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We propose to combine the Carleman estimate and the Newton method to solve an inverse problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse problem for determination of initial condition is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed. [ABSTRACT FROM AUTHOR]

• Thermoelastic inclusion problem with nonlinear electric current effects has been solved using complex function theory. • The shape, bluntness and orientation of the inclusion significantly affect the stress concentration around it. • Stress concentration around the quasi-hexagonal inclusion can be lower than that around a circular inclusion. The presence of inclusions in metal-based composites subjected to an electric current or a heat flux induces thermal stresses. Inclusion geometry is one of the important parameters in the stress distribution. In this study, the plane problem of an arbitrarily-shaped inclusion embedded in an infinite conductive medium is investigated based on the complex variable method. The shape of the inclusion is defined approximately by a polynomial conformal mapping function. Faber series and Fourier expansion techniques are used to solve the corresponding boundary value problems. The obtained results show that the shape, bluntness and rotation angle of the inclusion have a significant effect on the stress concentration around the inclusion induced by the far-field electric current. In addition, for the considered inclusion-matrix system under given electric loading, a lower amount of the Von Mises stress concentration than that around a circular inclusion could be achieved by appropriate selection of the inclusion shape and orientation. [ABSTRACT FROM AUTHOR]

Rotating beams are usually encountered in various dynamic analyses of rotation machineries, most of the current studies are mainly focused on the classical boundary conditions. From the viewpoint of engineering practice, elastic boundary supports will be of more significance. Motivated by such limitation, a Fourier series solution for vibration analyses of a rotating hub-beam with elastic boundary restraints and centrifugal stiffening is established by solving the governing differential equation and general boundary conditions, simultaneously. For elastic restraints at both ends, two types of boundary springs against translation and rotation are introduced to simulate the general boundary supports. Fourier series supplemented by boundary smoothed terms is constructed for the rotating beam displacement expression. Harmonic balance principle is employed to formulate the system characteristic matrix, from which all the modal information of rotating beam can be derived by solving a standard eigen-value problem. Numerical examples are then presented to verify the correctness and effectiveness of the developed model. The influence of boundary restraining stiffness and rotation speed on modal characteristics of rotating beam are investigated and addressed. The analytical solution established in this work can be used as the benchmark for other solution development, and provide an efficient tool for the parametric studies and/or optimization of dynamic characteristics of rotating beam structure, especially with complicated boundary conditions. [ABSTRACT FROM AUTHOR]

• Unlike the existing relevant works [12–18], the Fourier series and the singular integral equation method are applied to study a two-dimensional finite initially stressed neo-Hookean solid. Comparing with the mathematical technique of the references [12–18], this technique can study some problems of a finite initially stressed neo-Hookean solid, it can reveal the influence of material edge on stress field. For the singular integral equation method, it can indicate the singularity of stress field clearly and have a better numerical simulation effect than the reference [12]. • Comparing with the boundary conditions of the reference [12–18], the displacement's continuity boundary condition on the crack surface and horizontal displacement's density boundary condition on contact surface is considered. Using displacement's continuity boundary condition on the crack surface can obtain the same results as reference.[12] when the length L of initially stressed neo-Hookean solid tend to +∞. • Comparing with the references [32–33], the singular integral equation of the contact area and the singular integral equations of the cracking tip are solved together. It can reveal the coupling effect of crack and contact better. • For k = 1 and unchanged material property parameters, the incremental stress distribution on the initially stressed neo-Hookean base is very similar to the Cauchy stress distribution of the isotropic linear elastic base. The coupling problem of a cracked neo-Hookean substrate with initial stress under a rigid punch is examined in the current work. Fourier series is used to transform the mixed boundary value problem(MBVP) into singular integral equations (SIEs). The obtained SIEs is solved by using Gauss-Chebyshev collocation method. The stress distributions on the contact zone and crack surface, and related singularities are present. The numerical experiments examine how the punch position, punch width, crack length and propagation parameter affect the stress field. The result showed that: for k = 1 and unchanged material property parameters, the incremental stress distribution on the initially stressed neo-Hookean base is very similar to the Cauchy stress distribution of the isotropic linear elastic base. [ABSTRACT FROM AUTHOR]

Abstract The main purpose of this paper is to provide a semi analytical method to analyze the free vibration of spherical-cylindrical-spherical shell subject to arbitrary boundary conditions. The formulations are established based on energy method and Flügge thin shell theory. The displacement functions are expressed by unified Jacobi polynomials and Fourier series. The arbitrary boundary conditions are simulated by penalty method about spring stiffness. The final solutions of spherical-cylindrical-spherical shell are obtained by Rayleigh–Ritz method. To sufficient illustrate the effectiveness of proposed method, some numerical example about spring stiffness, Jacobi parameters etc. are carried out. In addition, to verify the accuracy of this method, the results are compared with those obtained by FEM, experiment and published literature. The results show that the proposed method has ability to solve the free vibration behaviors of spherical-cylindrical-spherical shell. Highlights • Free vibration of spherical-cylindrical-spherical shellstructureisinvestigated by using asemi-analytical method. • Thepaper presents a generalized and unified Jacobi−Ritz formulation. • The papergeneralizesthe selection of admissible displacement functions by using Jacobi polynomial. [ABSTRACT FROM AUTHOR]

Highlights • A 3-D formulation for vibration and damping analysis of sandwich shallow shells is presented. • Arbitrary boundary conditions including elastic ones are considered. • Frequency dependent viscoelastic models are discussed for various temperatures and boundary conditions. • New results for sandwich plate, spherical, cylindrical and hyperbolic paraboloidal shells are calculated. • 3-D modal shapes are depicted for the sandwich shallow shells. Abstract In this paper, we present an accurate three-dimensional formulation for the vibrations of the laminated and sandwich shallow shells. The sandwich structure is characterized by a thick viscoelastic core and two thin composite faces. Frequency dependent viscoelastic models are introduced in the sandwiches. Without any change in solution procedure, the formulation makes it quite easy to change the boundary conditions. The solution can be obtained by means of Rayleigh–Ritz process combined with the three-dimensional modified Fourier series which are actually assumed displacement functions. These functions, without need to meet the boundary conditions in advance, take the form of the three-dimensional Fourier series with several closed-form auxiliary functions which are supplemented to deal with the discontinuities at the boundaries in terms of displacements and its derivatives. Besides, only three assumed displacement variables are employed in the formulation which effectively reduces the computation cost. The reliability and accuracy of the method are demonstrated by numerical comparisons and examples with the constant viscoelastic models as well as the frequency dependent ones. Modal analysis and parametric studies are conducted to examine the influences of the boundary condition, dimension, lamination scheme, temperature and frequency dependence of the materials. [ABSTRACT FROM AUTHOR]

Highlights • Reviewed various analytical methods applicable for plates with general boundary conditions. • Applied finite integral transform to vibration of rotationally-restrained orthotropic plates. • Proposed a new formulation of finite integral transform. • Discussed numerical issues related to application of finite integral transform method. Abstract The exact series solutions of plates with general boundary conditions have been derived by using various methods such as Fourier series expansion, improved Fourier series method, improved superposition method and finite integral transform method. Although the procedures of the methods are different, they are all Fourier-series based analytical methods. In present study, the foregoing analytical methods are reviewed first. Then, an exact series solution of vibration of orthotropic thin plate with rotationally restrained edges is obtained by applying the method of finite integral transform. Although the method of finite integral transform has been applied for vibration analysis of orthotropic plates, the existing formulation requires of solving a highly non-linear equation and the accuracy of the corresponding numerical results can be questionable. For that reason, an alternative formulation was proposed to resolve the issue. The accuracy and convergence of the proposed method were studied by comparing the results with other exact solutions as well as approximate solutions. Discussions were made for the application of the method of finite integral transform for vibration analysis of orthotropic thin plates. [ABSTRACT FROM AUTHOR]

The aim of this paper is to put forward a semi-analytical solution to conduct the free vibration analysis of thick orthotropic annular sector plates with general boundary conditions, internal radial line and circumferential arc supports for the first time. Based on the Mindlin plate theory, stationary energy principle and Rayleigh-Ritz method, the governing eigenvalue equation is derived for thick orthotropic annular sector plates of different radii, sector angles, thickness ratio, line/arc supports and various boundary conditions along the radial and circumference edges. The main innovation point of this paper lies in breaking the classical boundary barrier by using an improved Fourier series to replace the traditional Fourier series. By comparing with those results in open reported literature, the present method proves to be of excellent accuracy and reliability. In addition, in order to enrich the research, some results involving thick orthotropic annular sector plates with various boundary conditions are presented firstly, which can be acted as benchmark data for the future research. [ABSTRACT FROM AUTHOR]

The intent of this paper is to firstly perform the vibration analysis of the functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates with arbitrary boundary conditions by means of the semi-analytical method which is proposed by the author’s team. In the material model, the distribution of the carbon nanotubes is uniform or functionally graded along with the thickness direction of structures and four types of the CNTs distribution are studied in this paper. The refined rule of mixtures approach containing the efficiency parameters is adopted to determine the properties of the composite media. The admissible displacement functions of the FG-CNTRC circular, annular and sector plates are uniformly expanded as the modified Fourier series which embodies a standard cosine Fourier series and several auxiliary functions which are introduced to eliminate the limit of the boundary conditions. On this foundation, the first-order shear deformation elasticity theory is employed to construct the energy expression of the FG-CNTRC circular, annular and sector plates. Then the Ritz-variational energy method is used to decide the natural frequencies and the associated mode shapes. To examine the convergence, accuracy, stability and efficiency of the computational model, the comprehensive studies are conducted. Based on that, some crucial parametric studies covering the influence of the geometrical parameters, CNTs distributions, volume fraction of CNTs and boundary conditions are also investigated in detail. [ABSTRACT FROM AUTHOR]

This paper establishes a plate-cavity system to study the vibro-acoustic coupling characteristics based on an improved Fourier series method (IFSM). The established coupling system consists of an acoustic cavity with rigid walls or impendence walls and a single or double thin laminated rectangular plate with various elastic boundary conditions. The energy principle is employed for the modeling of this coupling system. Under the framework, the displacement and sound pressure admissible functions are generally set, no relations to boundary conditions, to the combinations of Fourier cosine series and supplementary functions which can eliminate the discontinuous or jumping phenomenon in the boundaries. All the series expansion coefficients can be obtained through the Rayleigh-Ritz technique. The results obtained by presented method show good convergence and agreement by being compared with those of the literatures or finite element method (FEM). The natural characteristics of the plate-cavity coupling system are studied. The structure vibration response and acoustic pressure response are also investigated by applying structural force to the plate and applying the monopole source into the acoustic cavity. Furthermore, some new results are given, based on the various boundary conditions, anisotropic degrees, lamination schemes and cavity depths, which can be the benchmark for future research. [ABSTRACT FROM AUTHOR]

This paper presents new solutions for the load carrying capacity of a circular plate subjected to arbitrary rotational symmetric loadings. Simply supported and clamped boundary conditions are considered. The lower bound method of limit analysis has been used together with the half- range Fourier series expansion of the loading function. The resulting nonlinear equation has been solved numerically using the Newton–Raphson algorithm. Since the obtained solution satisfies all of the requirements for an exact solution, it is in fact the exact critical load factor. The results have been verified against available findings in the literature. Finally, several new loading types for which no solution is currently available have been analyzed. [ABSTRACT FROM AUTHOR]

Given a compact manifold M with boundary ∂ M , in this paper we introduce a global symbolic calculus of pseudo-differential operators associated to ( M , ∂ M ) . The symbols of operators with boundary conditions on ∂ M are defined in terms of the biorthogonal expansions in eigenfunctions of a fixed operator L with the same boundary conditions on ∂ M . The boundary ∂ M is allowed to have (arbitrary) singularities. As an application, several criteria for the membership in Schatten classes on L 2 ( M ) and r -nuclearity on L p ( M ) are obtained. We also describe a new addition to the Grothendieck–Lidskii formula in this setting. Examples and applications are given to operators on M = [ 0 , 1 ] n with non-periodic boundary conditions, and of operators with non-local boundary conditions. [ABSTRACT FROM AUTHOR]

The vibrational behaviours of single-walled carbon nanotubes (SWCNTs) bridged on a silicon channel are investigated using a three-segment Timoshenko beam model and a one-segment Timoshenko beam model with elastic boundaries together with molecular dynamics (MD) simulation. A modified Fourier series method (MFSM) is proposed to analyse the free vibration of the Timoshenko beam models with elastic boundary conditions. Explicit formulas are derived for the van der Waals (vdW) interaction coefficients between the SWCNTs and silicon substrates. The boundary elastic constants of the SWCNTs bridged on the silicon channel are obtained by fitting the bending curve of SWCNTs subjected to a static uniformly distributed lateral load simulated via the MD method. The MD simulations show that both the three-segment Timoshenko beam model and the one-segment Timoshenko beam model with elastic boundaries have a relatively good ability to predict the vibrational behaviours of SWCNTs bridged on a silicon channel. [ABSTRACT FROM AUTHOR]

In this investigation, an improved Fourier series method is presented for the free vibration analysis of the moderately thick laminated composite rectangular plate with non-uniform boundary conditions, a class of problems which are rarely attempted in the literatures. Under the current framework, the displacement and rotation functions are generally sought, regardless of boundary conditions, in spectral form, as a double Fourier cosine series and three supplementary functions. These supplementary functions are introduced to remove the potential discontinuities associated with the original displacement functions along the edges when they are viewed as periodic functions defined within the entire coordinates of laminate plate. The boundary conditions can be readily realized by setting the stiffness of the five types restraining springs. All the series expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz technique. Unlike most of the existing studies, the presented method can be readily and universally applied to a wide spectrum of plate vibration problems involving different boundary conditions, varying material, and geometric properties with no need of modifying the basic functions or adapting solution procedures. The excellent accuracy of the current result is validated by comparing the present results with those Finite Element Method (FEM) data. Numerous new results for free vibration of moderately thick rectangular plates with various multi-points supported and non-uniform partially supported boundary conditions are presented, which may serve as benchmark solution for future researches in the field. [ABSTRACT FROM AUTHOR]

A new analytical solution based on a higher-order beam theory for static, buckling and vibration of laminated composite beams is proposed in this paper. The governing equations of motion are derived from Lagrange’s equations. An analytical solution based on trigonometric series, which satisfies various boundary conditions, is developed to solve the problem. Numerical results are obtained to compare with previous studies and to investigate the effects of length-to-depth ratio, fibre angles and material anisotropy on the deflections, stresses, natural frequencies and critical buckling loads of composite beams with various configurations. [ABSTRACT FROM AUTHOR]

The existence of a crack in a rectangular plate makes the exact closed-form solutions for the vibrations of the plate intractable, if they even exist. This work presents analytical solutions for vibrations of horizontally or vertically cracked rectangular plates under various boundary conditions. The solutions are constructed by combining Fourier cosine series with domain decomposition. A rectangular plate with a side crack or an internal crack is divided into four or six rectangular sub-domains, respectively. The series solutions that satisfy the governing equations for the vibrations of a plate are firstly established for each sub-domain based on the classical plate theory. The solutions for each sub-domain are related to each other by satisfying the continuity conditions along the interconnection boundaries between the sub-domains. Finally, the boundary conditions of the cracked plate are imposed on the solutions. Comprehensive convergence studies are performed for intact plates and cracked plates with various boundary conditions, and the presented natural frequencies are compared with the published ones to confirm the correctness of the proposed solutions. Not like a typical energy method, which overpredicts true frequencies if the used shape functions do not exactly satisfy the governing equations for the problem under consideration, the convergence studies reveal that the present solutions converge from the lower bounds to the exact frequencies as the number of series terms increases. The present solutions are further applied to determine the first five frequencies of vibration of rectangular plates with side cracks and internal cracks of various lengths and locations. The results obtained under SSSS, CFFF, FSFS and FFFF boundary conditions are tabulated, and some of these are presented here for the first time. [ABSTRACT FROM AUTHOR]

In this paper, the free and forced vibration analysis of circular cylindrical double-shell structures under arbitrary boundary conditions is presented. This is achieved by employing the improved Fourier series method based on Hamilton’s principle. In the formulation, each displacement component of the cylindrical shells and annular plates is invariantly expanded as the superposition of a standard Fourier series with several supplementary functions introduced to remove the potential discontinuities of the original displacement and its derives at the boundaries. With the introduction of four sets of boundary springs at the coupling interfaces and end boundaries of the shell–plate combination, both elastic and rigid coupling and end boundary conditions can be easily obtained by assigning the stiffnesses of the artificial springs to certain values. The natural frequencies and mode shapes of the structures as well as frequency responses under forced vibration are obtained with the Rayleigh–Ritz procedure. The convergence of the method is validated by comparing the present results with those obtained by the finite element method. Several numerical results including natural frequencies and mode shapes are presented to demonstrate the excellent accuracy and reliability of the current method. Finally, a number of parameter studies concerning various end and coupling boundary conditions, different dimensions of shells and annular plates are also performed. [ABSTRACT FROM AUTHOR]

This study was motivated by the two-dimensional hydrodynamic slamming problem of a steep wave hitting a vertical wall. The fundamental problem considers dual impact on the wall at the lower and upper regions resembling the impact of a wave at the time of its breaking. The solution method results into a mixed-boundary value problem that involves a triplet of trigonometrical series which, to the author’s best knowledge, has not been investigated in the past. The formulation of the mixed-boundary value problem is generic and could be used in different fields as well. [ABSTRACT FROM AUTHOR]

Stress and electric field concentration around two arbitrarily-shaped holes in a finite electrostrictive solid subjected to uniform remote electric field are studied. The edge of the solid and the shapes of the holes are defined via several conformal mappings. The electro-elastic field of the solid as well as the electric fields both outside the solid and inside the holes are obtained on using conformal mapping techniques, Faber series and Fourier expansion method. Extensive numerical results are shown for elliptical, triangular, square, oval and rectangular holes in a square electrostrictive plane. The exact external electric field near the edge of the solid is found to be essentially non-uniform and its maximum appears to be much larger than the uniform remote electric field, while it is shown that the hoop stress nearby the point of maximum curvature on the hole's boundary increases rapidly as the point of maximum curvature approaches either the edge of the solid or the other hole. On the other hand, for a single centrally-located hole, the finite electrostrictive plane can be approximately treated as an infinite plane if the size of the finite plane is not less than six times the size of the hole, while for two interacting holes, the interaction between them is negligible when the distance between the two holes is larger than three times the size of the holes. [ABSTRACT FROM AUTHOR]

Point discretization methods such as the differential quadrature method (DQM) are well known to have difficulties in solving partial differential equations that involve the Dirac-delta function because the Dirac-delta function is a generalized singularity function and it cannot be discretized directly using the DQM. To overcome this difficulty, a simple differential quadrature methodology is proposed in this study, where the Dirac-delta function is expanded into a Fourier trigonometric series. By expanding the Dirac-delta function into a Fourier trigonometric series, this singular function is treated as non-singular functions, which can be discretized easily and directly using the DQM. The applicability of the proposed method is demonstrated by the in-plane vibration analysis of variable thickness circular arches traversed by a moving point load. The numerical results show that the proposed method is highly accurate and reliable. [ABSTRACT FROM AUTHOR]

This paper describes a unified solution for the vibration analysis of functionally graded material (FGM) doubly-curved shells of revolution with arbitrary boundary conditions. The solution is derived by means of the modified Fourier series method on the basis of the first order shear deformation shell theory considering the effects of the deepness terms. The material properties of the shells are assumed to vary continuously and smoothly along the normal direction according to general three-parameter power-law volume fraction functions. In summary, the energy functional of the shells is expressed as a function of five displacement components firstly. Then, each of the displacement components is expanded as a modified Fourier series. Finally, the solutions are obtained by using the variational operation. The convergence and accuracy of the solution are validated by comparing its results with those available in the literature. A variety of new vibration results for the circular toroidal, paraboloidal, hyperbolical, catenary, cycloidal and elliptical shells with classical and elastic boundary conditions as well as different geometric and material parameters are presented, which may serve as benchmark solution for future researches. Furthermore, the effects of the boundary conditions, shell geometric and material parameters on the frequencies are carried out. [ABSTRACT FROM AUTHOR]

The paper discusses membrane instability phenomena by a Fourier-related double scale approach that was introduced recently. This leads to a reduced-order model that is able to capture the main features of the wrinkles with few degrees of freedom. The paper focuses on the corresponding finite element procedure, its implementation and evaluation and applications to various cases of loading and boundary condition. The finite element model has first been implemented in a home-made code, the nonlinear system being solved by the Asymptotic Numerical Method (ANM), which has advantages of efficiency and reliability for stability analyses. It has also been implemented as a user element in a commercial software to evaluate the effectiveness of the reduction technique. Various loading cases were considered and the numerical tests establish that the reduced model can predict the wrinkling patterns, even when there are few wrinkles. The numerical results highlight the strong influence of a dimensionless parameter for wrinkling initiation. [ABSTRACT FROM AUTHOR]

Highlights: [•] A unified vibration analysis method is developed for truncated conical shells. [•] The proposed method is appropriate for general boundary conditions. [•] Effects of dimensional parameters are studied. [•] Influence of elastic restraint parameters is investigated. [•] New results for truncated conical shells with elastic restraints. [Copyright &y& Elsevier]

Bridging techniques between microscopic and macroscopic models are discussed in the case of wrinkling analysis. The considered macroscopic models are related to envelope equations of Ginzburg-Landau type, but generally, they are not valid up to the boundary. To this end, a multi-scale approach is considered: the reduced model is implemented in the bulk while the full model is applied near the boundary and these two models are coupled with the Arlequin method (Ben Dhia, 1998). This paper focuses on the definition of the coupling model and the transition between two scales. Especially, a new nonlocal bridging technique is presented and compared with another recent one (Hu et al., 2011). The present method can also be seen as a guide for coupling techniques involving other reduced order models. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider fractional sub-diffusion equation on two-dimensional unbounded domains and obtain an exact and some approximating artificial boundary conditions for it by a joint application of the Laplace transform and the Fourier series expansion. In order to test the derived artificial boundary condition, after reducing the main problem to an initial–boundary value problem on bounded domain by imposing the obtained boundary condition, we just use classical Crank–Nicolson method for space variables and L1 approximation for the fractional time derivative. Some numerical examples are given which confirm the effectiveness of the proposed artificial boundary conditions. [Copyright &y& Elsevier]

Abstract: Analytical approach used for critical load determination is based on Ritz energy technique in which two factors are crucial for the accuracy of results. First factor is deflection function. Herein, double Fourier series are used to represent buckled shape of the plates under arbitrary external loads. The second and the most important factor is adoption of adequate realistic stress distribution within plate prior to buckling. Based on the Baker&Pavlović&Tahan and Pavlović&Liu investigations, exact stress distributions were introduced herein. In that way, with the adequate deflection functions and precise stress distribution, Ritz energy method can produce highly accurate results. Through several examples for the plates with different boundary conditions, for which available literature offers very few analytical solutions, accuracy of the presented analytical approach is proved. Results obtained by analytical approach are reaffirmed by numerical finite-element runs. [Copyright &y& Elsevier]

Abstract: The iris biometric recognizes a human based on his/her iris texture, which is a stable and unique feature for every individual. A typical iris biometric system performs better for the ideal data, which is acquired under controlled conditions. However, its performance degrades when localizing iris in non-ideal data containing the noisy issues, e.g., the non-uniform illumination, defocus, and non-circular iris boundaries. This study proposes a reliable algorithm to localize iris in such images robustly. First, a small region containing the coarse location of iris is localized. Next, the pupillary boundary is extracted within this small region using an iterative-scheme comprising an adaptive binarization and a pupil location verification test. Following that, the limbic boundary is localized by reusing the Hough accumulator. The iris location is also verified through a gray-level test. After that, the pupillary and limbic boundaries are regularized by applying an enhanced method comprising a Radial-gradient operator (RGO), an error-transform (ET), and the Fourier series. Experimental results, obtained on the CASIA-IrisV3, CASIA-IrisV4, MMU V1.0, and MMU(new) V2.0 iris databases, show superiority of the proposed technique over some of the contemporary techniques. [Copyright &y& Elsevier]

Abstract: As far as the study of circular cylindrical shell is concerned, most of the existing works are limited to classical homogeneous boundary conditions. However, complicated boundary conditions, such as elastic and non-uniform boundary conditions are encountered in many engineering applications. Thus, it is of important practical significance to investigate the vibrations of cylindrical shell with this kind of boundary conditions. This paper is mainly concerned with the free vibrations of cylindrical shell with non-uniform elastic boundary constraints. The exact solution for the problem is obtained using improved Fourier series method, in which each of three displacements of the shell is represented by a Fourier series supplemented by several terms introduced to ensure and accelerate the convergence of the series expansions. The unknown expansions coefficients are treated as the generalized coordinates and determined using the Rayleigh–Ritz procedure. The change of the boundary conditions can be easily achieved by only varying the stiffness of the four sets of boundary springs at each end of the shell without the need of making any change to the solution procedure. The excellent accuracy of the current results is validated by comparison with the finite element method (FEM) results. Numerical results are presented to illustrate the effects of some interesting and practically important boundary restraints on free vibrations of circular cylindrical shell, including varying stiffness of boundary springs, point supported and partially supported boundary conditions. [Copyright &y& Elsevier]

Abstract: A general algorithm is presented to approximately solve a great variety of linear and nonlinear ordinary differential equations (ODEs) independent of their form, order, and given conditions. The ODEs are formulated as optimization problem. Some basic fundamentals from different areas of mathematics are coupled with each other to effectively cope with the propounded problem. The Fourier series expansion, calculus of variation, and particle swarm optimization (PSO) are employed in the formulation of the problem. Both boundary value problems (BVPs) and initial value problems (IVPs) are treated in the same way. Boundary and initial conditions are both modeled as constraints of the optimization problem. The constraints are imposed through the penalty function strategy. The penalty function in cooperation with weighted-residual functional constitutes fitness function which is central concept in evolutionary algorithms. The robust metaheuristic optimization technique of the PSO is employed to find the solution of the extended variational problem. Finally, illustrative examples demonstrate practicality and efficiency of the presented algorithm as well as its wide operational domain. [Copyright &y& Elsevier]

Abstract: Determination of material properties of orthotropic plates with general elastic boundary supports using the Inverse method is presented in this paper. The material properties were identified through updating four parameters in the governing equation of a symmetrically laminated thin plate. The sum of the squared difference of the natural frequencies obtained from modal testing was minimized using the Forward method. The displacement function was expressed as a 2-D Fourier cosine series supplemented with several terms in the form of 1-D series. A classical solution was derived by letting the series exactly satisfy the governing differential equation and all the boundary conditions at every field and boundary point. In the Inverse method, the series expansions for all the relevant derivatives of the eigen frequencies were obtained through term-by-term differentiations of the displacements. Many simulations were done but one numerical example is presented to demonstrate the accuracy and convergence of the solutions. [Copyright &y& Elsevier]

Abstract: Two multi-scale techniques will be applied in this paper to analyze buckling and wrinkling phenomena in sandwich structures. The first one is a Fourier-related analysis, where the unknowns are slowly variable Fourier coefficients of the basic fields. As shown in [19], it permits to describe the coupling between global and local buckling without describing the details of the spatial oscillations. This leads to a macroscopical model similar to Landau-Ginzburg envelope equations, but the latter is not able to account for local-global coupling and it is accurate only near the bifurcation. Nevertheless it is sometimes difficult to associate boundary conditions with this Fourier-related approach because the assumption of a nearly periodic response of the structure is not always valid near the boundary. In [17] it has been proposed to apply the initial model near the boundary, the macroscopical model in the bulk and to relate them by the Arlequin method. In this paper, we associate the Fourier-related analysis and the Arlequin method to study the instability of sandwich structures. [Copyright &y& Elsevier]

Abstract: A new higher order theory based analytical solution to the static analysis of general cross-ply plates is presented. The boundary-discontinuous generalized double Fourier series approach is used to solve highly coupled linear partial differential equations with the mixed type simply supported boundary conditions prescribed on the edges. The present results will provide data for the unsolved boundary conditions and provide benchmark comparisons for early design stages and verifications of numerical results such as finite element and boundary element. Analytical results are compared with finite element counterparts using commercially available software under uniformly distributed load. Present results are in good agreement with the finite element counterparts. The effects of important parameters such as lamination scheme, material property, thickness effects as well as their interactions are investigated in detail. [Copyright &y& Elsevier]

Abstract: Various macroscopic models to describe instability pattern formation are discussed in this paper. They are similar to the Ginzburg–Landau envelope equation, but they can remain valid away from the bifurcation and are based on the technique of Fourier series with slowly varying coefficients. We focus on two questions: the need to take phase changes into account and the boundary conditions to be associated with macroscopic models. The analysis is carried out on the basis of numerical simulations for the problem of a compressed beam on a nonlinear foundation that is quite similar to the well known Swift–Hohenberg equation. The first macroscopic model involves a real envelope so that the phase is assumed to be constant. The second model is also macroscopic and it is a sort of Ginzburg–Landau equation with a complex envelope. The third one follows from a multi-scaled approach with a numerical bridging between the full model near the boundary and a macroscopic model in the bulk. [Copyright &y& Elsevier]

Abstract: Rectangular plates on distributed elastic foundations are widely employed in footings and raft foundations of variety of structures. In particular, mounted columns and single footings may partially occupy the rectangular plate of any kind. This study deals with free vibration problem of thin rectangular plates on Winkler and Pasternak elastic foundation model which is distributed over a particular arbitrary area of the plate. Closed form solutions are developed through solving the governing differential equations of plates. Moreover, a novel mathematical approach is proposed to find the exact analytical solution of free vibration of plates with mixed or fully-clamped boundary conditions. Results are discussed in detail thorough parametric and verification studies. It was concluded that proposed mathematical method can successfully obtain the exact solution to free vibration problems of plates on partial elastic foundations with clamped and/or simply-supported boundary conditions. [Copyright &y& Elsevier]

Abstract: In this paper, wave propagation in generalized thermo elastic plate of polygonal cross-sectional plate is studied using the Fourier expansion collocation method. The equations of motion based on two-dimensional theory of elasticity is applied under the plane strain assumption of generalized thermo elastic plate of polygonal cross-sections composed of homogeneous isotropic material. The frequency equations are obtained by satisfying the boundary conditions along the surface of the polygonal plate using Fourier expansion collocation method. The numerical calculations are carried out for triangular, square, pentagonal and hexagonal cross sectional plates. Dispersion curves are drawn using the computed non-dimensional frequencies for longitudinal and flexural (symmetric and antisymmetric) modes of vibration. [Copyright &y& Elsevier]

Abstract: In this paper a high order theory for functionally graded (FG) axisymmetric cylindrical shells based on the expansion of the axisymmetric equations of elasticity for functionally graded materials (FGMs) into Fourier series in terms of Legendre''s polynomials is presented. Starting from the axisymmetric equations of elasticity, the stress and strain tensors, the displacement, traction and body force vectors are expanded into Fourier series in terms of Legendre''s polynomials in the thickness coordinate. In the same way the material parameters that describe the functionally graded material properties are also expanded into Fourier series. All equations of the linear elasticity including Hooke''s law are transformed into the corresponding equations for the Fourier series expansion coefficients. Then a system of differential equations in terms of the displacements and the boundary conditions for the Fourier series expansion coefficients is obtained. In particular the first and second order approximations of the exact shell theory are considered in more details. The obtained boundary-value problems are solved by the finite element method (FEM) with COMSOL Multiphysics and MATLAB software. Numerical results are presented and discussed. [Copyright &y& Elsevier]

Abstract: A semi-analytical solution for evaluation of field distributions around the surface of a conductive half space which contains a hidden long crack and excited by a three-dimensional (3D) current-carrying inducing coil at arbitrary frequency is presented. The governing Helmholtz equation is solved in three dimensions by separation of variables. The solution is obtained by developing a two-dimensional (2D) Fourier series model and using exponential functions in the third dimension. To expand all possible field components in the problem, we assume the conductor as a lossy material. The displacement current in the crack mouth is regarded to have a nonzero value accordingly. After imposing boundary conditions and using the mode matching technique, a linear system of AX=B is solved to obtain the unknown coefficients. The accuracy of the proposed modeling technique is demonstrated by comparing our results with those obtained numerically for a 3D inducing coil. [Copyright &y& Elsevier]

Abstract: Initial boundary value problems for the heat conduction equation with the constant and piecewise constant coefficients are considered in the paper. The Dirac delta function appears in the initial data. The considered problems are mathematical models of the temperature distribution arising from a pulse point heat source in homogeneous or two composite materials. The Fourier series expansion method is applied for solving these problems. The formal solutions are constructed in the form of the infinite series which are not classical functions. We show that these formal solutions are generalized functions (distributions). We prove that the partial sum of the constructed infinite series is a classical function which can be considered as a regularization of the generalized solution. The method of the computation of this regularized solution is suggested in the paper. The computational examples confirm the robustness of the proposed method. [Copyright &y& Elsevier]

Abstract: This study presents an analytical method which combines the superposition technique along with the solution structure theorem such that a closed-form solution of the hyperbolic heat conduction (HHC) equation can be obtained using fundamental mathematics. The method is applied to investigate two one-dimensional HHC problems with different boundary and initial conditions to study the effect of spatially-decaying and time-varying laser incidence at the surface. The methodology provides a convenient, accurate, and efficient solution to the HHC equation, which is applicable to a variety of HHC analyses for various engineering applications. The examples presented in this study can be used as benchmark problems for future numerical method validations. [Copyright &y& Elsevier]

Abstract: In the conventional semi-analytical finite strip analysis of folded plates, the boundary conditions and the intermediate support conditions must be satisfied a priori. The admissible functions used as the longitudinal part of the displacement functions are sometimes difficult to find, and they are valid for specific conditions only. In this paper, a general finite strip is developed for the static and vibration analyses of folded plate structures. The geometric constraints of the folded plates, such as the conditions at the end and intermediate supports, are modelled by very stiff translational and rotational springs as appropriate. The complete Fourier series including the constant term are chosen as the longitudinal approximating functions for each of the displacements. As these displacement functions are more general in nature and independent of one another, they are capable of giving more accurate solutions. The potential problem of ill-conditioned matrices is investigated and the appropriate choice of the very stiff springs is also suggested. The formulation is done in such a way to obtain a unified approach, taking full advantage of the power of modern computers. A few numerical examples are presented for comparison with numerical results from published solutions or solutions obtained from the finite element method. The results show that this kind of strips is versatile, efficient and accurate for the static and vibration analyses of folded plates. [Copyright &y& Elsevier]

Abstract: A two-dimensional analysis is presented for piezoelectric beam with variable thickness which is simply supported and grounded along its two ends. According to the governing equations of plane stress problems, the displacement solutions, which exactly satisfy the governing differential equations and the simply-supported boundary conditions at two ends of the beam, are derived. The unknown coefficients in the solution are then determined by using the Fourier sinusoidal series expansion to the boundary equations on the upper and lower surfaces of the beams. The present solutions show a good convergence and the numerical results are presented and compared with those available in the literature. The method could be applied to control engineering and other projects with highly accurate demand on stress and displacement analysis such as the design of micro-mechanical apparatuses. [Copyright &y& Elsevier]

Abstract: This paper presents an analytical method for solving the axisymmetric stress problem for a long hollow cylinder subjected to locally-distributed residual (incompatible) strains. This method is based on direct integration of the equilibrium and compatibility equations, which thereby have been reduced to the set of two governing equations for two key functions with corresponding boundary and integral conditions. The governing equations were solved by making use of the Fourier integral transformation. Application of the method is illustrated with an analysis of the welding residual stresses in a butt-welded thick-walled pipe. [Copyright &y& Elsevier]

Abstract: This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated. [Copyright &y& Elsevier]

Abstract: A finite element formulation is developed for the analysis of thin-walled pipes based on thin shell theory. The formulation starts with a Fourier series solution of the equilibrium equations developed in a companion paper and develops a family of exact shape functions for each mode. The shape functions developed are used in conjunction with the principle of stationary potential energy and yield a finite element that is exact within the assumptions of the underlying shell formulation. The stiffness matrix contribution for each mode n is observed to be fully uncoupled from those based on other modes m ≠ n. The resulting finite element is shown to be free from discretization errors normally occurring in conventional finite elements. The applicability of the solution is illustrated through examples with various loading cases and boundary conditions. A comparison with other finite element and closed form solutions demonstrates the validity and accuracy of the current finite element. [Copyright &y& Elsevier]

Abstract: The general form of the solution of the Airy function for the stress distributions that describe the non-linear effect developed from the large deflection of simply supported plates with movable edges are found by superposition of the Airy functions, which satisfy the large deflection condition and the boundary conditions of the edges. Each term of the Airy function consists of a particular solution and a homogeneous one. The particular solution satisfying the large deflection condition is classified into six cases, depending on the combinations of the modal numbers of the comparison functions. The corresponding homogeneous solution is found to make each Airy function satisfy the boundary condition by using the Fourier series method. The solution is applied to the non-linear analysis of the deflection of the simply supported plates with movable edges under transverse loading, and is verified by comparison with other investigation. [Copyright &y& Elsevier]

Abstract: This article presents some starting solutions corresponding to the oscillating flows of a Burgers’ fluid with fractional derivatives. The fractional calculus approach in the constitutive relationship model is introduced and a fractional Burgers’ model is built. The exact solutions for the oscillating motions of a fractional Burgers’ fluid due to cosine and sine oscillations of an infinite flat plate as well as those corresponding to an oscillating pressure gradient are established with the help of integral transforms (Fourier sine and Laplace transforms). In order to avoid lengthy calculations of residues and contour integrals, the discrete Laplace transform method has been used. The expressions for the velocity field and the resulting shear stress that have been obtained, presented under integral and series form in terms of the generalized G functions, satisfy all imposed initial and boundary conditions. Similar solutions for generalized Oldroyd-B, Maxwell and second grade fluids as well as those for ordinary Oldroyd-B, Maxwell, second grade and Newtonian fluids are obtained as limiting cases of our general solutions. Finally, the obtained solutions are graphically analyzed for variations of interesting flow parameters. [Copyright &y& Elsevier]

Abstract: In this study Free vibration analysis of vertical rectangular Mindlin plates resting on Pasternak elastic foundation and fully or partially in contact with fluid on their one side is investigated for different combinations of boundary conditions. The plate is assumed to be one of vertical rectangular walls of a container in contact with fluid. In order to analyze the interaction of the Mindlin plate with the elastic foundation and fluid system, three displacement components of the plate are expressed in the Ritz method by adopting a set of static Timoshenko beam functions satisfying geometric boundary conditions in a Cartesian co-ordinate system. The method of separation of variables and the method of Fourier series expansion is used to model fluid and to obtain the exact expression of the motion of fluid in the form of integral equations. The fluid domain is finite in depth and width but infinite in the length direction. To demonstrate the accuracy of the present solution, convergence study is first carried out and then a few comparison studies are carried out with the available data in the literature. Finally, natural frequencies of rectangular plates are presented in tabular and graphical forms for different fluid levels, foundation parameters, aspect ratios, thickness to width ratios and boundary conditions. [Copyright &y& Elsevier]

Abstract: We study the Poisson problem with zero boundary datum in a (finite) polyhedral cylinder with a non-convex edge. Applying the Fourier sine series to the equation along the edge and by a corner singularity expansion for the Poisson problem with parameter, we define the edge flux coefficient and the regular part of the solution on the polyhedral cylinder. We present a numerical method for approximating the edge flux coefficient and the regular part and show the stability. We derive an error estimate and give some numerical experiments. [Copyright &y& Elsevier]

Abstract: In this paper we introduce the boundary value problem of three-dimensional classical elasticity for an infinite body containing an elliptical crack. Using the method of simultaneous dual integral equations, the problem is transformed to the system of linear algebraic equations. Stress intensity factor is obtained in the form of the Fourier series expansion. Several solutions for specific cases of applied polynomial stress fields are derived and compared with existing results. Eligibility of the method for more complicated stress fields is demonstrated on the example of partially loaded elliptical crack. [Copyright &y& Elsevier]

Abstract: Extending the previous work on isotropic beams and plates by the second author [Li WL, et al. An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. J Sound Vib 2009;321:254–69], this paper describes an accurate analytical method for calculating the static deflections and modal characteristics of orthotropic plates with general elastic boundary supports. The displacement function is expressed as a 2-D Fourier cosine series supplemented with several terms in the form of 1-D series. The series expansions for all the relevant derivatives can be directly obtained through term-by-term differentiations of the displacement series. Thus, a classical solution can be derived by letting the series exactly satisfy the governing differential equation at every field point and all the boundary conditions at every boundary point, respectively. Several numerical examples are presented to demonstrate the excellent accuracy and convergence of the current solutions. [Copyright &y& Elsevier]