Harmonic Green Research Articles

Harmonic Green and Neumann functions for domains bounded by two intersecting circular arcs

ABSTRACT The parqueting-reflection principle is shown to work for constructing the harmonic Green functions and harmonic Neumann functions for a class of domains, which are bounded by two intersecting circular arcs in with a particular angle . With the explicit expressions of the harmonic Green and Neumann functions, we solve the Dirichlet and Neumann boundary problems to the Poisson equation in these domains by applying the Green and Neumann representation formulas.

Reduced modelling computation of layered soil's harmonic green functions

Ground-borne vibrations are disturbing to human beings. In order to model and reduce these vibrations, the calculation of the harmonic Green's-functions of the soil is highly required since the most effective numerical solution to predict ground-borne vibration is to couple the finite-element and the boundary-element methods. In this work, we elaborate a direct space-frequency formulation that is especially suitable of material exhibiting visco-elastic behavior. Moreover, an original formulation of the Proper Generalized Decomposition (PGD) reduction technique was undertaken in the frequency-space domain. This new approach highly shortens time computing costs. It is mainly valuable when a parametric study is concerned. Moreover, the PGD resolution allows providing an off-line solution, thus, the solution is calculated only once.

Harmonic Green functions for the Almaty apple

The parqueting reflection principle is suitable for constructing harmonic Green functions for certain plane domains with boundaries consisting of sub-arcs from circles and straight lines. The (continued) reflection at the boundary parts has to provide a parqueting of the complex plane. Occasionally, however, a parqueting of is not sufficient and multiple parqueting is required, e.g. for the disc sector with opening angle , where a double parqueting is needed, then compensated by applying the square root mapping. The parqueting-reflection principle is more convenient to apply than basing on the conformal invariance of the harmonic Green functions, see e.g., but sometimes the latter is applicable where the first is not. Here the situation is studied for the Almaty apple, a modified half disc. While the bigger of this cut disc is also requiring a double parqueting, for the smaller part a simple one suffices. However, for the bigger part a proper square root reduction to a simple parqueting is still to be determined in order to find the harmonic Green (and Neumann) function. The idea to use this disc cap for a parqueting of the complex plane has occured while two of the authors were visiting a monument of an apple in Almaty, the centre of an area full of wild apple trees considered as the locus of origin of the apple fruit.

Remark on Robin problem for Poisson equation

The two basic boundary value problems for the Poisson equation are the Dirichlet and the Neumann problems. Their related fundamental solutions are the harmonic Green and Neumann functions. A linear combination of these two boundary value problems is the Robin problem. The related fundamental solution, the Robin function, can be chosen as an interpolation between the Green and the Neumann functions. For plane domains this was done in [Begehr H, Vaitekhovich T. Modified harmonic Robin functions. Complex Variables Elliptic Equ. 2013;58:483–496] with complex notation. While the Dirichlet problem is unconditionally solvable this is in general not the case for the other ones. In explicit form, however, this was shown only for the unit disc. Here the solvability condition is given for any admissible plane domain. Proper alteration of the fundamental solution can make the solvability condition disappear. This is shown explicitly for the case of the upper half of the unit disc .

Time dependent flexural gravity waves in the presence of current

In the present study, a combined effect of current and compressive force on time dependent flexural gravity wave motion in both the cases of single and two-layer fluids is analyzed in finite and infinite water depths in two dimensions. The roots of the dispersion relation associated with the plane flexural gravity waves are analyzed via contour plots and by plotting various terms of the dispersion relation separately. The characteristic of plane flexural gravity waves is studied by analyzing the phase and group velocities along with the law of conservation of energy flux to understand the combined effect of current and compressive force on the wave motion. The integral form of the time dependent Green's function in the presence of current is obtained using the Laplace transform method and used in Green's identity to derive the time dependent velocity potential for the flexural gravity wavemaker problem. The time harmonic Green's function and velocity potentials are obtained as a special case from the time dependent problems. Numerical results are computed and analyzed in particular cases using the method of stationary phase to obtain the asymptotic results for Green's function and the deflection of ice sheet. The integral form of Green's function derived here will be suitable to deal with physical problem when the roots of the dispersion relation for the flexural gravity wave problem coalesces which were otherwise not possible in the eigenfunction expansion method used for time harmonic problems.

Harmonic Green's functions for flexural waves in semi-infinite plates with arbitrary boundary conditions and high-frequency approximation for convex polygonal plates

This paper provides a method for obtaining the harmonic Green's function for flexural waves in semi-infinite plates with arbitrary boundary conditions and a high frequency approximation of the Green's function in the case of convex polygonal plates, by using a generalised image source method. The classical image source method consists in describing the response of a point-driven polygonal plate as a superposition of contributions from the original source and virtual sources located outside of the plate, which represent successive reflections on the boundaries. The proposed approach extends the image source method to plates including boundaries that induce coupling between propagating and evanescent components of the field and on which reflection depends on the angle of incidence. This is achieved by writing the original source as a Fourier transform representing a continuous sum of propagating and evanescent plane waves incident on the boundaries. Thus, the image source contributions arise as continuous sums of reflected plane waves. For semi-infinite plates, the exact Green's function is obtained for an arbitrary set of boundary conditions. For polygonal plates, a high-frequency approximation of the Green's function is obtained by neglecting evanescent waves for the second and subsequent reflections on the edges. The method is compared to exact and finite element solutions and evaluated in terms of its frequency range of applicability.

An enhanced algorithm for evaluation of surface waves

A procedure based on the Stroh formalism is presented that yields the approximation for the planar harmonic Green's matrix function for piezoelectric half-space that is valid in the vicinity of the surface, or pseudosurface, wave number. This approximation can be directly exploited in the spectral-theoretical analysis of surface acoustic wave devices. Otherwise, the known characterization of the surface wave supporting substrate can be immediately inferred from it, like the wave velocity, piezoelectric coupling coefficient, the wave damping in the case of leaky pseudosurface waves, and the corresponding surface displacement velocities.

Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part I: a 2D model

AbstractThe Green's functions for a triclinic half‐space for embedded harmonic line load are considered. Corresponding displacement and stress fields are expressed in terms of double Fourier integrals. The first integral was evaluated using contour integration while the second one was computed through the Gauss–Legendre quadrature. The resulting Green's functions algorithm avoids repeated calculations of the same quantities and utilizes the vector computational features within MATLAB environment. Extensive testing of the results has been performed for both displacement and stress fields. The tests demonstrate the accuracy of the proposed procedure for evaluating the Green's functions without any restrictions upon material properties, frequency, and location of the source and observation points. Copyright © 2006 John Wiley & Sons, Ltd.

Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part II: a 3D model

AbstractThe displacement and stress Green's functions for a 3D triclinic half‐space with embedded harmonic point load is considered. The resulting displacement and stress fields are expressed in terms of triple Fourier integrals. The first integral was evaluated using contour integration and the 3D Green's functions were obtained as a superposition of 2D results over the azimuthal angle. The resulting algorithm developed for evaluation of the Green's functions avoids repeated calculations of the same quantities and it utilizes the vectorized manipulation within MATLAB environment. The algorithm places no restriction on material properties, frequency and location of source and observation points. Extensive testing of the numerical results was performed for both displacement and stress. The tests confirm the accuracy of the numerical results. Copyright © 2006 John Wiley & Sons, Ltd.

A modal solution for reflection and transmission at a Chiral–Chiral interface

A harmonic Green's function solution for a magnetic line source above and below a flat interface between two chiral materials is derived. The solution is expressed in terms of the characteristic right and left circularly polarized waves. The harmonic Green's function formulation is converted into a modal representation. The modal representation is suitable for the complete expansion of the electromagnetic fields above and below a rough interface between two chiral materials with laterally varying material properties. The modal expansion is written in terms of orthogonal-basis and reciprocal-basis functions, which have been used to formulate generalized Fourier transforms and derive the generalized telegraphists' equations for electromagnetic fields in irregular chiral media.PACS Nos.: 33.55.Ad, 78.20.Ek

Rigorous evaluation of the vertex effects on the frequency-dependent circuit parameters of an open-ended microstrip line

This study presents the original inclusion of right singular vertex conditions in integral equations solved by the method of moments for accurate evaluation of the frequency-dependent microstrip discontinuity models required in microwave integrated circuits (MIC) and monolithic microwave integrated circuits (MMIC) design. The 90/spl deg/ wedges singularity function for the current density, weighted with a novel sampling function for corner cells, and the harmonic Green's function for shielded structures, is accurately and efficiently integrated in the conical geometry of the 90/spl deg/ sector. The frequency-dependent effective length and excess equivalent capacitance of a shielded open-ended microstrip line are calculated to a higher accuracy with respect to previous two-dimensional and three-dimensional full wave analyses.

Far–field asymptotics of the Green's tensor for a transversely isotropic solid

The dynamic time–harmonic Green's tensor for a transversely isotropic homogeneous linearly elastic solid is studied. First, the representation of the Green's tensor as an integral over the unit sphere is obtained. It consists of three parts: quasi–longitudinal (P), shear–horizontal (SH) and quasi–shear (SV). Then, an exact analytical solution for the SH part in terms of elementary functions is derived. The complete far–field asymptotic approximation of P and SV parts is obtained next, using the uniform stationary phase method. For the P wave it involves the leading term of the ray series since there is only one arrival of this wave. The wave surface for the SV wave contains conical points and cuspidal edges. The asymptotic description applicable near these singular directions is derived involving the Airy and Bessel functions. The directions close to the points of tangential contact of the SH and SV sheets of the wave surface are also treated. At the end of the paper numerical results in both frequency and time domain are presented. They show that the agreement between the outputs of the asymptotic and direct numerical codes is very good throughout all regions but the former can be orders of magnitude faster.

Extreme limit in linear polarisation ratio in step-index single- mode fibres with a central dip

From Maxwell's equations, using a Fourier expansion, spatial harmonic balance and Green's functions, closed-form expressions are found for the longitudinal electric and magnetic fields, for two orthogonally polarised modes of a birefringent fibre with or without a central index dip. Propagation constants can then be calculated to the desired degree of accuracy. Corrections up to third order are found to be significant. The linear polarisation ratio in the core and in the cladding is almost insensitive to birefringence but exhibits large variations in the central dip, where it deteriorates as the birefringence increases.

Harmonic Green’s functions of a semi-infinite plate with clamped or free edges

Harmonic Green’s functions for a thin semi-infinite plate with clamped or free edges are developed starting from either simply supported or roller supported solutions and applying corrections to account for boundary excitation. This is achieved by connecting the solutions in terms of polar coordinates with the solutions in Cartesian coordinates. The formal solutions in the form of improper wave-number integrals are numerically evaluated using adaptive Clenshaw–Curtis integration. Alternate solutions obtained for each boundary condition compare well. Only harmonic point loads are considered in this article but the methodology may be extended to moment excitation and distributed loads. The methodology developed here will form the basis for advancing the ray tracing technique for vibration analysis of finite plates.

Strongly and uniformly convergent Green's function expansions

The convergence of Green's function expansions used in the exact analytical treatment of problems involving boundaries of different shapes is a property crucial in obtaining their solution. Existing expansions in most cases suffer from two serious setbacks: they do not converge uniformly in their region of validity, exhibiting a slow and conditional convergence near the source (singular) point and, even worse, they change expression when the field point moves past the source point. For such reasons they are unsuited for the solution of singular integral equations, in which values of the Green's function G at the source point do appear inside the integral. These inadequacies are met head-on by extracting the singular behavior in a closed-form term. Additional simple terms are also extracted to improve the convergence of the expansion of the remaining, non-singular part of G. The so-obtained new eigenfunction expansions for G converge uniformly over the whole region of their validity and very strongly (exponentially) near the source point. They are particularly suited for the solution of singular integral equations by the Carleman-Vekua method, otherwise known as the method of regularization by solving the dominant equation. These new expansions can be further subjected to a Watson transformation yielding a third expansion exhibiting strong convergence in regions where the convergence of the preceding series weakens, and vise versa. All these considerations are illustrated in this paper by means of a two-dimensional harmonic Green's function of a line source inside a rectangular shield, which is useful in a variety of shieldedline configurations. Extensions to different dielectric sublayers, to wave (Helmholtz) Green's functions, etc., are also discussed.

A note on superposition in Mindlin's problem

The problem of a half-space subjected to an interior concentrated force is posed as a superposition problem. For a free plane boundary it is shown that the boundary condition of traction may be expressed directly through the normal derivative of a harmonic Green's function by means of a linear operator. These circumstances at once generate the classical result. For a clamped boundary such a procedure becomes cumbersome and direct integration, in a manner proposed by Sanders, is performed.

On the theory of Kirchhoff's method for the determination of electrical conductivity

A measure of unification is achieved for the purpose of analyzing the steady current flows through elongated cylinders of general section which are activated by a pair of small electrodes on the end faces and are relevant to a scheme of Kirchhoff for determining the electrical conductivity of homogeneous materials; and this features, in a central role, the harmonic Green's function which satisfies a Neumann condition on the boundary of an infinitely long duct whose section duplicates that of the appropriate cylinder. The utility of a Green's function formalism is brought out with specific numerical details in the respective cases of circular and square cross-sections.

Biharmonic classification of Riemannian manifolds

In the study of the structure of the space of biharmonic functions it is often necessary to impose some nondegeneracy condition on the base manifold with respect to quasiharmonic functions (cf. [2], [4]). For this reason it is useful to introduce various quasiharmonically degenerate classes of Riemannian manifolds and to investigate relations among them. This is the purpose of the present note. 1. Quasiharmonic degeneracy. Let R be a noncompact orientable C?-manifold of dimension m _ 2 with C?? Riemannian metric ds2 = gij dxidxj. The corresponding Laplace-Beltrami operator is (1) A = g1/2jL()g1/2gii.) i=l Ax j=1 AXi where g = det (gj) and (gij) = (gij)-l. We call a function u quasiharmonic if Au = const $ 0, and denote by Q = Q(R) the class of quasiharmonic functions u on R normalized by Au = 1. Such functions are superharmonic on R. Following [5], we denote the classes of nonnegative, bounded, and Dirichlet finite functions by P, B, and D respectively, and we set BD = B n D. Similarly we write QX (X = P, B, D, or BD) for Q n X. We are interested in the question as to when QX = 0. 2. Characterization of null classes. We denote by OQx the class of Riemannian manifolds R on which QX(R) = 0, and by OG the class of parabolic manifolds. Let G(x, y) = GR(X, y) be the harmonic Green's function on R ? OG, and set G(x, y) = oo on R E OG. Denote by dy the Riemannian volume element (g(y))1/2 dyl ... dym. Received by the editors September 11, 1970. AMS 1970 subject classifications. Primary 31B30. 1 The work was sponsored by the U.S. Army Research Office-Durham, Grant DA-ARO-D-31-124-70-G7, University of California, Los Angeles. ? American Mathematical Society 1972 165 This content downloaded from 157.55.39.177 on Tue, 15 Nov 2016 03:53:17 UTC All use subject to http://about.jstor.org/terms 166 MITSURU NAKAI AND LEO SARIO [January THEOREM 1. The classes ?QX are characterized in terms of G(x, y) as follows: R e OQP if and only if fG(x, Y) dy = oo; R eOQB if and only if a = sup G(x, y) dy = oo; XER (2) x ReOQD if and only if b= G(x,y)dxdy = oo; R XR ReOQBD if andonlyif a=oo or b=oo. 3. Proof. Let u e Q(R) and take a regular subregion Q of R containing a given point x. Denote by HQ the harmonic solution of the Dirichlet problem on Q with boundary values u. By Stokes' formula, f [(u(y) Hu(y)) A8GQ(x, y) Gn(x, y) A/\g(u(y) H'(y))] dy (3) [(U(Y)HU(y)) * d Gn(x, y) AQ-aB -Gn(x, y) * dj(u(y) Hu(y))] with B a small geodesic ball about x of radius 8. On letting e -+ 0, we obtain the Riesz representation (4) u(x) = H(x) + Gn(x, y) dy on Q (cf., e.g., [1]). Again by Stokes' formula, D({ GQ(., y) dy) =f (fGn(x, y) dy /Xx{Gn(x, y) dy dx =f Gn(x, y) dx dy. We conclude that (6) DQ(u) = Dn(H) + f GQ(x, y) dx dy. nxn If SR G(x, y) dy exists, then it is of class C2, and AX G(x, y) dy = 1 (cf., e.g., [3]). On letting Q -+ R in (5), we obtain (7) DR G(, Y) dy) f?G(X, y) dx dy < o0. This content downloaded from 157.55.39.177 on Tue, 15 Nov 2016 03:53:17 UTC All use subject to http://about.jstor.org/terms 1972] QUASIHARMONIC CLASSIFICATION 167 Suppose there exists a u E QP. Since u is positive and superharmonic on R, h(x) = limn,R Hn(x) exists and a fortiori G(x, y) dy = yim GQ(X, Y) dy = u(x) h(x) < oo. If u E QB, then since IH(x)l < sup0n lul for every x E? Q, (4) implies {GQ(,Y)dY < 2supJul R and consequently a < oo. If u E QD, then (6) and (7) give b < oo. A fortiori, if u E QBD, then a < xo and b < oo. Conversely let v(x) = fR G(x, y) dy. If it is finite or bounded on R, then v E QP or QB. If b < oo, then v(x) is finite and (7) implies v E QD. Consequently, if a < oo and b < oo, then v E QBD. 4. Strict inclusion relations. By means of the characterizations in Theorem 1 we shall prove: THEOREM 2. The following strict inclusion relations are valid:

One- and Two-Dimensional Green's Functions for Electromagnetic Waves in Moving Simple Media

This paper considers the one- and two-dimensional Green's functions associated with electromagnetic radiation in a moving medium. The medium is assumed to be lossless, to have constant permittivity and permeability, and to move with constant velocity with respect to a given inertial coordinate system. For each source (line source or plane source) both the time-dependent and the harmonic Green's functions are found. The results are obtained by means of special transformations which convert the equations to be solved into standard forms. The significance of the results, in terms of one- and two-dimensional Cerenkov radiation, is discussed.

Bounds for the Two-Dimensional Discrete Harmonic Green's Function

Bounds for the Two-Dimensional Discrete Harmonic Green's Function