THE BOUNDARY COLLOCATION METHOD WITH MESHLESS CONCEPT FOR ACOUSTIC EIGENANALYSIS OF TWO-DIMENSIONAL CAVITIES USING RADIAL BASIS FUNCTION

In this paper, a meshless method for the acoustic eigenfrequencies using radial basis function (RBF) is proposed. The coefficients of influence matrices are easily determined by the two-point functions. In determining the diagonal elements of the influence matrices, two techniques, limiting approach and invariant method, are employed. Based on the RBF in the imaginary-part kernel, the method results in spurious eigenvalues which can be separated by using the singular value decomposition (SVD) technique in conjunction with the Fredholm alternative theorem. To understand why the spurious eigenvalues occur, analytical study in the discrete system by discretizing the circular boundary is conducted by using circulants. By using the SVD updating terms and documents, the true and spurious eigensolutions can be extracted out respectively. Several examples are demonstrated to see the validity of the present method.     

Analysis of mutiple-shepers radiation and scattering problems by using a null-field integral equation approach

A systematic approach using the null-field integral equation in conjunction with the degenerate kernel is employed to solve the multiple radiation and scattering problems. Our approach can avoid calculating the principal values of singular and hypersingular integrals. Although we use the idea of null-field integral equation, we can locate the point on the real boundary thanks to the degenerate kernel. The proposed approach is seen as one kind of semi-analytical methods, since the error is attributed from the truncation of spherical harmonics. Finally, the numerical examples including one and two spheres are given to verify the validity of proposed approach.     

Scattering of flexural wave in a thin plate with multiple circular inclusions by using the null-field integral equation approach

The subject of scattering flexural wave in a thin plate with multiple circular inclusions under the incident flexural wave is studied in this paper. A semi-analytical approach is proposed to solve this problem which can be decomposed into several interior circular inclusion problems and an exterior plate problem subject to the incident wave. The scattered field in the associated exterior problem is solved by using the null-field integral formulation in conjunction with degenerate kernels, tensor transformation and Fourier series. All dynamic kernels of plate in the direct formulation are expanded into degenerate forms to avoid the integral singularity and further the rotated degenerate kernels have been derived to consider the general case of multiple circular inclusions. The proposed results for an infinite plate with one circular inclusion are compared with the available analytical solutions to verify the validity of the proposed method. To demonstrate the generality of the proposed method, the cases of multiple inclusions are studied and their quasi-static results are verified by static data of FEM using ABAQUS. Numerical results indicate that the DMCF of two inclusions is apparently larger than that of one when two inclusions are close to each other. Fictitious frequency appearing in the exterior problem can be suppressed by using the more number of Fourier series terms. Numerical results show that the space between scatterers has the opposite effect on the near-field DMCF in comparison with the far-field scattering pattern.     

ON THE FALSE DEGENERACY OF THE HELMHOLTZ BOUNDARY INTEGRAL EQUATIONS

In this paper, the false degeneracy of the Helmholtz boundary integral equations is examined. A new theory to explain the false degeneracy of the Helmholtz boundary integral equations is given. In this proposed theory, a false degeneracy of the boundary integral equation is explained as finding a non-trivial source distribution such that it results in trivial field quantities inside the domain interested and non-trivial field quantities for its counter part, i.e., outside the domain interested. It is clearly explained that such a false degeneracy is independent of prescribed boundary conditions but dependent on the integral equation one selects. Moreover, the false degeneracy of the integral equation for the interested domain relates to the eigenproblem for its counter part. Under such a unified theory, the fictitious eigenvalue, spurious eigenvalue and pseudo-fictitious eigenvalue can be explained in a simple mathematical frame. It is concluded from our theoretical analysis that a multiply connected domain results in the pseudo-fictitious eigenvalue even the complex-valued formulations are used. In order to eliminate various kinds of false degeneracy, two methods are employed according to the previous research. A unified view of these two methods is provided such that they can be thought to be equivalent from mathematical point of view. Several numerical examples are given to show the validity of current approach.     

Singular value decomposition of energy matrices in theoretical atomic structure calculations

The singular value decomposition, SVD, is applied to the linear eigenvalue problem in atomic structure calculations. By comparing with recent calculations of energy levels in neutral Ca, it is shown that the SVD can give quite accurate results and much faster than normal diagonalization techniques, even of the Davidson type. However, the energy levels calculated in this approach are more strongly bound than the real eigenvalues, and this is ascribed to an artefact of the SVD, caused by the use of a discretized continuum in the calculations. This effect can lead to fairly large errors if there is strong CI present. The property of the linear eigenvalue problem that the spectrum is unchanged when a constant is added to the diagonal does not apply to the SVD. This means that it is impossible to solve the problems connected with a discretized continuum simply by shifting the spectrum.     

A new method for true and spurious eigensolutions of arbitrary cavities using the combined Helmholtz exterior integral equation formulation method

Integral equation methods have been widely used to solve interior eigenproblems and exterior acoustic problems (radiation and scattering). It was recently found that the real-part boundary element method (BEM) for the interior problem results in spurious eigensolutions if the singular (UT) or the hypersingular (LM) equation is used alone. The real-part BEM results in spurious solutions for interior problems in a similar way that the singular integral equation (UT method) results in fictitious solutions for the exterior problem. To solve this problem, a Combined Helmholtz Exterior integral Equation Formulation method (CHEEF) is proposed. Based on the CHEEF method, the spurious solutions can be filtered out if additional constraints from the exterior points are chosen carefully. Finally, two examples for the eigensolutions of circular and rectangular cavities are considered. The optimum numbers and proper positions for selecting the points in the exterior domain are analytically studied. Also, numerical experiments were designed to verify the analytical results. It is worth pointing out that the nodal line of radiation mode of a circle can be rotated due to symmetry, while the nodal line of the rectangular is on a fixed position.     

A new bifurcation in non-smooth continuous system inducing semi-limit cycle

In this paper, we presented a study on a non-smooth continuous system with emphasis on a special bifurcation. As the parameter varies, a series of concentric closed orbits appear near the equilibrium point. Moreover, the outermost closed orbit attracts all the trajectories outside. It is called as a semi-limit cycle as the trajectories at only one side of this orbit are attracted. By using the theory of generalized Jacobian matrix, it is revealed that this bifurcation can be featured by a pair of complex conjugate eigenvalues reaching exactly but not crossing the imaginary axis. The bifurcation can somewhat be considered to be a degenerate case of the Hopf bifurcation, in which the eigenvalues cross the imaginary axis totally. This study enriches the knowledge of bifurcation analysis for non-smooth dynamical systems.     

Solution of wave diffraction problems by the null-field method

Numerical realization of the null-field method for solving wave diffraction problems is investigated. It is shown that this method provides correct and stable algorithms only when the null-field surface encloses the set of singularities of the analytic continuation of the wave field inside the scatterer. Considering specific examples of diffraction problems, it is demonstrated that the null-field surface is best constructed by analytic deformation of the scatterer’s boundary.     

Restoration of the signal and its spectral characteristics with an irregular set of measurements

An extension of the algorithm for singular value decomposition (SVD)-based restoration of the signal is suggested that makes it possible to gain information on the dynamics of spectral lines (response of the medium) recorded in a single series of irregular measurements of the signal’s integral characteristics with the help of a sliding window. The size and displacement of the window fluctuate in time. SVD-based restoration is accomplished both directly from irregular measurement data and after interpolation of the data to the nodes of a regular grid. An algorithm of signal restoration that is based on spline interpolation is also proposed. When combined with SVD-based restoration, this algorithm allows almost complete elimination of spurious frequencies. In the case of integral measurements using an irregular-size window with its center displaced, the signal restoration quality and determination of the spectral line dynamics are shown to be no worse than in the case of fluctuation-free measurements. Analysis is performed both for a model signal and for the real response considered earlier in the physical experiment.     

Quantum speed limit for mixed states in a unitary system

Since the evolution of a mixed state in a unitary system is equivalent to the joint evolution of the eigenvectors contained in it, we could use the tool of instantaneous angular velocity for pure states to study the quantum speed limit (QSL) of a mixed state. We derive a lower bound for the evolution time of a mixed state to a target state in a unitary system, which automatically reduces to the quantum speed limit induced by the Fubini-Study metric for pure states. The computation of the QSL of a degenerate mixed state is more complicated than that of a non-degenerate mixed state, where we have to make a singular value decomposition (SVD) on the inner product between the two eigenvector matrices of the initial and target states. By combing these results, a lower bound for the evolution time of a general mixed state is presented. In order to compare the tightness among the lower bound proposed here and lower bounds reported in the references, two examples in a single-qubit system and in a single-qutrit system are studied analytically and numerically, respectively. All conclusions derived in this work are independent of the eigenvalues of the mixed state, which is in accord with the evolution properties of a quantum unitary system.     

A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations

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.     

Soliton and boundary condition induced fractional fermion number

We show that for a fermion in a bounded background potential in a finite box, eigenvalues of the total charge are independent of whether the potential is solitonic and depend only on the boundary condition: half-odd integral or integral for charge conjugation (C) invariant boundary conditions and an arbitrary fraction forC non-invariant boundary conditions. Fractional fermion numbers for infinite space Jackiw-Rebbi and Goldstone-Wilczek Hamiltonians are reproduced in finite space by using boundary conditions different from the periodic ones of Rajaraman and Bell.     

Exact solution for the fractional cable equation with nonlocal boundary conditions

The fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.     

A coupled-mode sound propagation model with complex effective depth

A coupled-mode sound propagation model with complex effective depth is presented,in order to involve the effect of branch line integral for acoustic field in a range-dependent waveguide.The equations of motion and continuity are used to obtain the coupled equations,which satisfy boundary conditions in the waveguide with varying topography and contain one coupling matrix.Meanwhile,the couplings between discrete and continuous spectrum are dealt with based on complex effective depth theory.Numerical simulations show that the accuracy of transmission loss is improved by the coupled mode model when eigenvalues of trapped modes are located near the branch point.The acoustic field in a non-horizontally stratified waveguide can be calculated efficiently and accurately by this model,and the energy corresponding to trapped modes,leaky modes and branch line integral can be considered adequately.     

Natural mode analysis of an acoustic cavity with multiple elliptical boundaries by using the collocation multipole method

This paper presents a semi-analytical approach to solve the eigenproblem of an acoustic cavity with multiple elliptical boundaries. To satisfy the Helmholtz equation in the elliptical coordinate system, the multipole expansion for the acoustic pressure is formulated in terms of angular and radial Mathieu functions. The boundary conditions are satisfied by uniformly collocating points on the boundaries. The acoustic pressure at each point is directly calculated in each elliptical coordinate system. In different coordinate systems, the normal derivative of the acoustic pressure is calculated by using the appropriate directional derivative, an alternative to the addition theorem. By truncating the multipole expansion, a finite linear algebraic system is derived. The direct searching approach is employed to determine the natural frequencies by using the singular value decomposition (SVD). Numerical results are widely discussed for several examples including an elliptical cavity, a confocal elliptical annulus cavity and an elliptical cavity with two elliptical cylinders. The accuracy and numerical convergence of the presented method is validated by comparison with available results from the analytical method and the commercial finite-element code ABAQUS. No spurious eigensolutions are found in the proposed formulation. Excellent accuracy and fast rate of convergence are the key features of the present method thanks to its semi-analytical feature.     

Asymptotically degenerate maximum eigenvalues of the eight-vertex model transfer matrix and interfacial tension

We obtain complex nonlinear integral equations for the two asymptotically degenerate maximum eigenvalues of the transfer matrix of the eight-vertex model. These are exact for a lattice of a finite numberN of columns. Solving the equations recursively gives an expansion of the eigenvalues aboutN = . Thus we can obtain the interfacial tension of the model, as well as rederiving our previous result for the free energy.     

Dynamics of the Smooth Positons of the Wadati-Konno-Ichikawa Equation

We discuss a modified Wadati-Konno-Ichikawa(m WKI) equation,which is equivalent to the WKI equation by a hodograph transformation.The explicit formula of degenerated solution of m WKI equation is provided by using degenerate Darboux transformation with respect to the eigenvalues,which yields two kinds of smooth solutions possessing the vanishing and nonvanishing boundary conditions respectively.In particular,a method for the decomposition of modulus square is operated to the positon solution,and the approximate orbits before and after collision of positon solutions are displayed explicitly.     

Quasiconfigurations and the theory of effective interactions

Perturbation theory is reformulated. Schrödinger's equation is recast as a non linear integral equation which yields by Neumann expansion a linked cluster series for the degenerate, quasi degenerate or non degenerate problem. An effective interaction theory emerges that can be formulated in a biorthogonal basis leading to a non Hermitian secular problem. Hermiticity can be recovered in a clear and rigorous way. As the mathematical form of the theory is dictated by the request of physical clarity the latter is obtained naturally. When written in diagrammatic many body language, the integral equation produces a set of linked coupled equations for the degenerate case. The classic summations (Brueckner, Bethe-Faddeev and RPA) emerge naturally. Possible extensions of nuclear matter theory are suggested.     

CUDA-accelerated simulation of multiple scattering using decoupling approximation

Abstract  

In this research a CUDA-accelerated simulation method of multiple scattering in dense participating media based on decoupling approximation is proposed. Simulation of multiple scattering is important, in that it visually illustrates how radiance transfers between gaseous particles. The existing radiance transfer equation has poor computational performance because of its complicated integral calculation of in-scattering in each radiance update. However, this computation can be avoided using previous cached results proposed in this paper. To construct the cached results, we first decouple the phase function in radiance transfer equation with singular value decomposition (SVD) approximation. SVD approximates the multivariate phase function as a sum of products of incident light and outgoing light of lower dimensionality. Then the incident light and outgoing light data of current radiance update are cached independently in two textures. The cached two textures are used to replace integral calculation of in-scattering in the next radiance update to improve computation performance. The proposed method is designed in a parallelized way so that the parallel computing power of CUDA can be fully exploited. The simulation results show that our method allows fast rendering of dynamic scenes while high accuracy is maintained.     

A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems

A solution of a non-homogeneous orthotropic elastic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions as well as the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to the homogeneous ones. Then by virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally presented, which can degenerate in a rather straightforward way to the solution for a homogeneous orthotropic cylindrical shell and isotropic solid cylinder as well as that for a non-homogeneous isotropic cylindrical shell. Using the present method, integral transform can be avoided. It is fit for a cylindrical shell with arbitrary thickness subjected to arbitrary thermal loads. It is also very convenient to deal with dynamic thermoelastic problems for different boundary conditions. Besides, the numerical calculation involved is very easy to be performed. Several examples are presented.