the coupled excess pore water pressure dissipation – deformation response of soft soils improved by pervious columns such as stone columns and soil- deep cement mixing columns reasonably well.

VISCOSITY AND HALL EFFECTS ON UNSTEADY MHD CASSON FLUID FLOW THROUGH A PARALLEL POROUS PLATE

This study investigates the temperature viscosity and Hall effects of a Casson fluid flow through a porous medium of a viscous incompressible fluid bounded by two parallel porous plates under the influence of thermal radiation and chemical reaction. A uniform suction and injection are applied perpendicular to the plates while the fluid motion is subjected to variable pressure gradient. The transformed conservation equations are solved analytically subject to physically appropriate boundary conditions by using perturbation and Eigenfunction expansion techniques. The influence of a number of emerging non-dimensional parameters namely, variable pressure gradient parameter, suction parameter, radiation parameter and Hartman number are examined. It is observed that the primary velocity profile is increased with increasing temperature dependent viscosity while increase in Hall parameter leads to increase in secondary velocity profile of the flow.

Eigenfunction Expansion (EFE) Analysis of Cylindrical Metasurfaces—Part II: Sectors and Multishells

This article presents a formulation of an eigenfunction expansion (EFE) technique for the analysis of multishell cylindrical metasurface (MS) configurations and tests the validity of zero-thickness sheet models for simulating practical structures. The formulation homogenizes the MS, modeling the surface as a zero-thickness discontinuity via the generalized sheet transition conditions (GSTCs) characterized by surface susceptibilities. Comparison with full-wave simulations in commercial solvers show that, for a thin deep subwavelength unit cell, single-shell arcs with increasing curvatures (up to at least half of a wavelength in radius) are accurately captured by the use of the model. The simulations also show that the edge diffraction ff the finite length arcs are well modeled. This is an important result as it indicates that susceptibilities extracted from full-wave unit-cell simulations assuming a flat periodic infinite surface are appropriate for use in high curvature finite surfaces. A second set of simulations showed that the EFE predicted multishell field distributions accurately when compared with unit-cell-based simulations. A final set of simulations for single-and double-shell configurations is used to demonstrate the limitations of the GSTC approach for unit cells of appreciable thickness.

TIME DEPENDENT PRESSURE GRADIENT EFFECTS ON UNSTEADY MHD COUETTE FLOW THROUGH A PARALLEL POROUS PLATE

In this paper, the unsteady MHD Couette flow through a porous medium of a viscous incompressible fluid bounded by two parallel porous plates under the influence of thermal radiation and chemical reaction is investigated. A uniform suction and injection are applied perpendicular to the plates while the fluid motion is subjected to time dependent pressure gradient. The transformed conservation equations are solved analytically subject to physically appropriate boundary conditions by using perturbation and Eigenfunction expansion techniques. The effects of some non-dimensional parameters are graphically represented and interpreted. It is observed that primary velocity is maximum when pressure gradient is time dependent as compared to when it is independent on time. Also, increase in temperature dependent pressure gradient leads to oscillation in primary velocity along distance y.

Employment of Fracture Mechanics Criteria for Accurate Assessment of the Full Set of Elastic Constants of Orthorhombic/Tetragonal Mono-Crystalline YBCO

The effect of elastic constants, cij, on the nature (easy or difficult) of a cleavage system in mono-crystalline YBa2Cu3O7−δ is investigated by employing a novel three-dimensional eigenfunction expansion technique, based in part on the separation of the thickness variable and partly on a modified Frobenius-type series expansion technique in conjunction with Eshelby–Stroh formalism. Out of the three available, complete sets of elastic constants, only the experimental measurements using resonant ultrasound spectroscopy merit serious attention, despite reported values of c12 and, to a lesser extent, c66 being excessively high. The present investigation considers six through-thickness crack systems weakening orthorhombic mono-crystalline Yttrium barium copper oxide (YBCO) plates. More importantly, the present investigation establishes sufficient conditions for crack path stability/instability, which entail a cleavage system being easy or difficult, i.e., whether a crack would propagate in its original plane/direction or deflect to a different one. This criterion of fracture mechanics is then employed for accurate determination of the full set of elastic constants of superconducting mono-crystalline YBCO. Finally, heretofore unavailable results pertaining to the through-thickness variations of stress intensity factors and energy release rates for a crack corresponding to symmetric and skew-symmetric hyperbolic cosine loads, which also satisfy the boundary conditions on the plate surfaces, bridge a longstanding gap.

Oblique water wave scattering by a floating bridge fitted with a rectangular porous structure and the resulting waveload mitigation

Scattering of oblique water waves by a floating bridge with rectangular porous walls fitted on its vertical sides in finite ocean depth is studied. An oblique water wave is assumed to interact with the structure and is partially reflected. The whole fluid region is split into five subregions in which two regions are porous regions, and the remaining three are water regions. The boundary value problems with respect to the velocity potentials are solved in all sub-regions by employing eigenfunction expansion technique. In order to figure out the reflection coefficients and hydrodynamic forces, a set of algebraic equations is formed from the appropriate matching conditions. Dispersion relation roots are analyzed and used in the system of linear equations. The effect of a number of parameters, such as the width of the porous wall, friction factor, angle of incidence, and draft of the bridge, on the reflection coefficients and hydrodynamic forces, is investigated. Porosity does not affect the reflection coefficient up to some certain value of the wavenumbers. The behavior of the reflection coefficient due to changes in the width of the porous wall is analyzed in order to choose appropriate width. It is noted that, because of the rectangular porous wall, a substantial amount of wave energy gets reflected back, and the waveload thus gets reduced on the floating bridge. All these observations lead to a suggestion of selection of suitable values of various parameters so that the reflection and forces acting on the floating bridge can be mitigated which will provide a safer working environment for the bridge. Graphs support all of our observations. The excellent agreement between our results and earlier results establish the validity of the present model.

Mitigation of Wave Impact on Sea Wall by a Floating Elastic Plate and a Porous Structure

AbstractThe purpose of the present study is to reduce the high wave load on a sea wall by utilizing an elastic plate (EP) kept at fixed distance from a porous structure (PS). Thin plate theory is used to model the flow past EP, while Sollit and Cross theory is used to model the flow past PS. A linear potential theory-based analytical solution to the current problem is developed using the eigenfunction expansion technique. To understand the effect of PS and EP in creating tranquility zone and minimum wave loads on the rigid wall, horizontal wave force on the wall, reflection coefficient, dissipation coefficient, and free surface elevation are computed and analyzed for different values of width and friction factor of PS, flexural rigidity and length of EP, angle of incidence, and distance between PS and EP, and the distance between EP and rigid wall. The study demonstrates that both structures considerably reduce the stress on the rigid wall and the wave reflection. It is found that the force on the wall shifted to the left as the width and frictional factor of PS increased. Furthermore, it is observed that PS effectively minimizes the free surface elevation in the region between EP and the wall. It is also found that an effective tranquility zone may be produced, which will put less wave force on the rigid wall, with sufficient spacing between PS and EP, and EP and the wall. The given model is expected to assist in preserving various coastal assets significantly.

An Analytical Solution for 2D Plane Strain Consolidation in Unsaturated Soils with Lateral and Vertical Semipermeable Drainage Boundaries under Time-Dependent Loading

This paper has proposed a closed-form solution for two-dimensional (2D) plane strain consolidation in unsaturated soils with lateral and vertical semipermeable drainage boundaries under time-dependent loading. By using the eigenfunction expansion technique, the governing equations are expanded into coupled ordinary differential equations, and the final solution of pore pressures is obtained by the integral transform method. The degraded proposed solution is then verified with the existing solution. The results show full agreement with the solutions, which indicates the proposed solution and program are completely correct. Meanwhile, the comparison between the analytical solutions and the finite-element simulations also demonstrates the correctness of the proposed solution. Subsequently, the effect of semipermeable parameters and loading parameters on the consolidation behavior has been investigated through the obtained analytical solution. And the distribution of consolidation profiles along two different directions has also been studied. It can be concluded that time-dependent loading changes the dissipation process obviously compared with constant loading.

An analytical study of scattering of water waves by a surface-piercing bottom-mounted compound porous cylinder placed on a porous sea-bed

The scattering problem of a bottom-mounted surface-piercing compound porous cylinder located on a porous sea-bed is theoretically investigated under the assumption of small amplitude wave theory. The compound cylinder is comprised of an impermeable upper cylinder and a porous lower cylinder beneath it. The boundary conditions on the porous boundaries follow Darcy’s law by assuming fine pores in the porous structure. The whole fluid region is split into three bounded and an unbounded sub-regions, within which the individual velocity potentials are found by using eigenfunction expansion technique. Further, utilization of the matching conditions along the boundaries of individual successive regions leads to a semi-analytical solution of the proposed problem. The impact of the non-dimensional porous-effect parameter of the cylindrical wall, the draft ratio, radius ratio and the sea-bed porosity on waveloads and wave run-ups are studied. The results show that suitable consideration of porosity and structure parameters enhance the efficiency of the proposed compound cylinder in mitigating wave impact. Furthermore, the hydrodynamic waveload acting on the lower as well as upper cylinder can be reduced by the suitable positioning of the annular spacing of the system, which will provide explicit information for the purpose of engineering design in the coastal area.

Structural performance of a submerged bottom-mounted compound porous cylinder on the water wave interaction in the presence of a porous sea-bed

Scattering problem of a submerged bottom-mounted compound porous cylinder located on a porous sea-bed is theoretically investigated under the assumption of linear potential flow theory. The compound cylinder is comprised of an impermeable inner cylinder and a porous outer cylinder. The boundary conditions on the porous boundaries follow Darcy's law by assuming fine pores in the porous structure. The whole fluid region is split into three bounded and unbounded sub-regions, within which the individual velocity potentials are found by using the eigenfunction expansion technique. Furthermore, utilization of the matching conditions along the boundaries of individual successive regions leads to a semi-analytical solution of the proposed problem. The impact of the non-dimensional porous-effect parameter of the cylindrical wall, the draft ratio, radius ratio, and the sea-bed porosity on wave loads and free-surface elevation are studied. In addition, the wave power dissipation for the system is calculated by integrating the power absorbed by the outer cylinder porous wall via direct method. Also, the far-field scattering coefficients are obtained with the help of asymptotic forms of Hankel functions in the plane wave representation form. Numerical results for the far-field scattering coefficient and power dissipation are investigated for various parameters. The theoretical model is verified by comparing it with the results of the conventional analytical work and experimental work. The results show that suitable consideration of porosity and structure parameters enhances the efficiency of the proposed compound cylinder in mitigating wave impact. Furthermore, the hydrodynamic wave load acting on the inner and outer cylinders can be reduced by the suitable positioning of the annular spacing of the system, which will provide explicit information for the purpose of engineering design in offshore and coastal regions.

Solution of One-Dimensional Contaminant Flow Problem Incorporating the Zero Order Source Parameter by Method of Eigen-Functions Expansion

A semi – analytical study of a time dependent one – dimensional advection – dispersion equation (ADE) with Neumann homogenous boundary conditions for studying contaminants flow in a homogenous porous media is presented. The governing equation which is a partial differential equation incorporates the advection, hydrodynamic dispersion, first order decay and a zero order source effects in the model formulation. The velocity of the flow was considered exponential in nature. The solution was obtained using Eigen function expansion technique after a suitable transformation. The results which investigate the effect change in the parameters on the concentration was discussed and represented graphically. The study revealed that as the zero order source coefficient increases, the contaminant concentration decreases with time.

Approximate analytical solution for transient heat and mass transfer across an irregular interface

Motivated by practical applications in heat conduction and contaminant transport, we consider heat and mass diffusion across a perturbed interface separating two finite regions of distinct diffusivity. Under the assumption of continuity of the solution and diffusive flux at the interface, we use perturbation theory to develop an asymptotic expansion of the solution valid for small perturbations. Each term in the asymptotic expansion satisfies an initial–boundary value problem on the unperturbed domain subject to interface conditions depending on the previously determined terms in the asymptotic expansion. Demonstration of the perturbation solution is carried out for a specific, practically-relevant set of initial and boundary conditions with semi-analytical solutions of the initial–boundary value problems developed using standard Laplace transform and eigenfunction expansion techniques. Results for several choices of the perturbed interface confirm the perturbation solution is in good agreement with a standard numerical solution.

Modeling Fire Spread in a Real-Time Coupled Atmospheric-Vegetation Fire: An Analytical Approach

The ability to analyse the rate of fire spread outbreak in a real-time coupled Atmospheric-vegetation fire has become increasingly vital as forest fire fighters are building diverse kinds of models to combat the dangers/effects of fire spread across a given fire vicinity. This paper theoretically examines the analysis of fire spread in a real fire environment. A partial differential equations (PDE) governing the phenomenon is presented. The analytical solution of the model is obtained via direct integration and eigenfunction expansion technique, which displays the influence of the parameters involved in the system. The effect of change in parameters such as Frank-Kamenetskii number, Radiation number, Peclet energy number and Activation energy number are presented graphically and discussed. The results obtained show that Frank-Kamenetskii number, Radiation number, Peclet energy number, and Activation energy number all reduced transient state temperature.

Mathematical Modeling of Effect of Pumping Rate on Contaminant Transport in Riverbank Filtration System

Riverbank filtration (RBF) is a natural technology that is used for river water treatment. This research seeks to investigate the effect of pumping rate on the transport of colloids in RBF. However, this work considered Dissolved Organic Matter (DOM) as a nutrient for bacteria. The mathematical model consists of groundwater flow equation and colloids concentration equations. The equations were solved analytically using parameter expanding method and Eigen function expansion techniques. The results obtained are presented graphically and discussed. It was observed that increase in pumping rate value enhance both the hydraulic head and concentration of colloids which slightly reduces the quality of pumped water from RBF.
 Keywords: Riverbank filtration, analytical model, colloids, hydraulic head and pumping rat

Theoretical Study on One-Dimensional Consolidation of Underconsolidated Marine Clay Induced by Ramp Loading

The position of maximum excess pore water pressure is found to be a critical indication in the design of horizontal drainage systems for saturated soil, controlled by the drainage capacity at the boundaries and the self-weight of soil both of which are often ignored. In sea reclamation works, however, the dredged clay used is mostly under consolidated and will undergo the self-weight consolidation. In this paper, an exponentially time-growing drainage boundary condition is adopted, based on which the analytical solution of one-dimensional consolidation of underconsolidated soil under a ramp loading is obtained by using the eigenfunction expansion technique. The obtained solution is verified and a parameter analysis is later carried out. It turns out that when considering the self-weight of the soil subjected to the ramp loading, the variation of the position of maximum excess pore water pressure over time depends heavily on the symmetry of the boundary conditions. The smaller the over consolidation ratio (OCR<1), the shorter the distance of the plane of maximum excess pore water pressure to the bottom surface of the soil layer. Moreover, the larger the loading rate, the higher the plane of maximum excess pore water pressure at the loading stage.