symmetries of the turbulent state the emphasis of this review is on fundamental properties degree of universality and symmetries of the turbulent state the central questions are which symmetries remain broken even when the symmetry breaking factor reaches zero and which symmetries in contrast emerge in the state of developed turbulence we shall see that time reversibility is broken in all cases since turbulence is a far from equilibrium state accompanied by dissipation as far as scale invariance is concerned we argue that it is always broken in direct cascades toward small scales no matter how far one goes away from the pumping scale in contrast inverse cascades become scale invariant as they go toward large scales moreover some properties of the inverse cascades seem to be conformal invariant and related to schramm loewner evolution a class of random curves that can be mapped to a 1d brownian walk

a newtonian procedure for the solutions of ex λax abstract newtonian iteration on a scalar function which is obtained by condensation of the pencil e a is used in conjunction with sturm sequence properties of the pencil to provide an infallible quadratically convergent algorithm for all eignevalues of a 1 e within any stated range the algorithm has been programmed for use on a desk top machine and large sparse matrix eigenvalue problems have been solved a brief outline of the essential steps in the program is given along with some particular solutions

higher order finite element methods and pointwise error estimates for elliptic problems on surfaces we define higher order analogues to the piecewise linear surface finite element method studied in g dziuk finite elements for the beltrami operator on arbitrary surfaces in partial differential equations and calculus of variations springer verlag berlin 1988 pp 142 155 and prove error estimates in both pointwise and l 2 based norms using the laplace beltrami problem on an implicitly defined surface gamma as a model pde we define lagrange finite element methods of arbitrary degree on polynomial approximations to gamma which likewise are of arbitrary degree then we prove a priori error estimates in the l 2 h 1 and corresponding pointwise norms that demonstrate the interaction between the pde error that arises from employing a finite dimensional finite element space and the geometric error that results from approximating gamma we also consider parametric finite element approximations that are defined on gamma and thus induce no geometric error computational examples confirm the sharpness of our error estimates

a progressive simultaneous iteration solution for dynamic element vibration analysis development and employment of dynamic elements can result in substantial gain in solution convergence for vibration analysis when compared with the usual finite element discretization efficient solution of the associated quadratic matrix eigenproblem is crucial in achieving a superior and relatively economical solution this paper first describes a novel eigensolution of the quadratic matrix equation by a progressive simultaneous iteration method free vibration analysis results of a rectangular prestressed membrane are next presented in detail that provides an assessment of relative solution convergence and computing expenses of the two idealization procedures

dissipation statistics of a passive scalar in a multidimensional smooth flow we compute analytically the probability distribution function p e of the dissipation field e 2 of a passive scalar advected by a d dimensional random flow in the limit of large peclet and prandtl numbers batchelor kraichnan regime the tail of the distribution is a stretched exponential for e ln p e d2e 1 3

structural vibration analysis by a progressive simultaneous iteration method this paper provides details of a newly developed structural eigenproblem solution technique that effects simultaneous convergence of a set of desired roots and associated vectors in an accurate and efficient fashion the procedure is based on a progressive simultaneous iteration method that enables effective computation of either the first few or a specified number of intermediate roots and associated vectors without having to compute any other numerical results pertaining to some practical problems are also presented that testify to the superior convergence characteristics of the present procedure when compared with other existing ones

least squares element method for boundary eigenvalue problems linear and non linear boundary eigenvalue problems are discretized by a new finite element like method the reason for the new construction principle is the non linear dependence of the dynamic stiffness element matrix on an eigenparameter the dynamic stiffness element matrix is evaluated for a fixed number of parameters and is then elementwise replaced by a polynomial in the eigenparameter by solving least squares problems a fast solver is introduced for the resulting non linear matrix eigenvalue problem it consists of a combination of bisection method and inverse iteration the superiority of the newconstructionprinciple in comparison with the finite or dynamic element method is demonstrated finally for some numerical examples

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

locality and stability of the cascades of two dimensional turbulence we investigate and clarify the notion of locality as it pertains to the cascades of two dimensional turbulence the mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions first introduced by l vov and procaccia as a point of departure we use a revised version of the system of hypotheses that was proposed by frisch for three dimensional turbulence we show that both the enstrophy cascade and the inverse energy cascade are local in the sense of non perturbative statistical locality we also investigate the stability conditions for both cascades we have shown that statistical stability with respect to forcing applies unconditionally for the inverse energy cascade for the enstrophy cascade statistical stability requires large scale dissipation and a vanishing downscale energy dissipation a careful discussion of the subtle notion of locality is given at the end of the paper

a variant of the subspace iteration algorithm for generalized eigenproblems a variant of the subspace iteration algorithm for the generalized eigenproblem kp mp is proposed the algorithm does not require the computation of the projected stiffness matrix and hence the projected eigenproblem turns out to be a standard eigenproblem as against the generalized eigenproblem of the classical algorithm this results in appreciable saving in the number of operations of the rayleigh ritz step and hence the solution time particularly when a large number of eigenvalues are to be computed the numerical experiments suggest that the proposed algorithm can yield a saving in solution time of up to 28 for certain eigenproblems copyright 2003 john wiley sons ltd

locality of turbulent cascades in this paper we establish sufficient conditions for locality of turbulent cascades by an exact analysis of the fluid equations no assumptions of homogeneity or isotropy are invoked and the results apply to individual realizations with no resort to any statistical averaging the only requirement is suitable regularity of the turbulent solutions of the fluid equations in the high reynolds number limit corresponding to holder continuity but non differentiability in space we use a smooth filtering approach to resolve the turbulent fields both in space and in scale we discuss several physical cascades to exemplify and clarify the analysis including joint energy and helicity cascades in three space dimensions and the dual inverse energy cascade and direct enstrophy cascade in two dimensions

onsager s conjecture on the energy conservation for solutions of euler s equation we give a simple proof of a result conjectured by onsager 1 on energy conservation for weak solutions of euler s equation

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

minimal surface computation using a finite element method on an embedded surface we suggest a finite element method for finding minimal surfaces based on computing a discrete laplace beltrami operator operating on the coordinates of the surface the surface is a discrete repres

universal direct cascade in two dimensional turbulence we consider the correlation functions of vorticity ensuremath omega in the region of the direct cascade in a steady two dimensional turbulence the nonlocality of the cascade in k space provides for logarithmic corrections to the expressions obtained by dimension estimates and the main problem is to take those logarithms into account our procedure starts directly with the euler equation rewritten in the comoving reference frame we express the correlation functions of the vorticity via the correlation functions of the pumping force and renormalized strain it enables us to establish a set of integrodifferential equations which gives a logarithmic renormalization of the vorticity correlation functions in the inertial interval we find the indices characterizing the logarithmic behavior of different correlation functions for example the two point simultaneous functions are as follows mathrm ensuremath omega mathit n mathbf r 1 mathrm ensuremath omega mathit n mathbf r 2 ensuremath sim mathit p 2 ln l mathbf r 1 mathbf r 2 ensuremath vert 2 mathit n 3 where l is the pumping scale we demonstrate that the form of those correlation functions is universal i e independent of the pumping the only pumping related value which enters the expressions is the enstrophy production rate mathit p 2 the contributions related to pumping rates mathit p mathit n of the higher order integrals of motion are demonstrated to be small in comparison with the ones induced by mathit p 2 we establish also the time dependence of the correlation functions the correlation time ensuremath tau in the comoving reference frame is the same for the vorticity and strain and is scale dependent ensuremath tau ensuremath propto mathrm ln 2 3 l r we reformulate our procedure in the diagrammatic language to reinforce the conclusions

a frequency dependent rectangular plate finite dynamic element for high precision transverse vibration analysis of simply supported plates the inclusion of the twist parameter is shown to be essential using a rational choice of parameters to form the deflection functions a constant term sixteen degrees of freedom rectangular element is developed which is shown to predict exactly the first mode frequency and mode shape when used in whole plate modelling

a subspace forward iteration method for solving the quadratic eigenproblem associated with the fde formulation this paper presents the development of an efficient and numerically stable algorithm for accurately computing the eigensolutions of the quadratic real symmetric eigenproblem arising in the finite dynamic element fde formulation closely related to the subspace forward iteration method the proposed scheme is well suited to extracting the lowest natural frequencies and associated mode shapes of large practical eigenproblems takes full advantage of the banded configuration of the stiffness inertia and dynamic correction matrices involved in the eigenproblem and ensures a monotonic convergence from above to the required eigenpairs a shifting technique for convergence acceleration and an eigenpair verification scheme are also presented numerical examples are shown demonstrating the excellent performance of the solution procedure

quadratic representation of a nonlinear dynamic stiffness matrix and related eigenvalue problems the approximate representation of an exact dynamic stiffness matrix k rho by a quadratic matrix formulation a rho b rho 2 c is studied theoretically in this paper the matrix formulation is formed by expressing the elements of k rho as parabolic functions based on choosing three fixed values of the eigenparameter rho the general bounding properties of the approximate eigenvalues provided by the quadratic matrix formulation are shown to exist provided that the three fixed values are below the lowest pole of the nonlinear stiffness matrix and that the three coefficient matrices a b and c are positive definite it is shown theoretically in this paper that the approximate eigenvalues are either upper or lower bounds of the corresponding exact ones of the exact dynamic stiffness matrix

mass matrices by minimization of modal errors a new approach to constructing mass matrices is presented based on expressing it through use of a variable parameter this allows the mass matrix to be adjusted in such a way that a simple eigenvalue problem get the best solution possible in terms of some error measure this procedure is used to create both diagonal mass matrices and mixed mass matrices 1997 john wiley sons ltd

local energy flux and the refined similarity hypothesis in this paper we demonstrate the locality of energy transport for incompressible euler equations both in space and in scale the key to the proof is the proper definition of a local subscale flux t r which is supposed to be a measure of energy transfer to length scales l at the space point r kraichnan suggested that for such a quantity the refined similarity hypothesis will hold which kolmogorov originally assumed to hold instead for volume averaged dissipation we derive a local energy balance relation for the large scale motions which yields a natural definition of such a subscale flux for this definition a precise form of the refined similarity hypothesis is rigorously proved as a big o bound the established estimate is t r o l3h 1 in terms of the local holder exponenth at the point r which is also the estimate assumed in the parisi frisch multifractal model our method not only establishes locality of energy transfer but it also clarifies the physical reason that convection effects which naively violate locality do not contribute to the subscale flux in fact we show that as a consequence of incompressibility such effects enter into the local energy balance only as the divergence of a spatial current therefore they describe motion of energy in space and cancel in the integration over volume we also discuss theorems of onsager eyink and constantinet al on energy conservation for euler dynamics particularly to explain their relation with the parisi frisch model the constantinet al proof may be interpreted as giving a bound on the total flux t d d r t r of the form t r o lz3h 1 wherez3 is the third order scaling exponent or besov index in agreement with the multifractal model finally we discuss how the local estimates are related to the results of caffarelli kohn nirenberg on partial regularity for solutions of navier stokes equations they provide some heuristic support to a scenario proposed recently by pumir and siggia for singularities in the solutions of navier stokes with small enough viscosity

partial eigensolution of damped structures by non linear optimization the partial solution of damped structures involves the determination of only a few of the resonant frequencies and corresponding damping characteristics and mode shapes the procedure outlined here uses a modification of the variable metric solution algorithm in which the dicision variables are elements of the complex eigenvector and the eigenvalue the quadratic eigenvalue problem is reformulated as a sequential unconstrained minimization problem both normality and orthogonality constraints are incorporated in the objective function the method is compared to an adaptation of the subspace iteration technique also applicable to the quadratic eigenvalue problem

exact results on stationary turbulence in 2d consequences of vorticity conservation we establish a series of exact results for a model of stationary turbulence in two dimensions the incompressible navier stokes equation in a periodic domain with stochastic force white noise in time essentially all of our conclusions follow from the simple consideration of the simultaneous conservation of energy and enstrophy by the inertial dynamics our main results are as follows 1 we show the blow up of mean energy as l02e for 0 when there is no ir dissipation at the large length scale l0 2 with an additional ir dissipation we establish the validity of the traditional cascade directions and magnitudes of flux of energy and enstrophy for 0 assuming finite mean energy in the limit 3 we rigorously establish the balance equations for the energy and vorticity invariants in the 2d steady state and the forward cascade of the higher order vorticity invariants assuming finite mean values 4 we derive exact inequalities for scaling exponents in the 2d enstrophy range as follows if l p l p then 2 23 and p 0 for p 3 if the minimum holder exponent of the vorticity hmin 0 then we establish a 2d analogue of the refined similarity hypothesis which improves these bounds the most novel and interesting conclusion of this work is the connection established between intermittency in 2d and negative holder singularities of the vorticity we show that the latter are necessary for deviations from the 1967 kraichnan scaling to occur

wavelet transforms of self similar processes this paper deals with the wavelet transforms of self similar random processes of the kind assumed in the kolmogorov 1941 theory of turbulence it is shown that after suitable rescaling the wavelet transform at a given position becomes a stationary random function of the logarithm of the scale argument in the transform the rescaling depends on the scaling exponent the statistical properties of the resulting random fluctuations such as its correlation function depend on the choice of the analyzing wavelet some implications are i the presence of fluctuations in log log plots of the absolute value of wavelet transforms versus the scale ii an estimate of the small scale fluctuations generalizing the iterated logarithm law of brownian motion iii an ultraviolet ergodic formula giving the scaling exponents in terms of zoom averages some observations are made on the issue of fluctuations observed in wavelet transforms of turbulence data

finite element procedures for computing normals and mean curvature on triangulated surfaces and their use for mesh refinement in this paper we consider finite element approaches to computing the mean curvature vector and normal at the vertices of piecewise linear triangulated surfaces in particular we adopt a stabilizat

numerical formulation for a higher order plane finite dynamic element the paper describes the development of an eight node plane rectangular finite dynamic element and presents detailed descriptions of the associated numerical formulation involving the higher order dynamic correction terms pertaining to the related stiffness and inertia matrices numerical test results of free vibration analyses are presented in detail for the newly developed eight node element and also the corresponding four node element in order to make a clear comparison of the relative efficiencies of the corresponding finite element and dynamic element procedures such results indicate a superior pattern of solution convergence of the presently developed dynamic element

an accelerated subspace iteration method the subspace iteration method is widely used for the computation of a few smallest eigenvalues and the corresponding eigenvectors of large eigenproblems under certain conditions this method exhibits slow convergence for improving the convergence of the method a few acceleration techniques are already available in the literature in this paper yet another method is presented the method has been applied for the computation of a few smallest eigenpairs of some typical eigenprobiems the results indicate that the acceleration method is efficient for large eigenproblems simplicity ease of computer implementation absence of parameters whose values are to be chosen based on experience and effectiveness for large problems are the attractive features of the proposed method

new concepts for finite element mass matrix formulations improved finite element formulations are presented for dynamic analysis the improvements are based on modifications of the finite element mass matrix and are of high importance since no additional computational effort is required compared with the necessary effort for the standard consistent mass matrix formulation formulations are presented for the one dimensional bar two dimensional membrane and the pure bending beam element free vibration analysis and forced vibration results are presented employing new mass matrices in general the frequency errors and stress errors are at least three times smaller than the corresponding consistent or lumped mass formulation errors accurate model data can be achieved for all the modes of relatively coarse meshes

the basic equations for the large eddy simulation of turbulent flows in complex geometry abstract the equations for large eddy simulation les of turbulent flows are derived by applying a filtering operation to the navier stokes equations les of inhomogeneous turbulent flows requires the use of filters with variable filter width the use of such filters invalidates the standard derivation of the basic equations for the filtered fields since the filtering operation in general does not commute with the operation of differentiation in this paper an alternate definition of the filtering operation based on the mapping function of the nonuniform grid is introduced it is shown that with this modified definition the filtering and differentiation operations commute up to an error which is second order in the filter width it is also shown that the commutation error can be expressed in terms of the filtered field and its derivatives as an asymptotic series in the square of the filter width these results are then applied to the navier stokes equations to derive the basic equations satisfied by the filtered fields

chebyshev filtering lanczos process in the subspace iteration method bathe s basic algorithm of subspace iteration for the solution of the symmetric eigenvalue problem is improved by including a chebyshev filtering mechanism to obtain satisfactory convergence for the largest eigenvalues a shifting strategy is adopted the shift factor is approximately computed by the lanczos process

besov spaces and the multifractal hypothesis parisi and frisch proposed some time ago an explanation for multiscaling of turbulent velocity structure functions in terms of a multifractal hypothesis i e they conjecture that the velocity field has local holder exponents in a range hmin hmax with exponents h occurring on a sets h with a fractal dimensiond h heuristic reasoning led them to an expression for the scaling exponentz p ofpth order as the legendre transform of the codimensiond d h we show here that a part of the multifractal hypothesis is correct under even weaker assumptions namely if the velocity field hasl p mean holder indexs i e if it lies in the besov spaceb p s then local holder regularity is satisfied ifs d p then the hypothesis is true in a generalized sense of holder space with negative exponents and we discuss the proper definition of local holder classes of negative index finally if a certain box counting dimension exists then the legendre transform of its codimension gives the scaling exponentz p and more generally the maximal besov index of order p ass p z p p our method of proof is derived from a recent paper of s jaffard using compactly supported orthonormal wavelet bases and gives an extension of his results we discuss implications of the theorems for ensemble average scaling and fluid turbulence

multilocality and fusion rules on the generalized structure functions in two dimensional and three dimensional navier stokes turbulence using the fusion rules hypothesis for three dimensional and two dimensional navier stokes turbulence we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences we call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator multilocality the resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry our main result is that the fusion rules hypothesis implies both locality and multilocality in both the ir and uv limits for the downscale energy cascade of three dimensional navier stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two dimensional navier stokes turbulence we stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term by term perturbative locality of diagrammatic theories or closure models that involve only two point correlation and response functions

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

development of a solid hexahedron finite dynamic element this paper presents the pertinent details of a newly developed solid rectangular hexahedron finite dynamic element involving the derivation of the higher order stiffness and inertia dynamic correction matrices numerical results of a test case are also presented which indicate that adoption of the dynamic elements significantly improves the solution convergence when compared with the related performance of the corresponding finite elements

inverse energy cascade in two dimensional turbulence deviations from gaussian behavior high resolution numerical simulations of stationary inverse energy cascade in two dimensional turbulence are presented deviations from gaussian behavior of velocity differences statistics are quantitatively investigated the level of statistical convergence is pushed enough to permit reliable measurement of the asymmetries in the probability distribution functions of longitudinal increments and odd order moments which bring the signature of the inverse energy flux no measurable intermittency corrections could be found in their scaling laws the seventh order skewness increases by almost two orders of magnitude with respect to the third thus becoming of order unity

a method for solving high order real symmetric eigenvalue problems a new method for solving structural dynamics problems has been proposed by the author in a separate paper 1 this method of solution produces a high order real symmetric eigenvalue problem of the form a b 2c 3d u 0 an algorithm for solving such an eigenvalue problem using simultaneous iteration is presented in this paper methods of accelerating the convergence and reducing the amount of computation are also described a numerical example is given in which the algorithm is used to calculate the eigenvalues and eigenvectors of a framed structure

free and forced vibration analysis using the smoothed finite element method sfem abstract smoothed finite element method sfem was recently proposed by the authors to improve the accuracy and convergence rate of the existing standard four node isoparametric finite element method fem by 2 2 quadrature in the static analysis in this work it is further extended to the free and forced analysis of two dimensional 2d dynamic problems it is found that sfem achieves more accurate results and higher convergence rates as compared with the corresponding finite elements in solving dynamic problems without increasing the computational cost as only shape function itself is involved in computing the field gradients and no coordinate transformation is required the method is very easy to implement nonlinear elastic forced vibration problem is also examined by using smoothed deformation gradients on each cell

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

high order unfitted finite element methods on level set domains using isoparametric mappings abstract we introduce a new class of unfitted finite element methods with high order accurate numerical integration over curved surfaces and volumes which are only implicitly defined by level set functions an unfitted finite element method which is suitable for the case of piecewise planar interfaces is combined with a parametric mapping of the underlying mesh resulting in an isoparametric unfitted finite element method the parametric mapping is constructed in a way such that the quality of the piecewise planar interface reconstruction is significantly improved allowing for high order accurate computations of unfitted domain and surface integrals we present the method discuss implementational aspects and present numerical examples which demonstrate the quality and potential of this method

a simple mesh generator in matlab creating a mesh is the first step in a wide range of applications including scientific computing and computer graphics an unstructured simplex mesh requires a choice of meshpoints vertex nodes

frequency and time domain analysis of linear systems with frequency dependent parameters abstract a modal analysis scheme is developed for systems with frequency dependent parameters which are assumed to be well represented by even order polynomials the systems with frequency dependent parameters are transformed into equivalent systems with real constant parameters which give rise to identical frequency response matrices two example problems are considered in order to investigate the dynamic characteristics of the systems with frequency dependent parameters by using the proposed method

acceleration of subspace iteration an algorithm is presented and discussed which gives 40 per cent reduction in the cost of the basic subspace iteration for eigenvalues

an improved subspace iteration method with shift for structures with multiple natural frequencies an efficient and numerically stable eigensolution method for structures with multiple natural frequencies is presented the proposed method is developed by improving the well known subspace iteration method with shift a major difficulty of the subspace iteration method with shift is that because of singularity problem a shift close to an eigenvalue cannot be used resulting in slower convergence in this paper the above singularity problem has been solved by introducing side conditions without sacrifice of convergence the proposed method is always non singular even if a shift is on a distinct eigenvalue or multiple ones this is one of the significant characteristics of the proposed method the non singularity is proved analytically the convergence of the proposed method is at least equal to that of the subspace iteration method with shift and the operation counts of the above two methods are almost the same when a large number of eigenpairs are required to show the effectiveness of the proposed method two numerical examples are considered

a new approach to the dynamic analysis of structures using fixed frequency dynamic stiffness matrices the dynamic analysis of structures by the standard finite element method introduces additional inaccuracies into the solution which are not present when the method is used for static analyses these inaccuracies can arise from two sources i the element formulation and ii the reduction of the size of the matrices by a static condensation i e using the guyan method1 2 the errors in both cases are caused by neglecting frequency dependent terms in the functions relating the displacements at any point in the structure to the displacements at certain fixed points i e nodes in the element formulation and masters in the condensation a new method of solution is proposed in this paper in which the frequency dependent terms are retained implicitly by using dynamic stiffness matrices defined at a number of fixed frequencies the dynamic stiffness matrices may be condensed efficiently to a relatively small number of master degrees of freedom using a front solution algorithm the final stage in the solution uses these matrices to synthesize a high order eigenvalue problem a method of solving such an eigenvalue problem of arbitrary order is described in a separate paper 3 numerical examples are given to show the accuracy and efficiency of the proposed method compared with conventional methods of solution

computation of three dimensional dendrites with finite elements starting from an initial seed crystal inside an undercooled liquid the solid phase begins to grow rapidly and develops unstable growth patterns some growth directions are preferred because of anisotropic parameters in the physical model this results in the development of dendrites the physical model includes the heat equation for both the liquid and solid phases the gibbs thomson law couples velocity and curvature of the interface and the temperature we describe a numerical method that enables us to compute dendritic growth of crystals in two and three space dimensions the method consists of two coupled finite element algorithms the first one solves the heat equation the other operates on a discretization of the free boundary and computes the evolution of this moving interface the two methods work with totally independent grids by using timedependent locally refined and coarsened adaptive meshes in both methods we are able to reach a spatial resolution necessary to compute dendritic growth in two and three space dimensions

stabilized finite element approximation of the mean curvature vector on closed surfaces the mean curvature vector of a surface is obtained by letting the laplace beltrami operator act on the embedding of the surface in bf r 3 in this contribution we develop a stabilized finite element approximation of the mean curvature vector of certain piecewise linear surfaces which enjoys first order convergence in l 2 the stabilization involves the jump in the tangent gradient in the direction of the outer co normals at each edge in the surface mesh we consider both standard meshed surfaces and so called cut surfaces that are level sets of piecewise linear distance functions we prove a priori error estimates and verify the theoretical results numerically

smooth finite element methods convergence accuracy and properties a stabilized conforming nodal integration finite element method based on strain smoothing stabilization is presented the integration of the stiffness matrix is performed on the boundaries of the finite elements a rigorous variational framework based on the hu washizu assumed strain variational form is developed we prove that solutions yielded by the proposed method are in a space bounded by the standard finite element solution infinite number of subcells and a quasi equilibrium finite element solution a single subcell we show elsewhere the equivalence of the one subcell element with a quasi equilibrium finite element leading to a global a posteriori error estimate we apply the method to compressible and incompressible linear elasticity problems the method can always achieve higher accuracy and convergence rates than the standard finite element method especially in the presence of incompressibility singularities or distorted meshes for a slightly smaller computational cost it is shown numerically that the one cell smoothed four noded quadrilateral finite element has a convergence rate of 2 0 in the energy norm for problems with smooth solutions which is remarkable for problems with rough solutions this element always converges faster than the standard finite element and is free of volumetric locking without any modification of integration scheme copyright 2007 john wiley sons ltd

two step component mode synthesis for the eigensolution of large systems component mode synthesis is a dynamic substructuring technique for approximate eigensolutions of large degree of freedom systems divisible into two or more components system synthesis using component modes results in approximate eigenparameters the approximation is improved by using more component modes in synthesis typically all component modes up to x times the frequency range of interest are required to ensure reasonable accuracy where x varies from 2 to 10 a two step component mode synthesis approach is presented the first step involves system synthesis using the minimum number of component modes required to obtain approximate cigenvalues up to a preselected frequency and the second step uses additional component modes in a convergence scheme operating on the system eigenparameters calculated in the first step this results in an infinite series eigenvalue problem that is solved by a modified shooting method a perturbation approach is adopted to derive the convergence scheme in which separate iterations are performed for each eigenvalue and eigenvector in each step selection of initial values for the convergence scheme is also presented in a few instances an eigenvalue nearly reached the convergence tolerance but then wandered away and converged to an adjacent value the algorithm was modified to automatically detect this problem and abort iteration with a small penalty in accuracy this two step approach provides the analyst with some insight into the convergence and accuracy of the eigenparameters without resolving an increasingly larger system over and over with more component modes numerical examples are included to demonstrate the applicability of the method

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality