Eigenfunction Expansion Method Research Articles

Accurate fracture analysis of multi-material V-notched Reissner plates under bending or twisting

An accurate fracture analysis of multi-material V-notched Reissner plates under bending or twisting is performed by a new coupling approach of isogeometric analysis (IGA) and eigenfunction expansion method. In this approach, the overall plate is divided into two regions: a singular region near the notch tip (near field) and a regular region far away from the notch tip (far field). The overall model is discretized by a T-spline-based IGA. Asymptotic solutions of singular stress fields are applied to the near field, and therefore, the large number of fundamental unknowns of control points is reduced into a small set of undetermined coefficients. The unknowns in the far field remain unchanged. Consequently, the computational cost is significantly reduced, and explicit expressions of singular stress components in the vicinity of notch tip are obtained without post processing. Comparisons are presented to demonstrate the accuracy and convergence of the present approach. Effects of key influencing factors on the singularity order and singular items of stresses are investigated also.

Optimal Temperature Rise Control for a Large-Scale Vertical Quench Furnace System

An optimal switching heating control strategy based on minimum margin (OS-MM) is proposed for high efficiency, low energy consumption, and high-uniformity fine control in a large-scale vertical quench furnace. This work was stimulated by the need for solving the contradiction between the heating rate and temperature overshoot to achieve a rapid rise in temperature with no overshoot. Based on a three-dimensional (3-D) transient temperature field model of the quenching furnace, the temperature field optimization control problem is transformed into a boundary optimization control problem of rapid heating, whereby it is rigorously proved that the fastest rise in temperature is attained by the full-power heating, which satisfies the bang-bang characteristics. The key parameter that determines the overshoot in the full-power heating mode is introduced and defined as the minimum margin in the temperature-rising process. The analytical expression for solving the minimum margin is calculated by the eigenfunction expansion method and using the strong metal thermal inertia of the internal temperature field, the OS-MM method is designed to stop the heating in advance at a calculated switching point to optimize full-power heating, thus conserving energy by reducing its consumption. The results of simulation and industrial experiments show that this strategy greatly shortens the temperature adjusting time, significantly reduces energy consumption, and effectively improves the control performance indices, such as overshoot and static error in the temperature holding period of the thermal treatment process, as compared to the existing heating methods.

Transient electro-osmotic flow in rotating soft microchannel

Exploiting secondary velocities produced by Coriolis forces in Lab-on-CD systems is key to achieving better transport in pharmaceutical and biomedical applications. We explore the transient behavior of velocities in rotating microchannels aided by a grafted polyelectrolyte layer (a soft layer). We further obtain an analytical solution for governing differential equations of the rotational electro-osmotic flow by the eigenfunction expansion method. We check and benchmark the solution with an in-house finite volume numerical code and also with results in literature for situation after transience has completed. We explore and discuss the effect of channel rotation, electro-osmosis, and polyelectrolyte layer on the oscillatory transient behavior of the flow velocities. We show that the size of the polyelectrolyte layer grafted to the walls aids in better control of the flow velocities and oscillations. We believe that controlled transient oscillatory behavior of velocities can be greatly used in Lab-on-CD based systems to manage the mass and momentum transport.

An Efficient Method for Predicting the Output Signal of an Eddy Current Probe Around the Edge of a Conductive Cylindrical Rod

We propose a semi-analytical model to predict the output signal of an eddy current probe around the edge of a conductive cylindrical rod. The proposed model is based on the truncated region eigenfunction expansion method and uses the second-order vector potential to formulate the resultant boundary problem. The use of appropriate boundary conditions on the truncation locations leads to less complex eigenvalue equations which remarkably increases the computation speed. The validity and computation efficiency of the proposed model are demonstrated by comparing the simulation results of several case studies with measurements and those obtained using a commercial finite element code.

Parametric investigations on dynamics of cracked thin rectangular plates, excited by a moving mass

Dynamic analysis of cracked thin rectangular plates subjected to a moving mass first is investigated in this paper. To this end, the eigenfunction expansion method is utilized to solve the governing equation. For the first time, the intact plate orthogonal polynomials in combination with admissible crack functions as a composition, are employed in the eigenfunction expansion method formulation, required professional computer programming to solve the equation. The proposed approach guarantees upper bound of the true solution, which is the property of an appropriate numerical solution. Parametric investigation is performed to determine the effect of the moving mass weight, the moving mass velocity, the crack length, and the crack angular orientation as well as the plate’s aspect ratio, on the dynamic response of cracked plates. The results confirm, that the moving mass has a greater impact than the moving load on the dynamic responses of cracked plates. Furthermore, there is nonlinear relation among enhancing the dynamic responses of the cracked plates with various boundary conditions, and magnifying the moving mass weights, elevating the moving mass velocities, lengthening the crack length, raising the inclined crack angels as well as augmenting the plates aspect ratios.

Attenuation of wave force on a floating dock by multiple porous breakwaters

The impact of an array of porous breakwaters for reducing wave forces on a floating dock is studied based on the small-amplitude water wave theory in finite water depth. The breakwater is approximated by a porous bottom-standing rectangular structure and an impermeable floating rectangular structure is placed at a finite distance on the lee-side representing a dock. The physical problem is solved analytically using Eigenfunction expansion method and numerically using a hybrid element multi domain boundary element method. Two scenarios are considered: (i) wave scattering by N porous structures and the floating dock and (ii) wave reflection by a dock attached to the impermeable wall in presence of N porous structures. The reflection, transmission and dissipation coefficients are evaluated and analyzed for various structural and wave parameters. The study reveals that the Bragg reflection occurs in the case of wave scattering by an array of porous structures in the absence of floating dock. It is also found that in the absence of dock N−2 sub-harmonic peaks occurs whereas, in the presence of dock the effect of Bragg resonance is clearly evident on hydrodynamic coefficients. Also, the study concludes that for four structures with larger height, 100% of wave energy can be dissipated and thus mitigate the wave force on the floating rigid dock. Furthermore, the height and width of the porous structure and spacing between the structures play a vital role to protect the floating platform.

Prediction approach for wave forces on large-scale offshore structures under linear wave action

This paper presents a numerical mothed that is implemented by applying higher-order boundary elements in the hybrid integral equation method for predicting the wave forces on arbitrarily-shaped offshore structures. In this study, the virtual cylindrical boundary (i.e., matching boundary) is set to divide the fluid region into inner and outer subdomains. The velocity potentials in the outer domain are expressed using the eigenfunction expansion method. In the inner domain, the simple Green's function is adopted as the fundamental solution and Green's second theorem is applied to derive the boundary integral equation (BIE). The velocity potentials in the two subdomains are matched on the matching boundary. The higher-order boundary elements are ultimately employed to discretize BIE and the velocity potentials in the inner domain are then solved. The wave forces on the truncated and stepped cylinders are calculated using the presented method, and the results match well with analytical results, which illustrates the validity of the presented method. The effects of element sizes and modal truncation numbers are further explored, and the interaction between waves and square columns is subsequently analyzed. Eventually, the proposed method is extended to the practical engineering application of wave diffractions from quasi-elliptic pile caps for sea-crossing bridges. The results show that the proposed method can deal with boundaries with complex geometries. In addition, satisfactory results can be obtained by setting relatively few elements and control points on boundaries, and this method can also effectively predict wave forces on the multi-column array.

Wave interaction with a rectangular bar in the presence of two trenches

A coupled model of wave interaction with a rectangular bar in the presence of two trenches are proposed to study in the context of linear potential theory. The physical problem is converted to a set of integral equations using the eigenfunction expansion method by matching pressure and velocity at the interface boundaries of the structure. After solving the integral equations, the analytical solutions utilize the multi-term Galerkin method involving the Gegenbauer polynomials of order 1/6 with suitable weights as basis functions. The convergence and correctness of the method are confirmed by comparing the results with the one available in the literature. Energy balance relations for the present problem are derived and used to check the accuracy of the computed results. The effects of various parameters are analyzed through different graphs for a rectangular bar in the presence of submerged rectangular trenches. It is found that the reflection coefficient is increasing with increasing the rectangular bar length, thickness, and trench thickness. It can be concluded that the physical parameters of the structure have a significant effect on the present study. Moreover, it is observed that the value of the reflection coefficient reduces for the presence of a single trench compared to the presence of multiple trenches. The present study may be useful for understanding the performance of submerged rectangular trenches to protect the rectangular bar from wave action.

An analytical model of an eddy‐current coil near the edge of a conductive plate

Truncated region eigenfunction expansion (TREE) method is applied to the analysis of impedance change of a coil in the vicinity of the edge of a metal plate. The analysis is carried out with the second order vector potentials (SOVPs) and provided for both cylindrical and rectangular coils. By full discretisation of the eigenvalues, double series solutions are obtained for the field and coil impedance variations. To simplify the solving procedure further, a novel approach for the determination of eigenvalues is proposed by means of the 1D finite element method (FEM), which is easy to implement and works well for the plate of non-magnetic material. Numerical results for the impedance variations of cylindrical and rectangular coils are compared with those of 3D FEM simulation, by which the efficiency and accuracy of the authors’ method are confirmed, and the former is much faster and more memory efficient.

Enhancement of backscattering cross section from an equivalent human cylinder under oblique incidence

An analysis about the enhancement of backscattering radar cross section from an equivalent human body cylinder using a specific type of coating is presented. The types of coatings considered for the human body are taken to inductive meta surface (MS), inductive MS followed by a dielectric layer and an epsilon negative (ENG) metamaterial. An eigenfunction expansion method has been used to derive analytical expressions for the scattered electric and magnetic fields. It is investigated that inductive MS and inductive MS followed by a dielectric layer type of coatings cannot be used to enhance the backscattering radar cross section from the human body. It is found that an ENG metamaterial coating having specific permittivity and coating thickness can be used to significantly enhance the back-scattering radar cross section from the human body under oblique incidence. Therefore, an ENG metamaterial coating is selected for the proposed model and finally, the modelling of the proposed design is suggested.

Incorporating injection stage into DFIT analysis for permeability estimation, and its significance

Diagnostic fracture injection tests (DFIT) have been broadly used in unconventional reservoirs to derive properties such as initial formation pressure, formation permeability, and closure pressure. DFIT involve pumping a small volume of untreated fluid into the formation to form a small crack. Formation permeability is usually obtained by modeling fluid leakoff during well shut-in. Prevailing studies assume constant fluid pressure boundary conditions on the fracture walls, constant leakoff coefficient, or both. However, the results deduced from these assumptions can introduce tremendous errors because the fluid pressure inside the fracture dissipates fast as fluid leaks off. Therefore, this study proposes a material balance approach to analyze DFIT data and obtain formation permeability. The proposed analysis incorporates fluid leakoff during both injection and well shut-in periods. According to the material balance relationship, volume of water injected during hydraulic fracturing should be equal to the summation of leakoff volume during hydraulic fracturing, leakoff volume during shut-in period and residual fracture volume. Thus, we propose leakoff models for both fracture propagation and shut-in periods. These models are then validated through a few synthetic cases. The model accounting for leakoff during fracture propagation is modified to more accurately calculate the leakoff volume. The model for the shut-in period is solved using the Fourier transform and the eigenfunction expansion method. The results suggest that the proposed DFIT analysis approach can provide a good estimation of the formation permeability. The proposed solution can be used in the field even when pressure data contain significant levels of noise. We showed that more accuracy can be achieved using this method compared to available solutions in the literature.

Unified Theory of Unsteady Planar Laminar Flow in the Presence of Arbitrary Pressure Gradients and Boundary Movement

A general unified solution of the plane Couette–Poiseuille–Stokes–Womersley incompressible linear fluid flow in a slit in the presence of oscillatory pressure gradients with periodic synchronous vibrating boundaries is presented. Oscillatory flow remains stable and laminar with no-slip boundary conditions applied. Eigenfunction expansion method is used to obtain the exact analytical solution of the general linear inhomogeneous boundary value problem. Fourier expansion of arbitrary harmonic pressure gradient and non-harmonic wall oscillations was used to calculate arbitrary driving of the fluid. In-house developed optimized computational fluid dynamics marching-in-time finite-volume method was used to test and verify all analytical results. A number of particular transients, steady-state and combined flows were obtained from the general analytical result. Generalized Stokes and Womersley flows were solved using the analytical computations and numerical experiments. The combined effects of periodic non-harmonic wall movements with oscillatory pressure gradients offers rich and interesting flow patterns even for a linear Newtonian fluid and may be particularly interesting for pumping-assist microfluidic devices. The main motivation for developing a unified solution of the unsteady laminar planar Couette–Stokes–Poiseuille–Womersley flow originates in a need for, but is not limited to, in-depth exploration of flow patterns in hemodynamic and microfluidic pumping applications.

Hydroelastic analysis of interaction between water waves and a floating laminated disk

This paper studies the interaction between water waves and a very large floating laminated disk in water of finite depth. The disk is a composite structure consisting of two surface sheets and a middle low-density elastic core layer. Based on the linear potential flow theory, an analytical solution of the hydroelastic problem is developed using the eigenfunction expansion method for the velocity potential of fluid motion. In the solution procedure, the laminated disk is regarded as double circular Euler sheets connected by a series of closely spaced and mutually independent vertical springs, and then an eighth-order differential equation of motion of the laminated disk is derived as the elastic boundary condition of the hydroelastic problem. An approximated model is then developed for the hydroelastic problem in shallow water. The deflection and bending moment of the disk and the free surface elevation near the disk are calculated, and it is found that the series solution for the velocity potential converges rapidly. Typical examples are presented to show the effects of different parameters, including wave frequency, the edge conditions of the disk, and the elastic coefficient of the core layer, on wave force, structural hydroelastic response, and wave field. Moreover, viscoelastic damping is introduced in the core layer, and its effect on the hydroelastic response is evaluated by adopting the complex stiffness method. The results indicate that the wave force on the laminated disk is larger than that on a corresponding rigid one over a very wide range of wave frequencies, and the local deformation of the lower sheet can be suppressed by designing a core layer with viscoelastic damping.

Scattering of Water Waves by Very Large Floating Structure in the Presence of a Porous Box

Abstract We investigate the effectiveness of a porous box in attenuating the structural response of a very large floating structure (VLFS). Assuming the water depth to be finite and small amplitude water wave theory, the physical problem is formulated by employing Darcy’s law for flow past a porous structure. The boundary value problem is reduced to a system of linear algebraic equations with the aid of matched eigenfunction expansion method. Further, these simultaneous equations are solved numerically to compute physical quantities. The mathematical model is validated through a comparison with the theoretical results available in the literature. The reflection, transmission, and dissipation coefficients, elastic plate deflection, forces acting on the box, and free-surface elevation are computed. The reflection, transmission, and dissipation coefficients exhibit an oscillatory pattern for large values of wavenumber irrespective of the structural parameters of the box. It is highlighted that a porous box with moderate values of the porous-effect parameter is effective in reducing strain and shear force on the VLFS. Further, the results on elevation and VLFS deflection depict that the suitable porous-effect parameter values for the box reduce the resultant wave amplitude of the elastic plate deflection as well as the amplitude in the lee side of the VLFS. The study reveals that the width and height of the porous box are critical toward the trapping of incident waves inside the box and dissipating the maximum amount of incident wave energy in reduction of wave transmission in the lee side of the structure and thereby attenuating the structural response of the VLFS.

A review of the analytical solution methods for the eigenvalue problems of rectangular plates

This paper presents a review of typical analytical solution methods for the eigenvalue problems of rectangular plates. Apart from the exact Levy and Navier methods, this paper mainly focuses on non Levy-type of solution methods, including the Kantorovich-Krylov method, the separation-of-variable methods, the series expansion-based methods, the symplectic eigenfunction expansion method and the dynamic stiffness method. For each method, the motivation factors, development history and relevant works are introduced first, then a rectangular thin plate vibration problem is taken as an example to describe solution procedures. All methods are compared in terms of mode function forms, the principles based on, features and application scopes. Numerical results showed that all methods can obtain highly-accurate frequencies and mode functions.

Scattering of oblique water waves by thick porous structure and thin elastic plate

The present study deals with the usefulness of the partial porous structure of finite width placed at a finite distance from the elastic plate to mitigate the hydrodynamic response of the elastic plate. Two different configurations of the porous structure namely bottom-standing and surface-piercing structures are analyzed. The elastic plate is modeled using thin plate theory, and the wave past the thick porous structure is based on Sollitt and Cross model, the problems are investigated based on the small-amplitude water wave theory in water of finite depth. With the aid of the eigenfunction expansion method, the associated boundary value problem is reduced to a system of linear algebraic equations, which is solved numerically. For both configurations, the effects of various system parameters such as wavenumber and angle of incidence are analyzed. The impacts of structural parameters such as length, width of the porous structure, porosity are investigated from the graphs plotted on the reflection coefficient, transmission coefficient and dissipation coefficient and hydrodynamic forces. Further, detailed analysis on free surface elevation, elastic plate deflection, shear force and strain is presented. The study reveals that with an increase in length, width, porosity, and friction factor of the porous structure the dissipation of wave energy increases which helps in diminishing the wave impact on the elastic plate.

Wave action of non-coaxial double-layer cylindrical structure

Based on the potential flow theory, the mathematical formula of the wave action of the non-coaxial double-layer cylinder structure is derived, and the effect caused by the non-coaxial of the upper and lower single cylinders is analyzed. In this paper, the water area is divided into two regions, and their respective coordinate systems are established in each region to describe the velocity potential function. The velocity potential of the two regions is linked by the matched eigenfunction expansion method (MEEM) at the interface of the two regions. The cross-sectional radius function of the interface is expressed by Fourier series. After verifying the accuracy of the mathematical formula, the effect of asymmetric structure is analyzed. The results show that when the relative angle of the two axes is 0°, the wave force on the structure can be reduced to a greater extent if the two axes are as far away as possible. At the same time, the effect of this reduction weakens with the increase of the relative angle, and even increases the wave force when α = 90°. When the lower-lawyer cylinder radius is 0.6 times the upper-lawyer cylinder radius, the effect of wave force reduction caused by the non-coaxial is the largest.

Influence of submerged annular rippled breakwater on hydrodynamic performance of column floating pier

Based on linear potential flow theory, a mathematical model of the interactions between a floating pier with a submerged annular rippled breakwater and linear monochromatic waves was established. The fluid domain was divided into four regions: an exterior region, a mild-slope region, a gap region, and an interior region. The modified mild slope equation (MMSE) was used to establish the velocity potential expression of the mild-slope region, where the rippled breakwater was located. The matched eigenfunction expansion method (MEEM) was then used to derive the semi-analytical solution of the diffraction and radiation problem of the floating pier under linear monochromatic waves. After the accuracy of the mathematical model was ensured by comparison with other results and convergence analysis, the effects of the amplitudes and lengths of the ripples, the distance between the floating pier and the breakwater, and the number of ripples of the submerged annular rippled breakwater was analyzed. The results showed that the increase in the ripple amplitude and number aggravated the fluctuations of the wave excitation forces and the hydrodynamic coefficients of the floating pier as the wavenumber and ripple length varied. The wave excitation force and hydrodynamic coefficient of the floating pier changed periodically with the increase in the distance between the breakwater and the floating pier. This period was only related to the wavenumber, which was slightly larger than half the wavelength. The valley value and the corresponding position of the valley that appeared with this periodic change were affected by the ripple amplitude and length of the submerged annular rippled breakwater.

Different approaches in scattering of water waves by two submerged porous plates over an elastic sea-floor

A hydroelastic model is presented here to look into the scattering of oblique water waves by two totally submerged vertical porous plates, placed at some distance from each other, in a homogeneous fluid flowing over an elastic sea-floor. Consideration of Euler–Bernoulli beam equation allows the elastic sea-floor to be approximated as a thin elastic plate whereas the porous plates follow the porous wave-maker theory. The complete analytical solution, under the assumption of small-amplitude theory and structural response, is obtained by employing eigenfunction expansion and least square method. Subsequently, numerical computation for the reflection and transmission coefficients and energy loss are carried out and discussed for different values of the elasticity of the sea-floor, porosity of the porous plates and other structural parameters. The present study establishes that reflection of the waves exhibits an oscillatory behaviour. It further shows that, due to an increase in the inertial effect of the porous plate, the minima in wave reflection occurs. The vertical porous plates are found to dissipate a significant portion of the wave energy when an increase in the inertial effect of the porous plates takes place. Furthermore, wave transmission decreases significantly due to the energy dissipation by the elastic sea-floor. Significant variation in the elastic specification of the sea-floor commands considerable influence when the propagating wave impinges upon the submerged vertical porous plates. In order to validate the present model, the obtained results are compared with available results which points towards a good agreement.

Analytical approach to the solution of short-crested wave interaction with V-shaped and arc-shaped breakwaters

The hydrodynamic performance of short-crested wave diffraction by bottom-mounted V-shaped and arc-shaped breakwaters was examined based on the linear wave theory and the eigenfunction expansion method, respectively. Both types of breakwaters are assumed to be thin, impermeable, vertical, and rigidly embedded in the seabed. With the help of a virtual interface, the fluid domains were subdivided into subdomains and the velocity potential in each subdomain was described by eigenfunctions. The linear algebraic equations that determine the unknown coefficients can be obtained using the boundary and matching conditions. The analytical models fully agree with previous predictions. The major factors including wave propagation direction, opening angle, and water depth that affect wave forces and run-ups were explored. The results showed that the wave patterns of short-crested waves near breakwaters were quite different from those of long-crested waves, and using the long-crested wave model can overestimate the wave-defending effects of the breakwaters in a short-crested sea. An analogous wave attenuation effect was observed for breakwaters with comparable structures, which means that engineering practice has a certain degree of substitutability between them.