Numerical Green Research Articles

Numerical Green’s Function for Steady-State Heat Conduction in Heterogeneous Media

Numerical Green's function for steady-state heat conduction problems is derived in a finite-sized medium that may contain inclusions (fibers) in the matrix phase. Green's function is approximated by employing the Galerkin method that uses permissible functions which satisfy the homogeneous boundary condition for the given geometry. The present approach allows physical fields in a medium that contain multiple inclusions to be expressed through isolated integrals semi-analytically while retaining all the relevant material parameters.

Nonsingular BEM for fracture modeling

A new three-dimensional formulation for the fracture mechanics problem based on a non-singular form of the hypersingular Somigliana stress identity (SSI) is presented. The resulting formulation permits the use of low-order numerical integration algorithms and allows the crack opening displacement variable to be modeled in a piecewise- C 1, α manner which is well suited to standard boundary element method (BEM) algorithms. The method is not limited to particular crack shapes and is sufficiently general that it holds excellent prospects for providing a general class of numerical Green's functions for this important class of problems.

Excitation of leaky modes on multilayer stripline structures

A quasi-analytical method for calculating the excitation of leaky modes on multilayer stripline structures by a finite source is presented in this paper. Simple sources such as an infinitesimal dipole near the conducting strip or a delta-gap feed on the conducting strip of the transmission line are considered. The method uses a numerically constructed Green's function for the source in the presence of the conducting strip, which is calculated from Fourier transform theory in terms of a one-dimensional Green's function for a line source in the presence of the conducting strip. The numerical Green's function involves a one-dimensional integration in the longitudinal wavenumber plane. The residue contributions from the poles of the Green's function define the excitation amplitudes of the leaky and bound modes that exist on the structure. The numerical Green's function is also used to numerically calculate the complete current on the strip excited by the source. The correlation between the leaky-mode current and the complete current is used to define the extent of the physical meaning of the leaky mode. The generalized pencil of functions (GPOF) method is used to study this correlation by resolving the complete current on the strip into exponential waves, which are then compared with the current of the leaky mode. The physical meaning of the leaky modes is also analytically examined by consideration of the branch cuts in the longitudinal wavenumber plane for the numerical Green's function integration. A path consistency is established as a necessary condition for the physical meaning of the leaky mode.

Numerical Green's function approach to finite-sized plate analysis

The eigenfunction expansion method is used to obtain numerical Green's functions to solve for deflection of irregular-shaped classical plates. The associated eigenvalue problem allows to express the Green's function as a series of eigenfunctions which are approximated by a series of polynomials that satisfy the homogeneous boundary conditions to which the plates are subjected. A computer algebra system (Mathematica) has been extensively used to construct the approximate Green's functions consisting of polynomials, reducing substantially the amount of work involved in the calculation and achievement of the solution.

Validation of a procedure for calculating broadband strong-motion time histories with empirical Green's functions

Abstract We quantitatively evaluate the validity of an approach to calculating strong ground motion time histories that involves the kinematic modeling of earthquake rupture by using recorded empirical Green's functions to constrain the propagation path and site contribution in the period range 0.05 to 2.0 sec, leaving only the contribution of the source to be specified. In addition, we use numerical Green's functions for long periods (> 10 sec) where the average crustal structure is assumed to be well known. In the intermediate band (2 to 10 sec) where site and path effects are still important but empirical Green's functions have small signal-to-noise ratios, we use an arbitrary smoothing method to connect the numerical and empirical Green's function synthetics. The evaluation presented here consists of comparing actual recordings of the Loma Prieta earthquake at 26 sites to syntheses from a simple source model similar to what is thought to have occurred during the Loma Prieta earthquake. We show that the standard error between observed and predicted response spectra is less than or equal to errors from other methods for periods between 0.05 and 0.4 sec and is significantly less than regression methods based on pre-Loma Prieta empirical strong-motion data at periods between 0.5 and 5.0 sec. The strong ground motion prediction methodology that we are validating provides an accurate, defendable means to characterize site and path effects and, therefore, allows the uncertainties in the predicted hazard to be due to unresolved issues about the earthquake source such as the geological constraints of a particular fault and details about the physics of earthquakes. This methodology produces accurate and realistic acceleration and displacement time histories, which are becoming increasingly important for modeling the nonlinear response of large, distributed structures and soft soils. The implication for strong ground motion prediction is that a range of representations of an earthquake source based on physically realistic parameters can be used to generate a suite of possible time histories that represent the range of ground motion for that specific site and earthquake. In addition, in the future, as more information about a particular fault and earthquakes in general becomes available, the uncertainty in ground-motion prediction should decrease.

Application of numerical Green's functions in the design of plates using symbolic computation

In order to solve even simple plate problems, one may need to set up quite an amount of computer codes for the implementation of general numerical methods such as finite element analysis. Using the semi-analytical approach introduced in this paper, the amount of time-consuming work may be reduced significantly. It would be even more convenient if the solutions were given in the forms of algebraic expressions instead of numerical values. A primary concern in the design of plates would be the deflections of plates due to loadings at various points inside the plates. The deflections can be obtained from an influence function known as the Green's function of the plates. However, except a few known simple cases, it is difficult to obtain Green's function of the plate for various shapes and boundary conditions analytically. To circumvent such difficulties in finding this function, a numerical Green's function is derived in this paper for the first time. Once the numerical Green's function is obtained the solution, a lateral deflection in this paper, may be calculated by carrying out simple integration. Also, since the numerical Green's function obtained in this paper is given in terms of algebraic polynomials, it is suitable for differentiation or integration without introducing numerical errors using symbolic computation. Utilizing the numerical Green's functions of plates, a simple and accurate yet flexible approach to the subject is addressed.

Green's function applied to solution of mindlin plates

Numerical Green's functions constructed via the eigenfunction expansion method are derived to solve for non-classical (Mindlin) plate problems. The associated eigenvalue problem enables the expression of the Green's function as a series of eigenfunctions which are approximated by a series of polynomials that satisfy the homogeneous boundary conditions to which the plates are subjected. In order to construct the approximate Green's function consisting of polynomials, a computer algebra system (Mathematica) has been extensively used reducing substantially the amount of algebra involved in the calculation. The results are favorably compared with values found in the literature.

Numerical Green's function technique for the analysis of screens perforated by multiply connected apertures

A technique for the analysis of perforated screens with multiply connected apertures (frequency selective surfaces) is presented. This type of structure offers higher flexibility with respect to screens perforated with simply connected apertures. In particular, it can be attractive when a response with two reflection bands separated by a transmission band is required. The solution is obtained in two steps. In the first, the Green's function of a reference structure (a screen with simply connected apertures in a stratified dielectric medium) is determined; in the second, this characterization is exploited to analyze the complete structure. Both steps are carried out by the method of moments in the spectral domain. Screens with tripole patches inside circular apertures are studied. Reflection and transmission characteristics of single and double screen structures are presented. Comparisons with another established method confirm the validity of the new formulation.

Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics.

We present mathematical results which dramatically enhance the computational efficiency of the phase-field method for modeling the solidification of a pure material. These results make it possible to resolve a smaller capillary length to interface thickness ratio and thus render smaller undercooling and three-dimensional computations accessible. Furthermore, they allow one to choose computational parameters to produce a Gibbs-Thomson condition with an arbitrary kinetic coefficient. The method is tested for dendritic growth in two dimensions with zero kinetic coefficient. Simulations yield dendrites with tip velocities and tip shapes which agree within a few percent with numerical Green's function solutions of the steady-state growth problem.

The slip distribution of the 1992 Nicaragua Earthquake from tsunami run‐up data

The 2 September 1992 Nicaragua tsunami was induced by an MS≃ 7–7.3 earthquake and caused about 170 victims and significant damage to the coastal areas. The aim of this work is to infer the slip distribution along the causative seismic fault from the available tsunami run‐up field data. We have subdivided the fault plane into five slip‐independent subfaults and computed the corresponding numerical Green's functions by integrating shallow‐water equations via a finite‐element technique. The slip distribution on the fault has been determined by applying a least‐squares procedure to the differences between the observed run‐up values and the computed maximum water levels along the coast. The main result is that the best fit is found for a nonuniform coseismic slip, with the highest displacement taking place in the southernmost part of the fault. The measure of the improvement is given by the global rms residual of the heterogeneous‐slip case being circa 25% smaller than that of the uniform case. An ancillary outcome of the analysis is the evaluation of the mean amplification factor of the tsunami wave investing the Nicaraguan coast, that, on assuming a rigidity of 1×1010 N/m², results to be 3.45.

Green's function for arbitrarily shaped dielectric bodies of revolution

In this paper, a solution is developed to calculate the electric field at one point in space due to an electric dipole exciting an arbitrarily shaped dielectric body of revolution (BOR). Specifically, the electric field is determined from the solution of coupled surface integral equations (SIE) for the induced surface electric and magnetic currents on the dielectric body excited by an elementary electric current dipole source. Both the interior and exterior fields to the dielectric BOR may be accurately evaluated via this approach. For a highly lossy dielectric body, the numerical Green's function is also obtainable from an approximate integral equation (AIE) based on a surface boundary condition. If this equation is solved by the method of moments, significant numerical efficiency over SIE is realized. Numerical results obtained by both SIE and AIE approaches agree with the exact solution for the special case of a dielectric sphere. With this numerical Green's function, the complicated radiation and scattering problems in the presence of an arbitrarily shaped dielectric BOR are readily solvable by the method of moments.

Open-boundary relaxation

The relaxation technique for computing the electrostatic potential in 2D is extended to open-boundary problems. Two methods are introduced for updating the outer boundary potential between iterations using a numerical Green's function. Comparisons with conventional grounded-boundary results show that greatly increased accuracy is obtainable with the open-boundary method. © 1992 John Wiley & Sons, Inc.

Transmission line matrix modeling of disperse wide-band absorbing boundaries with time-domain diakoptics for S-parameter extraction

A numerical modeling procedure based on Johns time-domain diakoptics approach with space interpolation techniques for efficient transmission-line matrix (TLM) analysis of two-dimensional microwave circuits is discussed. Frequency-dispersive boundaries are represented in the time domain by their characteristic impulse response or numerical Green's function (Johns matrix). Almost perfect wide-band absorbing boundary conditions have been obtained with this technique, permitting accurate characterization of waveguide discontinuities and compounds. The application of these techniques saves considerable computer run time and memory when compared with conventional TLM analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Matched‐field processing and the effect of realistic noise correlation models

In simulating acoustic source localization by matched‐field processing, it is desirable to use a realistic ambient noise model. Two general classes of noise model were considered: a planar isotropic noise source distribution and a line source distribution. The former was implemented based on the work of Kuperman and Ingenito [J. Acoust. Soc. Am. 67, 1988–1996 (1980)]. Their expressions for the correlation integrals were adapted to compute the correlation function for both simple and realistic propagation models, through the use of analytical and numerical Green's functions, respectively. The line source model involved the assumption of uncorrelated noise sources along an infinite line and was designed to model a shear zone in a shallow ice‐covered ocean. Using the corresponding Green's functions for a waveguide, the line model correlation function could be reduced to a summation over all mode pairs, each term of which involved an integration over wavenumber. In this paper, the line source correlation model is presented and correlation functions given by both noise models are illustrated. The effect of using these models on the localization of a discrete source by a grid search‐optimization procedure is then examined.

One-loop corrections to solitons in two-dimensional theories.

We have investigated quantum corrections to topological and nontopological solitons in several two-dimensional theories at the one-loop level utilizing a numerical Green's-function method. For the phi/sup 4/ kink and sine-Gordon soliton, as well as the kink coupled to fermions, we reproduce the masses of static solitons obtained analytically in the cases for which analytic results exist. The advantage of the Green's-function method is that it is suited to dealing with a wide range of theories and with most background-field configurations. This becomes apparent from our study of theories where tractable analytic methods of calculating quantum corrections at the one-loop level do not exist. We have used the method to investigate in detail the derivative and inverse-mass expansions of the one-loop, nonlocal actions of bosons and fermions for some typical nontopological background fields.

Solution of the Pauli master equation via the lanczos algorithm

We present a numerical Green's function approach to solving the Pauli master equation which utilizes the recursive residue generation method (RRGM) developed by Wyatt and co-workers. In most practical applications, only a few Green's function matrix elements are required to express all physical quantities of interest. For such systems, computation time reductions of 10 2–10 3 over direct diagonalization algorithms are achieved even for systems of moderate size. An application to a simple model for trapping in polymers is given.

Coupling of numerical Green's matrix and boundary integral equations for the elastodynamic analysis of inhomogeneous bodies on an elastic half-space

A coupled system of integral equations (of the domain and boundary types) is formulated for the elastodynamic response analysis of a locally inhomogeneous body on a homogeneous elastic half-space. The method uses the fundamental solution for homogeneous elastostatics in the inhomogeneous domain owing to the lack of a fundamental solution in inhomogeneous elastodynamics. The integral representation of displacements in the inhomogeneous domain is formulated with the help of this elastostatic fundamental solution by considering the term induced by the inhomogeneity of materials and the acceleration term as the body force term. Then the Green's matrix is obtained numerically from this integral representation and combined with the ordinary boundary integral equations, which are valid in the exterior homogeneous half-space. Some numerical examples show the efficiency and the versatility of this coupled method.

Technique for calculating the radiation and scattering characteristics of antennas mounted on a finite ground plane

The problem of an antenna mounted on a conducting body is formulated by using a modification of the isolated thin-wire kernel with the numerical Green's function for the body. This formulation is then employed for the investigation of the problem of a monopole antenna over a finite ground plane. The numerical Green's function for the plate is obtained by applying the finite-difference technique, which has previously been successfully applied to the thin-wire junction problem. A user-oriented Fortran program is developed, and extensive numerical results are reported.

MIS array potential calculation

A numerical Green's function technique is used to obtain the depletion layer profile of a two-dimensional MIS array. The depletion layer profile is approximated by a piecewise linear curve. Once the depletion layer profile is obtained, the surface potential is calculated by use of Green's function.