Boundary integral equations from Hamilton's principle for surface acoustic waves under periodic metal gratings

The procedure describes the derivation of boundary integral equations for surface acoustic waves propagating under periodic metal strip gratings with piezoelectric films. It takes into account the electrical and mechanical perturbations, including the effects of mass loading caused by the gratings with an arbitrary shape. First, an integral equation is derived with line integrals on the boundaries within one period. This derivation is based on Hamilton's principle and uses Lagrange's method of multipliers to alleviate the continuous conditions of the displacement and the electric potential on the boundaries. Second, boundary integral equations corresponding to each substrate, piezoelectric film, metal strip, and free space region are obtained from the integral equation using the Rayleigh-Ritz method for admissible functions. With this procedure, it is not necessary to make any assumptions for separation of the boundary conditions between two neighboring regions. Consequently, we clarify the theoretical basis for the analytical procedure using boundary integral equations for longitudinal LSAW modes.     

Propagation of longitudinal leaky surface waves under periodic SiO (2)/Al structure on Li(2)B(4)O(7) substrate

The properties of longitudinal leaky surface waves (LLSW) under a periodic SiO(2)/Al structure on Li(2)B(4)O (7) (LBO) substrate, were investigated theoretically and experimentally, in order to improve the high propagation losses of LLSWs under a periodic Al grating with the normalized thickness over 2%. In the theoretical analysis, the previously presented method based on the boundary integral equations for the periodic metal grating structure on the substrate was extended to include the dielectric layer. In the experiments, devices with Al electrodes recessed into a SiO(2) groove on LBO were fabricated, and the propagation losses of them were estimated. As a result, it was shown that, when the surface of the structure was flattened, the propagation losses were sufficiently low and the first Bragg stopband width decreased.     

Analysis of leaky-surface-wave propagating under periodic metal grating

A detailed field analysis is presented for a leaky surface wave propagating under a periodic metal grating, using a theory that neglects the effect of mass loading due to the grating. The approach is based on Floquet's theorem and the coupled equations of wave motion with unperturbed mechanical and perturbed (or periodic) electrical boundary conditions, yielding a general field solution applicable to any material and to arbitrary connections to the grating. As a key step, the periodic boundary equations are solved by combining them into a set of infinite homogeneous equations through algebraic treatment and performing orthogonal integration with respect to space harmonics. The advantage in using this method results from there being no need to use assumptions or complicated expressions anticipating an accurate solution if sufficient space harmonics are considered. It is shown that the theory proposed here can be directly extended to solve simpler SAW problems. An analysis is carried out for LiNbO(3) for both the leaky wave and Rayleigh wave, taking into account dispersion relations, propagation attenuation of the leaky wave, and other field distributions. Theoretical and experimental results for the width of the first stopband are discussed.     

Stress-intensity factors for a cruciform crack with equal arm in an infinite elastic strip of finite thickness

The stress-intensity factors and the crack energy for a cruciform crack with equal arm, in an infinite elastic strip of finite thickness have been determined by solving the corresponding boundary value problem by using the Fourier Transform technique. The dual integral equations generated by using the necessary boundary conditions have been reduced to solving a pair of integral equations of Fredholm type involving two unknown functions. The integral equations have been solved by a numerical technique and the formulae for determining the quantities of physical interest have been expressed in terms of the solutions of the integral equations. Numerical results for these physical quantities have been furnished at the end for different values of the thickness of the strip.     

New implementation of MLBIE method for heat conduction analysis in functionally graded materials

The meshless local boundary integral equation (MLBIE) method with an efficient technique to deal with the time variable are presented in this article to analyze the transient heat conduction in continuously nonhomogeneous functionally graded materials (FGMs). In space, the method is based on the local boundary integral equations and the moving least squares (MLS) approximation of the temperature and heat flux. In time, again the MLS approximates the equivalent Volterra integral equation derived from the heat conduction problem. It means that, the MLS is used for approximation in both time and space domains, and we avoid using the finite difference discretization or Laplace transform methods to overcome the time variable. Finally the method leads to a single generalized Sylvester equation rather than some (many) linear systems of equations. The method is computationally attractive, which is shown in couple of numerical examples for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

The numerical solutions of the periodic crack problems of anisotropic strips*

The numerical solutions of the periodic crack problems of anisotropic strip are investigated in this paper. Using the complex variable method the mathematical model, boundary value problems for analytic functions, is established. After constructing appropriate integral transformations the boundary value problems are transformed into integral equations. Employing Lobotto-Chebyshev quadrature formulas and Gauss quadrature formulas, the approximate analytical expressions of the stress intensity factors are obtained. And some numerical results are performed for several special cases.     

Dynamic behavior of a crack in a functionally graded piezoelectric strip bonded to two dissimilar half piezoelectric material planes

Summary. The dynamic behavior of a crack in a functionally graded piezoelectric material (FGPM) strip bonded to two half dissimilar piezoelectric material planes subjected to combined harmonic anti-plane shear wave and in-plane electrical loading was studied under the limited permeable and permeable electric boundary conditions. It was assumed that the elastic stiffness, piezoelectric constant and dielectric permittivity of the functionally graded piezoelectric layer vary continuously along the thickness of the strip. By using the Fourier transform, the problem can be solved with a set of dual integral equations in which the unknown variables are the jumps of the displacements and the electric potentials across the crack surfaces. In solving the dual integral equations, the jumps of the displacements and the electric potentials across the crack surfaces were expanded in a series of Jacobi polynomials. Numerical results illustrate the effects of the gradient parameter of FGPM, electric loading, wave number, thickness of FGPM strip and electric boundary conditions on the dynamic stress intensity factors (SIFs).     

Method of integral functionals for electromagnetic wave scattering from a double-periodic magnetodielectric layer

A numerical method in the frequency domain is developed for analyzing three-dimensional gratings using the concept of a double-periodic magnetodielectric layer. The method is based on the three-dimensional volume integral equations for the equivalent electric and magnetic polarization currents of the assumed periodic medium. The integral equations are solved by using the integral functionals related to the polarization current distributions and the technique of double Floquet-Fourier series expansion. Once the integral functionals are determined, the scattered fields outside the layer are calculated accordingly. The unit cell of the layer comprises several parallelepiped segments of materials characterized by the complex-valued relative permittivity and permeability of step function profiles. The arbitrary profiles of three-dimensional dielectric or metallic gratings can be flexibly modeled by adjusting the material parameters and sizes or locations of the parallelepiped segments in the unit cell. Numerical examples for various grating geometries and their comparisons with those presented in the literature demonstrate the accuracy and usefulness of the proposed method.     

Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction

Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings.     

Scattering of electromagnetic plane wave from a perfect electric conducting strip placed at free space–chiral interface

The scattering of electromagnetic plane wave from a perfect electric conducting strip of finite width is investigated in this study. The strip is placed at the planar interface of free space and chiral medium. The Kobayashi potential method is used to determine the scattering from the strip. The dual integral equations are acquired through the boundary conditions. These equations as well as the edge conditions are satisfied through the properties of Weber–Schafheitlin integrals and the Jacobi polynomials. The projection of the Jacobi polynomials is used to get the matrix equations which are solved numerically to determine the unknown coefficients. Monostatic and bistatic scattering widths are examined in the free space region of the geometry. The far-zone scattered fields are analyzed by changing the values of different parameters, i.e. chirality and incident angle.     

Boundary element method for band gap calculations of two-dimensional solid phononic crystals

A boundary element method (BEM) is developed to calculate the elastic band gaps of two-dimensional (2D) phononic crystals which are composed of square or triangular lattices of solid cylinders in a solid matrix. In a unit cell, the boundary integral equations of the matrix and the scatterer are derived, the former of which involves integrals over the boundary of the scatterer and the periodic boundary of the matrix, while the latter only involves the boundary of the scatterer. Constant boundary elements are adopted to discretize the boundary integral equations. Substituting the periodic boundary conditions and the interface conditions, a linear eigenvalue equation dependent on the Bloch wave vector is derived. Some numerical examples are illustrated to discuss the accuracy, efficiency, convergence and the computing speed of the presented method.     

A wavenumber domain boundary element method model for the simulation of vibration isolation by periodic pile rows

The wavenumber domain boundary element method (WDBEM) for the interaction between the half-space soil and periodic structures is important for the design of various periodic structures in civil engineering. In this study, a WDBEM model for the half-space soil and periodic pile rows is developed and used in the analysis of the vibration isolation via pile rows. To establish the model, the rigid-body-motion method for the estimation of the Cauchy type singular integrals involved in the WDBEM is established for the first time. In the proposed model, the half-space soil and periodic pile rows are treated as elastic media. Employing the spatial domain boundary integral equations for the half-space soil and pile rows as well as the sequence Fourier transform method, the wavenumber domain boundary integral equations for the soil and pile rows are derived. By using the obtained wavenumber domain boundary integral equations, WDBEM formulations for the half-space soil and periodic pile rows are established. Using the WDBEM formulations as well as the continuity conditions at the pile–soil interfaces, a coupled WDBEM model for the pile–soil system is derived. With the proposed WDBEM model, the influences of the pile length and the shear modulus of the half-space soil on the vibration isolation effect of pile rows are examined. Presented numerical results show that the isolation vibration effect of pile rows is enhanced considerably with increasing length of the piles. Besides, the isolation vibration effect of pile rows is weakened considerably with increasing shear modulus of the half-space soil. Moreover, as expected, multiple pile rows usually produce a better isolation vibration effect than a single pile row.     

A meshless solution to two-dimensional convection–diffusion problems

An integral equation domain decomposition method has been implemented in a meshless fashion. The method exploits the advantage of placing the source point always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green's identities and the remaining equations are the derivatives of the first equation in respect to space coordinates. Radial basis function interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). Dual reciprocity method (DRM) has been applied to convert the domain integrals into boundary integrals, though the approach is general and can be applied without the DRM. The accuracy and robustness of the method has been tested on a convection–diffusion problem. The results obtained using the current approach have been compared with previously reported results obtained using the finite element method (FEM), and the DRM multi-domain approach (DRM-MD) showing similar level of accuracy.     

Crack propagating in a functionally graded strip under the plane loading

In the present paper, a finite crack with constant length (Yoffe type crack) propagating in the functionally graded strip under the plane loading is investigated by means of the Schmidt method. By using the Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of the material properties, the thickness of the functionally graded strip, and speed of the crack propagating upon the dynamic fracture behavior.     

On the Moving Griffith Crack in a Nonhomogeneous Orthotropic Strip

A finite crack with constant length (Yoffe type crack) propagating in the functionally graded orthotropic strip under the plane loading is investigated by means of the Schmidt method. By using the Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties, the thickness of the functionally graded orthotropic strip and the speed of the crack propagating upon the dynamic fracture behavior.     

多角形域上第一类边界积分方程的高精度配置法

本文提出了一种多角区域上的第一类边界积分方程的高精度算法:离散之前采用特殊周期变换,消去边界积分方程未知函数在积分端点的奇异性,然后使用常元配置法求解,该方法在内点获得超收敛o(h3)。此外,通过Richardson整体外推,可进一步提高内点解的精度。实际计算结果表明,该方法优上同一问题的Galerkin方法甚至机械求积法。     

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.     

Complex variable boundary integral method for linear viscoelasticity: Part I—basic formulations

The basic formulations (direct and indirect) of the complex variable boundary integral method for linear viscoelasticity are presented. Complex variable temporal integral equations for the formulations are obtained for viscoelastic solids whose behavior in shear is governed by a Boltzmann model while the bulk behavior is purely elastic. The functions involved in the integral equations are the time-dependent complex boundary tractions and displacements for the direct approach and the unknown time-dependent complex density functions for the indirect approaches. The temporal integral equations give the displacements and stresses at a point inside a viscoelastic region in terms of time convolution and space integrals over the boundary of this region. The equations are valid for the boundaries of arbitrary shapes provided that these boundaries are sufficiently smooth. Complex variable temporal boundary equations are obtained by taking the inner point to the boundary. Numerical treatment of spatial and time convolution integrals involved in the boundary equations is discussed.     

Single edge-cracked orthotropic strip under three-point load

The problem of an orthotropic strip having a crack of unit length normal to one edge and subjected to a bending moment resulting from three-point loading is solved using integral transform method. The mixed boundary conditions lead to dual integral equations which are ultimately reduced to a Fredholm integral equation of second kind. The integral equation thus obtained is solved by the method developed by Fox and Goodwin. Numerical solutions for a fibre-reinforced composite material have been carried out to determine the stress intensity factor of an orthotropic medium. The same has been compared with the isotropic case.     

Maxwell equations in Fourier space: fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media.

We establish the most general differential equations that are satisfied by the Fourier components of the electromagnetic field diffracted by an arbitrary periodic anisotropic medium. The equations are derived by use of the recently published fast-Fourier-factorization (FFF) method, which ensures fast convergence of the Fourier series of the field. The diffraction by classic isotropic gratings arises as a particular case of the derived equations; the case of anisotropic classic gratings was published elsewhere. The equations can be resolved either through classic differential theory or through the modal method for particular groove profiles. The new equations improve both methods in the same way. Crossed gratings, among which are grids and two-dimensional arbitrarily shaped periodic surfaces, appear as particular cases of the theory, as do three-dimensional photonic crystals. The method can be extended to nonperiodic media through the use of a Fourier transform.