Kirchhoff Boundary Research Articles

A finite-element formulation for room acoustics simulation with microperforated panel sound absorbing structures: Verification with electro-acoustical equivalent circuit theory and wave theory

A simple frequency-domain finite-element method (FD–FEM) for sound field analyses inside rooms installed with microperforated panel (MPP) sound absorbing structures is described here. This method can also analyze sound absorbing structures composed of MPPs and permeable membranes (PM) simply by changing the only material parameters of MPP into those of PM. As the first stage of the study, the validity of the present FD–FEM is tested through the numerical experiments based on the impedance tube method for measuring the absorption characteristics at normal incidence. In the numerical experiments, we calculated the absorption characteristics of a single MPP absorber, a double-leaf MPP space absorber and a space absorber composed of MPP and PM by using the FD–FEM in two-dimensions, and the computed absorption characteristics are compared with those calculated by an electro-acoustical equivalent circuit theory or a wave theory based on Helmholtz–Kirchhoff boundary integral equation. The results showed that the presented FD–FEM can analyze the absorption characteristics at normal incidence of the MPP sound absorbing structures accurately with the simplicity of the formulation.

A frictionless contact problem for a flexible circular plate and an incompressible non-homogeneous elastic halfspace

In this paper we apply an energy method to examine the axisymmetric contact problem for a flexible circular plate in smooth contact an incompressible elastic halfspace, where the linear elastic shear modulus varies exponentially with depth. The approach adopted approximates the deflected shape of the plate by a power series expansion which satisfies the kinematics of deformation of the plate and the Kirchhoff boundary condition at the edge of the plate. The coefficients in the series are evaluated by making use of the principle of minimum potential energy. Results are obtained for the maximum deflection, the relative deflection and the maximum flexural moment in the circular plate. The results derived from the proposed procedure are compared with equivalent results derived from a computational procedure.

The fluid–structure interaction technique specialized to axially symmetric targets

The fluid–structure interaction technique provides a paradigm for solving scattering from elastic targets embedded in a fluid by a combination of finite and boundary element methods. In this technique, the finite element method is used to compute the target’s impedance matrix and the Helmholtz–Kirchhoff integral with the appropriate Green’s function is used to represent the field in the exterior medium. The two equations are coupled at the surface of the target by imposing the continuity of pressure and normal displacement. This results in a Helmholtz–Kirchhoff boundary element equation that can be used to compute the scattered field anywhere in the surrounding environment. This method reduces a finite element problem to a boundary element one with drastic reduction in the number of unknowns, which translates to a significant reduction in numerical cost. This method was developed and tested for general 3D targets. In this paper, the method is specialized to axially symmetric targets, which provides further reduction in numerical cost, and validated using benchmark solutions.

Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent

In this paper we investigate the following Kirchhoff type elliptic boundary value problem involving a critical nonlinearity: $$\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2dx)\Delta u=\mu g(x,u)+u^5, u>0& \text{in }\Omega, u=0& \text{on }\partial \Omega,\end{array}\right. {\rm {(K1)}}$$ here \({\Omega \subset \mathbb{R}^3}\) is a bounded domain with smooth boundary \({\partial \Omega, a,b \geq 0}\) and a + b > 0. Under several conditions on \({g \in C(\overline{\Omega} \times \mathbb{R}, \mathbb{R})}\) and \({\mu \in \mathbb{R}}\), we prove the existence and nonexistence of solutions of (K1). This is some extension of a part of Brezis–Nirenberg’s result in 1983.

A rigid-body model for diffraction imaging of solid objects: Theory and experimental results

Acoustical imaging via the method of diffraction tomography is typically applied to weakly scattering, fluid objects, wherein the first Born approximation holds. Nonetheless, the technique can be applied to strong scatterers in a meaningful way if an appropriate object function is considered. In this talk, the theoretical form of the object function for a rigid body is developed along with an inversion formula for centripetal, broadband data collection. Applying the latter to experimental, underwater echo data from a variety of objects, reconstructions are presented and interpreted in terms of a Kirchhoff boundary condition. The approach can potentially reveal the relative importance of different scattering mechanisms in the overall pressure field reflected from a body by how closely the rigid body object function is reconstructed. Morphological characteristics may also be identified for objects that are, for example, hidden or buried.

An inverse spectral theorem

We prove a substantial extension of an inverse spectral theorem of Ambarzumyan, and show that it can be applied to arbitrary compact Riemannian manifolds, compact quantum graphs and finite combinatorial graphs, subject to the imposition of Neumann (or Kirchhoff) boundary conditions.

Diffusion processes in thin tubes and their limits on graphs

The present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary, respectively, reflecting at the boundary, and corresponding processes in the limit where the thin tubular domains are shrinking to graphs. The methods we use are probabilistic ones. For shrinking, we use big potentials, respectively, reflection on the boundary of tubes. We show that there exists a unique limit process, and we characterize the limit process by a second-order differential generator acting on functions defined on the limit graph, with Kirchhoff boundary conditions at the vertices.

Three dimensional analyzes of scattering by pressure release sinusoidal surfaces. I. Theory.

Many previous analyzes of scattering by sinusoidal surfaces are based on the Helmholtz integral equation in two dimensions and assume that the incident energy is a plane wave. Some of these solutions are exact insofar as they do not use the Kirchhoff boundary value approximation. However, all of the analyzes are based on one or more commonly used scattering approximations. Analyses presented here are carried out in three dimensions and are based on an integral equation which is analogous to the Rayleigh integral formula. The scattering is restricted to the plane of incidence, but includes receiver grazing angles ranging from 0 to 180 deg. Both the Fresnel phase approximation and Gaussian source and receiver beam functions are used. One of the objectives of this study is to explore the validity of the Kirchhoff approximation, so shadowing and second order scattering are explicitly included. The surface heights and slopes are not restricted in any way, and the surface slope treatment is exact. The resulting expressions are used in the companion paper to explore various approximations and limiting cases that are often encountered in the scattering literature.

An efficient back‐scattering model for arbitrarily shaped objects.

A method is presented for efficiently computing the propagating pressure field back‐scattered from arbitrarily shaped, 3‐D objects. This is accomplished by drawing upon a previously reported relationship between the boundary condition on a 2‐D radiating aperture and the pressure propagating along an axis normal to the aperture, and the diffraction slice theorem, which relates the Fourier transform of an object function to its scattered pressure field. Together, these two results allow for the derivation of an integral formula that expresses the pressure field back‐scattered from an object as a 1‐D Fourier transform of its scattering amplitude. Use of this formula is demonstrated for computing the back‐scattered pressure field from a uniform sphere in the first Born approximation (weak scatterer) and with a Kirchhoff boundary condition (rigid scatterer); the results of which are compared to the corresponding rigorous partial wave expansions.

Modified Kirchhoff's theory of plates including transverse shear deformations

A primary flexure problem defined by Kirchhoff theory of plates in bending is considered. Significance of auxiliary function introduced earlier in the in-plane displacements in resolving Poisson–Kirchhoff's boundary conditions paradox is reexamined with reference to reported sixth order shear deformation theories, in particular, Reissner's theory and Hencky's theory. Sixth order modified Kirchhoff's theory is extended here to include shear deformations in the analysis.

Engineering Nanorobots: Chronology of Modeling Flagellar Propulsion

Nanorobots are propitious to swim or fly compared with crawling and walking because of issues with desirable characteristics of high velocity, efficiency, specificity, controllability, and a simple propagation mechanism that can be realized with miniaturized parts. Inspired by the fact that microorganisms existing in nature function expeditiously under these circumstances, researchers have shown a great interest to conceptualize, model, analyze, and make micro-/nanosized swimmers (nanorobots) that can move in body fluids for applications such as targeted drug delivery, nanomedication, and in-viscera nanosurgery. The present work compiles modeling of physics as investigated since 1951 of flagellar propulsion in engineering nanorobots. Existing theories in flagellar propulsion such as resistive force theory, slender body theory, Kirchhoff rod theory, bead model, and boundary element method as well as progress in designing the propulsion system of a nanorobot are summarized, and various interdisciplinary aspects of realizing nanorobots and issues in moving nanorobots have been presented chronologically.

Three‐dimensional scattering from pressure‐release rough surfaces.

In order to compare a variety of three‐dimensional (3‐D) rough surface scattering theories, the scattering of a spherical wave incident on a pressure‐release rough surface is modeled. Random surface realizations are computed from a spatial roughness power spectrum measured as part of the EVA sea test conducted in 2006. Scattering from these surfaces is computed using boundary and finite element methods. A singularity removal technique is applied to solve the Helmholtz–Kirchhoff boundary integral equation in 3‐D. This integral solution is compared with 3‐D finite elements and the 3‐D Kirchhoff approximation, to determine the range of validity of the models.

Sch’nol’s theorem for strongly local forms

We prove a variant of Sch’nol’s theorem in a general setting: for generators of strongly local Dirichlet forms perturbed by measures. As an application, we discuss quantum graphs with δ- or Kirchhoff boundary conditions.

Three-Dimensional Wake Field Computations Based on Scattered-Field Time Domain Boundary Element Method

We present time domain 3-D wake field calculations based on Scattered-field Time Domain Boundary Element Method (S-TDBEM), which is retarded Kirchhoff's boundary integral equations of scattered electromagnetic fields formulated on interior region problem and has no numerical dispersion in all spatial direction. We propose two main types of S-TDBEM schemes: full 3-D scheme for general 3-D geometries and a so-called 2.5-D scheme for transverse wake field calculation in axis-symmetric structures. Several numerical examples are demonstrated for typical accelerator structures: elliptical pillbox and the TESLA 9-cell cavities. The developed 3-D/2.5-D S-TDBEM codes are verified by comparison with the modal analysis and the finite integration technique (FIT) in the numerical examples. Finally, as one of the most meaningful applications of the S-TDBEM, we demonstrate wake field simulations of a curved trajectory bunch with the developed 3-D S-TDBEM code.

Numerical Simulation of Acoustic Imaging Using a Combination of Finite Difference Time Domain and Boundary Integral Equation Methods

A numerical method based on the combination of the finite difference time domain (FDTD) and boundary integral equation (BIE) methods is proposed and applied to the simulation of acoustic imaging. The sound field is divided into two domains; one is the small domain, which is solved by the FDTD method to include wave properties such as diffraction, and the other is the large domain, which is analyzed by the boundary integral equation method to obtain a response at the required point to avoid numerical dispersion error. The reflected waves from the targets are calculated by the Kirchhoff boundary integral equation method. Some numerical demonstrations are carried out for a two-dimensional sound fields in water. It is confirmed that the combination method causes very little numerical dispersion error and is suitable for the simulation of large scale acoustic imaging.

Time Domain Boundary Element Analysis of Wake Fields in Long Accelerator Structures

We present an explicit time domain boundary element method (TDBEM) scheme with moving window technique for short-range wake field simulations of long accelerator structures. The proposed scheme is formulated by Kirchhoff's boundary integral equation of the scattered electromagnetic field in interior region problems. Implementation of a moving window technique in the framework of TDBEM is achieved by taking into account the causality and the retardation properties of the boundary integral equation. A parallelization algorithm for this moving window implementation is also proposed. The proposed TDBEM code with the moving window technique is applied to several practical examples of long accelerator structures. Numerical results obtained with the TDBEM code are compared with those of several finite integration codes.

Diffraction at a single ideally conducting slit

Maxwell's equations are solved when light, incident normally, is diffracted by a long single slit whose jaws are ideally conducting and of negligible thickness. Expressions, sums over Mathieu functions, are given for the and fields in the plane of the slit, and in the (Fraunhofer) far field, for both polarization possibilities. Graphs of these fields are presented, computed using Mathematica. Comparisons are made with the ‘usual’ optical assumptions (Kirchhoff boundary conditions, St Venant's hypothesis). The far field agrees excellently, as is well known from experiment; though agreement is best if the obliquity factor is set equal to 1. Some field components in the plane of the slit agree well with Kirchhoff, others less; we try to explain why these differences do not affect the far field. The diffracted field, and the total energy transmitted, become strongly polarization dependent for narrow slits (≲0.4λ), and Kirchhoff then breaks down seriously.

A theorem on boundary functions for quantum shutters

We prove for one-dimensional time-dependent quantum absorbing (and reflecting) slits that for right-moving incident waves, the Laplace transform of the boundary function must have singular points at the complex roots of s ± i ( i ɛ / ℏ ) = 0 . We test our result against the exact case of the Moshinsky absorbing (and reflecting) shutter, and the agreement is perfect. In the same Moshinsky problem, when the approximated Kirchhoff boundary condition is used, the transmitted wave is a superposition of right- and left-moving Moshinsky packets. Neglecting the wrong directed wave components we get the exact solution.

Vectorial thin-element approximation: a semirigorous determination of Kirchhoff's boundary conditions

A semirigorous model is presented that bridges the gap between classical scalar diffraction theory on the one hand and fully rigorous simulation models on the other. It falls back on the basic ideas of scalar diffraction theory, especially the Kirchhoff approximation. In contrast to classical techniques, however, the boundary values are determined by rigorous methods of the stratified medium theory in the scope of a fully vectorial formulation. By this means the proposed approach takes vertical rigorous coupling effects inside the grating into account while neglecting the lateral ones. We therefore call this method semirigorous and use the name vectorial thin-element approximation. A direct comparison with rigorous coupled-wave analysis as a fully rigorous simulation model allows a detailed discussion of its range of validity and demonstrates a reduction of computation time of the order of 3 magnitudes. In addition, it also reveals deeper insight into the details of the electrodynamics inside diffracting structures. Some examples will demonstrate this benefit.

Incorporating mask topography edge diffraction in photolithography simulations

In deep ultraviolet lithography simulations, conventional application of Kirchhoff's boundary conditions on the mask surface provides the so-called "thin-mask" approximation of the object field. Current subwavelength lithographic operation, however, places a serious limitation on this approximation, which fails to account for the topographical, or "thick-mask," effects. In this paper, a new simulation model is proposed that is theoretically founded on the well-established physical theory of diffraction. This model relies on the key result that diffraction effects can be interpreted as an intrinsic edge property, and modeled with just two fixed parameters: width and transmission coefficient of a locally determined boundary layer applied to each chrome edge. The proposed model accurately accounts for thick-mask effects of the fields on the mask, greatly improving the accuracy of aerial image simulations in photolithography, while maintaining a reasonable computational cost.