An advanced BEM for electromagnetic wave scattering problems with axisymmetric dielectric particles

An advanced boundary element/fast Fourier transform (FFT) methodology for solving axisymmetric electromagnetic wave scattering problems with general, non-axisymmetric boundary conditions is presented. The incident field as well as the boundary quantities of the problem are expanded in complex Fourier series with respect to the circumferential direction. Each of the expanding coefficients satisfies a surface integral equation which, due to axisymmetry, is reduced to a line integral along the surface generator of the body and an integral over the angle of revolution. The first integral is evaluated by discretizing the meridional line of the body into isoparametric elements and employing Gauss quadrature. The integration over the angle of revolution is performed simultaneously for all the expanding coefficients through the FFT. The singular integrals are computed directly with high accuracy. Representative numerical examples demonstrate the accuracy of the proposed boundary element formulation.     

Static and harmonic BEM solutions of gradient elasticity problems with axisymmetry

An advanced boundary element method/fast Fourier transform (BEM/FFT) methodology for treating static and time harmonic axisymmetric problems in linear elastic structures exhibiting microstructure effects, is presented. These microstructure effects are taken into account with the aid of a simple strain gradient elastic theory proposed by Aifantis and co-workers [Aifantis (1992), Altan and Aifantis (1992), Ru and Aifantis (1993)]. Boundary integral representations of both static and dynamic gradient elastic problems are employed. Boundary quantities, classical and non-classical (due to gradient terms) boundary conditions are expanded in complex Fourier series in the circumferential direction and the problem is decomposed into a series of problems, which are solved by the BEM by discretizing only the surface generator of the axisymmetric body. The BEM integrations are performed by FFT in the circumferential directions simultaneously for all Fourier coefficients and by Gauss quadrature in the generator direction. All the strongly singular integrals are computed directly by employing highly accurate three-dimensional integration techniques. The Fourier transform solution is numerically inverted by the FFT to provide the final solution. The accuracy of the proposed boundary element methodology is demonstrated by means of representative numerical examples.The authors acknowledge with thanks for the support provided by I.K.Y. through the program IKYDA 2002 (scientific cooperation between the University of Patras, Greece and the Ruhr-University Bochum, Germany).     

Strategy for solving semi-analytically three-dimensional transient flow in a coupled <Emphasis Type="Italic">N</Emphasis>-layer aquifer system

Efficient strategies for solving semi-analytically the transient groundwater head in a coupled N-layer aquifer system , i = 1, ... , N, with radial symmetry, with full z-dependency, and partially penetrating wells are presented. Aquitards are treated as aquifers with their own horizontal and vertical permeabilities. Since the vertical direction is fully taken into account, there is no need to pose the Dupuit assumption, i.e., that the flow is mainly horizontal. To solve this problem, integral transforms will be employed: the Laplace transform for the t-variable (with transform parameter p), the Hankel transform for the r-variable (with transform parameter α) and a particular form of a generalized Fourier transform for the vertical direction z with an infinite set of eigenvalues (with the discrete index m). It is possible to solve this problem in the form of a semi-analytical solution in the sense that an analytical expression in terms of the variables r and z, transform parameter p, and eigenvalues of the generalized Fourier transform can be given or in terms of the variables z and t, transform parameter α, and eigenvalues . The calculation of the eigenvalues and the inversion of these transformed solutions can only be done numerically. In this context the application of the generalized Fourier transform is novel. By means of this generalized Fourier transform, transient problems with horizontal symmetries other than radial can be treated as well. The notion of analytical solution versus numerical solution is discussed and a classification of analytical solutions is proposed in seven classes. The expressions found in this paper belong to Class 6, meaning that the transformed solutions are written in terms of eigenvalues which depend on one transform parameter (here p or α). Earlier solutions to the transient problem belong to Class 7, where the eigenvalues depend on two transform parameters. The theory is applied to three examples.     

Determination of Nonstationary Temperatures with the Help of Improved Formulas of the Inverse Laplace Transformation

We propose a method for the improvement and optimization of the algorithm of finding a function whose Laplace transform is known with the help of a numerical–analytic method based on the representation of the original function in the form of Fourier series. The relations obtained by the method of boundary elements are used to determine three-dimensional nonstationary temperature fields in bodies with cavities. As an example, we consider a problem of nonstationary heat conduction for a half space z > 0 containing an ellipsoidal cavity heated by a concentrated heat source with intensity Q. The process of heat exchange between the body and media washing the plane boundary and filling the cavity obeys the Newton law.     

Axisymmetric elasticity solutions for a uniformly loaded annular plate of transversely isotropic functionally graded materials

Summary The axisymmetric problem of a functionally graded, transversely isotropic, annular plate subject to a uniform transverse load is considered. A direct displacement method is developed that the non-zero displacement components are expressed in terms of suitable combinations of power and logarithmic functions of r, the radial coordinate, with coefficients being undetermined functions of z, the axial coordinate. The governing equations as well as the corresponding boundary conditions for the undetermined functions are deduced from the equilibrium equations and the boundary conditions of the annular plate, respectively. Through a step-by-step integration scheme along with the consideration of boundary conditions at the upper and lower surfaces, the z-dependent functions are determined in explicit form, and certain integral constants are then determined completely from the remaining boundary conditions. Thus, analytical elasticity solutions for the plate with different cylindrical boundary conditions are presented. As a promising feature, the developed method is applicable when the five material constants of a transversely isotropic material vary along the thickness arbitrarily and independently. A numerical example is finally given to show the effect of the material inhomogeneity on the elastic field in the annular plate.     

Analytical solutions describing the consolidation of a multi-layered soil under circular loading

Analytical solutions describing the consolidation of a multi-layered soil under circular loading are presented. From the governing equations of saturated poroelastic soil in a cylindrical coordinate system, the eighth-order state-space equation of consolidation is obtained by eliminating the variation of time t using the Laplace transform together with the technique of Fourier expansions with respect to the coordinate θ and the Hankel transform with respect to coordinate r. The solution of the eighth-order state-space equation is derived directly by using the Laplace transform and its inversion of the z-domain. Based on the continuity between layers and the boundary conditions, the transfer-matrix method is utilized to derive the solutions for the consolidation of a multi-layered soil under circular loading in the transformed domain. By the inversion of the Laplace transform and the Hankel transform, the analytical solutions in the physical domain are obtained. A numerical analysis based on the solutions is carried out by a corresponding program.     

Non-homogeneous Steady-State Heat Conduction Problem in a Thin Circular Plate and Its Thermal Stresses

The present paper deals with the determination of the displacement and thermal stresses in a thin circular plate defined as 0 ≤ r ≤ b, 0 ≤ z ≤ h under a steady temperature field, due to a constant rate of heat generation within it. A thin circular plate is insulated at the fixed circular boundary (r = b), and the remaining boundary surfaces (z = 0, z = h) are kept at zero temperature. The governing heat conduction equation has been solved by using an integral transform technique. The results are obtained in series form in terms of modified Bessel functions. The results for displacement and stresses have been computed numerically and are illustrated graphically.     

A mixed semi analytical solution for functionally graded (FG) finite length cylinders of orthotropic materials subjected to thermal load

A simplified and accurate analytical cum numerical model is presented here to investigate the behavior of functionally graded (FG) cylinders of finite length subjected to thermal load. A diaphragm supported FG cylinder under symmetric thermal load which is considered as a two dimensional (2D) plane strain problem of thermoelasticity in (r, z) direction. The boundary conditions are satisfied exactly in axial direction (z) by taking an analytical expression in terms of Fourier series expansion. Fundamental (basic) dependent variables are chosen in the radial coordinate of the cylinder. First order simultaneous ordinary differential equations are obtained as mathematical model which are integrated through an effective numerical integration technique by first transforming the boundary value problem into a set of initial value problems. For FG cylinders, the material properties have power law dependence in the radial coordinate. Effect of non homogeneity parameters and orthotropy of the materials on the stresses and displacements of FG cylinder are studied. The numerical results obtained are also first validated with existing literature for their accuracy. Stresses and displacements in axial and radial directions in cylinders having various l/r _i and r _o/r _i ratios parameter are presented for future reference.     

Displacement discontinuity formulation for modeling cracks in orthotropic shear deformable plates

A displacement discontinuity formulation is presented for modeling cracks in orthotropic Reisnner plates. Fundamental solutions for displacement discontinuity are derived for the first time using a Fourier transform method. Boundary integral equations are presented in terms of discontinuity rotations on the crack surfaces for opening mode problems. As the fundamental solutions have singularity of O (1/r ²), Chebyshev polynomials of the second kind are used to evaluate the integral equations. By solving for coefficients of the Chebyshev polynomials, the stress intensity factors at the crack tips are obtained directly. Comparisons are made with solutions using the finite element method to demonstrate that the displacement discontinuity method is an efficient and accurate method for solving crack problems in orthotropic Reissner plates.     

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W ₂(y) and W ₃(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.     

Numerical techniques on improving computational efficiency of spectral boundary integral method

Numerical techniques are suggested in this paper, in order to improve the computational efficiency of the spectral boundary integral method, initiated by Clamond & Grue [D. Clamond and J. Grue. A fast method for fully nonlinear water‐wave computations. J. Fluid Mech. 2001; 447 : 337–355] for simulating nonlinear water waves. This method involves dealing with the high order convolutions by using Fourier transform or inverse Fourier transform and evaluating the integrals with weakly singular integrands. A de‐singularity technique is proposed here to help in efficiently evaluating the integrals with weak singularity. An anti‐aliasing technique is developed in this paper to overcome the aliasing problem associated with Fourier transform or inverse Fourier transform with a limited resolution. This paper also presents a technique for determining a critical value of the free surface, under which the integrals can be neglected. Numerical tests are carried out on the numerical techniques and on the improved method equipped with the techniques. The tests will demonstrate that the improved method can significantly accelerate the computation, in particular when waves are strongly nonlinear. Copyright © 2015 John Wiley & Sons, Ltd.     

On scattering of an SH-wave by a corner comprised of two different elastic materials

The problem of the scattering of plane SH-waves by a corner comprised of two different elastic materials is considered. This problem is the analogy of the dielectric wedge problem in electromagnetics. A Fourier transform method is used to derive equations which can readily be solved numerically. The unknowns are the Fourier transforms of the displacements and tractions on the interfaces of the wedge. Fredholm integral equations of the second kind with simple continuous kernels are obtained after taking a physically reasonable representation of the unknown quantities. Numerical results for the diffraction coefficients are presented. The method can be generalized for incidence of Stonely, plane P-, or SV-waves.     

A new boundary integral equation method for analysis of cracked linear elastic bodies

Abstract

A novel integral equation method is developed in this paper for the analysis of two‐dimensional general anisotropic elastic bodies with cracks. In contrast to the conventional boundary integral methods based on reciprocal work theorem, the present method is derived from Stroh's formalism for anisotropic elasticity in conjunction with Cauchy's integral formula. The proposed boundary integral equations contain boundary displacement gradients and tractions on the non‐crack boundary and the dislocations on the crack lines. In cases where only the crack faces are subjected to tractions, the integrals on the non‐crack boundary are non‐singular. The boundary integral equations can be solved using Gaussian‐type integration formulas directly without dividing the boundary into discrete elements. Numerical examples of stress intensity factors are given to illustrate the effectiveness and accuracy of the present method.     

Axisymmetric crack problem for a hollow cylinder imbedded in a dissimilar medium

The analytical solution for the linear elastic problem of flat annular crack in a transversely isotropic hollow cylinder imbedded in a transversely isotropic medium is considered. The hollow cylinder is assumed to be perfectly bonded to the surrounding medium. This structure, which can represent a cylindrical coating-substrate system, is subjected to uniform crack surface pressure. Because of the geometry and the loading, the problem is axisymmetric. The z = 0 plane on which the crack lies, is also a plane of symmetry. The composite media consisting of the hollow cylinder and the surrounding medium extends to infinity in z and r directions. The mixed boundary value problem is formulated in terms of the unknown derivative of the crack surface displacement by using Fourier and Hankel transforms. By extending the crack to the inner surface and to the interface, the cases of surface crack and crack terminating at the interface are obtained. Asymptotic analyses are performed to derive the generalized Cauchy kernel and associated stress singularities. The resulting singular integral equation is solved numerically. Stress intensity factors for various crack configurations, crack opening displacements and stresses along the interface and on z = 0 plane are presented for sample material combinations and geometric parameters.     

An algorithmically consistent macroscopic tangent operator for FFT‐based computational homogenization

The present work addresses a multiscale framework for fast‐Fourier‐transform–based computational homogenization. The framework considers the scale bridging between microscopic and macroscopic scales. While the macroscopic problem is discretized with finite elements, the microscopic problems are solved by means of fast‐Fourier‐transforms (FFTs) on periodic representative volume elements (RVEs). In such multiscale scenario, the computation of the effective properties of the microstructure is crucial. While effective quantities in terms of stresses and deformations can be computed from surface integrals along the boundary of the RVE, the computation of the associated moduli is not straightforward. The key contribution of the present paper is the derivation and implementation of an algorithmically consistent macroscopic tangent operator which directly resembles the effective moduli of the microstructure. The macroscopic tangent is derived by means of the classical Lippmann‐Schwinger equation and can be computed from a simple system of linear equations. This is performed through an efficient FFT‐based approach along with a conjugate gradient solver. The viability and efficiency of the method is demonstrated for a number of two‐ and three‐dimensional boundary value problems incorporating linear and nonlinear elasticity as well as viscoelastic material response.     

An efficient and high‐order frequency‐domain approach for transient acoustic–structure interactions in three dimensions

This paper is concerned with numerical solution of the transient acoustic–structure interaction problems in three dimensions. An efficient and higher‐order method is proposed with a combination of the exponential window technique and a fast and accurate boundary integral equation solver in the frequency‐domain. The exponential window applied to the acoustic–structure system yields an artificial damping to the system, which eliminates the wrap‐around errors brought by the discrete Fourier transform. The frequency‐domain boundary integral equation approach relies on accurate evaluations of relevant singular integrals and fast computation of nonsingular integrals via the method of equivalent source representations and the fast Fourier transform. Numerical studies are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermal stress intensity factors of an infinite orthotropic layer with a crack

Stress intensity factors are determined for a crack in an infinite orthotropic layer. The crack is situated parallel to the plane surfaces of the layer. Stresses are solved for two kinds of the boundary conditions with respect to temperature field. In the first problem, the upper surface of the layer is heated to maintain a constant temperature T ₀, while the lower surface is cooled to maintain a constant temperature –T ₀. In the other problem, uniform heat flows perpendicular to the crack. The surfaces of the crack are assumed to be insulated. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the crack, the difference in temperature at the crack surfaces and differences in displacements are expanded in a series of functions that vanish outside the crack. The unknown coefficients in each series are evaluated using the Schmidt method. Stress intensity factors are then calculated numerically for a steel layer that behaves as an isotropic material and for a tyrannohex layer that behaves as an orthotropic material.     

Radial transport in a porous medium with Dirichlet, Neumann and Robin-type inhomogeneous boundary values and general initial data: analytical solution and evaluation

The analytical solution is presented to the convection–diffusion equation describing the concentration of solutes in a radial velocity field due to extracting groundwater from or injecting water into an aquifer with arbitrary initial concentration data F(r), with r the radial distance, and an inhomogeneous mixed boundary condition G(t), with t the time, at the well radius r = r ₀. The analytical solution is obtained with a generalized Hankel transformation or with a Laplace transformation. The Hankel transformation turns out to be easier for G = 0, F ≠ 0, while the Laplace transformation is easier for F = 0, G ≠ 0. Both techniques can, however, deal with the full problem. The representation found by the generalized Hankel transform can also be found by the Laplace transform, through modification of the contour through the complex plane in the Bromwich integral for the inverse Laplace transform to the real axis. In practice, the numerical evaluation of the integral representation is difficult, due to the oscillating behavior of the integrands. A more appropriate numerical inversion procedure is also suggested, which circumvents the integration of the oscillating integrands, by an alternative modification of the contour in the Bromwich integral such that the new contour follows the steepest descent path starting from a saddle point at the real axis.     

Pricing financial claims contingent upon an underlying asset monitored at discrete times

Exotic option contracts typically specify a contingency upon an underlying asset price monitored at a discrete set of times. Yet, techniques used to price such options routinely assume continuous monitoring leading to often substantial price discrepancies. A brief review of relevant option-pricing methods is presented. The pricing problem is transformed into one of Wiener–Hopf type using a z-transform in time and a Fourier transform in the logarithm of asset prices. The Wiener–Hopf technique is used to obtain probabilistic identities for the related random walks killed by an absorbing boundary. An accurate and efficient approximation is obtained using Padé approximants and an approximate inverse z-transform based on the trapezoidal rule. For simplicity, European barrier options in a Gaussian Black–Scholes framework are used to exemplify the technique (for which exact analytic expressions are obtained). Extensions to different option contracts and options driven by other Lévy processes are discussed.     

Calculation of diffraction patterns on a spatial surface

An approximated formulation of the Fresnel function is put forward and is used in the approximate evaluation of the Fresnel diffraction integral. By comparing the approximate formulation with the experimental measurements and calculations in the fast Fourier transform (FFT) method of the diffraction integral, we demonstrate that the proposed method is sufficiently accurate for calculating the Fresnel diffraction. For the diffraction field calculation on a spatial surface, the calculation speed of this method is usually higher than that of the FFT method.