polynomials for infinite domain spectral elements in this note it is found that laguerre type elements may be used in conjunction with legendre type elements to resolve more accurately regions of outflow and infinite boundaries

the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

a functional equation for the specular reflection of rays this paper aims to generalize the radiosity method when applied to specular reflection within the field of thermics the radiosity method is also called the standard procedure the integral equation for incident energy which is usually derived for diffuse reflection is replaced by a more appropriate functional equation the latter is used to solve some specific problems and it is shown that all the classical features of specular reflection for example the existence of image sources are embodied within this equation this equation can be solved with the ray tracing technique despite the implemented mathematics being quite different several interesting features of the energy field are presented

elastic plastic analysis of arbitrary heterogeneous materials with the voronoi cell finite element method abstract in this paper a voronoi cell finite element method is developed to solve small deformation elastic plasticity problems for arbitrary heterogenous materials dirichlet tessellation of microstructural representative materials results in a network of arbitrary sided polygons called voronoi cells each voronoi cell encompasses one second phase heterogeneity at most these are natural elements for the microstructure representing the basic structural elements of the material in this paper formulations are developed for directly considering voronoi cells as elements in a finite element model without any further dissection furthermore a composite voronoi cell finite element method is developed to account for the presence of the second phase within each polygonal element various numerical elastic plastic examples are executed for validating the effectiveness of this formulation finally studies are conducted to understand the effect of size shape and distribution of second phase on the averaged and true local responses of representative material elements

two scale analysis of heterogeneous elastic plastic materials with asymptotic homogenization and voronoi cell finite element model in this paper a multiple scale finite element model vcfem homo has been developed for elastic plastic analysis of heterogeneous porous and composite materials by combining asymptotic homogenization theory with the voronoi cell finite element model vcfem vcfem for microstructural modeling originates from dirichlet tessellation of representative material elements at sampling points in the structure structural modeling is done by the general purpose finite element code abaqus and interfacing with the microscale vcfem analysis is done through the user subroutine in abaqus for material constitutive relation umat asymptotic homogenization in umat generates macroscopic material parameters for abaqus following the macroscopic analysis a local vcfem analysis is invoked to depict the true evolution of microstructural state variables various numerical examples are executed for validating the effectiveness of vcfem homo and the effect of size shape and distribution of heterogeneities on local and global response is examined

a graph theory approach to identify resonant and non resonant transmission paths in statistical modal energy distribution analysis statistical modal energy distribution analysis smeda extends classical statistical energy analysis sea to the mid frequency range by establishing power balance equations between modes in different subsystems this circumvents the sea requirement of modal energy equipartition and enables applying smeda to the cases of low modal overlap locally excited subsystems and to deal with complex heterogeneous subsystems as well yet widening the range of application of sea is done at a price with large models because the number of modes per subsystem can become considerable when the frequency increases therefore it would be worthwhile to have at one s disposal tools for a quick identification and ranking of the resonant and non resonant paths involved in modal energy transmission between subsystems it will be shown that previously developed graph theory algorithms for transmission path analysis tpa in sea can be adapted to smeda and prove useful for that purpose the case of airborne transmission between two cavities separated apart by homogeneous and ribbed plates will be first addressed to illustrate the potential of the graph approach a more complex case representing transmission between non contiguous cavities in a shipbuilding structure will be also presented

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

voronoi cell finite element model based on micropolar theory of thermoelasticity for heterogeneous materials in this paper a new voronoi cell finite element model is developed for solving steady state heat conduction and micropolar thermoelastic stress analysis problems in arbitrary heterogeneous materials the method is based on the natural discretization of a multiple phase domain into basic structural elements by dirichlet tessellation tessellation process results in a network of polygons called voronoi polygons in this paper formulations are developed for treating these polygons as elements in a finite element mesh furthermore a composite voronoi cell finite element model is developed to account for the presence of a second phase inclusion within a polygonal element various numerical examples are executed for validating the effectiveness of this model in the analysis of the temperature and stress fields for micropolar elastic materials effective material properties are derived for microstructures containing different distributions of second phase

space average and wave interference in vibration conductivity plane wave interference and space averages play a significant role in the derivation of some vibration conductivity equations that are becoming more and more popular in modelling vibroacoustic problems particularly the thermal approach and the modified vibration conductivity equations are here considered with the aim of establishing similarities and or differences between them and stating relevant consequences it is shown by formal developments that the thermal equation is obtained under the assumption of performing appropriate space averages while the modified vibration conductivity equation does not need in some cases this condition it is discussed however that in practical applications the conditions of validity of both approaches are quite similar

a complete orthonormal system of functions in l 2 infty infty space a new complete orthonormal system of functions in the l 2 infty infty space is introduced the system consists of two sequences composed of odd and even functions respectively unlike hermite and laguerre sets of functions which behave exponentially at infinity the new system exhibits asymptoticbehavior x 1 for the odd sequence and x 2 for the even one formulae representing products derivatives etc in series in the system are developed a nonlinear differential equation with a requirement for summability of the square of its solutions instead of boundary conditions is solved the example displays most of the features of the new method important for essentially nonlinear problems

spectral element methods for elliptic problems in nonsmooth domains engineering applications frequently require the numerical solution of elliptic boundary value problems in irregularly shaped domains for smooth problems spectral element methods have proved very successful since they can accommodate fairly complicated geometries while retaining a rapid rate of convergence geometric singularities however often give rise to singular solutions the accuracy of the spectral element methods is then degraded and they offer no apparent advantage over low order finite element methods in many cases however the singular structure of the solution is known and its form may be exploited by the spectral element method among the various ways of doing so through supplementary basis functions eigenfunction expansions and graded meshes the method of auxiliary mapping proves to be particularly effective for certain simple cases the problem is transformed to a coordinate system in which the solution is analytic and exponential convergence is recovered even when this is not possible the singularity is usually much weaker after mapping so that other treatments are more effective in the new coordinate system in this paper we study different ways of treating singularities and in particular the method of auxiliary mapping coupled with the use of supplementary basis functions error estimates are presented explaining why the combined approach is more effective and these estimates are confirmed through a number of numerical experiments for the laplace poisson and helmholtz equations

a new computational method for transient dynamics including the low and the medium frequency ranges this paper deals with a new computational method for transient dynamic analysis which enables one to cover both the low and medium frequency ranges this is a frequency approach in which the low frequency part is obtained through a classical technique while the medium frequency part is handled through the variational theory of complex rays vtcr initially introduced for vibrations preliminary examples are shown

post buckling analysis of shallow shells by the field boundary element method the non linear field boundary element technique is applied to the analysis of snap through phenomena in thin shallow shells the equilibrium path is traced by using the arc length method and the solution strategy is discussed in detail the results show that as compared to the approaches based on the popular symmetric variational galerkin finite element formulation the current approach based on an unsymmetric variational petrov galerkin field boundary element formulation gives a faster convergence while using fewer degrees of freedom the illustrative numerical examples deal with post buckling responses of several shallow shells with different geometries

a hybrid strain finite element for plates and shells this paper describes a new hybrid strain finite element for geometrically linear analysis of plates and shells the formulation for materially linear problems is based on a mixed variational principle having strain and displacement rates as independently varied fields to facilitate extension to materially non linear analysis the variational principle is used in rate form and the plane stress assumption is not imposed a priori in the formulation when desired the plane stress assumption is imposed in a manner similar to imposition of fixed boundary conditions the element geometry is based on a nine noded quadrilateral with biquadratic interpolation of the displacement field the strain field is based on a bilinear interpolation with additional non linear terms chosen on the basis of eigenvalue analysis the element passes the required patch tests does not display either membrane or shear locking effects and is free of communicable spurious modes numerical results are presented for elastic analyses of several standard plate and shell test problems

derivation of statistical energy analysis from radiative exchanges abstract this paper is concerned with the derivation of sea equations from structural ray equations rays are assumed to be uncorrelated leading to the additivity of energy inside all subsystems the energy density is the sum of a direct field from driving forces a reflected field from the boundary and a transmitted field from adjacent subsystems assuming a rain on the roof excitation and a compact shape for subsystems actual and fictitious sources on the boundary are found to be constant furthermore if the attenuation of rays during a mean free path normalized attenuation factor is light the field becomes diffuse i e homogeneous and isotropic the net exchanged power between two adjacent subsystems is then proportional to the difference of energy densities and therefore to the difference of modal energies the derived proportionality coefficient is consistent with the well known formula for coupling loss factor in terms of transmission factors these results are illustrated by a numerical simulation for a multi plate system finally the validity domains of sea and the ray theory are discussed and particularly the diffuse field assumption

accurate extreme value based frequency response bounding for structures with a small number of highly random parameters abstract a modified extreme value based methodology is discussed for computing statistical bounds associated with the magnitude of the frequency response of a specified number of structures with high levels of random parameter uncertainty the methodology intended for small numbers of uncertain parameters is capable of constructing accurate statistical bounds in terms of quantiles associated with the extreme value distribution quantiles can be constructed for an ensemble of structural responses across the entire frequency range without using monte carlo simulation to test the methodology statistical bounds for the energy of an l shaped structure with low and high levels of uniformly distributed length and thickness variability are obtained i via direct integration using an ansys finite element model and ii via statistical energy analysis sea comparisons are shown with bounds obtained using monte carlo simulation the merit of direct integration for computing bounds associated with the responses of an ensemble of structures with high levels of random parameter uncertainty is demonstrated by its simplicity high accuracy and absence of statistical scatter

a dual algorithm for the topology optimization of non linear elastic structures dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of lagrange multipliers associated with the constraints to date dual algorithms have been applied only for linear structures here we extend this methodology to the case of non linear structures the perimeter constraint is used to make the topology problem well posed we show that the proposed algorithm yields a value of perimeter that is close to that specified by the user we also address the issue of manufacturability of these designs by proposing a variant of the standard dual algorithm which generates designs that are two dimensional although the loading and the geometry are three dimensional

passive vibration suppression of flexible space structures via optimal geometric redesign acomputationalframework is presented for the design of large flexible space structures with nonperiodicgeometries to achieve passive vibration suppression the present system combines an approximationmodel management framework ammf developed for evolutionary optimization algorithms eas with reduced basis approximate dynamic reanalysis techniques a coevolutionary genetic search strategy is employed to ensure that design changes during the optimization iterations lead to low rank perturbations of the structural system matrices for which thereduced basis methods give high quality approximations the k means algorithm is employed for cluster analysis of the population of designs to determine design points at which exact analysis should be carried out the fitness of the designs in an ea generation is then approximated using reduced basis models constructed around the points where exact analysis is carried out results are presented for the optimal design of a two dimensional cantilevered space structure to achieve passive vibration suppression it is shown that significant vibration isolation of the order of 50 db over a 100 hz bandwidth can be achieved further it is demonstrated that the ammf can potentially arrive at a better design compared to conventional approaches when a constraint is imposed on the computational budget available for optimization

alternative ways for formulation of hybrid stress elements an element stiffness matrix can be derived by the conventional potential energy principle and indirectly also by generalized variational principles such as the hu washizu principle and the hellinger reissner principle the present investigation has the objective to show an approach which is concerned with the formulation of incompatible elements for solid continuum and for plate bending problems by the hellinger reissner principle it is found that the resulting scheme is equivalent to that considered by tong 1982 for the construction of hybrid stress elements in tong s scheme the inversion of a large flexibility matrix can be avoided it is concluded that the introduction of additional internal displacement modes in mixed finite element formulations by the hellinger reissner principle and the hu washizu principle can lead to element stiffness matrices which are equivalent to the assumed stress hybrid method

plane isoparametric hybrid stress elements invariance and optimal sampling formulation and applications of the hybrid stress finite element model to plane elasticity problems are examined conditions for invariance of the element stiffness are established for two dimensional problems the results of which are easily extended to three dimensional cases next the hybrid stress functional for a 3 d continuum is manipulated into a more convenient form in which the location of optimal stress strain sampling points can be identified to illustrate these concepts 4 and 8 node plane isoparametric hybrid stress elements which are invariant and of correct rank are developed and compared with existing hybrid stress elements for a 4 node element lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation alternative 8 node elements are considered and the best invariant element is shown to be one in which the stress compatibility equations are invoked results are also presented which demonstrate the validity of the optimal sampling points the effects of reduced orders of numerical integration and the behaviour of the elements for nearly incompressible materials

a hybrid finite element method for stokes flow part ii stability and convergence studies abstract in part i we have presented a hybrid finite element method based on an assumed stress field which has the features i the unknowns in the final system of finite element equations are a the nodal velocities and b the constant term in the arbitrary pressure field over each element ii exact integrations were performed for each element in the following we present studies of stability and convergence of the above hybrid finite element method

least squares spectral elements applied to the stokes problem least squares spectral element methods are based on two important and successful numerical methods spectral hp element methods and least squares finite element methods in this respect least squares spectral element methods seem very powerful since they combine the generality of finite element methods with the accuracy of the spectral methods and also have the theoretical and computational advantages of the least squares methods these features make the proposed method a competitive candidate for the solution of large scale problems arising in scientific computing in order to demonstrate its competitiveness the method has been applied to an analytical problem and the theoretical optimal and suboptimal a priori estimates have been confirmed for various boundary conditions moreover the exponential convergence rates typical for a spectral element discretization have also been confirmed the comparison with the classical galerkin spectral element method revealed that the least squares spectral element method is as accurate as the galerkin method for the smooth model problem

transport of phase space densities through tetrahedral meshes using discrete flow mapping discrete flow mapping was recently introduced as an efficient ray based method determining wave energy distributions in complex built up structures wave energy densities are transported along ray trajectories through polygonal mesh elements using a finite dimensional approximation of a ray transfer operator in this way the method can be viewed as a smoothed ray tracing method defined over meshed surfaces many applications require the resolution of wave energy distributions in three dimensional domains such as in room acoustics underwater acoustics and for electromagnetic cavity problems in this work we extend discrete flow mapping to three dimensional domains by propagating wave energy densities through tetrahedral meshes the geometric simplicity of the tetrahedral mesh elements is utilised to efficiently compute the ray transfer operator using a mixture of analytic and spectrally accurate numerical integration the important issue of how to choose a suitable basis approximation in phase space whilst maintaining a reasonable computational cost is addressed via low order local approximations on tetrahedral faces in the position coordinate and high order orthogonal polynomial expansions in momentum space

power flow between two continuous one dimensional subsystems a wave solution abstract the power flow between two point coupled wave bearing subsystems is determined by using a wave approach the subsystems are one dimensional in that each supports just one wave mode and are described by a reflection coefficient from which in certain circumstances the modal overlap and modal density can be defined as continuous functions of frequency the coupling element is described by a reflection coefficient and attention is focused on the case of conservative coupling time harmonic point forces or rain on the roof excitations may act on the subsystems uncertainty exists in the properties of the subsystems which are assumed to be drawn from ensembles of similar subsystems the exact power flow and the ensemble statistics are seen to depend on the magnitudes of the subsystem and coupling reflection coefficients for some particular cases the maximum and minimum power flows are examined together with the ensemble mean and geometric mean the cumulative probability distribution and the probability density function for a given power incident upon the coupling the ensemble mean power flow through the coupling is always less than that expected from a normal sea wave approach due to power re radiation the differences are largest when the damping is light and the coupling strong for a given level of excitation power re radiation effects are also apparent for the ensemble mean power flow from an infinite subsystem to a finite subsystem in the opposite case the ensemble mean power flow from a finite to an infinite subsystem is greater than that between two infinite subsystems due to power re injection the differences are largest for light damping and for weak coupling finally the case where there is uncertainty in one subsystem only is considered

efficient exceedance probability computation for randomly uncertain nonlinear structures with periodic loading abstract a method is developed to compute low level response amplitude exceedance probabilities associated with uncertain nonlinear structures with random parameters and deterministic periodic forcing emphasis is focused on accurate and efficient computation in the tails of the exceedance probability distribution function associated with the largest possible response of one displacement variable for unspecified forcing frequency and normally distributed parameters this gives a measure of system reliability when a large amplitude response exceedance of a specified threshold is designated as the mode of failure the method exploits the first order reliability method form in which the failure surface is constructed via the harmonic balance method hbm this combined approach is tested on a duffing oscillator with harmonic forcing and up to three uncertain parameters for which the frequency of multiple solution maximum amplitude is found directly and the probability computed via the hasofer lind reliability index the accuracy of the proposed hbm form in the tails of the amplitude exceedance probability is shown for the duffing example to be acceptably accurate whereas the efficiency is shown to be around 1000 times faster than direct integration and around 200 times faster than monte carlo simulation

development and testing of stable invariant isoparametric curvilinear 2 and 3 d hybrid stress elements abstract linear and quadratic serendipity hybrid stress elements are examined in respect of stability coordinate invariance and optimality a formulation based upon symmetry group theory successfully addresses these issues in undistorted geometries and is fully detailed for plane elements the resulting least order stable invariant stress polynomials can be applied as astute approximations in distorted cases through a variety of tensor components and variational principles a distortion sensitivity study for two and three dimensional elements provides favourable numerical comparisons with the assumed displacement method

comparison of an integral equation on energy and the ray tracing technique in room acoustics this paper deals with a comparison of two room acoustic models the first one is an integral formulation stemming from power balance and the second is the ray tracing technique with a perfectly diffuse reflection law the common assumptions to both models are the uncorrelated wave hypothesis and the perfectly diffuse reflection law the latter allows the use of these methods for nondiffuse fields beyond the validity domain of sabine s formula comparisons of numerical simulations performed with the softwares rayon and ceres point out that these results are close to each other and finally a formal proof is proposed showing that both methods are actually equivalent

error estimation of hermite spectral method for nonlinear partial differential equations hermite approximation is investigated some inverse inequalities imbedding inequalities and approximation results are obtained a hermite spectral scheme is constructed for burgers equation the stability and convergence of the proposed scheme are proved strictly the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains

efficient modeling of flat and homogeneous acoustic treatments for vibroacoustic finite element analysis direct field formulations abstract this paper is concerned with the development of a simplified model for noise control treatments to speed up finite element analysis in vibroacoustic applications the methodology relies on the assumption that the acoustic treatment is flat and homogeneous moreover its finite lateral extent is neglected this hypothesis is justified by short wavelength and large dissipation which suggest that the reflected field emanating from the acoustic treatment lateral boundaries does not substantially affect its dynamic response under these circumstances the response of the noise control treatment can be formally obtained by means of convolution integrals involving simple analytical kernels i e green functions such fundamental solutions can be computed efficiently by the transfer matrix method however some arbitrariness arises in the formulation of the mathematical model resulting in different baffling conditions at the two ends of the treatment to be considered thus the paper investigates the possibility of different formulations i e baffling conditions within the same hybrid finite element transfer matrix framework seeking for the best strategy in terms of tradeoff between efficiency and accuracy numerical examples are provided to show strengths and limitations of the proposed methodology

multiple scale computational model for damage in composite materials abstract in this paper a hierarchical multiple scale computational model is proposed to concurrently predict evolution of variables at the structural and microstructural scales as well as to track the incidence and propagation of microstructural damage the microscopic analysis is conducted with the voronoi cell finite element model vcfem while a conventional displacement based fem code executes the macroscopic analysis adaptive schemes and mesh refinement strategies are developed to create a hierarchy of computational sub domains with varying resolution such hierarchy allow for differentiation between non critical and critical regions and help in increasing the efficiency of computations through preferential zoom in regions coupling between the scales for regions with periodic microstructure is accomplished through asymptotic homogenization whereas regions of nonuniformity and non periodicity are modeled by true microstructural analysis with vcfem various numerical examples are executed for validating the effectiveness of the model for different microstructural configurations

an approach to the theoretical background of statistical energy analysis applied to structural vibration rayleigh s classical approach to the study of vibration of systems having a finite number of degrees of freedom is applied to the problem of coupling of subsystems in a complicated structure in order to probe the regions of applicability of the approach to vibration analysis usually known as statistical energy analysis sea the classical method has advantages of simplicity and rigor over previous approaches to the background of sea in certain cases and provides extensions and simplifications in several areas of the theory it also suggests modifications to sea modeling strategy depending on the type of coupling involved even when that coupling is weak so that earlier analyses might be thought to apply

uncertainty propagation in sea for structural acoustic coupled systems with non deterministic parameters abstract considering limited available information on uncertainties in structural acoustic coupled systems two methods namely the vertex method and the legendre orthogonal polynomial based method for predicting their dynamic behavior are developed based on the statistical energy analysis sea approach for the vertex method an efficient program for determining coordinates of all vertices of the rectangular spanned by entries of the involved interval input vector is coded which is well suited for an interval input vector in arbitrary dimension instead of calculating the extremum of the response of interest a method for determining its minimal and maximal point vectors dimension by dimension with respect to uncertain parameters is proposed based on the legendre orthogonal polynomial approximation following the theoretical analysis of the accuracy and efficiency of the proposed methods their validation is performed by one numerical example and two applications

a model and numerical method in the medium frequency range for vibroacoustic predictions using the theory of structural fuzzy in linear dynamical analysis of complex mechanical systems the structural fuzzy is defined as the set of minor subsystems that are connected to the master structure but are not accessible by classical modeling the notion of master structure is presently extended to others elements such as an external dense compressible fluid strongly coupled with the primary structure for the low frequency lf dynamical analysis the modeling of the structural fuzzy is commonly made with a system of masses if the lf modeling of the structural fuzzy is applied in the medium frequency mf domain there are some large differences between calculations and experiment it is therefore necessary to take into account internal degrees of freedom of the structural fuzzy a global probabilistic modeling of the structural fuzzy is proposed to improve the calculated estimates of the mf vibrations into the master structure and of the far field radiated by itself in or out of context of the acoustic scattering this paper reviews the author s previous work and introduces the type ii probabilistic constitutive law in this paper 1 a probabilistic modeling of the structural fuzzy is presented 2 two probabilistic constitutive laws of the structural fuzzy are constructed 3 the modeling in the mf range for vibroacoustic predictions using theory of structural fuzzy is developed 4 the fuzzy solution in the mf range is studied and 5 numerical simulations on standard structures and on a submerged complex industrial structure are shown

harmonic green s functions for flexural waves in semi infinite plates with arbitrary boundary conditions and high frequency approximation for convex polygonal plates abstract this paper provides a method for obtaining the harmonic green s function for flexural waves in semi infinite plates with arbitrary boundary conditions and a high frequency approximation of the green s function in the case of convex polygonal plates by using a generalised image source method the classical image source method consists in describing the response of a point driven polygonal plate as a superposition of contributions from the original source and virtual sources located outside of the plate which represent successive reflections on the boundaries the proposed approach extends the image source method to plates including boundaries that induce coupling between propagating and evanescent components of the field and on which reflection depends on the angle of incidence this is achieved by writing the original source as a fourier transform representing a continuous sum of propagating and evanescent plane waves incident on the boundaries thus the image source contributions arise as continuous sums of reflected plane waves for semi infinite plates the exact green s function is obtained for an arbitrary set of boundary conditions for polygonal plates a high frequency approximation of the green s function is obtained by neglecting evanescent waves for the second and subsequent reflections on the edges the method is compared to exact and finite element solutions and evaluated in terms of its frequency range of applicability

response variance prediction in the statistical energy analysis of built up systems in the statistical energy analysis sea of high frequency noise and vibration a complex engineering structure is represented as an assembly of subsystems the response of the system to external excitation is expressed in terms of the vibrational energy of each subsystem and these energies are found by employing the principle of power balance strictly the computed energy is an average taken over an ensemble of random structures and for many years there has been interest in extending the sea prediction to the variance of the energy a variance prediction method for a general built up structure is presented here closed form expressions for the variance are obtained in terms of the standard sea parameters and an additional set of parameters alpha k that describe the nature of the power input to each subsystem k and alpha ks that describe the nature of the coupling between subsystems k and s the theory is validated by comparison with monte carlo simulations of plate networks and structural acoustic systems

solving the stationary liouville equation via a boundary element method energy distributions of linear wave fields are in the high frequency limit often approximated in terms of flow or transport equations in phase space common techniques for solving the flow equations in both time dependent and stationary problems are ray tracing and level set methods in the context of predicting the vibro acoustic response of complex engineering structures related methods such as statistical energy analysis or variants thereof have found widespread applications we present a new method for solving the transport equations for complex multi component structures based on a boundary element formulation of the stationary liouville equation the method is an improved version of the dynamical energy analysis technique introduced recently by the authors it interpolates between standard statistical energy analysis and full ray tracing containing both of these methods as limiting cases we demonstrate that the method can be used to efficiently deal with complex large scale problems giving good approximations of the energy distribution when compared to exact solutions of the underlying wave equation

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

on the vibrational conductivity approach to high frequency dynamics for two dimensional structural components abstract the vibrational conductivity approach to high frequency dynamics is applied here to the case of two dimensional structural components with this approach an equation which is analogous to the steady state heat conduction equation is used to model the spatial variation of the energy density of the structure it is shown that the standard derivation of the method is incomplete in the sense that the steady state heat conduction equation admits a class of solution which is not encompassed by the assumed form of response and a revised derivation is presented it is further shown that the associated boundary conditions for coupled structural components may be expressed in terms of the coupling loss factors which are used in statistical energy analysis a variational principle for the differential equation and the associated boundary conditions is then derived which permits the application of the rayleigh ritz technique to coupled problems and it is shown that a one term rayleigh ritz solution leads to standard statistical energy analysis the method is applied to a single plate and to two coupled plates and it is found that the technique is unable to provide an accurate estimate of the direct field due to point loading this difficulty is traced to the fact that the assumptions employed in the derivation become self contradictory for this case finally the usefulness or otherwise of the approach as an engineering tool is discussed

multiple scale analysis of heterogeneous elastic structures using homogenization theory and voronoi cell finite element method abstract this paper deals with the development of a multiple scale finite element method by combining the asymptotic homogenization theory with voronoi cell vcfem for microstructural modeling the voronoi cell finite element model originates from dirichlet tessellation of a representative material element or a base cell in the microstructure homogenized material coefficients for a global displacement finite element model are generated by vcfem analysis using periodic boundary conditions on the base cell following the macroscopic analysis the local vcfem analysis is implemented to depict the true evolution of microstructural stresses and strains various numerical examples are executed for validating the effectiveness of vcfem macro micro modeling of elastic materials the effect of size shape orientation and distribution of heterogeneities on the local and global response are examined

polygon scaled boundary finite elements for crack propagation modelling summary an automatic crack propagation modelling technique using polygon elements is presented a simple algorithm to generate a polygon mesh from a delaunay triangulated mesh is implemented the polygon element formulation is constructed from the scaled boundary finite element method sbfem treating each polygon as a sbfem subdomain and is very efficient in modelling singular stress fields in the vicinity of cracks stress intensity factors are computed directly from their definitions without any nodal enrichment functions an automatic remeshing algorithm capable of handling any n sided polygon is developed to accommodate crack propagation the algorithm is simple yet flexible because remeshing involves minimal changes to the global mesh and is limited to only polygons on the crack paths the efficiency of the polygon sbfem in computing accurate stress intensity factors is first demonstrated for a problem with a stationary crack four crack propagation benchmarks are then modelled to validate the developed technique and demonstrate its salient features the predicted crack paths show good agreement with experimental observations and numerical simulations reported in the literature copyright 2012 john wiley sons ltd

orthogonal rational functions on a semi infinite interval abstract by applying a mapping to the chebyshev polynomials we define a new spectral basis the rational chebyshev functions on the semi infinite interval denoted by tl n y continuing earlier work by the author and by grosch and orszag we show that these rational functions inherit most of the good numerical characteristics of the chebyshev polynomials orthogonality completeness exponential or infinite order convergence matrix sparsity for equations with polynomial coefficients and simplicity seven numerical examples illustrate their versatility the charney stability problem of meteorology for example is solved to show the feasibility of applying spectral methods to a semi infinite atmosphere for functions that are singular at both endpoints such as k 1 y one may combine rational chebyshev functions with a preliminary mapping to obtain a single exponentially convergent expansion on y 0 finally we successfully generalize the wkb method to obtain for the j 0 bessel function an amplitude phase approximation which is convergent rather than asymptotic and is accurate not merely for large y but for all y even the origin

nonconforming least squares spectral element method for european options several methods have been proposed in the literature for solving the black scholes equation for european options the method proposed in the current study achieves spectral accuracy in both space and time the method is based on minimization of a functional given in terms of the sum of squares of the residuals in the partial differential equation and initial condition in different sobolev norms and a term which measures the jump in the function and its derivatives across inter element boundaries in appropriate fractional sobolev norms to obtain values of the solution and its derivatives the initial condition is mollified and the computed solution is post processed error estimates are obtained for this method specific numerical examples are given to show the efficiency of this method

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

study on the validity region of energy finite element analysis abstract the validity domain of energy finite element analysis efea is studied in this paper the validity region and criterion of efea are studied theoretically from the formation of reverberant plane wave field one of the main assumptions of efea the studies are acquired by virtue of the equation of radiative energy transfer method a similar wave method that can express the direct field and its conversion relationship with reverberant field exactly the result shows that the sea criterion of diffuse field derived by le bot can be used as a good indicator for the efea validity numerical simulations on a rectangular plate with different physical parameters are applied to validate the criterion the validity region and the diagrams of validity of efea are assessed and discussed some noteworthy conclusions about efea are drawn

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

on the suppression of zero energy deformation modes based on the hellinger reissner principle and the deformation energy due to assumed stresses and displacements the problem of the kinematic deformation modes in assumed stress hybrid mixed finite elements has been examined basic schemes are developed for the choice of assumed stress terms that will suppress all kinematic deformation modes quadrilateral membrane and axisymmetric elements and three dimensional hexahedral elements are used to illustrate the suggested procedure

finite element analysis of plane couple stress problems using first order stress functions a complementary energy based variational principle using first order stress functions is developed for plane linear elastic couple stress problems the principle is analogous to that used in a total potential energy based mindlin reissner thick plate bending analysis and as such is a generalization of the classical analogy between plate stretching and plate bending traction boundary conditions are enforced using a lagrange multiplier technique the resulting c0 finite element equilibrium stress model is validated by investigating the reduction of the stress concentration factor associated with a small hole in a field of uniform tension

on the existence and stability conditions for mixed hybrid finite element solutions based on reissner s variational principle abstract the extensions of reissner s two field stress and displacement principle to the cases wherein the displacement field is discontinuous and or the stress field results in unreciprocated tractions at a finite number of surfaces interelement boundaries in a domain as for instance when the domain is discretized into finite elements is considered the conditions for the existence uniqueness and stability of mixed hybrid finite element solutions based on such discontinuous fields are summarized the reduction of these global conditions to local element level and the attendant conditions on the ranks of element matrices are discussed two examples of stable invariant least order elements a four node square planar element and an eight node cubic element are discussed in detail

boundary element dynamical energy analysis a versatile method for solving two or three dimensional wave problems in the high frequency limit dynamical energy analysis was recently introduced as a new method for determining the distribution of mechanical and acoustic wave energy in complex built up structures the technique interpolates between standard statistical energy analysis and full ray tracing containing both of these methods as limiting cases as such the applicability of the method is wide ranging and additionally includes the numerical modelling of problems in optics and more generally of linear wave problems in electromagnetics in this work we consider a new approach to the method with enhanced versatility enabling three dimensional problems to be handled in a straightforward manner the main challenge is the high dimensionality of the problem we determine the wave energy density both as a function of the spatial coordinate and momentum or direction space the momentum variables are expressed in separable polar coordinates facilitating the use of products of univariate basis expansions however this is not the case for the spatial argument and so we propose to make use of automated mesh generating routines to both localise the approximation allowing quadrature costs to be kept moderate and give versatility in the code for different geometric configurations

discussions on on the number of modes required for statistical energy analysis based calculations in ref 1 it was argued using two subsystems that for the applicability of statistical energy analysis sea what is important is the presence of large number of modal pairs in the interacting subsystems and not large number of modes in each subsystem it was further claimed that even if there is only one mode present in a particular subsystem if the interacting subsystem has several modes in the frequency band sea is applicable the purpose of this discussion is to highlight that the conclusions made in ref 1 cannot be generalized to using sea for the analysis of a system with more than two subsystems and such generalization might incur substantial errors

specular and diffuse reflections of rays in coupled thin plates at high frequencies the flexural energy distribution in two right angled point excited thin plates at high frequencies is investigated by means of an integral energy flow approach the fields of energy averaged over time and frequency are described by the superposition of uncorrelated cylindrical waves stemming from both boundaries and direct sources specular and diffuse laws are considered for the reflection and transmission of rays giving rise to two kinds of energy equations the diffuse law leads to a fredholm integral equation over the boundary sources while the specular law is shown to allow an image source solution when the plates have identical propagation properties the algorithm for computing the image position magnitude and directivities is described then some comparisons between the results from the both energy formulations and also from the statistical energy analysis and the numerical solution of the equations of motion are performed with two damped plates at high frequency the non diffuse pattern of the averaged flexural energy fields is well described by the energy flow approaches

a space averaged quadratic method for predicting one dimensional power transfers a method which is based on space averaged quadratic variables and does not assume the restrictive assumptions of the statistical energy analysis sea such as high overlap or damping conditions is described the results are spatially more detailed than an sea response and of a lower computational cost than the one of the usual displacement formulation some of the quadratic variables used like energy densities or structural intensity retain a strong energy meaning even when time and space averaged their governing equation is derived for a homogeneous and isotropic medium with hysteretic damping the main originality of this work lies in considering complex structural intensity and establishing appropriate energetic boundary conditions for both active and reactive space averaged structural intensities by using the usual boundary conditions for the displacement field and the stress tensor the numerical examples prove that this space averaged quadratic method is well suited to describe global energy transfers along one dimensional dissipative structures in a frequency range for which the overlap is too low to obtain a quadratic response from the sea and solving the wave equation by using the finite element method fem would require more elements besides describing junctions with impedances makes the resonant behaviour of the system still accessible on frequency responses

analysis of cosserat materials with voronoi cell finite element method and parametric variational principle abstract the elastic plastic analysis of heterogeneous cosserat materials is carried out with the voronoi cell finite element method vcfem under small deformation the parametric complementary energy principle of the cosserat theory is developed by using the parametric variational principle the quadratic programming model and finite element formulation for the vcfem are established compared with the conventional finite element method fem the newly developed method requires no tedious iterative procedures and has no convergence problems based on the developed method the influence of microscopic heterogeneities on the overall mechanical responses of heterogeneous materials is studied numerical examples are given to demonstrate the validity and efficiency of the method developed

least squares spectral element preconditioners for fourth order elliptic problems abstract in this paper we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions d 2 3 using spectral element methods these preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert for second order elliptic problems this technique has proven to be successful and performs better than other preconditioners in the framework of least squares methods we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them numerical results for the condition number reflects the effectiveness of the preconditioners

wave derivation of the energetics of driven coupled one dimensional dynamic systems abstract the equation of statistical energy analysis was originally derived on the basis of modal concepts in this paper a derivation of the equation on the basis of wave concepts is outlined although the new formalism is limited at present to the analysis of coupled one dimensional dynamic systems it carries details that were a priori forfeited by the previous formalism the wave approach to sea as developed in the paper reveals a formalism that can be naturally used for the analysis of net power and intensity in coupled dynamic systems this aspect of the formalism is briefly discussed

a scaled boundary polygon formulation for elasto plastic analyses this study presents a novel scaled boundary polygon formulation to model elasto plastic material responses in structures the polygons have flexible mesh generation capabilities and are more accurate than standard finite elements especially for problems with cracks and notches shape functions of arbitrary n sided polygons are constructed using the scaled boundary finite element method these shape functions are conforming and linearly complete when modeling a crack strain singularities are analytically modeled without enrichment standard finite element procedures are used to formulate the stiffness matrix and residual load vector the nonlinear material constitutive matrix and the internal stresses are approximated locally in each polygon by a polynomial function the stiffness matrix and the residual load vector are matrix power integrals that can be evaluated analytically even when a strain singularity is present standard nonlinear equation solvers e g the modified newton raphson algorithm are used to obtain the nonlinear response of the structure the proposed formulation is validated using several numerical benchmarks

energy transfer for high frequencies in built up structures this paper is concerned with the prediction of vibrations at high frequencies in built up structures in the high frequency range the dynamical behaviour of structures is driven by energy transfer rather than modal aspects the description of systems in terms of energy is thus well suited and leads to closed form equations of an integral type on energy variables the numerical solution of these equations is achieved with appropriate software called ceres and the results are compared with experimental measurements on a multi plate structure

a monolithic approach to fluid structure interaction using space time finite elements abstract the paper presents a simultaneous solution procedure for fluid structure interaction problems the structural motion is described by geometrically nonlinear elastodynamics the fluid is modeled by the incompressible navier stokes equations the space time finite element method is applied to both continua leading to an almost uniform discretization in which velocity variables are used for fluid and structure in order to enforce momentum conservation and geometric continuity at the interface a weighted residual formulation of coupling conditions is introduced the discretized model equations for fluid structure and coupling conditions are formulated in a single equation system the nonlinear system and the fluid mesh movement are solved in a single iteration loop two dimensional examples of membrane flutter and vortex excited plate vibrations demonstrate the efficiency of the methodology

preconditioners for spectral element methods for elliptic and parabolic problems in this paper we propose preconditioners for spectral element methods for elliptic and parabolic problems these preconditioners are constructed using separation of variables and are easy to invert moreover they are spectrally equivalent to the quadratic forms which they are used to approximate

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

a localized mixed hybrid method for imposing interfacial constraints in the extended finite element method xfem the paper proposes an approach for the imposition of constraints along moving or fixed immersed interfaces in the context of the extended finite element method an enriched approximation space enables consistent representation of strong and weak discontinuities in the solution fields along arbitrarily shaped material interfaces using an unfitted background mesh the use of lagrange multipliers or penalty methods is circumvented by a localized mixed hybrid formulation of the model equations in a defined region in the vicinity of the interface the original problem is re stated in its auxiliary formulation the availability of the auxiliary variable enables the consideration of a variety of interface constraints in the weak form the contribution discusses the weak imposition of dirichlet and neumann type interface conditions as well as continuity requirements not fulfilled a priori by the enriched approximation the properties of the proposed approach applied to two dimensional linear scalar and vector valued elliptic problems are investigated by studying the convergence behavior

a material based finite element analysis of heterogeneous media involving dirichlet tessellations abstract in this paper a new finite element formulation has been developed for analysis of heterogeneous media in which the second phase is randomly dispersed within the matrix it introduces a tessellation based mesh generation technique to account for the arbitrariness in location shape and size of the second phase an assumed stress hybrid formulation has been implemented to utilize the resulting voronoi polygons as elements in a finite element model the effect of the second phase has been accounted for by introducing a transformation strain based constitutive formulation numerical examples have been conducted to validate the model

power flow boundary element analysis for multi domain problems in vibrational built up structures abstract a power flow boundary element methodology for the treatment of multi domain problems is developed to predict the vibrational responses of coupled structures in medium to high frequency ranges in the proposed methodology a matrix equation is formulated by utilizing the power flow coupling relationships at the interface of the domains this analysis method is successfully applied to the treatment of simply supported coupled beams and coupled plates the vibration analysis of these structures is then considered for all wave components the vibrational energy density and intensity of two coupled structures obtained through numerical simulations are compared with those of the original power flow analysis and these results show good agreement

extension of the variational theory of complex rays to shells for medium frequency vibrations a new approach to the calculation of vibrations of weakly damped elastic structures in the medium frequency range called the variational theory of complex rays vtcr is being developed here the extension of this theory to shells of relatively small curvature is considered numerical examples of structures made of plates and shells demonstrate the capabilities of the vtcr

validity diagrams of statistical energy analysis abstract this paper is concerned with the validity domain of statistical energy analysis sea which is defined in terms of four criteria the mode count n and the modal overlap m must be high the normalized attenuation factor m and the coupling strength must be small the application of dimensional analysis on the governing equations of plates gives the space of dimensionless parameters in which the validity domain of sea must be delimited this domain is discussed on the basis of geometry of the surfaces delimiting it the diagrams of validity of sea are introduced and discussed a numerical simulation on a couple of rectangular plates coupled along one edge illustrates the theoretical approach

a spectral method for unbounded domains a spectral method for an unbounded domain is presented rational basis functions which are algebraically mapped legendre functions are used for expansion in the radial direction of polar coordinates r or r z they satisfy the pole condition exactly at the coordinate singularity and their behavior asr is suitable for expanding smooth functions which decay algebraically or exponentially asr the method is not stiff when it is applied to initial value problems despite the presence of the coordinate singularity solenoidal vector fields are treated efficiently by the toroidal and poloidal decomposition which reduces the number of dependent variables from 3 to 2 examples include the computation of vortex dynamics in two and three dimensions

iterative methods for p version finite elements preconditioning thin solids abstract we present new preconditioning strategy for thin 3d p version elements used to model plate and shell structures as well as the microstructure of composite materials the strategy is incorporated in an adaptive preconditioner suitable for large real world problems the method is shown to perform well for several large real world problems including a model of the skin of an aircraft with over 1 6 million degrees of freedom and a highly accurate model of a 28 ply laminated fiber plate

adaptivity and convergence in the voronoi cell finite element model for analyzing heterogeneous materials in this paper an adaptive voronoi cell finite element model is presented for analyzing micromechanical response of composites and porous materials both elastic and elastic plastic materials are considered two error measures viz a traction reciprocity error and an error in the kinematic relation are formulated as indicators of the quality of vcfem solutions based on a posteriori evaluation of these error measures element adaptation is executed in two consecutive stages in the first stage displacement function adaptations on the element boundaries and matrix inclusion void interfaces are carried out to minimize the corresponding traction reciprocity errors this is accomplished through a sequence of h refinement and spectral p enrichment strategy in an optimal displacement direction following this an enrichment of matrix and inclusion stress functions or enrp adaptation is conducted to reduce the error in kinematic relations the complete process improves convergence characteristics of the vcfem solution numerical analysis is conducted to examine the potential of the resulting vcfem code in analyzing microstructures with different distributions sizes and shapes of heterogeneities the method is seen to perform very well for the wide variety of problems solved

exponentially accurate nonconforming least squares spectral element method for elliptic problems on unbounded domain abstract non conforming approximation methods are becoming increasingly popular because of the potential to apply to multi material and multi model analysis for both bounded and unbounded domains in this paper we present a least square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain the method gives exponential accuracy and shows superior performance when compared to other numerical methods differentiability estimates and the main stability estimate theorem using a non conforming spectral element method are also discussed the exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples

a quantitative criterion validating coupling power proportionality in statistical energy analysis the response of two general spring coupled elements is investigated to develop a unifying approach to the weak coupling criterion in statistical energy analysis sea first the coupled determinis