continuation of acoustic near fields this paper deals with the analytic continuation of a coherent pressure field specified on a finite sheet located close to and conformal to the surface of a vibrator this analytic continuation is an extension or extrapolation of the given measured field into a region outside and tangential to the original finite sheet and is based on the green s function the transfer function relating acoustic quantities on the two conformal surfaces the continuation of the measured pressure field is an inverse problem that requires the use or regularization theory especially when noise is present in the data an iteration algorithm is presented that is successful in continuing the pressure field into the tangential sheet the results are accurate close to the original boundary and taper decay toward zero with distance away from it the algorithm is tested on numerical and experimental data from a point driven rectangular plate results show the successful extrapolation continuation of this data into an area nearly double that of the original pressure field this algorithm is not limited to planar surfaces and can be applied to arbitrarily shaped surfaces

a numerical study on the solution of the cauchy problem in elasticity this work deals with the cauchy problem in two dimensional linear elasticity the equations of the problem are discretized through a standard fem approach and the resulting ill conditioned discrete problem is solved within the frame of the tikhonov approach the choice of the required regularization parameter is accomplished through the generalized cross validation criterion on this basis a numerical experimentation has been performed and the calculated solutions have been used to highlight the sensitivity to the amount of known data the noise always present in the data the regularity of boundary conditions and the choice of the regularization parameter the aim of the numerical study is to implicitly device some guidelines to be used in the solution of this kind of problems

new candidates for arbitrage free stock price models via generalized conditional symmetry method abstract in this paper a new perspective upon generating arbitrage free stock price models is proposed the generalized conditional symmetry gcs method is applied to the governing second order 1 1 partial differential equation which does contain a rational parameter p drawn from the interval 1 2 1 we investigate the conditions that yield the concerned equation admitting a special class of second order gcss the determining system is solved in several special cases and from invariance surface condition associated to each of the gcs operator for all values of p some invariant solutions are pointed out new candidate models for arbitrage free stock price are derived

mutations of puzzles and equivariant cohomology of two step flag varieties we introduce a mutation algorithm for puzzles that is a three direction analogue of the classical jeu de taquin algorithm for semistandard tableaux we apply this algorithm to prove our conjectured puzzle formula for the equivariant schubert structure constants of two step flag varieties this formula gives an expression for the structure constants that is positive in the sense of graham thanks to the equivariant version of the quantum equals classical result our formula specializes to a littlewood richardson rule for the equivariant quantum cohomology of grassmannians

3 d sound source reconstruction and field reprediction using the helmholtz integral equation abstract a method for reconstruction of the boundary conditions of an arbitrarily shaped three dimensional 3 d sound source from a finite number of field pressures is presented and the associated field is repredicted by using the helmholtz integral equation to minimize the reconstruction error due to noise in the field pressures a filtering method is proposed and it is verified that numerically singular values which correspond to the higher order modes of the evanescent like components of the source velocities are eliminated for arbitrarily shaped sources with complex boundary conditions the reconstruction is performed accurately when the size of each element is less than a quarter wavelength and the field measurement positions are equally or randomly distributed on the surface of a sphere in the vicinity of the source our technique can be used in computer aided loudspeaker design that reconstructs loudspeaker diaphragm velocities from the finite number of field pressures and repredicts the variation of the sound field procuced by a redesigned loudspeaker cabinet on which the reconstructed diaphragm is mounted

direct solution of ill posed boundary value problems by radial basis function collocation method numerical solution of ill posed boundary value problems normally requires iterative procedures in a typical solution the ill posed problem is first converted to a well posed one by assuming the missing boundary values the new problem is solved by a conventional numerical technique and the solution is checked against the unused data the problem is solved iteratively using optimization schemes until convergence is achieved the present paper offers a different procedure using the radial basis function collocation method we demonstrate that the solution of certain ill posed problems can be accomplished without iteration this method not only is efficient and accurate but also circumvents the stability problem that can exist in the iterative method copyright 2005 john wiley sons ltd

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 computational acoustic field reconstruction process based on an indirect boundary element formulation the objective of the work presented in this paper is to develop a computational capability based on the indirect boundary element method ibem for evaluating appropriate velocity boundary conditions on an assembly of piston type sources in order to recreate a prescribed acoustic field information for the acoustic pressure of the original acoustic field at certain field points constitutes the input to the developed process the velocities on the piston type sources are computed from transfer functions evaluated between the field points where the acoustic pressure of the original field is prescribed and the velocity boundary condition on each element of the generic source the ibem is employed for computing the transfer functions in order to accommodate the presence of openings in the model and radiation from both sides of the piston type sources simulating the generic source as a thin surface that radiates from both sides eliminates the presence of irregular frequencies in the analysis a singular value decomposition svd solver is integrated with the ibem computations in order to evaluate the velocity boundary conditions from the transfer functions the number of field points where the acoustic pressure is defined can be considerably smaller than the number of elements where the velocity is computed the solution that demonstrates the smallest magnitude is selected from all possible solutions an algorithm is also developed for identifying the optimum field points where the acoustic pressure of the original field must be prescribed the optimum field points are selected from a set of prescribed candidate points the number of optimum points is considered smaller than the number of elements where the velocity is computed the properties of the transformation matrix and the quality of the reconstruction depend on the location of the field points thus the selection of the optimum points is based on achieving the highest possible orthogonality among the vectors that comprise the range of the transformation matrix several validation and application cases are presented

primal dual weak galerkin finite element methods for elliptic cauchy problems abstract the authors propose and analyze a well posed numerical scheme for a type of ill posed elliptic cauchy problem by using a constrained minimization approach combined with the weak galerkin finite element method the resulting euler lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable and is thus an example of the primal dual weak galerkin finite element method this new primal dual weak galerkin algorithm is consistent in the sense that the system is symmetric well posed and is satisfied by the exact solution a certain stability and error estimates were derived in discrete sobolev norms including one in a weak l 2 topology some numerical results are reported to illustrate and validate the theory developed in the paper

the detection of the source of acoustical noise in two dimensions we consider the problem of detecting the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin mathematically this field satisfies the helmholtz equation in this paper we consider the model two dimensional case we show that any regular solution of this equation admits a unique representation by a single layer potential so that the problem is reduced to the solution of a linear integral equation of the first kind we prove uniqueness of reconstruction and obtain a sharp stability estimate finally for two geometries and sources of noise simulating the cabin of the aircraft and two engines we give results of the numerical solution of this integral equation comparing regularization by the truncated singular value decomposition and the conjugate gradient method

combined helmholtz equation least squares method for reconstructing acoustic radiation from arbitrarily shaped objects a combined helmholtz equation least squares chels method is developed for reconstructing acoustic radiation from an arbitrary object this method combines the advantages of both the hels method and the helmholtz integral theory based near field acoustic holography nah as such it allows for reconstruction of the acoustic field radiated from an arbitrary object with relatively few measurements thus significantly enhancing the reconstruction efficiency the first step in the chels method is to establish the hels formulations based on a finite number of acoustic pressure measurements taken on or beyond a hypothetical spherical surface that encloses the object under consideration next enough field acoustic pressures are generated using the hels formulations and taken as the input to the helmholtz integral formulations implemented through the boundary element method bem the acoustic pressure and normal component of the velocity at the discretized nodes on the surface are then determined by solving two matrix equations using singular value decomposition svd and regularization techniques also presented are in depth analyses of the advantages and limitations of the chels method examples of reconstructing acoustic radiation from separable and nonseparable surfaces are demonstrated

conditional lie backlund symmetries and differential constraints for inhomogeneous nonlinear diffusion equations due to linear determining equations abstract the method of linear determining equations is developed to study conditional lie backlund symmetries for evolution equations which is more general than the classical determining equations for lie s generators as an application of this approach the complete classification is presented for the inhomogeneous nonlinear diffusion equations which admit the second order and third order conditional lie backlund symmetries several examples are given to illustrate the corresponding symmetry reductions due to the compatibility of the invariant conditions and the governing equations

on a quasi reversibility regularization method for a cauchy problem of the helmholtz equation in this paper we consider the cauchy problem for the helmholtz equation in a rectangle where the cauchy data is given for y 0 and boundary data are for x 0 and x p the solution is sought in the interval 0

measurement of structural intensity using boundary element method based nearfield acoustical holography the regularization method for measurement of structural intensity si using boundary element method bem based nearfield acoustical holography nah is proposed spatial derivatives of normal displacement are necessary to obtain the structural intensity the derivative operations amplify high wavenumber component of measurement noise and contaminate the measurement result of si to overcome this difficulty regularization method for measurement of si using fast fourier transform based nah has been introduced in this paper this regularization method is modified for the bem based nah the bem based nah avoids the aperture replication problem therefore measurement aperture for bem based nah can be set smaller than that for fft based nah the effectiveness of the proposed method is demonstrated by experiments

boundary elements and the analog equation method for the solution of elastic problems in 3 d non homogeneous bodies abstract the application of the boundary element method bem to elastostatic problems involving 3d non homogeneous materials like functionally graded materials fgms is presented in this paper the analog equation method aem is used to transform the original problem into a new problem with unknown fictitious source but known fundamental solution by means of this transformation a system of uncoupled domain boundary integral equations d bies is first obtained combining standard boundary element discretization and radial basis functions rbfs approximation for the fictitious source the application of the original differential operator to the displacement bie provides the extra equations to compute the unknown fictitious source the boundary character of the method is maintained since the integrals involved in the equations are limited only to the boundary since the rbfs are selected in such a way that the corresponding analog equation could be solved analytically numerical examples for three dimensional problems in continuously non homogeneous isotropic and linear elastic fgms are presented and discussed

the use of partially measured source data in near field acoustical holography based on the bem in applying the conformal near field acoustical holography nah to actual source identification problems it is often possible to determine the velocity at certain points of the source surface in advance this partially known velocity data would reduce the problem size and permit better reconstruction accuracy in this paper the effectiveness of using partially measured source data in the conformal nah is investigated which uses the boundary element method a vibro acoustic transfer matrix and measured field pressure data which is involved with the boundary integral equation are reorganized in order to deal with the partially measured surface velocities for a baffled vibrating panel simulations were performed by varying the number of velocity known nodes in addition the effect of measurement error is investigated for two extreme positioning methods of velocity known nodes without regularization the reconstructed error can be reduced considerably by employing some of the source data and this err

a conjectural peterson isomorphism in k theory abstract we state a precise conjectural isomorphism between localizations of the equivariant quantum k theory ring of a flag variety and the equivariant k homology ring of the affine grassmannian in particular relating their schubert bases and structure constants this generalizes peterson s isomorphism in co homology we prove a formula for the pontryagin structure constants in the k homology ring and we use it to check our conjecture in few situations

an operator marching method for inverse problems in range dependent waveguides abstract for large scale inverse problems in acoustics and electromagnetics numerical schemes based on direct methods e g fem and meshless methods often result in huge linear systems and thus are not feasible in terms of computing speed and memory storage this work proposes the inverse fundamental operator marching method based on the dirichlet to neumann map for solving large scale inverse boundary value problems in range dependent waveguides truncated singular value decomposition is employed to solve ill conditioned linear systems arising in marching process and the number of propagating modes in the waveguide assumes the role of a regularization parameter numerical results show that the method is computationally efficient highly accurate stable with respect to data noise for retrieving the propagating part of the starting field it particularly suits to long range wave propagation in slowly varying stratified waveguide

identification of the number and locations of acoustic sources the identification of the number and locations of acoustic sources is a fundamental problem of engineering acoustics the method of multiple signal classification music has been widely used to solve this problem the main drawback of the music method is that the unavoidable measurement noise should be identical for all the channels of measurement system this requirement generally cannot be met in practical application in this work a method was proposed to identify the number and locations of multiple incoherent sources the proposed method was based on the traditional music method however without the basic assumption about the identity of the measurement noise besides a procedure was proposed to improve the calculation efficiency of identification the numerical and experimental results demonstrated that the identification accuracy of the proposed method was far better than that of the traditional music method the results also demonstrated that the proposed calculation procedure was very efficient only the data at a few frequencies should be used for identification

exact solutions to nonlinear equations with quadratic nonlinearity based on the concepts of the generalized conditional symmetry and of the invariant subspace a constructive way to derive nonlinear equations with quadratic nonlinearity that admit exact solutions as well as many new examples is presented this type of solution found by galaktionov describes the blowup phenomena examples of exact solutions that do not correspond to the invariant subspace are also presented and explained by the nonlinear generalized conditional symmetries

dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition abstract the dual formulation of the multiple reciprocity method mrm is employed to solve the acoustic mode of a cavity with a thin partition in order to avoid the spurious eigenvalues and non uniqueness of the solution in the case of zero thickness for the partition the hypersingular equation for mrm is considered to determine the invariant quantity of influence coefficients more efficiently an objectivity concept i e a frame of indifference is used by means of co ordinate transformation a generalized eigenvalue problem is derived and transformed into a standard eigenvalue problem through the state space formulation the spurious roots in mrm are examined and filtered out by using the hypersingular formulation three examples including finite thickness of the partition zero thickness and no partition are shown to check the validity of the proposed method also the analytical solution if available the fem results obtained by petyt and by abaqus and experimental measurements are compared with those of the proposed method and it is found that the agreement between them is very good

fast fourier transform and singular value decomposition formulations for patch nearfield acoustical holography nearfield acoustical holography nah requires the measurement of the pressure field over a complete surface in order to recover the normal velocity on a nearby concentric surface the latter generally coincident with a vibrator patch nah provides a major simplification by eliminating the need for complete surface pressure scans only a small area needs to be scanned to determine the normal velocity on the corresponding small area concentric patch on the vibrator the theory of patch nah is based on 1 an analytic continuation of the patch pressure which provides a spatially tapered aperture extension of the field and 2 a decomposition of the transfer function pressure to velocity and or pressure to pressure between the two surfaces using the singular value decomposition svd for general shapes and the fast fourier transform fft for planar surfaces inversion of the transfer function is stabilized using tikhonov regularization and the morozov discrepancy principle experimental results show that root mean square errors of the normal velocity reconstruction for a point driven vibrator over 200 2700 hz average less than 20 for two small concentric patch surfaces 0 4 cm apart reconstruction of the active normal acoustic intensity was also successful with less than 30 error over the frequency band

use of nonsingular boundary integral formulation for reducing errors due to near field measurements in the boundary element method based near field acoustic holography in the conformal near field acoustic holography nah using the boundary element method bem the transfer matrix relating the vibro acoustic properties of source and field depends solely on the geometrical condition of the problem this kind of nah is known to be very powerful in dealing with the sources having irregular shaped boundaries when the vibro acoustic source field is reconstructed by using this conformal nah one tends to position the sensors as close as possible to the source surface in order to get rich information on the nonpropagating wave components the conventional acoustic bem based on the kirchhoff helmholtz integral equation has the singularity problem in the close near field of the source surface this problem stems from the singular kernel of the green function of the boundary integral equation bie and the singularity can influence the reconstruction accuracy greatly in this paper the nonsingular bie is introduced to the nah calculation and the holographic bie is reformulated the effectiveness of nonsingular bem has been investigated for the reduction of reconstruction error through interior and exterior examples it is shown that the resolution of predicted field pressure could be improved in the close near field by employing the nonsingular bie because the bem based nah inevitably requires the field pressure measured in the close proximity to the source surface the present approach is recommended for improving the resolution of the reconstructed source field

bayesian approach of nearfield acoustic reconstruction with particle filters this paper demonstrates that inverse source reconstruction can be performed using a methodology of particle filters that relies primarily on the bayesian approach of parameter estimation in particular the proposed approach is applied in the context of nearfield acoustic holography based on the equivalent source method esm a state space model is formulated in light of the esm the parameters to estimate are amplitudes and locations of the equivalent sources the parameters constitute the state vector which follows a first order markov process with the transition matrix being the identity for every frequency domain data frame filtered estimates of the state vector obtained are assigned weights adaptively the implementation of recursive bayesian filters involves a sequential monte carlo sampling procedure that treats the estimates as point masses with a discrete probability mass function pmf which evolves with iteration the weight update equation governs the evolution of this pmf and depends primarily on the likelihood function and the prior distribution it is evident from the simulation results that the inclusion of the appropriate prior distribution is crucial in the parameter estimation

a regularization method for a cauchy problem of the helmholtz equation we investigate a cauchy problem for the helmholtz equation a modified boundary method is used for solving this ill posed problem some holder type error estimates are obtained the numerical experiment shows that the modified boundary method works well

identification of source velocities on 3d structures in non anechoic environments theoretical background and experimental validation of the inverse patch transfer functions method in noise control identification of the source velocity field remains a major problem open to investigation consequently methods such as nearfield acoustical holography nah principal source projection the inverse frequency response function and hybrid nah have been developed however these methods require free field conditions that are often difficult to achieve in practice this article presents an alternative method known as inverse patch transfer functions designed to identify source velocities and developed in the framework of the european silence project this method is based on the definition of a virtual cavity the double measurement of the pressure and particle velocity fields on the aperture surfaces of this volume divided into elementary areas called patches and the inversion of impedances matrices numerically computed from a modal basis obtained by fem theoretically the method is applicable to sources with complex 3d geometries and measurements can be carried out in a non anechoic environment even in the presence of other stationary sources outside the virtual cavity in the present paper the theoretical background of the iptf method is described and the results numerical and experimental for a source with simple geometry two baffled pistons driven in antiphase are presented and discussed

reconstruction of vibroacoustic fields in half space by using hybrid near field acoustical holography in this paper we examine the accuracy and efficiency of reconstructing the vibroacoustic quantities generated by a vibrating structure in half space by using hybrid near field acoustic holography nah and modified helmholtz equation least squares hels formulations in hybrid nah we combine modified hels with an inverse boundary element method ibem to reconstruct a vibroacoustic field the main advantage of this approach is that the majority of the input data can be regenerated but not measured thus the efficiency is greatly enhanced in modified hels we expand the field acoustic pressure in terms of outgoing and incoming spherical waves and specify the corresponding expansion coefficients by solving a system of equations obtained by matching the assumed form solution to the measured acoustic pressure here the system of equations is ill conditioned and tikhonov regularization is implemented through singular value decomposition svd and the generalized cross validation gcv method numerical examples of a dilating and oscillating spheres and finite cylinder are demonstrated test results show that hybrid nah can yield a more accurate reconstruction than does a modified hels but a modified hels is more efficient than is hybrid nah work supported by nsf

data extrapolation method for boundary element method based near field acoustical holography a new data extrapolation method for boundary element method bem based near field acoustical holography nah is proposed to reduce an error of the reconstructed result obtained from the pressure measured on an aperture small compared with the structure the finiteness of the measurement aperture is a serious impediment to actual large scale implementation of nah because nah requires the measurement of the pressure field over a complete surface of structure to eliminate the requirement the wave number space data extrapolation method for fast fourier transform fft based nah has been proposed in this paper the extension of this data extrapolation method to bem based nah is presented the effectiveness of this method is demonstrated by experiments the experiment results confirm that the reconstruction error is sufficiently suppressed by the proposed method

on the multiple microphone method for measuring in duct acoustic properties in the presence of mean flow nowadays the two microphone method is accepted as the standard as specified in astm e1050 90 for measuring in duct acoustic properties however research results on using the least square method with multiple measurement points and broadband excitation have been reported for enhancing the frequency response of the two microphone method in this paper the effects of varying the relative measurement positions on errors in the estimation of the acoustic quantities is studied for the multiple microphone method both the theoretical and experimental results show that among possible sensor positioning configurations the equidistant positioning of sensors yields the smallest error within the effective measurement frequency range in addition it is noted that the measurement accuracy can be increased and the effective frequency range can be widened by increasing the number of equidistant sensors measurement examples are shown and the results support the findings

effect of sensor proximity over the non conformal hologram plane in the near field acoustical holography based on the inverse boundary element method near field acoustical holography nah is a useful tool for the identification and visualization of vibro acoustic sources in particular nah can be applied to many practical sources having irregular shape if the inverse boundary element method bem is employed once the relation between the source and the radiated field is defined in the transfer matrix modeled by the bem the reconstruction of acoustic parameters on the source surface can be conducted by the multiplication of the inverse transfer matrix and the field data measured over the hologram surface the usual practical way to measure the field data radiated from an irregular shaped source is to adopt a regular hologram surface which can be a flat cylindrical or spherical shape for the measurement ease then the hologram surface is not conformal to the source surface and the resulting transfer matrix becomes further ill posed than the conformal case to investigate the effect of sensor proximity and distance variability on the reconstruction error simulation and measurement were conducted for an interior problem comprised of a parallelepiped rigid box with a vibrating end plate flat tilted and randomized hologram surfaces were adopted in the test it was shown that the reconstruction error is greatly affected by the conditioning of transfer matrix which is related to the positioning of sensor to improve the reconstruction accuracy an investigation was conducted to find proper field points among a large number of overdetermined field points the number of field data was reduced gradually under various reduction schemes using condition number effective independence value and sensor distance it was demonstrated that the quality of reconstruction result given by the non conformal measurement can be improved by removing some field points that contribute to the ill conditioning of the inverse problem a small improvement of the reconstruction accuracy was observed by reducing the field points in the overdetermined situation however further reduction of the field points becoming an underdetermined situation yielded a drastic improvement of the reconstruction accuracy

on identification of a wave number for helmholtz equation abstract this note is concerned with inverse evaluation of an unknown wave number in a helmholtz equation the algorithm assumes an initial guess for the unknown function and obtains corrections to the guessed value the updating stage is accomplished by generating a set of functions that satisfy some of the required boundary conditions we refer to this space as proper solution space the correction to the assumed value can then be obtain by imposing the remaining boundary conditions we consider the evaluation of real and complex wave numbers a number of numerical examples are used to study the applicability and effectiveness of the procedure

reconstruction of vibro acoustic fields using hybrid nearfield acoustic holography hybrid nearfield acoustical holography is utilized to reconstruct the vibro acoustic fields generated by an arbitrary structure vibrating in an unbounded fluid medium this hybrid nah is based on the modified helmholtz equation least squares method that expresses the field acoustic pressure in terms of expansions of outgoing and incoming waves the expansion coefficients are determined by solving an over determined linear system of equations obtained by matching the assumed form solution to a finite number of acoustic pressures measured in the field via least squares the measurements are taken over a conformal surface around the source at close range so that the evanescent waves can be captured once the expansion coefficients are specified enough field acoustic pressures are regenerated on a conformal surface around the source and are taken as input to the helmholtz integral formulation implemented numerically by the boundary element method the acoustic pressure and normal velocity on the source surface are reconstructed using singular value decomposition modified tikhonov regularization and generalized cross validation method this hybrid nah enables one to obtain a satisfactory reconstruction cost effectively because the majority of the input data are calculated but not measured numerical examples of reconstructing the vibro acoustic quantities on the surfaces of an engine block and a highly elongated cylinder are demonstrated

application of bem boundary element method based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries a method employing holograms that conform with arbitrarily shaped sources has been developed for enhancing conventional near field acoustic holography which has been limited to sources with simple geometries e g planar or cylindrical surfaces four holography transformation algorithms have been developed based on acoustic holography theory and the boundary element method bem singular value decomposition svd has been incorporated into the algorithms in order to alleviate the ill posed nature frequently encountered in backward reconstruction of a field a pulsating sphere a cylinder with spherical endcaps and a vibrating piston set in a rigid sphere have been adopted in a numerical simulation for verifying the algorithms satisfactory agreement has been achieved between the holographically transformed results and the analytical solutions

a herglotz wavefunction method for solving the inverse cauchy problem connected with the helmholtz equation this paper is concerned with the cauchy problem connected with the helmholtz equation on the basis of the denseness of herglotz wavefunctions we propose a numerical method for obtaining an approximate solution to the problem we analyze the convergence and stability with a suitable choice of regularization method numerical experiments are also presented to show the effectiveness of our method

regularization of the cauchy problem for the helmholtz equation by using meyer wavelet in this paper we investigate a cauchy problem associated with helmholtz type equation in an infinite strip this is a classical severely ill posed problem i e the solution if it exists does not depend continuously on the data or cauchy data a small perturbation in the data can cause a dramatically large error in the solution for 0

regularization methods for near field acoustical holography the reconstruction of the pressure and normal surface velocity provided by near field acoustical holography nah from pressure measurements made near a vibrating structure is a linear ill posed inverse problem due to the existence of strongly decaying evanescentlike waves regularization provides a technique of overcoming the ill posedness and generates a solution to the linear problem in an automated way we present four robust methods for regularization the standard tikhonov procedure along with a novel improved version landweber iteration and the conjugate gradient approach each of these approaches can be applied to all forms of interior or exterior nah problems planar cylindrical spherical and conformal we also study two parameter selection procedures the morozov discrepancy principle and the generalized cross validation which are crucial to any regularization theory in particular we concentrate here on planar and cylindrical holography these forms of nah which rely on the discrete fourier transform are important due to their popularity and to their tremendous computational speed in order to use regularization theory for the separable geometry problems we reformulate the equations of planar cylindrical and spherical nah into an eigenvalue problem the resulting eigenvalues and eigenvectors couple easily to regularization theory which can be incorporated into the nah software with little sacrifice in computational speed the resulting complete automation of the nah algorithm for both separable and nonseparable geometries overcomes the last significant hurdle for nah

reconstructing the vibro acoustic quantities on a highly non spherical surface using the helmholtz equation least squares method this paper presents helpful guidelines and strategies for reconstructing the vibro acoustic quantities on a highly non spherical surface by using the helmholtz equation least squares hels this study highlights that a computationally simple code based on the spherical wave functions can produce an accurate reconstruction of the acoustic pressure and normal surface velocity on planar surfaces the key is to select the optimal origin of the coordinate system behind the planar surface choose a target structural wavelength to be reconstructed set an appropriate stand off distance and microphone spacing use a hybrid regularization scheme to determine the optimal number of the expansion functions etc the reconstructed vibro acoustic quantities are validated rigorously via experiments by comparing the reconstructed normal surface velocity spectra and distributions with the benchmark data obtained by scanning a laser vibrometer over the plate surface results confirm that following the proposed guidelines and strategies can ensure the accuracy in reconstructing the normal surface velocity up to the target structural wavelength and produce much more satisfactory results than a straight application of the original hels formulations experiment validations on a baffled square plate were conducted inside a fully anechoic chamber

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

regularization method for the application of k space data extrapolation to near field acoustical holography a regularization method for applying the k space data extrapolation method to near field acoustical holography nah is proposed the influence of the finiteness of measurement aperture is a serious impediment to the nah reconstruction of structural vibration when the size of the measurement aperture is smaller than that of the structure the k space data extrapolation method has been proposed to reduce the influence of this finiteness an appropriate regularization method is necessary to apply the k space data extrapolation method if the measured pressure is contaminated by noise some regularization methods for nah have been proposed however it is difficult to determine an appropriate k space filter if the finiteness of the measurement aperture becomes an issue an estimation method of appropriate regularization filter for applying the k space data extrapolation method is proposed to solve this difficulty the effectiveness of this method is demonstrated by simulations the simulation results confirm

sound source reconstruction using inverse boundary element calculations whereas standard boundary element calculations focus on the forward problem of computing the radiated acoustic field from a vibrating structure the aim in this work is to reverse the process i e to determine vibration from acoustic field data this inverse problem is brought on a form suited for solution by means of an inverse boundary element method since the numerical treatment of the inverse source reconstruction results in a discrete ill posed problem regularization is imposed to avoid unstable solutions dominated by errors in the present work the emphasis is on tikhonov regularization and parameter choice methods not requiring an error norm estimate for choosing the right amount of regularization several parameter choice strategies have been presented lately but it still remains to be seen how well these can handle industrial applications with real measurement data in the present work it is demonstrated that the l curve criterion is robust with respect to the errors in a real measurement situation in particular it is shown that the l curve criterion is superior to the more conventional generalized cross validation gcv approach for the present tire noise studies

sound source identification in a noisy environment based on inverse patch transfer functions with evanescent green s functions abstract a modified inverse patch transfer function iptf method is used to reconstruct the normal velocities of the target source in a noisy environment the iptf method simplifies the helmholtz integral equation to one term by constructing a green s function satisfying neumann boundary conditions for an enclosure which is generally constructed by slowly convergent modal expansions the main objective of the present work is to provide an evanescent green s function to improve the convergence of calculations a brief description of the iptf method and two sets of green s functions for a rectangular cavity are presented firstly in simulations both the green s functions are used to calculate the condition numbers of impedance matrices describing the relation between source and measurement patches and the time cost of calculation based on the two sets of green s functions at 450 hz is compared double pressure measurements are then employed as the input data instead of pressure and velocity measurements the normal velocities of two baffled loudspeakers are reconstructed by the combination of a measurement method and a green s function in the presence of a disturbing source in the frequency range of 50 1000 hz in addition the double pressure measurements are examined by an experiment the precise identification of the sources indicates that the double pressure measurements are capable of localizing sources in a noisy environment it is also found that the reconstruction with the evanescent green s functions is slightly better than that with the modal expansions

nearfield acoustic holography nah ii holographic reconstruction algorithms and computer implementation the basic theory treating steady state acoustic radiation problems in the nearfield has been presented in several articles on nearfield acoustic holography in this article the approximations and assumptions necessary to reduce the infinite and continuous convolution integrals encountered in these problems to a finite and discrete form suitable for high speed numerical processing are illuminated theoretically and tested numerically to evaluate the convolution integrals two assumptions are made first the boundary field may be replaced with a patchwise constant field for reasonably small patches and second the field is negligible outside of a finite region with these two assumptions the problem reduces to one of representing the green s functions six methods of sampling or representing the green s functions are developed and these are compared theoretically and numerically

the fourier bessel method for solving the cauchy problem connected with the helmholtz equation this paper is concerned with the cauchy problem connected with the helmholtz equation we investigate the denseness of fourier bessel functions and propose a numerical method for approximating the solution to the problem the convergence and stability are analyzed with truncated singular value decomposition method numerical experiments are also presented to show the effectiveness of our method

new conditional symmetries and exact solutions of reaction diffusion convection equations with exponential nonlinearities abstract a complete description of q conditional symmetries for a class of reaction diffusion convection equations with exponential diffusivities is derived it is shown that all the known results for reaction diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa the symmetries obtained for constructing exact solutions of the relevant equations are successfully applied the exact solutions are compared with those found earlier by means of different approaches an application of the solutions for solving a boundary value problem arising in population dynamics is presented

on the reconstruction of the vibro acoustic field over the surface enclosing an interior space using the boundary element method the vibrational velocity sound pressure and acoustic power on the vibrating boundary comprising an enclosed space are reconstructed by the boundary element method based on the measured field pressures the singular value decomposition is used to obtain the inverse solution in the least square sense and to express the acoustic modal expansion between the measurement and source fields in general such an inverse operation has been considered an ill posed problem having a divergence phenomenon involved with extremely small measurement errors the ill conditioned nature of the acoustic inverse problem is caused by the singularity of the transfer matrix which produces nonradiating wave components in order to minimize the singularity and to also reduce the number of measurement points optimal measurement positions are determined by the effective independence method regularization methods are used to stabilize the reconstructed field by suppressing nonradiating components resulting in the singular transfer

reconstruction of the electric field of the helmholtz equation in three dimensions in this paper we rigorously investigate the truncation method for the cauchy problem of helmholtz equations which is widely used to model propagation phenomena in physical applications the truncation method is a well known approach to the regularization of several types of ill posed problems including the model postulated by reginska and reginski 2006 under certain specific assumptions we examine the ill posedness of the non homogeneous problem by exploring the representation of solutions based on fourier mode then the so called regularized solution is established with respect to a frequency bounded by an appropriate regularization parameter furthermore we provide a short analysis of the nonlinear forcing term the main results show the stability as well as the strong convergence confirmed by the error estimates in l 2 norm of such regularized solutions besides the regularization parameters are formulated properly finally some illustrative examples are provided to corroborate our qualitative analysis

a meshless method for the numerical solution of the cauchy problem associated with three dimensional helmholtz type equations in this paper the application of the method of fundamental solutions to the cauchy problem associated with three dimensional helmholtz type equations is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for under equally and over determined cauchy problems in a piecewise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

equivariant quantum schubert calculus abstract we study the t equivariant quantum cohomology of the grassmannian we prove the vanishing of a certain class of equivariant quantum littlewood richardson coefficients which implies an equivariant quantum pieri rule as in the equivariant case this implies an algorithm to compute the equivariant quantum littlewood richardson coefficients

acoustical inverse problems regularization direct definition of filter factors using signal to noise ratio abstract acoustic imaging aims at localization and characterization of sound sources using microphone arrays in this paper a new regularization method for acoustic imaging by inverse approach is proposed the method first relies on the singular value decomposition of the plant matrix and on the projection of the measured data on the corresponding singular vectors in place of regularization using classical methods such as truncated singular value decomposition and tikhonov regularization the proposed method involves the direct definition of the filter factors on the basis of a thresholding operation defined from the estimated measurement noise the thresholding operation is achieved using modified filter functions the originality of the approach is to propose the definition of a filter factor which provides more damping to the singular components dominated by noise than that given by the tikhonov filter this has the advantage of potentially simplifying the selection of the best regularization amount in inverse problems theoretical results show that this method is comparatively more accurate than tikhonov regularization and truncated singular value decomposition

a potential function method for the cauchy problem of elliptic operators abstract in this paper we deal with the cauchy problem of elliptic operators through the use of a single layer potential function we devise a numerical method for approximating the solution of the cauchy problem of elliptic operators which are well known to be highly ill posed in nature the method is based on the denseness of single layer potential functions convergence and stability estimates are then given with some examples for numerical verification on the efficiency of the proposed method it has been observed that the use of more cauchy data will greatly improve the accuracy of the approximate solutions

boundary infinite elements for the helmholtz equation in exterior domains a novel approach to the development of infinite element formulations for exterior problems of time harmonic acoustics is presented this approach is based on a functional which provides a general framework for domain based computation of exterior problems special cases include non reflecting boundary conditions such as the dtn method a prominent feature of this formulation is the lack of integration over the unbounded domain simplifying the task of discretization the original formulation is generalized to account for derivative discontinuities across infinite element boundaries typical of standard infinite element approximations continuity between finite elements and infinite elements is enforced weakly precluding compatibility requirements various infinite element approximations for two dimensional configurations with circular interfaces are presented implementation requirements are relatively simple numerical results demonstrate the good performance of this scheme 1998 john wiley sons ltd

helmholtz equation least squares method for reconstructing the acoustic pressure field a method using spherical wave expansion theory to reconstruct acoustic pressure field from a vibrating object is developed the radiated acoustic pressures are obtained by means of an expansion of independent functions generated by the gram schmidt orthonormalization with respect to the particular solutions to the helmholtz equation on the vibrating surface under consideration the coefficients associated with these independent functions are determined by requiring the assumed form of solution to satisfy the pressure boundary condition at the measurement points the errors incurred in this process are minimized by the least squares method once these coefficients are specified the acoustic pressure at any point including the source surface is completely determined in this paper this method is used to reconstruct the surface acoustic pressures based on the measured acoustic pressure signals in the field it is shown that this method can be applied to both separable and nonseparable geometries and the

comparison of regularization methods for solving the cauchy problem associated with the helmholtz equation in this paper several boundary element regularization methods such as iterative conjugate gradient tikhonov regularization and singular value decomposition methods for solving the cauchy problem associated to the helmholtz equation are developed and compared regularizing stopping criteria are developed and the convergence as well as the stability of the numerical methods proposed are analysed the cauchy problem for the helmholtz equation can be regularized by various methods such as the general regularization methods presented in this paper but more accurate results are obtained by classical methods such as the singular value decomposition and the tikhonov regularization methods copyright 2004 john wiley sons ltd

design of an optimal wave vector filter for enhancing the resolution of reconstructed source field by near field acoustical holography nah in near field acoustical holography using the boundary element method the reconstructed field often diverges due to the presence of small measurement errors in order to handle this instability in the inverse problem the reconstruction process should include some form of regularization for enhancing the resolution of source images the usual method of regularization has been the truncation of wave vectors associated with small singular values although the determination of an optimal truncation order is difficult in this article an iterative inverse solution technique is suggested in which the mean square error prediction is used a statistical estimation of the minimum mean square error between measured pressures and the model solution is required for yielding the optimal number of iterations the continuous curve of an optimal wave vector filter is designed for suppressing the high order modes that can produce large reconstruction errors experimental results from a baffled radiator reveal that the reconstruction errors can be reduced by this form of regularization by at least 48 compared to those without any regularization in comparison to results using the optimal truncation method of regularization the new scheme is shown to give further reductions of truncation error of between 7 and 39 for the example in this article

implicit methods of solution to integral formulations in boundary element method based nearfield acoustic holography two integral equation methods are considered which can be inverted to provide the surface velocity and or pressure given a measurement of the pressure on an imaginary surface in the nearfield of a vibrating or scattering body this problem is central to nearfield acoustical holography nah the integral equations are discretized using boundary element methods bems the integral equation methods considered are 1 an indirect formulation method based on the single layer integral equation and 2 a direct formulation method based on a system of equations derived from the helmholtz kirchhoff integral equation the formation of integral equations from the mentioned methods will not involve the explicit inversion of matrices but instead will require this inversion to be done implicitly since these methods back track the sound field from the measurement surface to the surface of the source vibrator they are ill posed in nature and tikhonov regularization are used to stabilize the reconstruction problems a

an alternating iterative algorithm for the cauchy problem associated to the helmholtz equation in this paper the iterative algorithm proposed by kozlov et al comput maths math phys 31 1991 45 for obtaining approximate solutions to the ill posed cauchy problem for the helmholtz equation is analysed the technique is then numerically implemented using the boundary element method bem the numerical results confirm that the iterative bem produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data an efficient stopping regularising criterion is also proposed

dual reciprocity boundary element method solution of the cauchy problem for helmholtz type equations with variable coefficients in this paper the application of the dual reciprocity boundary element method drbem to the cauchy problem for helmholtz type equations with variable coefficients is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for cauchy problems in smooth and piecewise smooth geometries as well as simply and doubly connected domains the accuracy convergence and stability of the proposed numerical method with respect to various approximating functions various drbem discretizations and various levels of noise added in the boundary data respectively are also analysed

hybrid near field acoustic holography hybrid near field acoustical holography nah is developed for reconstructing acoustic radiation from an arbitrary object in a cost effective manner this hybrid nah is derived from a modified helmholtz equation least squares hels formula that expands the acoustic pressure in terms of outgoing and incoming waves the expansion coefficients are determined by solving an overdetermined linear system of equations obtained by matching the assumed form solution to measured acoustic pressures through the least squares measurements are taken over a conformal surface around a source at close range so that the evanescent waves can be captured next the modified hels is utilized to regenerate as much acoustic pressures on the conformal surface as necessary and take them as input to the helmholtz integral formulation implemented numerically by boundary element method bem the acoustic pressures and normal velocities on the source surface are reconstructed by using a modified tikhnov regularization tr with its regularization parameter determined by generalized cross validation gcv method results demonstrate that this hybrid nah combines the advantages of hels and inverse bem this is because a majority of the input data are regenerated but not measured thus the efficiency of reconstruction is greatly enhanced meanwhile the accuracy of reconstruction is ensured by the helmholtz integral theory and modified tr together with gcv method provided that hels converges fast enough on the measurement surface numerical examples of reconstructing acoustic quantities on the surface of a simplified engine block are demonstrated work supported by nsf

equivariant quantum cohomology of the odd symplectic grassmannian the odd symplectic grassmannian mathrm ig mathrm ig k 2n 1 parametrizes k dimensional subspaces of mathbb c 2n 1 which are isotropic with respect to a general necessarily degenerate symplectic form the odd symplectic group acts on mathrm ig with two orbits and mathrm ig is itself a smooth schubert variety in the submaximal isotropic grassmannian mathrm ig k 2n 2 we use the technique of curve neighborhoods to prove a chevalley formula in the equivariant quantum cohomology of mathrm ig i e a formula to multiply a schubert class by the schubert divisor class this generalizes a formula of pech in the case k 2 and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring

new q conditional symmetries and exact solutions of some reaction diffusion convection equations arising in mathematical biology abstract a theorem giving a complete description of q conditional symmetries of a class of nonlinear reaction diffusion convection equations is proved furthermore the q conditional symmetries obtained and the method of additional generating conditions are applied for finding exact solutions of the generalized fisher fitzhugh nagumo and kolmogorov petrovskii piskunov equations the symmetries and solutions constructed are compared with those obtained by other authors in particular it was established that the known travelling wave solutions of these equations are particular cases of more general non lie solutions the relation between q conditional symmetries and generalized conditional symmetries is also shown

local patch acoustic holography methods in enclosed spaces the paper compares reconstruction accuracy and computational cost of three patch acoustic holography algorithms applicable in the enclosed spaces for prediction of sound field near sources algorithms under investigation were dl sonah ibem and classical nah with spatial transformations and hologram aperture enlargement all of the selected algorithms take an advantage of using double layer microphone array for measurement of sound pressure field the reconstruction accuracy of all algorithms is determined by using of simple 3d model of curved radiating surface on the basis of calculation of difference between true acoustic quantity pressure velocity or intensity values very close to the surface and predicted sound field at same positions all these methods use different calculation procedure to obtain predicted sound field near source surface thus comparison of effectiveness of these algorithms including prediction accuracy to computational cost ratio is useful to determination their applicability in

fading regularization mfs algorithm for the cauchy problem associated with the two dimensional helmholtz equation abstract in this paper we combine the fading regularization method with the method of fundamental solutions mfs and investigate its application to the cauchy problem for the two dimensional helmholtz equation we present a numerical reconstruction of the missing data on an inaccessible part of the boundary from the knowledge of overprescribed noisy data taken on the remaining accessible boundary part for both smooth and piecewise smooth two dimensional geometries the accuracy convergence stability and efficiency of the proposed numerical algorithm as well as its capability to deblur the noisy data are validated by three numerical examples

exact interaction of solitary waves for certain nonintegrable equations physically interesting exact solutions are constructed for a large class of nonlinear nonintegrable evolution equations these solutions describe the interaction of traveling waves they exhibit rich phenomenology including breaking of solitary waves generalized conditional symmetries and bilinearity are used to derive these exact results

reconstructing interior acoustic pressure fields via helmholtz equation least squares method this paper extends the helmholtz equation least squares hels method previously developed by wang and wu j acoust soc am 102 2020 2032 1997 to reconstruction of acoustic pressure fields inside the cavity of a vibrating object the acoustic pressures are reconstructed through an expansion of the acoustic modes generated by the gram schmidt orthonormalization with respect to the particular solutions to the helmholtz equation such an expansion is uniformly convergent because the selected acoustic modes consist of a uniformly convergent series of legendre functions the coefficients associated with these acoustic modes are determined by requiring the assumed form solution to satisfy the pressure boundary condition at the measurement points the errors incurred in this process are minimized by the least squares method numerical examples of partially vibrating spheres and cylinders with various half length to radius aspect ratios subject to different frequency excitations are demonstrated the recon

reconstruction of vibroacoustic responses of a highly nonspherical structure using helmholtz equation least squares method the vibroacoustic responses of a highly nonspherical vibrating object are reconstructed using helmholtz equation least squares hels method the objectives of this study are to examine the accuracy of reconstruction and the impacts of various parameters involved in reconstruction using hels the test object is a simply supported and baffled thin plate the reason for selecting this object is that it represents a class of structures that cannot be exactly described by the spherical hankel functions and spherical harmonics which are taken as the basis functions in the hels formulation yet the analytic solutions to vibroacoustic responses of a baffled plate are readily available so the accuracy of reconstruction can be checked accurately the input field acoustic pressures for reconstruction are generated by the rayleigh integral the reconstructed normal surface velocities are validated against the benchmark values and the out of plane vibration patterns at several natural frequencies are compared with the natural modes of a simply supported plate the impacts of various parameters such as number of measurement points measurement distance location of the origin of the coordinate system microphone spacing and ratio of measurement aperture size to the area of source surface of reconstruction on the resultant accuracy of reconstruction are examined