Eigenfunction Expansion Approach Research Articles

Hydrodynamic performance of the arc-shaped bottom-mounted breakwater

The problem of the hydrodynamic interaction with the arc-shaped bottom-mounted breakwaters is investigated theoretically. The breakwater is assumed to be rigid, thin, impermeable and vertically located in a finite water depth. The fluid domain is divided into two sub-regions of inner and outer by an auxiliary circular interface. Linear theory is assumed and the eigenfunction expansion approach is used to determine the wave field. In order to examine the validity of the theoretical model, the analytical solutions are compared to agree well with published results with the same parameters. Numerical results including wave amplitude, surge pressure, and wave force are presented with different model parameters. The major factors including wave parameters, structure configuration, and water depth that affect the surge pressure, wave forces, and wave amplitudes are discussed and illustrated by some graphs and cloud maps.

Transient response of general one-dimensional distributed systems through eigenfunction expansion with an implicit filter scheme

The current work is concerned with eigenfunction expansion of the transient response of general one-dimensional non-self-adjoint distributed dynamic systems. The main goal of the current work is to propose an alternative eigenfunction expansion approach for transient analysis of general one-dimensional non-self-adjoint systems which avoids complex-valued eigenvalue problems. With the implicit filter scheme proposed here one splits the solution into two additive components, namely, a filtered solution for which all boundary conditions are homogeneous and a carefully selected filter. Moreover, the eigenvalue problem associated with the differential equation for the filtered solution is self-adjoint and, therefore, all eigenvalues are real-valued and the eigenfunctions satisfy the classical orthogonality relations. The unknown time-dependent coefficients of the filter are then solved simultaneously with the modal co-ordinates of the filtered solution in order to satisfy the original boundary conditions. The proposed approach is demonstrated on a distributed system composed of a cantilever beam with end mass, viscous damper and linear spring.

A two-dimensional gas flow model for layered municipal solid waste landfills

Recovery and utilisation of landfill gas (LFG) can not only reduce the greenhouse effect but also permit the generation of electricity. A good understanding of gas migration from the waste body to the LFG extraction system is required to permit efficient gas recovery. This paper presents an analytical two-dimensional gas flow model to predict the distribution of gas pressure, the CH4 emission flux, the distance of influence and recovery efficiency in landfills. The model is indicative of the flow towards a combined extraction system of vertical wells and horizontal gravel-filled trenches. Moreover, the model has a horizontal layered structure to accommodate anisotropy of municipal solid waste (MSW) and vertical variations in both gas generation rate and permeability. The relevant governing equations for multiple homogenous layers were combined using continuity conditions of gas pressure and flux at the layer interfaces, subjected to realistic boundary conditions, and then solved using an eigenfunction expansion approach. The solution was compared with data available in the literature, and a parametric evaluation was performed to understand the roles of the relevant variables. The results show that, for a non-homogeneous model, the maximum pressure appears at the upper layer of the landfill rather than at the bottom. The location of the maximum pressure depends on the assumptions regarding the vertical distributions of the LFG generation rate and permeability. Additionally, the results can be used to form preliminary estimates of LFG recovery efficiency and overall landfill stability for different waste compositions.

Geometrically-induced singularities in functionally graded magneto-electro-elastic wedges

Asymptotic solutions are proposed for geometrically-induced magneto-electro-elastic (MEE) singularities at the vertex of a rectilinearly polarized wedge that is made of functionally graded magneto-electro-elastic (FGMEE) materials. The material properties are assumed to vary along the thickness of the wedge, and the direction of polarization may not be parallel or perpendicular to the thickness direction. An eigenfunction expansion approach with the power series solution technique and domain decomposition is adopted to establish the asymptotic solutions by directly solving the three-dimensional equations of motion and Maxwell׳s equations in terms of mechanical displacement components and electric and magnetic potentials. Since the direction of polarization can be arbitrary in space, the in-plane components of displacement, electric and magnetic fields are generally coupled with the out-of-plane components. The solutions are presented here for the first time. The correctness of the proposed solutions is confirmed by comparing the orders of MEE singularities with the published results for wedges under anti-plane deformation and in-plane electric and magnetic fields. The proposed solutions are further employed to investigate the effects of the direction of polarization, vertex angle, boundary conditions and material-property gradient index on the MEE singularities in BaTiO3–CoFe2O4 wedges.

A semispectral approach for the efficient calculation of scattering matrices in quasi-1D quantum systems and transmission coefficients for the Landauer formula

This paper proposes the semispectral method for the calculation of the scattering matrices for the charge carriers in quasi-one-dimensional quantum systems described by the Schrodinger equation. An efficient and accurate calculation method is achieved by adding of reference solutions at selected energy points to the eigenfunction expansion of the Green's function. A numerical simulation of the quantum wire with the square cross-section in transverse electric field was performed by different methods. The example problem of a quantum wire in a transverse electric field is used to compare the semispectral method with alternative approaches. We find that the semispectral method reliably converges and is significantly faster than the direct solution while the eigenfunction expansion approach has convergence issues. These results allow us to propose the semispectral method as a universal and efficient approach to calculation of the transmission coefficients for the Landauer formula as well as other scattering-based entities.

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.

On bimodal flutter behavior of a flexible airfoil

The dynamic aeroelastic behavior of an elastically supported airfoil is studied in order to investigate the possibilities of increasing critical flutter speed by exploiting its chord-wise flexibility. The flexible airfoil concept is implemented using a rigid airfoil-shaped leading edge, and a flexible thin laminated composite plate conformally attached to its trailing edge. The flutter behavior is studied in terms of the number of laminate plies used in the composite plate for a given aeroelastic system configuration. The flutter behavior is predicted by using an eigenfunction expansion approach which is also used to design a laminated plate in order to attain superior flutter characteristics. Such an airfoil is characterized by two types of flutter responses, the classical airfoil flutter and the plate flutter. Analysis shows that a significant increase in the critical flutter speed can be achieved with high plunge and low pitch stiffness in the region where the aeroelastic system exhibits a bimodal flutter behavior, e.g., where the airfoil flutter and the plate flutter occur simultaneously. The predicted flutter behavior of a flexible airfoil is experimentally verified by conducting a series of systematic aeroelastic system configurations wind tunnel flutter campaigns. The experimental investigations provide, for each type of flutter, a measured flutter response, including the one with indicated bimodal behavior.

Cnoidal Wave-Induced Seepage Forces on Large Porous Vertical Circular Cylinder Resting on Porous Elastic Seabed

The influence of the porosity of the structure on the shallow water wave-Induced seepage force on the bottom of porous vertical circular cylinder resting on porous elastic seabed has been investigated. Based on the shallow water diffracted wave theory and Biot consolidation theory on wave-induced seepage pressure, the analytical solutions to first order cnoidal wave diffraction by porous vertical circular cylinder and wave-induced seepage pressure are obtained by the eigenfunction expansion approach. Numerical results show that cnoidal wave-induced uplift and moment may have same order of magnitude as the horizontal cnoidal wave force and moment , and the body porosity of the structure may lead to a reduction both in direct cnoidal wave forces and in the cnoidal wave induced seepage moment. Compared with Airy wave theory, in certain shallow water conditions, the shallow water wave theory can more effectively reflect wave nonlinearity effect in wave load prediction.

Numerical analysis of the singularities for a magneto-electro-elastic notch under anti-plane deformation

Due to the capability of the conversion between mechanical, electric and magnetic energy, magneto-electro-elastic (MEE) materials can be applied to design sensors, actuators and other smart products. The notch and interfacial edge problem can always be encountered in these products. The singularity phenomenon is that the stress, electric displacement and magnetic induction may become theoretical infinite due to the materials or geometrical discontinuities at the vertex of notch. In practices, the mechanical failure or dielectric breakdown may initiate from the notch apex due to this high singularity. The present paper is trying to establish a new method to study the singularity at the vertex of notch encountered in MEE structures. Basing on the asymptotic assumption for the physical field near the notch tip, the singularities for a MEE notch under anti-plane deformation are investigated by the eigenfunction expansion approach. The characteristic differential equations respects to the singularities of the MEE notch are built from the equilibrium equations and Maxell equations. At the same time, the mechanical and electric boundary conditions together with the interfacial continuous conditions are expressed by the combination of the singularity orders and characteristic angle functions. Thus, the evaluation of the notch singularity orders is transformed into the problem of solving the ordinary differential equations under the designated boundary conditions. The singularity orders and the corresponding characteristic angle functions can be achieved by applying the interpolate matrix method to solve the obtained characteristic differential equations. The achieved results can provide the guidelines for the selection of notch angles and materials to reduce the singularity for the structure design of MEE products.

In This Issue

In This Issue

Three-dimensional singular stress fields near the circumferential junction corner line of an island/substrate system either free-standing or fully/partially bonded to a rigid block

The primary motivation of the present investigation stems from the need for understanding the three-dimensional singular stress fields in the vicinity of the re-entrant circumferential bimaterial junction corner line between a thin film island and the underlying substrate, either free-standing or fully/partially bonded to a super-rigid block. A three-dimensional eigenfunction expansion approach is employed for determination of the singular stress field in the vicinity of the re-entrant circumferential bimaterial junction corner line. The island-substrate bimaterial system is subjected to extension/bending (mode I), shear/twisting (mode II) and torsional (mode III) far field loadings. Both the island and substrate materials are assumed to be isotropic and elastic. The boundary conditions, prescribed on either side of the interfacial bond line between the island and substrate materials (free–free, fixed–fixed, free–fixed and fixed–free), are exactly satisfied. Numerical results include the dependence of the lowest eigenvalue(s) (or most severe order(s) of stress singularity) on the wedge aperture angle of the island material (looking at a axisymmetric cross-section). Variation of the same with respect to the shear moduli ratio of the constituent island and substrate materials is also an important part of the present investigation. Hitherto unobserved interesting and physically meaningful conclusions in regards to the mixed free–fixed type singularity and delamination type flaw sensitivity of an island-substrate bimaterial system are also presented.

Singularity analysis for a magneto–electro–elastic body of revolution

Three-dimensional asymptotic solutions are obtained for magneto–electro–elastic singularities in bodies of revolution that are made of a single magneto–electro–elastic (MEE) material or bi-materials that consist of an MEE material and an elastic material or piezoelectric material. The solutions are obtained by combining an eigenfunction expansion approach with the power series solution method to solve three-dimensional equilibrium equations and Maxwell’s equations in terms of mechanical displacement components and electric and magnetic potentials. The MEE material is assumed to be transversely isotropic and its polarization direction is not necessarily parallel to the axis of revolution. The polarization direction, which is not along the axis of revolution, yields and complicates non-axisymmetric solutions. The solutions are validated by comparing the present characteristic values of the asymptotic solutions, which are related to the orders of singularities of stress, electric displacement and magnetic flux, to the published ones for a piezoelectric body of revolution because no results have been published for an MEE body of revolution. The developed solutions are further employed to examine the effects of the direction of polarization, the configuration of the body of revolution and the material components on the orders of the singularities in bodies of revolution that comprise a single MEE material (BaTiO3–CoFe2O4) and bonded MEE/isotropic elastic (BaTiO3–CoFe2O4/Si), MEE/piezoelectric (BaTiO3–CoFe2O4/PZT-5H), or MEE/MEE BaTiO3–CoFe2O4VI=50%/BaTiO3–CoFe2O4VI=20% materials. These results are published here for the first time.

An Eigenfunction Expansion Method in the Stress Concentration Analysis

This paper deals with the traditional stress concentration problems based on the eigenfunction expansion approach. Due to the completeness property of the eigenfunction space obtained by the previous researches, the solution of an arbitrary problem can be expressed by their linear combination. Thus the original problem is transformed into finding the combination of these eigenfuctions satisfying boundary conditions. By applying adjoint symplectic relationships of the ortho-normalization, the combination can be obtained numerically. Numerical results in tensional problems show that stress concentration appears when one of the ends of the solid is clamped. The concentration is seriously confined near the boundary of the fixed, and decrease rapidly with the distance of the boundarys.

Three-dimensional analyses of stress singularities at the vertex of a piezoelectric wedge

The electroelastic singularities at the vertex of a rectilinearly polarized piezoelectric wedge are investigated using three-dimensional piezoelasticity theory. An eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions at the vertex of the wedge by directly solving the three-dimensional equilibrium and Maxwell’s equations in terms of the displacement components and electric potential. This study is the first to address the problems in which the polarization direction of the piezoelectric material is not necessarily either parallel to the normal of the mid-plane of wedge or in the mid-plane. The correctness of the proposed solution is verified by convergence studies and comparison with the published results that are based on generalized plan strain assumption. The solution is further employed to study comprehensively the effect of the direction of polarization on the electroelastic singularities of wedges that contain a single material (PZT-5H), bounded piezo/isotropic elastic materials (PZT-5H/Si), or piezo/piezo materials (PZT-5H/PZT-4).

Evaluating callable and putable bonds: An eigenfunction expansion approach

We propose an efficient method to evaluate callable and putable bonds under a wide class of interest rate models, including the popular short rate diffusion models, as well as their time changed versions with jumps. The method is based on the eigenfunction expansion of the pricing operator. Given the set of call and put dates, the callable and putable bond pricing function is the value function of a stochastic game with stopping times. Under some technical conditions, it is shown to have an eigenfunction expansion in eigenfunctions of the pricing operator with the expansion coefficients determined through a backward recursion. For popular short rate diffusion models, such as CIR, Vasicek, 3/2, the method is orders of magnitude faster than the alternative approaches in the literature. In contrast to the alternative approaches in the literature that have so far been limited to diffusions, the method is equally applicable to short rate jump–diffusion and pure jump models constructed from diffusion models by Bochner's subordination with a Lévy subordinator.

Three-dimensional singular stress field near the interfacial bond line of a tapered jointed plate either free-standing (notch) or (fully/partially) attached to a super-rigid inclusion (antinotch)

A three-dimensional eigenfunction expansion approach is employed for determining singular stress fields in the vicinity of interfacial bond line of a tapered jointed plate either free-standing (notch) or (fully/partially) attached to a super-rigid inclusion (antinotch). The tapered jointed bi-material plate is subjected to extension/bending (mode I) and shear/twisting (mode II) far field loading (extension and shear being in the plane of the cross-section and bending/twisting about mid-width). Hitherto unavailable results, pertaining to the width-wise variations of stress intensity factors corresponding to symmetric and skew-symmetric (about mid-width) cosine function loads, in the vicinity of the interfacial bond line, are presented.

Geometrically induced stress singularities in a piezoelectric body of revolution

An eigenfunction expansion approach is combined with a power series solution technique to establish the asymptotic solutions for geometrically induced electroelastic singularities in a piezoelectric body of revolution, with its direction of polarization not parallel to the axis of revolution. The asymptotic solutions are obtained by directly solving the three-dimensional equilibrium and Maxwell’s equations in terms of displacement components and electric potential. When the direction of polarization is not along the axis of revolution, the assumption of axisymmetric deformation that is often made in the published literature is not valid, and the direction of polarization and the circular coordinate variable can substantially affect the singularities. The numerical results related to singularity orders are shown in graphical form for bodies of revolution that comprise a single material (PZT-4 or PZT-5H) or bonded piezo/piezo (PZT-4/PZT-5H) or piezo/isotropic elastic (PZT-4/Al or PZT-5H/Al) materials. This is the first study to present results for the direction of polarization not along the axis of revolution.

A three-dimensional eigenfunction expansion approach for singular stress field near an adhesively-bonded scarf joint interface in a rigidly-encased plate

A three-dimensional eigenfunction expansion approach for determination of the singular stress field in the vicinity of an adhesively-bonded scarf joint interface in a plate, with its top and bottom surfaces being encased, fully or partially, between infinitely rigid blocks is presented. The plate is subjected to extension/bending (mode I) and in-plane shear/twisting (mode II) far-field loading. Both the adhesive layer and plate materials are assumed to be isotropic and elastic. The boundary conditions that are prescribed on the end-faces (free, fixed and lubricated) of the plate as well as those, prescribed at the bottom or top surface of the scarf-bonded plate on either side of the interface between the plate and adhesive layer materials (fixed-fixed, free-fixed and fixed-free), are exactly satisfied. Numerical results include the dependence of the lowest eigenvalue (or most severe stress singularity) on the wedge aperture angle of the plate material. Variation of the same with respect to the shear moduli ratio of the constituent plate and adhesive layer materials is also an important part of the present investigation. Hitherto unobserved interesting and physically meaningful conclusions in regards to the fixed edge singularity and delamination type flaw sensitivity of an adhesively bonded plate surface are also presented. Finally, hitherto unavailable results, pertaining to the through-width variations of stress intensity factors corresponding to symmetric and skew-symmetric sinusoidal loads that also satisfy the boundary conditions on the end-faces of the adhesively bonded plate, in the vicinity of the scarf joint interface, under investigation, bridge a longstanding gap in the bonded joint stress singularity/fracture mechanics literature.

Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions

The separation of variables ( SOV) method has recently been applied to solve time-dependent heat conduction problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solution in polar ( r- θ) coordinate system are always real numbers (unlike in the case of similar multidimensional Cartesian problems where they may be imaginary) allowing one to obtain the solution analytically. However, the SOV method cannot be applied when the boundary conditions and/or the source terms are time-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heat sources. In this paper, we present an alternative approach using the finite integral transform method to solve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundary conditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary condition in the angular direction is used. After integral transformation and subsequent weighted summation over the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinary differential equations ( ODEs) are formed for the transformed temperatures. Solutions of ODEs are then inverted to obtain the temperature distribution in each layer. Since the proposed solution requires the same eigenfunctions as those in the similar problem with time-independent sources and/or boundary conditions—a problem solved using the SOV method—it is also “free” from imaginary eigenvalues.

Potential-flow-based numerical study for the response of floating offshore structures with perforated columns

Perforated piles are used in near-shore breakwaters for coastal protection and deep water platforms for oil exploration. Emerged and submerged perforated cylindrical structures reduce wave–structure interactions and scouring problems considerably, but their use on a floating structure is limited. Mathematical model is developed to introduce perforated walls in floating structures and method of solution is discussed. Eigenfunction expansion approach is used to derive velocity potentials based on which hydrodynamic forces on perforated cylinders are obtained. An example tension leg platform having perforated caissons is chosen; length of the perforated region and its location on tension leg platform are varied to determine their influence on its motion. For increase in perforation length, fluid–structure interaction reduces, causing decrease in surge and pitch responses. However, increase in perforated length in the lower part of column does not significantly increase these responses; heave remains relatively unaffected regardless of the length of perforation in the columns.