Eigenfunction Series Expansion Research Articles

Hydroelastic interaction between water waves and an array of circular floating porous elastic plates

Abstract

A High-Frequency Uniform Asymptotic Solution for Electromagnetic Field Scattering by a PEC Wedge Including Grazing Incidence Scenarios

An asymptotic solution for the vector problem of scattering by a perfectly electrically conducting (PEC) wedge is developed and formulated as a modification of that obtained by employing the uniform asymptotic theory (UAT) of diffraction approach. This modified UAT (MUAT) solution is validated via comparison with the eigenfunction series expansion of the exact solution of the problem for varying excitation and observation scenarios, therefore evidencing to correct the standard UAT results which fail for excitation at grazing incidence.

Modelling the electric field in reactors yielding cold atmospheric\u2013pressure plasma jets

The behavior of the electric field in Cold Atmospheric–Pressure Plasma jets (CAPP jets) is important in many applications related to fundamental science and engineering, since it provides crucial information related to the characteristics of plasma. To this end, this study is focused on the analytic computation of the electric field in a standard plasma reactor system (in the absence of any space charge), considering the two principal configurations of either one–electrode or two–electrodes around a dielectric tube. The latter is considered of minor contribution to the field calculation that embodies the working gas, being an assumption for the current research. Our analytical technique employs the cylindrical geometry, properly adjusted to the plasma jet system, whereas handy subdomains separate the area of electric activity. Henceforth, we adapt the classical Maxwell’s potential theory for the calculation of the electric field, wherein standard Laplace’s equations are solved, supplemented by the appropriate boundary conditions and the limiting conduct at the exit of the nozzle. The theoretical approach matches the expected physics and captures the corresponding essential features in a fully three–dimensional fashion via the derivation of closed–form expressions for the related electrostatic fields as infinite series expansions of cylindrical harmonic eigenfunctions. The feasibility of our method for both cases of the described experimental setup is eventually demonstrated by efficiently incorporating the necessary numerical implementation of the obtained formulae. The analytical model is benchmarked against reported numerical results, whereas discrepancies are commented and prospective work is discussed.

An extension of general identities for 3D water-wave diffraction with application to the Diffraction Transfer Matrix

Interaction theories are used in numerous branches of physics to efficiently evaluate wave scattering by multiple obstacles. An example of these interaction theories is the direct matrix method introduced by Kagemoto and Yue [1], which enables fast computation of three-dimensional water-wave multiple-scattering problems. The building block of interaction theories is a mathematical operator that encapsulates the mapping between incident and scattered waves. This operator is generally referred to as T-matrix and satisfies both reciprocity and energy identities. In some branches of physics, such as acoustics and electromagnetism, these identities are well established; in hydrodynamics, however, they have only been derived for a T-matrix that maps two-dimensional incident and scattered water waves. In three dimensions, water waves can be represented as a series expansion of cylindrical eigenfunctions. In this paper, we use this representation of water waves to derive the reciprocity and energy identities satisfied by the T-matrix of the direct matrix method, known as Diffraction Transfer Matrix (dtm). The identities derived herein represent an extension of existing general relations between two diffraction solutions. We show that this extension can be applied to verify the accuracy of the dtm entries, thereby increasing the reliability of existing schemes for computing the dtm. We present results for the dtm of two geometrically different isolated obstacles, as well as for the dtm of an asymmetric array. Finally, we demonstrate that the results presented herein can be extended to floating bodies found in a wide range of ocean engineering problems.

Wave radiation from a truncated cylinder of arbitrary cross section

In this paper, a semi-analytical model for solving wave radiation from a truncated cylinder of arbitrary cross section is presented based on linear potential flow theory. The water domain is divided into the interior domain beneath the cylinder and the exterior domain outside the vertical cylinder column. Radiated spatial potentials in these subdomains are expressed as a series expansion of eigen-functions using the method of separation of variables. The continuity conditions for pressure and velocity are satisfied at the interface of the two domains, where the Fourier series expansion method is employed to deal with the radius function associated terms. Therefore, the unknown coefficients in the radiated potential expressions are determined by means of the eigen-function matching method. Hydrodynamic coefficients of the truncated cylinder are evaluated directly based on the radiated spatial potentials. Case studies on wave radiation from a truncated cylinder with “cosine” and “circular” cross sections show a good agreement between the semi-analytical results of added-mass/radiation damping and numerical ones/published data. The validated semi-analytical model is then adopted to study the hydrodynamic characteristics of truncated cylinders with “circular”, “cosine”, “elliptical” and “quasi-elliptical” sections. For the latter case, the influence of draft on wave radiation is also investigated.

Nonlinear PTO Effect on Performance of Vertical Axisymmetric Wave Energy Converter Using Semi-Analytical Method

Abstract The wave energy, as a clean and non-pollution renewable energy sources, has become a hot research topic at home and abroad and is likely to become a new industry in the future. In this article, to effectively extract and maximize the energy from ocean waves, a vertical axisymmetric wave energy converter (WEC) was presented according to investigating of the advantages and disadvantages of the current WEC. The linear and quadratic equations in frequency-domain for the reactive controlled single-point converter property under regular waves condition are proposed for an efficient power take-off (PTO). A method of damping coefficients, theoretical added mass and exciting force are calculated with the analytical method which is in use of the series expansion of eigen functions. The loads of optimal reactive and resistive, the amplitudes of corresponding oscillation, and the width ratios of energy capture are determined approximately and discussed in numerical results.

A singular element for reissner plate bending problem with V-shaped notches

A novel singular finite element is presented to study Reissner plate bending problem with V-shaped notches. Firstly, the expressions of the first four order asymptotic displacement field around the tip of V-shaped notch are derived systematically using William’s eigenfunction series expansion method. Subsequently, the expressions are used to construct a novel displacement mode singular finite element, which can well depict the characteristic of singular stress field around the tip of V-shaped notch having arbitrary opening angle. Combining with conventional finite elements, the novel element can be applied to solve Reissner plate bending problem with V-shaped notches, and mode I, mode II and mode III stress intensity factors can all be determined directly by the coefficients of the asymptotic expansion terms. Finally, four numerical examples are performed to demonstrate the performance of the present method, and the influences of the size and number of export nodes of the singular element on the calculation of bending stress intensity factors are also discussed. Numerical results show that the singular element method is a practical and effective numerical technique for obtaining directly and synchronously mode I, mode II and mode III stress intensity factors without other numerical techniques, such as extrapolation.

Electromagnetic Low‐Frequency Dipolar Excitation of Two Metal Spheres in a Conductive Medium

This work concerns the low‐frequency interaction of a time‐harmonic magnetic dipole, arbitrarily orientated in the three‐dimensional space, with two perfectly conducting spheres embedded within a homogeneous conductive medium. In such physical applications, where two bodies are placed near one another, the 3D bispherical geometry fits perfectly. Considering two solid impenetrable (metallic) obstacles, excited by a magnetic dipole, the scattering boundary value problem is attacked via rigorous low‐frequency expansions in terms of integral powers (ik)n, where n ≥ 0, k being the complex wave number of the exterior medium, for the incident, scattered, and total non‐axisymmetric electric and magnetic fields. We deal with the static (n = 0) and the dynamic (n = 1, 2, 3) terms of the fields, while for n ≥ 4 the contribution has minor significance. The calculation of the exact solutions, satisfying Laplace’s and Poisson’s differential equations, leads to infinite linear systems, solved approximately within any order of accuracy through a cut‐off procedure and via numerical implementation. Thus, we obtain the electromagnetic fields in an analytically compact fashion as infinite series expansions of bispherical eigenfunctions. A simulation is developed in order to investigate the effect of the radii ratio, the relative position of the spheres, and the position of the dipole on the real and imaginary parts of the calculated scattered magnetic field.

Electro-osmotic flow through a two-dimensional screen-pump filter

The electro-osmotic flow driven by a screen pump, composed of a line array of evenly spaced identical rectangular solid blocks, is investigated under the Debye-Hückel approximation. The geometry of the screen pump is determined by the spacing and aspect ratio of the solid blocks. A constant surface zeta potential is assumed on the block surface. The method of eigenfunction series expansion is applied to solve analytically for the applied electric field, electric charge potential in the fluid, and flow field. Because of the low Reynolds number, Stokes equations are applied for the flow. The analytic result is first confirmed by comparing with the exact solution of the electro-osmotic flow in an infinite channel. Then different geometries of the screen pump and the effect of the electrokinetic width are computed for their influence on the flow rate. Recirculating eddies and reversing flow are found even though the applied electric driving field is unidirectional.

A semi-analytic approach to angular momentum transport in stellar radiative interiors

We address the problem of angular momentum transport in stellar radiative interiors with a novel semi-analytic spectral technique, using an eigenfunction series expansion, that can be used to derive benchmark solutions in hydromagnetic regimes with very high Reynolds number (10^7 - 10^8). The error arising from the truncation of the series is evaluated analytically. The main simplifying assumptions are the neglect of meridional circulation and of non-axisymmetric magnetic fields. The advantages of our approach are shown by applying it to a spin-down model for a 1 M_sun main-sequence star. The evolution of the coupling between core and envelope is investigated for different values of the viscosity and different geometries and values of the poloidal field. We confirm that a viscosity enhancement by 10^4 with respect to the molecular value is required to attain a rigid rotation in the core of the Sun within its present age. We suggest that a quadrupolar poloidal field may explain the short coupling time-scale needed to model the observed rotational evolution of fast rotators on the ZAMS, while a dipolar geometry is indicated in the case of slow rotators. Our novel semi-analytic spectral method provides a conceptually simple and rigorous treatment of a classic MHD problem and allows us to explore the influence of various parameters on the rotational history of radiative interiors.

Elastic Stress and Magnetic Field Concentration Near the Vertex of a Soft-Ferromagnetic 2D Compound Wedge

Abstract The problem of elastic stress and magnetic field concentration near the vertex of a compound wedge is modeled and investigated. The wedge is made of two isotropic dielectric soft-ferromagnetic materials and is immersed in a static magnetic field. The technique of eigenfunction series expansion is applied on the components of the elastic displacement field and the induced magnetic potentials near the vertex. It is shown that in this region, the magnetic susceptibility and the applied magnetic field have a strong influence on the elastic stress and magnetic field concentration. The results are instrumental toward actively controlling the stress concentration intensity via the applied magnetic field.

Eddy currents induced in a finite length layered rod by a coaxial coil

We describe the calculation of eddy currents in a two-layer conducting rod of finite length excited by a coaxial circular coil carrying an alternating current. The calculation uses the truncated region eigenfunction expansion (TREE) method. By truncating the solution region to a finite length in the axial direction, we can express the magnetic vector potential as a series expansion of orthogonal eigenfunctions instead of as a Fourier integral. The restricted domain can be arbitrarily large to yield results that are as close to the infinite domain results as desired. Integral form solutions for an infinite rod are well known and relatively simple. For a finite length cylindrical conductor, however, additional boundary conditions must be satisfied at the ends. We do this by comparing series expansions term by term to match the solutions across the end of the cylinder. We derive closed-form expressions for the electromagnetic field in the presence of a finite two-layer rod. A special case of the solution is that for a conductive tube. We illustrate the end effect by calculating the coil impedance variation with respect to the axial location of the coil. The results are in very good agreement with those obtained by using a two-dimensional finite-element code.

New improved series expansion for solving the moving oscillator problem

A new method able to evaluate the dynamic stress response of an elastic beam subject to moving oscillators is presented. The proposed procedure improves the convergence and accuracy of the conventional eigenfunction series expansion of beam response taking into account the gravitational, inertial and damping effects due to the moving oscillators. The improvement of the conventional solution is obtained by means of an extension to continuous systems of the dynamic correction method, originally proposed for discretized structures. The proposed method is able to account for the truncated higher order eigenfunctions by adding a pseudo-static term to the conventional series expansion. Numerical results are presented to demonstrate the capability of the method to accurately determine the discontinuity and jump in bending moment and shear force distributions, respectively.

Methods for Calculating Bending Moment and Shear Force in the Moving Mass Problem

Two methods able to capture with different levels of accuracy the discontinuities in the bending moment and shear force laws in the dynamic analysis of continuous structures subject to a moving system modeled as a series of unsprung masses are presented. The two methods are based on the dynamic-correction method, which improves the conventional series expansion by means of a pseudostatic term, and on an eigenfunction series expansion of the continuous system response, which takes into account the effect of the moving masses on the structure, respectively.

Acoustic control in a tractor cabin using two optimally designed Helmholtz resonators

A virtual design methodology is developed to minimize the noise in enclosures with optimally designed, passive, 20 acoustic absorbers (Helmholtz resonators). A series expansion of eigenfunctions is used to represent the acoustic=20 absorbers as external volume velocities, eliminating the need for a solution of large matrix eigenvalue problems. A determination of this type (efficient model/reevaluation approach) significantly increases the design possibilities when optimization techniques are implemented. As a full-scale demonstration, the acoustic response from 90–190 Hz of a tractor cabin was investigated. The lowest cabin mode proposes a significant challenge to a noise control engineer since its anti-node is located near the head of the operator and often generates unacceptable sound-pressure levels. Exploiting the low-frequency capability of Helmholtz resonators, lumped parameter models of these resonators were coupled to the enclosure via an experimentally determined acoustic model of the tractor cabin. The virtual design methodology uses gradient optimization techniques as a post-processor for the modeling and analysis of the unmodified acoustic interior to determine optimal resonator characteristics. Using two optimally designed Helmholtz resonators, potential energy was experimentally reduced by 3.4 and 10.3 dB at 117 and 167 Hz, respectively.

Inversion methods for Lamb and Rayleigh wave surface holography

Lamb and Rayleigh wave surface holographies have advantages for nondestructive testing of concrete and steel structures that is based on the use of generated surface acoustic waves that propagate over the concrete and steel and are reflected by external cracks. When the surface wave field is observed by a circular array of surface wave transducers, the proposed method that is called ‘‘eigenfunction series expansion (EFSE)’’ can reconstruct exact images of the scatterers within the array. This holographic imaging system has the following features: (1) An inversion via expansion in eigenfunction series produces much higher resolution than one which a back propagation produces. (2) An iso-azimuthal signal processing enables to design a first image reconstruction procedure. In this study, the computational process in EFSE and wave fields near the cracks are discussed and their physical meanings are investigated through FEM simulations. The surface wave transfer function by EFSE permits an effective regularization of the source field with cracks: this is possible because the transfer function includes several wave modes at the cracks and surface wave fields. In particular, it is clearly shown that the reconstructed image can be improved by adopting EFSE method that suppresses the spurious lobes.

TUNING A WINE GLASS VIA MATERIAL TAILORING — AN APPLICATION OF A METHOD FOR OPTIMAL ACOUSTIC DESIGN

This paper describes a method for optimally designing a structure to “best fit” a specified set of acoustic characteristics, e.g., sound spectrum or radiated power. The method links the disciplines of structural dynamics, acoustics and optimization into a unified methodology. The design variables include, for example, the addition of masses or multiply-tuned resonators to the structure as well as distributions of stiffeners or constrained damping layers. In all cases, the design variables are introduced as external forces (via their impedances) in the equation for the structure that is given as a series expansion of eigen functions. This step eliminates the need for solution of large matrix eigenvalue problems. An acoustic program POWER is used to assess the radiated sound power as a function of the design variables. Various search engines are used within the computer program MATLAB® to determine which design variables give the ‘best fit’ to the acoustic specifications. To illustrate the design method, a wine glass is tuned optimally to move the first four eigenvalues into harmonic relationships. The design variables are small masses that are added to the upper surface of the wine glass. Comparison of the wine glass's radiated sound power with and without the optimal masses indicates an excellent agreement between the specified and measured spectra.

Laminar Heat Transfer in a Liquid Flowing in a Diverging Conical Annular Channel with a Varied Inner-Wall Temperature

The problem of convective heat transfer in a divergent slow liquid flow in a constant-width conical annular channel is analyzed for the case of first-kind thermal boundary conditions and linear variation of the inner-wall temperature along the channel length. The problem is solved by eigenfunction-series expansion. The spatial distribution of temperature is represented as the sum of two infinite series in confluent hypergeometric functions of the transverse coordinate that are multiplied by an exponential function of the axial coordinate and the angular opening of the cone. The solution is interesting in that it is the superposition of two solutions, each having its own eigenfunctions and eigenvalues.

Experimental analysis of crack propagation stability in single edge notch specimens

An experimental technique for analyzing the crack propagation stability of single edge notch specimen is discussed. The testing rig and its calibration procedure are described. Analyzed are the crack path under biaxial loading and its correlation with theoretical predictions. The mode II influence on the crack path is also presented. Use is made of the computer program ANGCRK. Fracture criteria are applied to determine the stability of crack growth. Stability is assumed to prevail when the second term of the eigenfunction series expansion is negative. Otherwise, the crack growth could be unstable.

Beam theory for strongly orthotropic materials

This paper presents a displacement-based model for orthotropic beams under plane linear elasticity based on the only kinematic assumption of transverse inextensibility. Any given axial and transverse loading as well as boundary conditions at the beam ends are considered. The solution is decomposed into the principal and the residual part (corresponding to the interior and the boundary problems) which are obtained by series expansions of polynomial functions and eigenfunctions, respectively. It is proved that the proposed one-dimensional theory gives both interior and boundary exact two-dimensional elasticity solutions for strongly orthotropic materials, i.e. for ratio between shear modulus and axial Young modulus approaching zero. For isotropic and orthotropic materials the accuracy of the beam model is also analysed and compared with that of other theories. In particular, the complementary energy error of the interior solution with respect to two-dimensional elasticity is evaluated, the asymptotic estimate of the characteristic decay length of end effects given in Choi and Horgan [J. Appl. Mech. ASME, 44, 424–430 (1977)] by two-dimensional analysis is reobtained and the range of validity of boundary solution is discussed. The numerical results presented agree very well with exact and finite element solutions even in the neighbourhood of clamped cross-sections, where the solution is mainly governed by the boundary problem.