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• Exact solution of the problem of Eshelby's inhomogeneity in strain gradient elasticity theory. • Green's function technique and integral equation developed for arbitrary boundary conditions. • Analytical expressions of mean strain inside inclusion. • Analytical expressions of effective elastic properties with strain gradient. • Effect of the scale parameter, the contrast of elastic behavior and the volume fraction of inhomogeneity. A new micromechanical approach to deal with the problem of Eshelby's inhomogeneity is developed for the prediction of the effective properties of composite materials according to the strain gradient elasticity theory. The method is based on the Green's function technique leading to an integral equation of the heterogeneous elastic problem. Within the simplified strain gradient elasticity theory, the integral equation for an infinite heterogeneous medium subjected to non-homogeneous boundary conditions is acquired. Thanks to this integral equation, the exact solution of Eshelby's inhomogeneity problem is detailed for spherical inhomogeneity and isotropic elastic behavior. From the expression of strain localization relations, the effective elastic properties of a two-phase composite material are then predicted through Mori Tanaka's homogenization scheme. To test the relevance of the suggested approach, its predictions are compared with results issued from some reference models and experimental data. [ABSTRACT FROM AUTHOR]

In the solution of frequency- and time-domain integral equations, the singularity cancellation transformations are well-known for the treatments of the singular integrals involving Green’s functions and their gradients. The cancellation transformations for strongly near-singular integrals become inaccurate and inefficient for an extremely deformed triangular patch, and this phenomenon is referred to as shape-dependent problem in this article. As the degree of the singularity increases, the shape-dependent problem of strong near-singularity is more severe than that of the weakly near-singular integrals. Furthermore, if the source triangle is deformed, the accuracy of the cancellation methods decreases when no near singularity exists. In this work, we first investigate the reason for these problems via the theoretical analysis with numerical verification. Second, an updated framework of singularity cancellation methods for strongly near-singular integrals is proposed, which has a fast and consistent convergence rate for both regular and irregular triangles. Third, some numerical experiments are presented to illustrate the effectiveness of the theoretical framework and the proposed transformations. [ABSTRACT FROM AUTHOR]

Uncertainty quantification related to nonlinear inverse scattering problems often involves the posterior covariance matrix that cannot be effectively estimated but still need to be considered as an important part of the nonlinear inverse scattering problem. The objective of this work is to show how to take advantage of a Bayesian framework to estimate the uncertainty of velocity reconstruction in the nonlinear inverse scattering imaging. The key ingredients are twofold. On the one hand, we rely on a Kalman Filter method, equipped with an optimization scheme, to solve the inverse problem. We use the distorted Born iterative method to formulate the sensitivity kernel. It directly uses an explicit representation of the data sensitivity function in terms of Green functions, rather than the indirect optimization approach based on the adjoint state method, in which the Green’s functions are based on integral equations. On the other hand, we use the direct envelope methods to provide the initial guess (large-scale smooth model) to overcome the nonlinearity in the inverse scattering problems. We first investigate the uncertainty of velocity imaging in high contrast media. Then we show by means of Lippmann-Schwinger-type equations that the uncertainty of multiparameter inverse scattering problems can be addressed. Numerical results dealing with the uncertainty of nonlinear inverse scattering problems in the case of both isotropic and anisotropic media highlight the important role played by uncertainty quantification. [ABSTRACT FROM AUTHOR]

We propose to compute a sparse approximate inverse Cholesky factor L of a dense covariance matrix \Theta by minimizing the Kullback--Leibler divergence between the Gaussian distributions \scrN (0, \Theta) and \scrN (0,L \top L 1), subject to a sparsity constraint. Surprisingly, this problem has a closed-form solution that can be computed efficiently, recovering the popular Vecchia approximation in spatial statistics. Based on recent results on the approximate sparsity of inverse Cholesky factors of \Theta obtained from pairwise evaluation of Green's functions of elliptic boundary-value problems at points \{xi\} 1\leq i\leq N \subset Rd, we propose an elimination ordering and sparsity pattern that allows us to compute \epsilon -approximate inverse Cholesky factors of such \Theta in computational complexity \scrO (N log(N/\epsilon)d) in space and \scrO (N log(N/\epsilon)2d) in time. To the best of our knowledge, this is the best asymptotic complexity for this class of problems. Furthermore, our method is embarrassingly parallel, automatically exploits low-dimensional structure in the data, and can perform Gaussian-process regression in linear (in N) space complexity. Motivated by its optimality properties, we propose applying our method to the joint covariance of training and prediction points in Gaussian-process regression, greatly improving stability and computational cost. Finally, we show how to apply our method to the important setting of Gaussian processes with additive noise, compromising neither accuracy nor computational complexity. [ABSTRACT FROM AUTHOR]

We apply the integral equation method to the diffraction of a waveguide wave on an impedance inductive cylinder in a rectangular waveguide. The Green's function determining the kernel of the relevant integral equation is constructed for a plane waveguide using the reflection method and the Poisson transform. [ABSTRACT FROM AUTHOR]

This paper shows how the classical representation techniques for the solution of elasticity problems, based on the Green's functions, can be generalized to second-gradient continua focusing on the specific case of pantographic lattices. As these last are strongly anisotropic, the fundamental solutions of isotropic second-gradient continua involving bi-Helmholtz-type operators are not applicable. More specifically we establish the analytical fundamental solution for the linearized equations governing the equilibrium of pantographic 2D continua in the neighbourhood of the reference configuration. Moreover, by means of found novel Green's functions, it is shown that it is possible to solve aforesaid equilibrium equations by using Fredholm integral equations. It is seen that an approximated analytical solution for the standard bias test for pantographic 2D continua can be found by using judiciously the found analytical fundamental solutions. The micro-macro-asymptotic identification allows for a clear and satisfactory physical interpretation of the obtained analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the source strength identification problem for the three-dimensional inverse heat conduction equations. The problem is to determine an unknown heat source strength from the measurement data for a specified location. In this process, the direct problem is solved by applying the Green's function method. Then, this problem can be converted into a Volterra integral equation of the first kind. Further, the Tikhonov and truncated singular value decomposition regularization methods are developed to identify the unknown source strength based on the discrepancy principle for choosing the regularization parameter. Finally, numerical examples are presented to show the feasibility and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Electromagnetic simulation of fractures has a growing importance in geophysical exploration, where the formation is usually inhomogeneous and modeled as a layered medium (LM). In this paper, we have developed the first integral-equation based solver to simulate the scattering of fractures straddling an LM. We refer to this solver as LM-thin dielectric sheet (TDS)-surface integral equation (SIE), since it is built on the TDS-based SIE and the LM Green’s function (LMGF). Compared with the traditional finite element method (FEM) and volume integral equation (VIE), LM-TDS-SIE achieves excellent efficiency by taking advantage of the multiscale feature of fracture, rather than being restricted by its volumetric mesh. Good accuracy but the lower computational cost of LM-TDS-SIE is well demonstrated by investigating several typical examples, where reference results are obtained by the commercial software COMSOL. In addition to fractures, LM-TDS-SIE can also be used to simulate the scattering from other types of TDS. [ABSTRACT FROM AUTHOR]

The translation matrix ($T$ -matrix) solution to the 3-D problem of scattering of the electromagnetic waves from a dielectric layered medium with random rough interfaces is presented. The solution is based on the coupled vector integral equation of the electric field using the dyadic periodic Green’s function as a kernel. It is shown that the $T$ -matrix solution conserves energy, the bistatic scattering pattern is regular even in cases, where guided modes within the layered medium are excited, and also that the method’s results coincide with those of the second-order Small Perturbation Method (SPM2) in the small height limit. One of the advantages of the $T$ -matrix method compared to the SPM2 is the wide validity range of the solution, which is also studied. [ABSTRACT FROM AUTHOR]

This paper presents a new methodology for the solution of problems of two- and threedimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in the presence of an arbitrary number of penetrable layers. Relying on the use of certain slow-rise windowing functions, the proposed windowed Green function approach efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multilayer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (Bruno et al. 2016 SIAM J. Appl. Math. 76, 1871-1898. (doi:10.1137/15M1033782)), is fast, accurate, flexible and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper, the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on the use of Sommerfeld integrals. [ABSTRACT FROM AUTHOR]

We present a fast direct solver for two-dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of Lippmann{Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad-tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require O(N3/2) work, where N denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both low and high frequency regimes. [ABSTRACT FROM AUTHOR]

Release of pollution into rivers is required to be handled with special consideration to environmental standards. For this purpose, it is essential to specify the contribution of each pollution source in contamination of water resources. In this study, a mathematical model is proposed for determining locations and concentration release histories of polluting point sources using measured downstream river concentrations via an inverse problem framework. The inverse solution is based on the integral equation obtained from applying the Green's function method on the one-dimensional advection-dispersion contaminant transport equation. Discretization of this integral equation results in a linear, over-determined and ill-posed system of algebraic equations that are solved by using the Tikhonov regularization method. Several examples and some real field data are investigated to illustrate the abilities of the proposed model. Results imply that the proposed method is effective and can identify the pollution sources in rivers with acceptable accuracy. [ABSTRACT FROM AUTHOR]

To support the recent progress in 3-D integration based on through-silicon via (TSV) technology, an improved electromagnetic modeling method for TSVs is presented. In the framework of the mixed-potential integral equations combined with cylindrical modal basis functions, the proposed method can extract the effects of depletion capacitances and a finite substrate. To include the effects of depletion region generated by an external dc bias voltage, an additional capacitive cell is employed around a TSV. The proposed method also considers the effect from the finite silicon substrate accurately by employing the multilayered Green’s functions. To reduce the computational cost for calculations involving Green’s functions, a method to approximate Green’s functions over localized intervals when computing partial potential coefficients is presented. The proposed method is validated for simple TSV examples and shows an improved accuracy with the acceptable usage of memory and simulation time. In addition, a $10 \times 10$ TSV array is modeled using different design parameters, showing the capability for dealing with larger size problems using this method. [ABSTRACT FROM AUTHOR]

Using the recently developed method based on the rigorous theory of singular and hypersingular integral equations (IEs) and Nystrom-type discretization, a paraboloidal reflector assisted with a conical shield is investigated. To decrease the computation time we propose a new method of calculation of the Modal Green's Functions in the quasi-optical range. The far and near fields of the studied reflector are analyzed in the transmission and reception cases, and reveal fine features that escape asymptotic analysis. Elementary numerical optimization of the shield-assisted paraboloidal antenna is performed. [ABSTRACT FROM AUTHOR]

Abstract: We propose a new algorithm to compute the teleseismic P and S receiver function synthetics for a multilayered Cartesian structure with anisotropic flat layers. The algorithm is based on the first-order perturbation theory in which the layered background structure is assumed one-dimensional with isotropic variations in vertical direction. Anisotropic velocity perturbations acting as secondary sources constitute the heterogeneities in the medium. The total wavefield is solved using a convolutional type integral equation along with the Green's function of the one-dimensional reference medium extracted using the reflectivity method. The integral equation involves a five-fold integration in space and wavenumber domains. Four of these integrals are achieved analytically and the fifth integral, which is spatial integral in the vertical direction, is performed numerically for which the Born single scattering approximation greatly suffices. The proposed algorithm is demonstrated on some selected numerical examples adapted from published work in the literature. [Copyright &y& Elsevier]

In this paper the Green's functions (GFs) of a thick microstrip in the spatial-domain is computed based on expanding the corresponding spectral-domain functions over inverse quadric (IQ) radial basis functions (RBFs). The scattered data interpolation ability of RBFs is exploited for efficient sampling of the Sommerfeld integration path (SIP), passing from close vicinity of singularities in the complex kp plane. By this, the information content of spectral-domain GFs is preserved, which makes it possible to compute the far-fields accurately. Thus, the method can be applied to the analysis of electrically large structures near layered media. The proposed method is direct with only one approximation level. [ABSTRACT FROM AUTHOR]

In eddy current testing, a flaw in a metal is detected when it gives rise to a change in the electrical impedance of the probe that induces current in the material. Theoretical models and computer codes have been developed to predict the probe signals as an aid to improving inspections and the interpretation of measurements. Model calculations can be efficient for a restricted class of problems in which the conductor geometry is simple, such as an infinite plate or tube. The computational cost is usually low in such cases because dedicated Green's kernels are available, allowing numerical approximations of integral equations to be found using only a few unknowns to represent the field in the flaw region. In this study, the aim has been to perform eddy current calculations on corner cracks efficiently using an approximate Green's function for a conductive quarter-space, thereby extending the class of problems that benefit from the use of a dedicated kernel. The properties of the kernel mean that numerical solutions based on boundary or volume elements can be found for an edge crack by rendering as a discrete approximation only the field at the surface of the flaw or the field within it respectively. Volume element calculations have been carried out to determine the field at a corner crack and from it the probe response. Comparisons of the calculated probe impedance due to edge notches show good agreement with experimental measurements. [ABSTRACT FROM AUTHOR]

A modeling technique to analyze the radiation onto arbitrary 1-D periodic metallizations residing on a microstrip substrate is presented. In particular, straight and meandering lines are being studied. The method is based on a boundary integral equation, more specifically on a mixed potential integral equation (MPIE), that is solved by means of the method of moments. A plane wave excites the microstrip structure, and according to the Floquet–Bloch theorem, the analysis can be restricted to one single unit cell. Thereto, the MPIE must be constructed using the pertinent 1-D periodic layered medium Green’s functions. Here, these Green’s functions are obtained in closed form by invoking the perfectly matched layer paradigm. The proposed method is applied to assess the radiation onto 1) a semi-infinite plate, 2) a straight microstrip line, and 3) a serpentine delay line. These three types of examples clearly illustrate and validate the method. Also, its efficiency, compared to a previously developed fast microstrip analysis technique, is demonstrated. [ABSTRACT FROM PUBLISHER]

Numerical solutions for electromagnetic (EM) integral equations rely on the discretization of integral domains and the use of meshes for geometric description. Meshing geometries is very tedious, especially for complicated structures with many details (tiny parts) and geometric discontinuities (corners or edges), and remeshing could be required in many scenarios. To reduce the costs of generating quality meshes, meshless or mesh-free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in EM community. The meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally. In this work, we first address the traditional meshless scheme for solving EM integral equations based on the moving least square (MLS) approximation for unknown currents and the use of background meshes for integrations, and then develop a novel meshless scheme by applying the Green's lemma to the EM surface integral equations with flat domains. The novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain. Since only the domain boundaries are discretized and no background meshes are needed, the scheme is truly meshless. Numerical examples for EM scattering by flat-surface objects are presented to demonstrate the effectiveness of the novel scheme. [ABSTRACT FROM PUBLISHER]

In this paper, the response of a transversely Isotropie half-space under the punch action of a set of rigid concentric annuli frictionless contacts is considered. By virtue of a compact potential representation and Hankel transforms, a set of ring-load Green's functions for the axisymmetric equations of equilibrium are derived and shown to be expressible in terms of standard elliptic integrals. With the aid of a rigorous yet highly efficient numerical method, the integral equation is solved for the multi-interval singular mixed boundary value problem. Detailed solutions to illustrate the performance of the computational approach and the influence of the degree of anisotropy and contact conditions on the mechanics problem are presented. [ABSTRACT FROM AUTHOR]

A method is described for solving the integral transfer equation in an infinite medium and in a homogeneous sphere with a point source at its center. Isotropic, coherent scattering is assumed. The Ambartsumian function is used for a semi-infinite medium. A nonintegrable singularity of the primary source function is eliminated. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the authors develop an algorithm for solving the time-dependent heat conduction equation in an analytical, exact fashion for a two-component domain. The Green’s function approach is used to get the formal solution of the problem. As an intermediate result an integral-equation for the temperature history at the domain interface is formulated which can be solved analytically. The Green’s function approach in conjunction with the integral-equation method is very useful in cases were strong discontinuities or jumps occur. The system parameters and the initial conditions of the investigated problem give rise to two jumps in the temperature field. In conjunction to the analytical solution a pure numerical solution is obtained by using the FEM (finite element method) , and compared with the solution obtained by the integral-equation approach. [Copyright &y& Elsevier]

A general solution of the Schrödinger equation in the potential representation has been obtained in the form of integral equations. In this representation, the wave function for positive and negative energies or bound states can be expressed as a product of the unperturbed solution for model potential and the function which depends on the additional potential or potential perturbation. Here we have proved that this method is equivalent to the method of variation of constants for negative energies. The linearly independent solutions of Schrödinger equation for harmonic oscillator potential have been obtained for derivation of integral equations, which are used for finding eigenfunctions and eigenvalues for Woods–Saxon potential. Eigenvalues obtained by numerical iterations of these integral equations are in good agreement with results obtained by the discretization method. The kernels of the obtained integral equations are proportional to the perturbation or difference of Woods–Saxon and harmonic oscillator potentials. [ABSTRACT FROM AUTHOR]

The quasi-Green's function method is used to solve the free vibration problem of clamped thin plates on the Winkler foundation. Quasi-Green's function is established by the fundamental solution and the boundary equation of the problem. The function satisfies the homogeneous boundary condition of the problem. The mode-shape differential equation of the free vibration problem of clamped thin plates on the Winkler foundation is reduced to the Fredholm integral equation of the second kind by Green's formula. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The numerical results show the high accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

A novel integral equation method for efficient and accurate modeling of power-ground planes with narrow slots is provided in this paper. The equivalence principle is used to extend finite-size power-ground planes to infinite parallel planes. The discontinuities in the original power-ground planes, including the slot, via clearance, and boundary, are represented by equivalent electric and magnetic currents. Coupled integral equations are created for these equivalent currents. By making use of the modal field distribution between the power and ground planes, the complex surface integrals are reduced to simple one-dimensional line integrals. This greatly reduces the computing time while good accuracy is preserved. Thereafter, the paths of integration are discretized into many segments, which are equivalent to network ports, such that an equivalent microwave network of the power-ground planes with slots is derived through the solution of the integral equations. This equivalent network is used for power integrity analysis. By comparing the proposed method with measurement results and available full-wave methods, its accuracy and efficiency are verified. [ABSTRACT FROM PUBLISHER]

In this article, I outline how integral equations and Green's functions are used in the courses that I teach, and how I believe my students benefit from exposure to these topics. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green’s function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green’s function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green’s function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums. [ABSTRACT FROM AUTHOR]

Abstract: A general approach is presented for proving existence and uniqueness of solutions to the singular boundary value problem The proof is constructive in nature, and could be used for numerical generation of the solution. The only restriction placed on is that it not be a singular function of the independent variable ; singularities in are easily avoided. Solutions are found in finite regions where , using an integral equation whose Green’s function contains an adjustable parameter that secures convergence of the Picard iterative sequence. Methods based on the theory are developed and applied to a set of problems that have appeared previously in published works. [Copyright &y& Elsevier]

Abstract: In this paper, an algorithm is proposed for the solution of second-order boundary value problems with two-point boundary conditions. The Green’s function method is applied first to transform the ordinary differential equation into an equivalent integral one, which has already satisfied the boundary conditions. And then, the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems. Numerical examples demonstrate the efficiency and reliability of the algorithm developed, it is quite accurate and readily implemented for both linear and nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions. Furthermore, the lower order approximation is of higher accuracy for most cases. Some other extended applications of this algorithm are also exhibited. [Copyright &y& Elsevier]

A generic integral equation and method of moments formulation is presented for laterally bounded stratified media including planar metallization. The main asset of the developed approach is its flexibility, as it encompasses generic lateral boundary conditions and explicitly applies to any linear subsectional basis functions with constant surface divergence. This includes the rooftop functions on rectangular and triangular supports currently proposed in standard method of moment meshers. This approach provides closed expressions for the coupling integrals appearing in the method of moments matrix elements. These formulas are based of Green's functions modal expansions and in the possibility, conclusively demonstrated in this paper to transform the surface integrals into contour integrals allowing an efficient and systematic implementation of the procedure. Full derivations are presented for several lateral boundary conditions, including rectangular and circular metallic cavities and periodic structures. Numerical examples including the analysis of real-life planar boxed circuits are presented. In all cases the obtained results compare favourably with other existing techniques. [ABSTRACT FROM AUTHOR]

The paper discusses the Green's function method of the Stokes flow equation in the presence of a slip wall. The far field of the Green's function is obtained and interpreted in terms of singularities. It is found that the flux for the stokeslet near a slip wall is finite and has a form similar to the one corresponding to the no-slip condition. The reciprocity formula of this Green's function is proved for a more general boundary shape. The singularity method is then developed for the motion of a solid sphere near a plane surface with slip condition on the surface. The results are compared with exact solutions. [ABSTRACT FROM PUBLISHER]

A novel method based on hybrid integral equation for electromagnetic emission and susceptibility modeling of power-ground planes with a finite size is proposed in this paper. First, the entire computational domain is divided into the internal subdomain including the substrate sandwiched between the power and ground planes and the external subdomain surrounding the power-ground planes. The internal subdomain is modeled by using the integral equation with dyadic Green's functions of the rectangular cavity, while the free-space Green's function is used to model the outside subdomain. These two kinds of integral equations are coupled through the equivalent electric and magnetic currents that are placed on the interfaces. The key advantage of this proposed method is its ability to model both emission and susceptibility problems. This method can accurately simulate the impedance of the power-ground planes and the radiated field. In particular, it is able to predict the induced electric currents inside the power-ground planes that result from external interference sources. Therefore, this technique is able to provide a complete solution for the electromagnetic compatibility analysis of the system-level package. [ABSTRACT FROM AUTHOR]

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green’s function. To solve the integral-differential equations, expansion method is used to obtain Green’s function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green’s function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells [ABSTRACT FROM AUTHOR]

We used the Green's function method to describe the multiple scattering of surface waves generated by interactions with the complex 3-D earth structures. The basic process involves an integral equation of convolutional type. An efficient multilevel fast multipole method is utilized to accelerate the matrix-vector products that correspond to the integrals on the horizontal plane. This new algorithm is shown to be quite successful when compared to its more traditional complement (e.g. direct integration method, DIM) with particularly the large number of data points. The fast execution is achieved through well-organized truncated multipole expansions, functions grouping and translation operators. In general, the new algorithm has a fruitful logarithmic time complexity as opposed to an uncomfortable exponential time complexity attainable with the traditional algorithms. This algorithm requires the user provide some important parameters to operate. One of them is the truncation number of the infinite series associated with the multipole expansions for which two linear relationships are derived for an automatic determination. The other central parameter is the clustering number that is used to group data points in the data structure. We showed that the clustering number around 5 is mostly an optimum value providing a minimum run time. However, greater clustering numbers (i.e. ∼10) become necessary for an optimum operation when the number of data points gets real large. In order to test the convergence of the current algorithm we compared our numerical results with analytical solutions provided for cylindrical obstacles by other researchers. A half-space model in which the total wavefield is represented by a finite number of propagating modes describes the reference structure. Not all waves are trapped in this representation since leaking modes and body waves downward radiating into the half-space can exist. The comparisons with the exact solutions revealed that the quality of vertical component amplitudes is reasonably well while some amplitude discrepancies on the horizontal components particularly near and inside the heterogeneities exist. Extra fast half-space velocities complementing the deficiencies on the modal structure greatly help reduce the discrepancies on the horizontal components. [ABSTRACT FROM AUTHOR]

By using the fundamental equations of axisymmetric shallow shells of revolution, the nonlinear bending of a shallow corrugated shell with taper under arbitrary load has been investigated. The nonlinear boundary value problem of the corrugated shell was reduced to the nonlinear integral equations by using the method of Green's function. To solve the integral equations, expansion method was used to obtain Green's function. Then the integral equations were reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral equations become nonlinear algebraic equations. Newton's iterative method was utilized to solve the nonlinear algebraic equations. To guarantee the convergence of the iterative method, deflection at center was taken as control parameter. Corresponding loads were obtained by increasing deflection one by one. As a numerical example, elastic characteristic of shallow corrugated shells with spherical taper was studied. Calculation results show that characteristic of corrugated shells changes remarkably. The snapping instability which is analogous to shallow spherical shells occurs with increasing load if the taper is relatively large. The solution is close to the experimental results. [ABSTRACT FROM AUTHOR]

A method of fast computation of the mixed potential Green's functions (MPGFs) in spatial domain for cylindrically stratified media has been developed. Based on the convergence behavior of the MPGFs in spectral domain, the discrete complex image method (DCIM) and the extrapolation technique are employed to perform the inverse Fourier transform, which is different from that of planarly stratified media. In order to evaluate the summation of cylindrical eigenmode from order of negative infinity to order of positive infinity, the Green's functions in spectral domain are rearranged so that acceleration technique can be well applied to speed up the convergence of summation process. For the integral along the deformed Sommerfeld integral path with respect to kz, DCIM is used to speed up the computation. After the quasistatic components are completely subtracted from the spectral domain Green's functions, the inverse transform of the remaining parts has an approximated analytical form. The proposed method can be used to accurately and efficiently calculate the spatial domain MPGFs, which are useful for characterizing conformal patch structures mounted on cylindrically stratified media. [ABSTRACT FROM AUTHOR]

The method of complex function and the method of Green's function are used to investigate the problem of SH-wave scattering by radial cracks of any limited length along the radius originating at the boundary of an elliptical hole, and the solution of dynamic stress intensity factor at the crack tip was given. A Green's function was constructed for the problem, which is a basic solution of displacement field for an elastic half space containing a half elliptical gap impacted by anti-plane harmonic linear source force at any point of its horizontal boundary. With division of a crack technique, a series of integral equations can be established on the conditions of continuity and the solution of dynamic stress intensity factor can be obtained. The influence of an elliptical hole on the dynamic stress intensity factor at the crack tip was discussed. [ABSTRACT FROM AUTHOR]

This paper presents an efficient technique for evaluating Green's functions associated to layered media, when formulated as Sommerfeld integrals in the space domain. The key step in the formulation is that Sommerfeld integrals are computed choosing a suitable integration path which is closed through the imaginary axis of the complex spectral plane. It is shown that with this original choke of the integration contour, the numerical effort usually involved in the evaluation of Sommerfeld integrals can be greatly reduced, specially when large source-observer distances are involved. One asset of this technique is that it can be easily incorporated into integral equation based CAD packages for the efficient analysis of complex printed microwave circuit and antennas. In addition, the theoretical developments needed to set up the numerical algorithm throw a new light on the asymptotic behavior of the layered media Green's functions for large source-observer distances. [ABSTRACT FROM AUTHOR]

In this Letter, the mixed potential integral equation (MPIE) method is used in conjunction with the complex images technique to carry out a full-wave analysis of a multiconductor coplanar waveguide (CPW) embedded in a layered substrate. Special attention is paid to the correct application of the method so as to maintain high accuracy and stability of the numerical results. It is shown that the procedure only provides a robust and accurate approach when the complex images expansion variable is properly chosen and the singularities of the Green's functions are properly handled. Numerical problems arise in any other case. In particular, it is demonstrated that the two-level sampling scheme used by some authors is not enough to overcome the numerical problems. However, the two-level sampling method is shown to be essential to deal with the presence of extremely thin substrate layers such as those found in multichip-module-deposition (MCM-D) technologies. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 34: 198–203, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10416 [ABSTRACT FROM AUTHOR]

This paper is devoted to the calculation of the capacitance of monolithic microwave integrated-circuit (MMIC) passive components. The method of moments (MoM) is used to solve the integral equation for the electrostatic problem with the help of the spatial-domain multilayered media Green's function that can be obtained using the discrete complex image method (DCIM). As an application, the capacitance of several MMIC passive components, including capacitors and interconnects, is calculated. The numerical experiments show the efficiency and accuracy of the present method. © 2002 John Wiley & Sons, Inc. Microwave Opt Technol Lett 32: 216–219, 2002. [ABSTRACT FROM AUTHOR]

A set of closed-form Green's functions is used in a mixed-potential integral-equation technique for the efficient modeling of planar microstrip structures of arbitrary shape on an electrically thin substrate. Various MMIC and microstrip antenna structures are simulated. It is shown that excellent accuracy at a substantial reduction in total simulation time is achieved for a wide range of frequencies and substrate parameters. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 20: 342–345, 1999. [ABSTRACT FROM AUTHOR]

An integral equation‐based nonoverlapping domain decomposition method (IE‐NDDM) is proposed for efficient and accurate scattering analysis of electrically large PEC targets in a half‐space environment. Taking the condition that there are null fields as well as current inside PEC objects in the original problem, a novel explicit transmission condition is proposed to ensure continuity of electric currents between adjacent subdomains. Moreover, to deal with this situation that targets located in half‐space environment, the stratified‐medium dyadic Green's function is adopted. Numerical examples are provided to demonstrate the correctness and the performance of the proposed method. In detail, an analysis about electromagnetic scattering respect to the altitude of the target away from half‐space interface, the incident direction of plane wave, and the material parameters of the lower half‐space is presented; furthermore, the convergence behavior of the proposed method is studied. [ABSTRACT FROM AUTHOR]

This article deals with the novel method for finding solutions for the initial-boundary value problems (IBVPs), which is called the Sawangtong's Green function homotopy perturbation method, shortly called SGHPM. The SGHPM is a method which combines the homotopy perturbation method with Green's function method. The convergence analysis for the SGHPM is shown. Furthermore, some examples are presented to illustrate the validity of the proposed method and to ensure that SGHPM is a technique which is powerful and efficient for finding approximate analytic solutions of IBVPs. [ABSTRACT FROM AUTHOR]