weierstrass formula and zero finding methods classical weierstrass formula 29 has been often the subject of investigation of many authors in this paper we give some further applications of this formula for finding the zeros of polynomials and analytic functions we are concerned with the problems of localization of polynomial zeros and the construction of iterative methods for the simultaneous approximation and inclusion of these zeros conditions for the safe convergence of weierstrass method depending only on initial approximations are given in particular we study polynomials with interval coefficients using an interval version of weierstrass method enclosures in the form of disks for the complex valued set containing all zeros of a polynomial with varying coefficients are obtained we also present weierstrass like algorithm for approximating simultaneously all zeros of a class of analytic functions in a given closed region to demonstrate the proposed algorithms three numerical examples are included

on approximate zeros and rootfinding algorithms for a complex polynomial on donne des criteres pour qu un nombre complexe soit une racine approchee d un polynome f par la methode de newton ou d euler d ordre k on construit des familles d algorithmes surement convergents

on the determination of the safe initial approximation for the durand kerner algorithm abstract in this paper applying the majorant function method we present a new proof of the convergence of the durand kerner method and obtain a greater computable radius estimation of safe initial discs this result is an improvement of all results now available furthermore combining the kuhn algorithm for solving all zeros of polynomials we obtain a discriminant to get safe initial discs finally we compare the complexity between the durand kerner algorithm and several known results and the numerical results show the superiority of our result

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

use of moving distributed mass element for the dynamic analysis of a flat plate undergoing a moving distributed load this paper presents the theory regarding a moving distributed mass element so that the dynamic responses of a rectangular plate subjected to a moving distributed mass with the effects of inertia force coriolis force and centrifugal force considered can be easily determined in which the property matrices of the moving distributed mass element are derived by means of the principle of superposition and the definition of shape functions and the overall property matrices of the entire vibrating system are determined from the combination of the last element property matrices of the moving distributed mass element and those of the plate itself since the property matrices of the moving distributed mass element have something to do with the instantaneous position and the distribution of the moving mass they are time dependent and so are the overall property matrices of the entire vibrating system based on the last concept the equations of motion for a rectangular plate subjected to a moving distributed mass are established and solved to yield the dynamic responses of the entire vibrating system numerical results reveal that the factors such as the contact area between the moving distributed mass and the plate moving load speed inertia force coriolis force and centrifugal force affect the vibration characteristics of the plate to some degree copyright 2006 john wiley sons ltd

dynamics of multi span continuous straight bridges subject to multi degrees of freedom moving vehicle excitation abstract the paper presents an analytical approach to the problem of vehicle bridge dynamic interaction starting from early studies based on a simply supported beam interacting with a lumped mass moving at constant speed in recent years researchers have improved the models of both the bridge and the vehicle on this basis the bridge is modelled here as a multi span continuous isotropic plate its response to external loads is defined by applying the mode superposition principle and takes into account both flexural and torsional mode shapes the latter being usually neglected in the literature the plate is considered proportionally damped and its modes are computed by means of the rayleigh ritz method the scheme adopted for the vehicle consists of a seven degrees of freedom system moving at constant speed over the isotropic rough bridge surface the numerical investigation based on these analytical models refers to a three span bridge and includes the importance of torsional mode shapes of road surface irregularities and of vehicle speed

point estimation of simultaneous methods for solving polynomial equations a survey ii the construction of computationally verifiable initial conditions which provide both the guaranteed and fast convergence of the numerical root finding algorithm is one of the most important problems in solving nonlinear equations smale s point estimation theory from 1981 was a great advance in this topic it treats convergence conditions and the domain of convergence in solving an equation f z 0 using only the information of f at the initial point z 0 the study of a general problem of the construction of initial conditions of practical interest providing guaranteed convergence is very difficult even in the case of algebraic polynomials in the light of smale s point estimation theory an efficient approach based on some results concerning localization of polynomial zeros and convergent sequences is applied in this paper to iterative methods for the simultaneous determination of simple zeros of polynomials we state new improved initial conditions which provide the guaranteed convergence of frequently used simultaneous methods for solving algebraic equations ehrlich aberth s method ehrlich aberth s method with newton s correction borsch supan s method with weierstrass correction and halley like or wang zheng method the introduced concept offers not only a clear insight into the convergence analysis of sequences generated by the considered methods but also explicitly gives their order of convergence the stated initial conditions are of significant practical importance since they are computationally verifiable they depend only on the coefficients of a given polynomial its degree n and initial approximations to polynomial zeros

formulation implementation and validation of multiple connection and free edge constraints in an indirect boundary element formulation the indirect boundary element method ibem can be used in numerical simulations for computing the noise radiated from vibrating structures or the combined structural acoustic response of a system the primary variable on the surface of the model is defined as the difference in the pressure between its two sides when the boundary element model contains multiple elements connected along the same edge or openings then the definition of the primary variables must be adapted in order to reflect the complexity of the geometry in order to accommodate the new definitions the connectivity of the boundary element model must be updated and constraint equations must be generated and introduced in the boundary element system of equations special attention is required when multiple connections and free edges occur in the same physical location this work presents a mathematical formulation associated with the multiple connection and the free edge constraints its numerical implementation an algorithm for the automated update of the boundary element model and validation of all the developments

some simultaneous iterations for finding all zeros of a polynomial with high order convergence in this paper using lagrange interpolation with special nodes we construct some iterations for simultaneously finding all zeros of an algebraic polynomial with high order convergence

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

approximate zeros of quadratically convergent algorithms smale s condition for a point to be an approximate zero of a func tion for newton s method is extended to the general quadratically convergent iterative algorithm it is shown in which way the bound in the condition is affected by the characteristics of the algorithm this puts the original condition of smale for newton s method in a more general perspective the results are also discussed in the light of numerical evidence

a fast multi level boundary element method for the helmholtz equation abstract most recently we have developed a novel multi level boundary element method mlbem for the solution of the steady heat diffusion equation involving asymptotically decaying non oscillatory log singular and strongly singular kernels this hierarchical approach generalizes the pioneering work of brandt and lubrecht on multi level multi integration mlmi and c cycle multi grid to the broader class of mixed boundary value problems the result is a fast accurate and efficient boundary element method the present paper extends this new computational methodology to the solution of the helmholtz equation involving oscillatory log singular and strongly singular kernels for two dimensional problems we consider a direct boundary element formulation and due to the nature of the fundamental solutions split the corresponding boundary integral equation into real and imaginary parts then we introduce double noded corners to facilitate a patch by patch application of the mlmi algorithm for fast matrix vector and matrix transpose vector multiplications within bi conjugate gradient methods the performance of the proposed fast mlbem is investigated using a numerical example that possesses an exact solution for wave numbers 20 and below we demonstrate that the fast mlbem algorithm for the helmholtz equation is robust accurate and exceptionally efficient

on the flexural vibration of rectangular plates approached by using simple polynomials in the rayleigh ritz method abstract sets of simple polynomials are proposed for use as admissible functions in the rayleigh ritz method for the study of the flexural vibration of rectangular plates the plates may be subject to a number of different complicating factors including the presence of in plane forces concentrated masses point and line spring supports and special orthotropy of the plate material the sets of polynomials are shown to be equivalent to the orthogonally generated polynomials used very successfully in earlier studies but are more easily generated and the subsequent evaluation of the integrals in the resulting eigenvalue problem are more simply performed numerical results are presented for several problems illustrating the applicability of the method and its accuracy

three dimensional vibration of rectangular plates effects of thickness and edge constraints abstract a continuum three dimensional ritz formulation is presented for the free vibration analysis of homogeneous isotropic thick rectangular plates the method is applied specifically to investigate the effects of boundary constraints and thickness ratios on the vibration responses of these plates a general switching of transverse vibratory motions to surface parallel modes at the higher thickness ratio is manifested vividly through three dimensional deformed mode shape plots the present formulation is based on the linear three dimensional small strain elasticity energy principle in the solution process sets of self generating polynomial shape functions are used to derive the governing eigenvalue equation in essence these functions comprise a class of orthogonal polynomial functions and a basic function which is chosen to satisfy the essential geometric boundary conditions at the outset they are found very suitable for the computationally intensive task of 3 d modelling due to the ease of generation and numerical implementation several numerical examples have been computed to demonstrate the accuracy and efficiency of the method the influences of plate thickness and boundary constraints on the overall vibratory characteristics of thick plates are also discussed in detail

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

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

vibration analysis of multi span plates having orthogonal straight edges abstract an accurate numerical technique is developed for flexural vibration of rectangular and l shaped plates continuous over rigid supports with arbitrary number of spans a set of orthogonally generated two dimensional plates function is used in the rayleigh ritz method to determine the natural frequencies of the plate systems several numerical examples are presented and the results obtained where possible are compared with the known solutions in the literature the present method is proven to be extremely versatile in solving a broad class of the aforementioned problems

stochastic dynamic finite element analysis of bridge vehicle system subjected to random material properties and loadings abstract this paper performs the statistical dynamic analysis on the bridge vehicle interaction problem with randomness in material properties and moving loads the bridge is modeled as a laminated composite beam with gaussian random elastic modulus and mass density of material with random moving forces on top the mathematical model of the bridge vehicle system is established based on the finite element model in which the gaussian random processes are represented by the karhunen loeve expansion some statistical response such as the mean value and standard deviation of the deflections of the beam are obtained and checked by monte carlo simulation

the theory of smale s point estimation and its applications abstract the main result of this paper is that we exact smale s point estimation theory i e without assuming k p z 1 p k z k k 2 being bounded by the point estimation convergence theorem of the ne method is set up by making use of the majorizing method the proof of the theorem is simple and precise while the required point estimation conditions are weaker than all those of known point estimation convergence theorems another result of this paper is an application of the above new theory to the durand kerner method we compare the point estimation conditions for the durand kerner method with other known point estimation conditions numerical results show that our results have evident advantages

on the guaranteed convergence of the square root iteration method the construction of initial conditions which guarantee the convergence of the applied iterative method is one of the most important problems in solving nonlinear equations which attracted the great attention for many years in this paper we give a precise convergent analysis of the ostrowski like method of the fourth order for the simultaneous determination of polynomial zeros using a procedure based on smale s point estimation theory and some recent results related to the localization of complex polynomial zeros we state initial conditions which enable both the guaranteed and fast convergence of this method these conditions are computationally verifiable since they depend only on polynomial coefficients its degree and initial approximations which is of practical importance

numerical experiments on the determination of natural frequencies of transverse vibrations of rectangular plates of non uniform thickness abstract a comparison of natural frequencies of transverse vibration of thin rectangular plates of non uniform thickness with different combinations of boundary conditions is presented in this paper as obtained by using the following methodologies 1 the rayleigh ritz method with characteristic orthogonal polynomial shape functions 2 the rayleigh ritz method with a shape function which includes two exponents that are to be determined by minimizing the fundamental frequency coefficient 3 the optimized kantorovich method which has been proposed rather recently 4 the finite element method the results are presented in tabular form and discussed

natural frequencies of arbitrarily shaped plates using the rayleigh ritz method together with natural co ordinate regions and normalized characteristic orthogonal polynomials abstract a rayleigh ritz method mapping arbitrarily shaped plates on to a natural co ordinate plane using serendipity functions is presented the displacement function uses normalized beam characteristic orthogonal polynomials the kinetic and strain energy equations are mapped from cartesian co ordinates to natural co ordinates and the rayleigh coefficient is minimized flexural free vibration analyses of various shaped plates is performed results are in excellent agreement with those found in the literature this paper corrects the theory previously presented on this topic

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

dynamic response of a beam with a crack subject to a moving mass abstract an iterative modal analysis approach is developed to determine the effect of transverse cracks on the dynamic behavior of simply supported undamped bernoulli euler beams subject to a moving mass the presence of crack results in higher deflections and alters the beam response patterns in particular the largest deflection in the beam for a given speed takes longer to build up and a discontinuity appears in the slope of the beam deflected shape at the crack location crack effects become more noticeable as crack depth increases the effect of the inertia force due to the moving mass is in general qualitatively similar and additive to the effect of the crack the exact effect of crack and mass depends on the speed time crack size crack location and the moving mass level other approximate methods namely a stationary mass model and a single iteration technique are also evaluated the stationary mass approach is useful for light moving masses