accuracy analysis and optimization of the method of auxiliary sources mas for scattering by a circular cylinder this paper presents a rigorous accuracy analysis of the method of auxiliary sources mas when applied to scattering problems a benchmark canonical geometry consisting of a perfectly conducting infinite circular cylinder is chosen for clarity and simplicity for this particular structure it is shown that the mas square matrix can be inverted analytically yielding exact mathematical expressions for the discretization error and the condition number of the pertinent linear system it is also demonstrated that the error increases very abruptly for source locations associated with the characteristic eigenvalues of the scattering geometry precisely as predicted in theory various plots depict comparisons between analytical and computational data for the boundary condition error and all occurring discrepancies are fully explained among several important results of the analysis the fundamental mas question concerning the optimal location of the auxiliary sources is thoroughly investigated and resolved on the grounds of error minimization

the collocation points of the fundamental solution method for the potential problem we propose an algorithm for computing harmonic functions in a two dimensional domain with prescribed dirichlet data our algorithm is a variant of what is called a fundamental solution method 1 3 this method requires us to select 2n points in the two dimensional plane n of which are called collocation points and the remaining n are called charge points representing the position of the singularities of the fundamental solution it is known that there exists a set of 2n points by which the error is exponentially small 1 3 6 however these papers are concerned mainly with existence and as far as the authors know few fast and reliable algorithms are known for good position of the points in this paper we propose a new rule for the position of the points and examine its efficiency by numerical experiments the new rule uses fft effectively

simulation of scattered fields some guidelines for the equivalent source method abstract three different approaches of the equivalent source method for simulating scattered fields are compared two of them deal with monopole sets the other with multipole expansions in the first monopole approach the sources have fixed positions given by specific rules while in the second one esga the optimal positions are determined via a genetic algorithm the pros and cons of each of these approaches are discussed with the aim of providing practical guidelines for the user it is shown that while both monopole techniques furnish quite good pressure field reconstructions with simple source arrangements esga requires a number of monopoles significantly smaller and with equal number of sources yields a better precision as for the multipole technique the main advantage is that in principle any precision can be reached provided the source order is sufficiently high on the other hand the results point out that the lack of rules for determining the proper multipole order necessary for a desired precision may constitute a handicap for the user

a comparison of high order polynomial and wave based methods for helmholtz problems the application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation two common strategies to address this issue are to use high order polynomial shape functions e g hp fem or to use physics based or trefftz methods where the shape functions are local solutions of the problem typically plane waves both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite element methods but they have yet to be compared in this paper a high order polynomial method p fem with lobatto polynomials and the wave based discontinuous galerkin method are compared for two dimensional helmholtz problems a number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties performance and conditioning it is generally assumed that a wave based method naturally provides better accuracy compared to polynomial methods since the plane waves or bessel functions used in these methods are exact solutions of the helmholtz equation results indicate that this expectation does not necessarily translate into a clear benefit and that the differences in performance accuracy and conditioning are more nuanced than generally assumed the high order polynomial method can in fact deliver comparable and in some cases superior performance compared to the wave based dgm in addition to benchmarking the intrinsic computational performance of these methods a number of practical issues associated with realistic applications are also discussed

green function discretization for numerical solution of the helmholtz equation abstract a discretization method based on free space green function interpolation with a minimuml2norm property has been developed for numerical solution of the helmholtz equation it has been shown to produce accurate solutions with less than three points per wave which implies a manyfold improvement in solution efficiency over previous methods the accuracy of the method does not break down formally until the grid resolution approaches two points per wave

on the holographic reconstruction of vibroacoustic fields using equivalent sources and inverse boundary element method in using the near field acoustical holography based on the inverse boundary element method bem for the reconstruction of vibroacoustic source parameters an enormous number of measurements required in practice have limited the extended application of this method to obtain the sufficient field data with a small number of actual measured data the regeneration method of the radiated field using the multipoint equivalent sources is attempted in this paper in using the equivalent source method a vibrating source can be represented by distributed spherical sources inside the actual source surface in this paper the radiated field is expressed with a series of spherical hankel functions and spherical harmonics for suppressing the adverse effect of high order spherical functions spatial filtering of coefficients and wave vector components by a regularization scheme is adopted restored field data appended with actual measured data can be used as input for the inverse bem to reconstruct the source field

an experimental comparison of various methods of nearfield acoustic holography abstract an experimental comparison of four different methods of nearfield acoustic holography nah for planar acoustic sources is presented in this study the four nah methods considered in this study are based on 1 spatial fourier transform 2 equivalent sources model 3 boundary element methods and 4 statistically optimized nah two dimensional measurements were obtained at distances ranging from one half to sixteen microphone spacings away from a sound source and the nah methods were used to reconstruct the sound field at the source surface reconstructed particle velocity and acoustic pressure fields presented in this study showed that the equivalent sources model based algorithm along with tikhonov regularization provided the best localization of the sources reconstruction errors were found to be smaller for the equivalent sources model based algorithm and the statistically optimized nah algorithm the effects of hologram distance on the performance of various algorithms are discussed in detail the study also compares the computational time required by each algorithm to calculate the reconstruction four different regularization parameter choice methods were compared the l curve method provided more accurate reconstructions than the generalized cross validation and the morozov discrepancy principle at larger measurement distances the performance of fixed parameter regularization was comparable to that of the l curve method

a generalized internal source density method for the forward and backward projection of harmonic pressure fields from complex bodies a generalized internal source density gisd method is presented to address the forward and backward projection of harmonic pressure fields from complex three dimensional bodies the gisd approach is based on decomposing the field on a closed surface of revolution in the fluid into a summation of circumferential orders where the pressure field for each order is associated with an internal linear distribution of ring sources on the axis of revolution of the surface both dirichlet and neumann boundary value problems may be addressed using the approach in which the axial variation of each ring source distribution is formulated as the solution of a minimum mean square error problem which is subsequently solved using svd methods the resulting linear distributions of ring sources can be simply used to determine the entire pressure and velocity fields of a complex harmonically vibrating body in particular the pressure and or normal velocity on a vibrating body can be readily obtained from the pressure field over a closed surface local intensity and radiated power can thus be simply obtained far field pressures are simply obtained from the axial fourier transforms of the ring source distributions several examples are presented to illustrate the behaviour of the source distributions for spherical geometries the analytical results obtained via the gisd method agree with those obtained via classical methods

detecting non uniqueness of solutions to biharmonic integral equations through svd we consider the singular values of an integral operator and of a corresponding square matrix derived from the integral operator by means of a quadrature formula and a collocation the integral operator and also the matrix depend on a real parameter which may also enter the singular values of the operator and the matrix when a singular value drops to zero for a certain critical value of the parameter the corresponding homogeneous integral equation or matrix equation has a nontrivial solution based on several examples with biharmonic integral operators we conjecture that the order of approximation of the critical value for the matrix is at least equal to the order of the quadrature formula used it is therefore possible with a reasonable accuracy to detect such critical values for the integral operator simply through a singular value decomposition of the matrix derived by a quadrature and collocation

dynamic solution of unbounded domains using finite element method discrete green s functions in frequency domain in this paper we present a new approach for finite element solution of time harmonic wave problems on unbounded domains as representatives of the wave problems discrete green s functions are evaluated in finite element sense the finite element mesh is considered to be of repeatable pattern cell constructed in rectangular co ordinates the system of fe equations is therefore reduced to a set of well known dispersion equations by using a spectral solution approach the spectral wave bases are constructed directly from the fe dispersion equations radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level dirichlet neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem a new discrete transformation method recently proposed by the authors is used to satisfy the boundary conditions comprehensive studies are made for showing the validity accuracy and convergence of the solutions the results of the benchmark problems indicate that the proposed method can be used to evaluate discrete green s functions whose analytical forms are not available

stability and convergence of the method of fundamental solutions for helmholtz problems on analytic domains the method of fundamental solutions mfs is a popular tool to solve laplace and helmholtz boundary value problems its main drawback is that it often leads to ill conditioned systems of equations in this paper we investigate for the interior helmholtz problem on analytic domains how the singularities charge points of the mfs basis functions have to be chosen such that approximate solutions can be represented by the mfs basis in a numerically stable way for helmholtz problems on the unit disc we give a full analysis which includes the high frequency short wavelength limit for more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued which in turn depends on both domain shape and boundary data using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10 1 1 on a wide variety of analytic domains at high frequencies of order only 3 points per wavelength are needed which compares very favorably to boundary integral methods

goodness of fit tests for log linear models in sparse contingency tables abstract the asymptotic normality of the likelihood ratio goodness of fit statistic is demonstrated for testing the fit of log linear models with closed form maximum likelihood estimates in sparse contingency tables unlike the traditional chi squared theory the number of categories in the table increases as the sample size increases but not all of the expected frequencies are required to become large some results of a small monte carlo study are presented the traditional chi squared approximation is reasonably accurate for the pearson statistic for many sparse tables but cases are presented for which it fails the normal approximation can be much more accurate than the chi squared approximation for the likelihood ratio statistic but the bias of estimated moments is a potential problem for very sparse tables

an improved accuracy version of the method of auxiliary sources for computational electromagnetics the method of auxiliary sources mas is normally applicable to electromagnetic problems involving structures of significant thickness so that an adequately large distance between source and collocation points is guaranteed for thin or open geometries the accuracy of the method is depleted due to numerical instabilities caused by the highly singular terms of the dyadic green s function dgf in this paper a modified mas mmas is developed to circumvent this particular difficulty higher order terms of the dgf are numerically calculated by introducing a canonical grid where derivatives can be accurately computed via a discrete scheme unlike standard mas where the dgf analytical problematic expression is invoked instead this procedure is equivalent to the involvement of auxiliary currents and charges in the solution instead of the elementary source fields used in standard mas

on convergence and inherent oscillations within computational methods employing fictitious sources recent studies of the method of auxiliary sources mas and the method of fundamental solutions mfs have shown that it is possible to have convergence of the field which is the finally desired quantity together with divergence of the fictitious sources which are intermediate quantities the aforementioned studies concern a simple scattering problem where the scattered field satisfies the helmholtz equation and the fictitious sources are placed symmetrically in the present paper we extend the previous results in two directions one having to do with the geometry and the other with frequency more precisely we first show that the above convergence divergence phenomena can also occur when the fictitious sources are placed asymmetrically and then study the nature of this divergence in detail second we show that all findings carry over to the case of laplace s equation we additionally point out certain differences of the discussed phenomena with the well known ones of internal resonances

a forward advancing wave expansion method for numerical solution of large scale sound propagation problems abstract in this paper a forward advancing field discretization method suitable for solving the helmholtz equation in large scale problems is proposed the forward wave expansion method fwem is derived from a highly efficient discretization procedure based on interpolation of wave functions known as the wave expansion method wem the fwem computes the propagated sound field by means of an exclusively forward advancing solution neglecting the backscattered field it is thus analogous to methods such as the one way parabolic equation method pem usually discretized using standard finite difference or finite element methods these techniques do not require the inversion of large system matrices and thus enable the solution of large scale acoustic problems where backscatter is not of interest calculations using fwem are presented for two propagation problems and comparisons to data computed with analytical and theoretical solutions and show this forward approximation to be highly accurate examples of sound propagation over a screen in upwind and downwind refracting atmospheric conditions at low nodal spacings 0 2 per wavelength in the propagation direction are also included to demonstrate the flexibility and efficiency of the method

calculation of sound radiation in infinite domain using a meshless method a meshless method coupling with a variable order infinite acoustic wave envelope element for sound radiation calculation in infinite domain is presented with the aim of accurately calculating the acoustic radiation and improving computational efficiency it is based on using the element free galerkin method in the inner region enclosing the radiator and a variable order infinite acoustic wave envelope element in the outer region for the proper modeling of the pressure amplitude decay the details are provided for the derivation and implementation of this method the factors of influencing the performance of the method which include the shape function constructing the number of integration points the weight functions and the support domain are discussed a hybrid adaptive gauss legendre quadrature is devised to obtain good integration accuracy the suitable radius of the support domain for the acoustic field calculation in free space is also determined by use of numerical experiments a complex structure is designed for simulation to validate the method the results illustrate the accuracy applicability and effectiveness of this method

enhanced nearfield acoustic holography for larger distances of reconstructions using fixed parameter tikhonov regularization this paper evaluates the performance of various regularization parameter choice methods applied to different approaches of nearfield acoustic holography when a very nearfield measurement is not possible for a fixed grid resolution the larger the hologram distance the larger the error in the naive nearfield acoustic holography reconstructions these errors can be smoothed out by using an appropriate order of regularization this study shows that by using a fixed manual choice of regularization parameter instead of automated parameter choice methods reasonably accurate reconstructions can be obtained even when the hologram distance is 16 times larger than the grid resolution

generalizing effective spatial bandwidth for pole location in mas the elliptic cylinder the location of poles in the method of auxiliary sources for 2 d scatterers can be assisted using the effective spatial bandwidth ebw of the incident and scattered fields previously only circular boundaries were considered here ebw is extended to noncircular boundaries calculations presented for a cylinder with elliptical cross section verify ebw as an additional tool for pole placement for more general geometries

discontinuous galerkin methods with plane waves for time harmonic problems a general framework for discontinuous galerkin methods in the frequency domain with numerical flux is presented the main feature of the method is the use of plane waves instead of polynomials to approximate the solution in each element the method is formulated for a general system of linear hyperbolic equations and is applied to problems of aeroacoustic propagation by solving the two dimensional linearized euler equations it is found that the method requires only a small number of elements per wavelength to obtain accurate solutions and that it is more efficient than high order drp schemes in addition the conditioning of the method is found to be high but not critical in practice it is shown that the ultra weak variational formulation is in fact a subset of the present discontinuous galerkin method a special extension of the method is devised in order to deal with singular solutions generated by point sources like monopoles or dipoles aeroacoustic problems with non uniform flows are also considered and results are presented for the sound radiated from a two dimensional jet

a particular solution trefftz method for non linear poisson problems in heat and mass transfer this paper proposes a computational procedure based on the trefftz method for the solution of non linear poisson problems the problem is solved by finding an approximate particular solution to the poisson equation and using boundary collocation to solve the resulting laplace equation this method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions the solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency a quasi newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme the efficacy of the new formulation has been demonstrated for two classes of problems viz the thermal explosion problem and the diffusion reaction problem in a partially wetted catalyst pellet a brief error analysis of the method as well as future research directions is presented

a symmetric trefftz dg formulation based on a local boundary element method for the solution of the helmholtz equation a general symmetric trefftz discontinuous galerkin method is built for solving the helmholtz equation with piecewise constant coefficients the construction of the corresponding local solutions to the helmholtz equation is based on a boundary element method a series of numerical experiments displays an excellent stability of the method relatively to the penalty parameters and more importantly its outstanding ability to reduce the instabilities known as the pollution effect in the literature on numerical simulations of long range wave propagation

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

the forward projection of harmonic pressure fields using the generalized internal source density method a previously developed generalized internal source density gisd method is extended to address the forward projection of harmonic pressure fields from normal velocity fields on a closed surface of revolution in a fluid the gisd approach is based on decomposing the normal velocity field on the closed surface into a summation of circumferential orders where the pressure field for each order is associated with an internal linear source distribution of ring sources on the axis of revolution of the surface the linear source distributions are formulated as integral equations of the first kind which are then solved using singular value decomposition methods far field pressures are simply related to the fourier transforms of the linear source distributions numerical results are presented to illustrate the forward projection of harmonic fields using the gisd method for both dirichlet and neumann problems the numerical results for spherical and finite cylindrical vibrators illustrate the usefulness of the gisd method over a wide range of frequencies for both acoustic radiation and scattering problems

the method of fundamental solutions for inverse boundary value problems associated with the steady state heat conduction in anisotropic media in this paper the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady state heat conduction in an anisotropic medium since the resulting matrix equation is severely ill conditioned a regularized solution is obtained by employing the truncated singular value decomposition while the optimal regularization parameter is chosen according to the l curve criterion numerical results are presented for both two and three dimensional problems as well as exact and noisy data the convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries and decreasing the amount of noise added into the input data respectively are analysed a sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed the numerical results obtained show that the proposed numerical method is accurate convergent stable and computationally efficient and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady state heat conduction copyright 2005 john wiley sons ltd

a numerical solution for the general radiation problem based on the combined methods of superposition and singular value decomposition in the method of wave superposition the acoustic field due to a complex radiator is expressed in terms of a fredholm integral equation of the first kind called the superposition integral equation in general fredholm integral equations of the first kind are ill posed and therefore difficult to solve numerically in this paper it will be shown that a simple collocation procedure when combined with the singular value decomposition can yield accurate results for the numerical solution of the superposition integral as an example of the application of the method the acoustic radiation from a circular cylinder will be analyzed using this numerical procedure and compared to the exact solution it is shown that for this problem the accuracy of the numerical solution can be judged by evaluating how well the superposition solution approximates the specified boundary condition on the surface of the radiator an example is also given of a problem which has no exact solution in this situation it is suggested without proof that the accuracy of the numerical solution can be judged in a similar manner by evaluating the error in the superposition solution s satisfaction of the boundary condition

acoustic scattering by a modified werner method a modified integral werner method is used to calculate pressure scattered by an axisymmetric body immersed in a perfect and compressible fluid subject to a harmonic acoustic field this integral representation is built as the sum of a potential of a simple layer and a potential of volume it is equivalent to the exterior helmholtz problem with neumann boundary condition for all real wave numbers of the incident acoustic field for elastic structure scattering problems the modified werner method is coupled with an elastodynamic integral formulation in order to account for the elastic contribution of the displacement field at the fluid structure interface the resulting system of integral equations is solved by the collocation method with a quadratic interpolation the introduction of a weighting factor in the modified werner method decreases the number of volume elements necessary for a good convergence of results this approach becomes very competitive when it is compared with other integral methods that are valid for all wave numbers a numerical comparison with an experiment on a tungsten carbide end capped cylinder allows a glimpse of the interesting possibilities for using the coupling of the modified werner method and the integral elastodynamic equation used in this research

a multilayer method of fundamental solutions for stokes flow problems the method of fundamental solutions mfs is a meshless method for the solution of boundary value problems and has recently been proposed as a simple and efficient method for the solution of stokes flow problems the mfs approximates the solution by an expansion of fundamental solutions whose singularities are located outside the flow domain typically the source points i e the singularities of the fundamental solutions are confined to a smooth source layer embracing the flow domain this monolayer implementation of the mfs monolayer mfs depends strongly on the location of the user defined source points on the one hand increasing the distance of the source points from the boundary tends to increase the convergence rate on the other hand this may limit the achievable accuracy this often results in an unfavorable compromise between the convergence rate and the achievable accuracy of the mfs the idea behind the present work is that a multilayer implementation of the mfs multilayer mfs can improve the robustness of the mfs by efficiently resolving different scales of the solution by source layers at different distances from the boundary we propose a block greedy qr algorithm bgqra which exploits this property in a multilevel fashion the proposed multilayer mfs is much more robust than the monolayer mfs and can compute stokes flows on general two and three dimensional domains it converges rapidly and yields high levels of accuracy by combining the properties of distant and close source points the block algorithm alleviates the overhead of multiple source layers and allows the multilayer mfs to outperform the monolayer mfs

analysis of an infinite current source above a semi infinite lossy ground using fictitious current auxiliary sources in conjunction with complex image theory techniques the canonical problem of an infinitely long electric current line radiating above a lossy infinite half space is examined the solution is based on the method of auxiliary sources mas in this method one can in general apply a numerical solution by introducing sets of fictitious current sources whose fields are elementary analytical solutions to the boundary value problem in order to approximately describe the actual electromagnetic em fields in each domain in general the convergence rate and the accuracy of the mas solution depend on the spatial distributions of the fictitious current sources sets and their locations in regard to the singularities of the actual em field simulated by each set here both the accuracy and the convergence rate of the method are examined investigating complex image approximations in order to optimally choose the auxiliary sources placements it is proved that the convergence rate and the accuracy of the method are significantly improved by utilizing the complex images as locations of the auxiliary sources the main contribution of the paper consists in the application of mas to an open structure which involves lossy dielectrics excited by a nonuniform em field as well as in the optimal choice of the locations of the fictitious current sources according to complex image techniques

oscillations caused by several retarded and advanced arguments in this paper we study the oscillatory behavior of equations of the forms y t i 1npiy t i 0 and y t i 1npiy t i 0 where piand i i 1 2 n are positive constants we prove that each one of the following conditions 1 pi i1 e for some i i 1 2 n 2 i 1npi 1 e where min 1 2 n 3 i 1npi 1 n i 1n i 1 e or 4 1 n i 1n pi i 1 2 21 e implies that every solution of and oscillates a generalization in the case where the coefficientspi i 1 2 n are positive and continuous functions is also presented

on the level error after bartlett adjustment of the likelihood ratio statistic summary it is well known that bartlett adjustment reduces level error of the likelihood ratio statistic from order n l to order n3 2 in the present note we show that level error of the adjusted statistic is actually order n2

acoustic radiation impedances and impulse responses for elliptical cylinders using internal source density and singular value decomposition methods abstract a new internal source distribution method is combined with the singular value decomposition method to determine the acoustic impedance and associated impulse response for an infinite cylinder with a plane of symmetry and a normal velocity with a specified circumferential variation an internal linear distribution of unknown monopole and or dipole line sources along the plane of symmetry is determined to match the specified harmonic normal velocity on the surface in a least mean square error sense after the discretization of the source distributions the subsequent solution of a set or sets of linear algebraic equations for the internal source distributions is obtained using the singular value decomposition method line integrals for the associated surface pressure the acoustic impedance and the corresponding impulse response are then simply evaluated by quadrature methods numerical results for the acoustic impedances and associated impulse responses are presented for circular cylinders vibrating with various natural in vacuo modal shapes numerical results for acoustic impedances and associated impulse responses for elliptical cylinders vibrating with various natural in vacuo modal shapes are then presented and their characteristics are discussed and compared to the corresponding results for circular cylinders

asymptotic stability of the fundamental solution method abstract in a previous paper 1988 we studied the numerical stability of the fundamental solution method applied to the dirichlet problem of laplace s equation and presented a practical scheme for examining the stability of the method using the singular value decomposition in the present paper we give an asymptotic estimate of the stability with respect to the number of collocation points

a meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks the numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods based on global interpolation by analytic functions show low accuracy in order to circumvent this issue a meshfree modification of the method of fundamental solutions is developed where the approximation basis is enriched by an extra span of corner adapted non smooth shape functions the high accuracy of the new method is illustrated by solving several boundary value problems for the helmholtz equation modelling physical phenomena from the fields of room acoustics and acoustic resonance

improved likelihood ratio statistics for generalized linear models summary this paper gives a general formula for the expected value of the likelihood ratio statistic for generalized linear models corrected up to terms of order n 1 where n is the size of the sample the application of this formula to some models is illustrated

improved likelihood ratio statistics for covariance selection models summary the expected value of the likelihood ratio statistic for covariance selection models when this statistic has a closed form is derived correct to terms of order n 2 n being the sample size then a bartlett type correction factor is available for such statistics by simulation the practical use of such a correction factor is illustrated and is compared with an alternative correction factor in a special case the null density of the corrected statistic is also considered

wave field singularity aspects in large size scatterers and inverse problems it is known that any scattered wave field carrying energy into infinity must have source singularity centers within a bounded space otherwise the scattered field should be identically equal to zero everywhere kupradze 1935 in this paper attention is paid to localization of these singularities under the assumption that every scattered wave is determined uniquely by its own singularities investigations have shown that these singularities are distributed as bright centers and the distance between them depends on frequency to determine the position localization of the scattered wave field singularities the functions describing converging and diverging waves are used based on these concepts and the method of auxiliary sources an efficient numerical method to reconstruct a field up to its singularities is suggested the localization of singularities is used for partial representation of the scattered fields which reduces significantly the number of unknowns in describing the scattering process and leading into optimized inverse scattering problem solutions

acoustic axisymmetric radiation and scattering from bodies of revolution using the internal source density and fourier methods a brief review is presented of a previously developed mean square error method to solve the neumann boundary value problem corresponding to acoustic axisymmetric harmonic radiation and rigid body scattering from bodies of revolution of arbitrary shape the method is based on the use of an internal line monopole source distribution along the axis in contrast to earlier work the pressures are simply expressed as line integrals of the source distributions which in the far field reduce to fourier transform relationships exact expressions for the source strength distributions for a sphere are developed using the fourier transform relationships the source strength distributions involve spatial derivatives of dirac delta functions and thus have a vanishingly small region of support about the center of the sphere for cylindrically shaped bodies with large aspect ratios the spatial characteristics of the source strength distributions are shown to be more closely matched to the normal velocity distribution the use of a discrete series of dirac delta functions to represent the continuous line source distributions is presented for spherical radiation and scattering problems

numerical implementation of the lumped parameter model for the acoustic power output of a vibrating structure in a previous paper a lumped parameter model for the acoustic radiation from a vibrating structure was defined by dividing the surface of the structure into elements expanding the acoustic field from each of the elements in a multipole expansion and truncating all but the lowest order terms in the expansion here the lumped parameter model is implemented numerically by requiring the boundary condition for the normal surface velocity to be satisfied in a lumped parameter sense this alleviates the difficulties typically encountered in enforcing the boundary condition leading to a relatively simple numerical solution with well defined convergence properties the basis functions for the numerical analysis are taken as the acoustic fields of discrete simple dipole and tripole sources located at the geometrical centers of the surface elements the different source types are used to represent the radiation from different kinds of surface elements simple sources for elements in the plane of an infinite baffle dipole sources for very thin structures which deform only in bending and tripole sources for elements associated with parts of a structure enclosing a finite volume the convergence of the numerical solution for the power output as a function of both frequency and element size is demonstrated through several example problems

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

a meshless method for some inverse problems associated with the helmholtz equation in this paper a new numerical scheme based on the method of fundamental solutions is proposed for the numerical solution of some inverse boundary value problems associated with the helmholtz equation including the cauchy problem since the resulting matrix equation is badly ill conditioned a regularized solution is obtained by employing truncated singular value decomposition while the regularization parameter for the regularization method is provided by the l curve method numerical results are presented for problems on smooth and piecewise smooth domains with both exact and noisy data and the convergence and stability of the scheme are investigated these results show that the proposed scheme is highly accurate computationally efficient stable with respect to the noise in the data and convergent with respect to decreasing the amount of data noise and increasing the distance between the physical and fictitious boundaries and could be considered as a competitive alternative to existing methods for these problems

boundary knot method for 2d and 3d helmholtz and convection diffusion problems under complicated geometry the boundary knot method bkm of very recent origin is an inherently meshless integration free boundary type radial basis function collocation technique for the numerical discretization of general partial differential equation systems unlike the method of fundamental solutions the use of non singular general solution in the bkm avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain the purpose of this paper is to extend the bkm to solve 2d helmholtz and convection diffusion problems under rather complicated irregular geometry the method is also first applied to 3d problems numerical experiments validate that the bkm can produce highly accurate solutions using a relatively small number of knots for inhomogeneous cases some inner knots are found necessary to guarantee accuracy and stability the stability and convergence of the bkm are numerically illustrated and the completeness issue is also discussed

design of two dimensional rectangular array beamformers with partial adaptivity this paper considers the problem of designing a two dimensional 2 d rectangular array beamformer with partial adaptivity using the eigenstructure of the signal data received by a 2 d rectangular array beamformer we first show that the optimal weight matrix when using full adaptivity can be obtained from a set of singular vectors then the design problem of using partial adaptivity is formulated as a result the optimal solution for the partially adaptive weight matrix can be found by solving two basic problems namely determining the proper dimension of the partially adaptive weight matrix and the set of the singular vectors we develop the detection formulas for the information theoretic criteria aic and mdl to find the proper dimension next an efficient method is presented so that the optimal solution for the set of the singular vectors can be found analytically we also investigate the required computational complexity it is shown that 2 d partially adaptive beamforming using the proposed technique requires less computational complexity than 2 d fully adaptive beamforming using conventional techniques moreover computer simulations demonstrate that the proposed 2 d partially adaptive technique provides satisfactory array performance when compared with conventional fully adaptive techniques

green s function discretization scheme for sound propagation in nonuniform flows a new frequency domain discretization technique that permits the numerical solution of the equation governing the propagation of small disturbances within nonuniform potential flows is described the method can be applied successfully to the numerical solution of aeroacoustic problems because a good accuracy is preserved up to 3 4 points per period for three dimensional unstructured meshes the discretization scheme is based on a local interpolation formula that is strictly joined to the physics of wave propagation because it is constructed with the superimposition of elementary sources that are local solutions of the local convective wave equation the method presents aspects in common with both finite difference and finite element methods with the peculiarity that the local interpolation formula can be interpreted as a specific shape function introduced to take advantage of the physics of the problem the method is applied to the potential equation linearized around an arbitrary aerodynamic mean flow and several comparisons with theoretical results as well as several convergence tests are conducted to show that a good accuracy is preserved up to 3 4 points per period also for irregular meshes with both random and systematic distortions numerical calculations are presented for three dimensional problems with both uniform and nonuniform flow and comparisons are made with theoretical and other available numerical results with the appropriate generalizations the method can be regarded as a new efficient general approach for the discretization of generic partial differential equations

bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator for rather general parametric models a simple connection is established between the bartlett adjustment factor of the log likelihood ratio statistic and the normalizing constant c of the formula c i i 1 2l for the conditional distribution of a maximum likelihood estimator as applied to the full model and the model of the hypothesis tested this leads to a relatively simple demonstration that division of the likelihood ratio statistic by a suitable constant or estimated factor improves the chi squared approximation to its distribution various expressions for these quantities are discussed in particular for the case of a one dimensional parameter an approximation to the constants involved is derived which does not require integration over the sample space

an equivalent source technique for calculating the sound field inside an enclosure containing scattering objects the equivalent source method has previously been used to calculate the exterior sound field radiated or scattered from bodies in the free field in this paper the method is used to calculate the internal pressure field for an enclosure which can have arbitrary boundary conditions and may include internal objects which scatter the sound some of the equivalent source positions are chosen to be the same as the first order images of the source inside the enclosure some are positioned within the scattering objects and the remainder are positioned on a spherical surface some distance outside the enclosure the normal velocity on the surfaces of the scattering objects and the enclosure walls is evaluated at a larger number of positions than there are equivalent sources the sum of the squared difference between this velocity and that expected because of the admittance of the boundary is minimized by adjusting the strengths of the equivalent sources the convergence of the method is checked by evaluating the velocity at a larger number of monitoring positions example results are presented for the sound field and frequency response inside a damped rectangular enclosure which compare very well with the conventional modal model the effect of having rigid spheres inside the enclosure are then investigated and it is found that the effect is significant even some distance from the spheres and at frequencies for which the size of the sphere is small compared to a wavelength finally the effect of a nonlocally reacting boundary condition is illustrated by assuming that one of the walls of the enclosure is an elastic plate

towards a stable two dimensional time domain source model solution by use of a combined source formulation the application of the source model technique to the solution of two dimensional 2 d transient scattering problems directly in the time domain is described special attention is given to the accuracy and the stability of the numerical solution scheme a detailed comparison is made between the more often used single type source solution scheme and a newly proposed combined source solution scheme it is found that the combined source scheme can offer a significant advantage in terms of stability without any appreciable loss in accuracy over the single type source scheme

a mapping relationship based near field acoustic holography with spherical fundamental solutions for helmholtz equation abstract in the procedure of the near field acoustic holography nah based on the fundamental solutions for helmholtz equation fs the number of fs and the measurement setup to obtain their coefficients are two crucial issues to the successful reconstruction the current work is motivated to develop a framework for the nah which supplies a guideline to the determination of the number of fs as well as an optimized measurement setup a mapping relationship between modes on surfaces of boundary and hologram is analytically derived by adopting the modes as fs in spherical coordinates thus reconstruction is converted to obtain the coefficients of participant modes on holograms in addition an integral identity is firstly to be derived for the modes on convex surfaces which is useful in determining the inefficient or evanescent modes for acoustic radiation in free space to determine the number of fs adopted in the mapping relationship based nah mrs based nah two approaches are proposed to supply reasonable estimations with criteria of point wise pressure and energy respectively a technique to approximate a specific degree of mode on patches by a set of locally orthogonal patterns is explored for three widely used holograms such as planar cylindrical and spherical holograms which results in an automatic determinations of the number and position of experimental setup for a given tolerance numerical examples are set up to validate the theory and techniques in the mrs based nah reconstructions of a cubic model demonstrate the potential of the proposed method for regular models even with corners and shapers worse results for the elongated cylinder with two spherical caps reveal the deficiency of the mrs based nah for irregular models which is largely due to the adopted modes are fs in spherical coordinates the nah framework pursued in the current work provides a new insight to the reconstruction procedure based on the fs in spherical coordinates

on two types of convergence in the method of auxiliary sources the method of auxiliary sources mas is often applied to problems involving an externally illuminated smooth perfect electric conductor pec one seeks to approximately satisfy the boundary condition on the pec surface using auxiliary sources located inside the pec surface usually the underlying auxiliary surface is also smooth and closed the currents on the auxiliary sources mas currents are the initial unknowns once they have been found one can easily determine the field due to them mas field at all points external to the auxiliary surface and in particular at all points external to the pec scatterer we show that in the limit n it is possible to have a convergent mas field together with divergent mas currents and that this phenomenon is accompanied by an abrupt behavior of the limiting value of the mas field we show this possibility through an analytical study of a two dimensional scattering problem involving a circular cylinder in which mas fields and currents can be determined explicitly the analytical study proceeds from first principles it involves fundamental electromagnetics and relatively simple mathematical manipulations numerical results supplement the analytical study and demonstrate the nature of the aforementioned divergence our study sheds light on other aspects of mas in particular it establishes interesting similarities and differences between mas and its continuous version and reveals many similarities between mas currents and numerical solutions of hallen s integral equation with the approximate kernel

on the phenomenon of oscillations in the method of auxiliary sources when one applies the method of auxiliary sources to scattering problems involving perfect conductors one first seeks fictitious auxiliary currents located inside the conductor and then determines the field from these currents for a simple two dimensional problem involving an infinite circular cylinder illuminated by an electric current filament it has recently been shown analytically that it is possible to have divergent auxiliary currents to make this statement precise one must properly normalize the currents together with a convergent field it was also shown through numerical investigations that the aforementioned divergence appears as abnormal rapid oscillations in the present paper we investigate such phenomena in more detail with particular emphasis on oscillations for a perfectly conducting ground plane illuminated by an electric current filament we once again demonstrate the possibility of having divergent oscillating currents producing a convergent field we develop an asymptotic formula for the oscillating current values which sheds light on the nature of the oscillations we revisit the circular cylinder problem to develop a similar asymptotic formula we also discuss roundoff errors and possible generalizations to scatterers of other shapes the present study is to a great extent analytical with the analytical predictions confirmed and supplemented by numerical results

a meshless integration free and boundary only rbf technique based on the radial basis function rbf nonsingular general solution and dual reciprocity method drm this paper presents an inherently meshless integration free boundary only rbf collocation technique for numerical solution of various partial differential equation systems the basic ideas behind this methodology are very mathematically simple in this study the rbfs are employed to approximate the inhomogeneous terms via the drm while nonsingular general solution leads to a boundary only rbf formulation for homogenous solution the present scheme is named as the boundary knot method bkm to differentiate it from the other numerical techniques in particular due to the use of nonsingular general solutions rather than singular fundamental solutions the bkm is different from the method of fundamental solution in that the former does not require the artificial boundary and results in the symmetric system equations under certain conditions the efficiency and utility of this new technique are validated through a number of typical numerical examples completeness concern of the bkm due to the sole use of the nonsingular part of complete fundamental solution is also discussed

on the use of svd improved point matching in the current model method em scattering an approach which uses the singular value decomposition svd to improve the accuracy of the numerical solution obtained with fictitious current models is introduced in this approach the svd is essentially facilitating a systematic way to optimally reduce the generalized inverse matrix used in the solution to a submatrix of smaller rank this reduction strikes a balance between the fulfillment of the boundary conditions at the matching points and that between them clearly the boundary conditions errors at the matching points are no longer strictly zero however the previously discernible errors between the matching points are markedly suppressed the approach is efficacious not only when the impedance matrix is inherently singular or highly ill conditioned but also when this matrix is entirely well conditioned it can be generalized and implemented in any method of moments code which uses point matching for testing the approach has been incorporated into an existing solution based on the current model method for the problem of scattering from periodic sinusoidal surfaces and is shown to render the solution more accurate

efficient electromagnetic scattering computation using the random auxiliary sources method for multiple composite 3 d arbitrary objects the electromagnetic em scattering problem from three dimensional 3 d arbitrary composite objects is proposed using the random auxiliary sources ras method based on direct application of the boundary conditions with the uniqueness theorem and the use of random equivalent problems concept more degrees of freedom to the sources positions are added resulting in significantly efficient solutions with lower memory requirements the technique does not require any singularity treatment due to placing the equivalent sources away from the boundaries while boundary conditions are not enforced exactly an iterative framework is introduced that can achieve an acceptable level of error in their satisfaction for an arbitrary randomly generated set of equivalent sources the presented technique promises a significant reduction in the execution time and memory requirements compared to the surface equivalent based method of moments mom the solution stability repeatability and numerical noise susceptibility are investigated thoroughly through this work also a novel edge correction scheme has been implemented to extend the capabilities of this procedure to structures with sharp edges the results of the presented technique are compared to series solutions for conducting spheres and a commercially available mom code for arbitrarily shaped objects and combinations of different materials

blind separation of sound sources from the principle of least spatial entropy abstract the aim of the paper is to offer a method for separating incoherent and compact sound sources which may overlap in both the space and frequency domains this is found of interest in acoustical applications involving the identification and ranking of sound sources stemming from different physical origins the principle proceeds in two steps the first one being reminiscent to source reconstruction e g as in near field acoustical holography and the second one to blind source separation specifically the source mixture is first expanded into a linear combination of spatial basis functions whose coefficients are set by backpropagating the pressures measured by an array of microphones to the source domain this leads to a formulation similar but no identical to blind source separation in the second step these coefficients are blindly separated into uncorrelated latent variables assigned to incoherent virtual sources these are shown to be defined up to an arbitrary rotation a unique set of sound sources is finally recovered by searching for that rotation by conjugate gradient descent in the stiefel manifold of unitary matrices which maximizes their spatial compactness as measured either by their spatial variance or their spatial entropy this results in the proposal of two separation criteria coined least spatial variance and least spatial entropy respectively the same concept of spatial entropy which is central to the paper is also exploited in defining a new criterion the entropic l curve dedicated to determining the number of active sound sources the idea consists in considering the number of sources that achieves the best compromise between a low spatial entropy as expected from compact sources and a low statistical entropy as expected from a low residual error the proposed methodology is validated on both laboratory experiments and numerical data and illustrated on an industrial example concerned with the ranking of sound sources on a diesel engine at the same time its robustness to the estimated number of active sources is demonstrated

analytic continuation considerations when using generalized formulations for scattering problems a generalized operator equation has been developed for a simple scattering problem for which a closed form solution exists the operator equation has been solved numerically via the method of moments using spatially impulsive fictitious sources as expansion functions together with a simple point matching testing procedure the study focused on the convergence and accuracy of the solution and examined how they are dependent on the location and distance of the fictitious sources relative to the area containing the singularities of the actual field simulated by these sources as expected it was found that when the actual field simulated by the fictitious impulsive sources has singularities lying between the physical boundary and the closed surface over which the sources are placed the impulsive expansion does not yield a uniformly convergent solution in this case instabilities are encountered as the number of sources increases in the sense that a small improvement of the boundary error requires a considerable change in the currents the moment matrix is then difficult to invert and easily susceptible to large round off errors conversely if the actual field has no singularities lying between the physical boundary and the closed surface over which the sources are placed the impulsive expansion does yield a uniformly convergent solution to any degree of precision

fmm accelerated source model technique for many scatterer problems motivated by the lack of scalability of the source model technique smt to many scatterer problems a new smt based method for 3 d many scatterer problems is developed the large time complexity of the smt is reduced by deriving an implicit compact representation of the scattering problem via a fast multipole formulation which divides the interactions into far and near ones while grouping and approximating the far interactions this results in significant acceleration of classical smt which is demonstrated in a variety of 3 d many body scattering problems the improvement is notable already at the minimal number of unknowns necessary and grows with the size of the problem

basis function considerations for the method of moments using the fictitious current model in the method of moments mom using the fictitious current model fcm an impulsive basis function is principally employed to expand the fictitious currents for various electromagnetic scattering problems in this paper we introduce the subdomain basis functions and consider the dependence of the solution accuracy and computation time of the mom fcm on the basis functions by means of a canonical two dimensional 2 d scattering problem

numerical investigation on convergence of boundary knot method in the analysis of homogeneous helmholtz modified helmholtz and convection diffusion problems abstract this paper concerns a numerical study of convergence properties of the boundary knot method bkm applied to the solution of 2d and 3d homogeneous helmholtz modified helmholtz and convection diffusion problems the bkm is a new boundary type meshfree radial function basis collocation technique the method differentiates from the method of fundamental solutions mfs in that it does not need the controversial artificial boundary outside physical domain due to the use of non singular general solutions instead of the singular fundamental solutions the bkm is also generally applicable to a variety of inhomogeneous problems in conjunction with the dual reciprocity method drm therefore when applied to inhomogeneous problems the error of the drm confounds the bkm accuracy in approximation of homogeneous solution while the latter essentially distinguishes the bkm mfs and boundary element method in order to avoid the interference of the drm this study focuses on the investigation of the convergence property of the bkm for homogeneous problems the given numerical experiments reveal rapid convergence high accuracy and efficiency mathematical simplicity of the bkm