a multilevel computational strategy for handling microscopic and macroscopic instabilities this paper presents a numerical technique to deal with instability phenomena in the context of heterogeneous materials where buckling may occur at both macroscopic and or microscopic scales we limit ourselves to elastic materials but geometrical nonlinearity is taken into account at both scales the proposed approach combines the multilevel finite element analysis fe2 and the asymptotic method anm in that framework the unknown nonlinear constitutive relationship at the macroscale is found by solving a local finite element problem at the microscale in contrast with fe2 the use of the asymptotic development allows to transform the nonlinear microscopic problems into a sequence of linear problems thus a direct analogy with classical linear homogenization can be made to construct a localisation tensor at each step of the asymptotic development and an explicit macroscopic constitutive relationship can be constructed at each step furthermore the salient features of the anm allow treating instabilities and limit points in a very simple way at both scales the method is tested and illustrated through numerical examples involving local instabilities which have ignificant influence on the macroscopic behavior

bifurcation points and bifurcated branches by an asymptotic numerical method and pade approximants the aim of this work is to develop a reliable and fast algorithm to compute bifurcation points and bifurcated branches it is based upon the asymptotic numerical method anm and pade approximants the bifurcation point is detected by analysing the poles of pade approximants or by evaluating along the computed solution branch a bifurcation indicator well adapted to anm several examples are presented to assess the effectiveness of the proposed method that emanate from buckling problems of thin elastic shells especially problems involving large rotations are discussed copyright 2004 john wiley sons ltd

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

asymptotic numerical methods for unilateral contact new algorithms based upon the asymptotic numerical method anm are proposed to solve unilateral contact problems anm leads to a representation of a solution path in terms of series or pade approximants to get a smooth solution path a hyperbolic relation between contact forces and clearance is introduced three key points are discussed the influence of the regularization of the contact law the discretization of the contact force by lagrange multipliers and prediction correction algorithms simple benchmarks are considered to evaluate the relevance of the proposed algorithms

extracting spectral envelopes formant frequency matching between sounds on different and modulated fundamental frequencies the four experiments reported here measure listeners accuracy and consistency in adjusting a formant frequency of one or two formant complex sounds to match the timbre of a target sound by presenting the target and the adjustable sound on different fundamental frequencies listeners are prevented from performing the task by comparing the absolute or relative levels of resolved spectral components experiment 1 uses two formant vowellike sounds when the two sounds have the same f0 the variability of matches within subject standard deviation for either the first or the second formant is around 1 3 which is comparable to existing data on formant frequency discrimination thresholds with a difference in f0 variability increases to around 8 for first formant matches but to only about 4 for second formant matches experiment 2 uses sounds with a single formant at 1100 or 1200 hz with both sounds on either low or high fundamental frequencies the increase in variability produced by a difference in f0 is greater for high f0 s where the harmonics close to the formant peak are resolved than it is for low f0 s where they are unresolved listeners also showed systematic errors in their mean matches to sounds with different high f0 s the direction of the systematic errors was towards the most intense harmonic experiments 3 and 4 showed that introduction of a vibratolike frequency modulation fm on f0 reduces the variability of matches but does not reduce the systematic error the experiments demonstrate for the specific frequencies and fm used that there is a perceptual cost to interpolating a spectral envelope across resolved harmonics

an asymptotic numerical method to compute the postbuckling behaviour of elastic plates and shells in this paper we apply an asymptotic numerical method for computing the postbuckling behaviour of plate and shell structures the bifurcating branch is sought in the form of polynomial expansions and it is determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can easily be computed by using recurrent formulas in comparison with a more classical step by step procedure the method is rapid and automatic however the polynomial expansions have a radius of convergence which limits the validity of the solution to a neighbourhood of the bifurcation point in the present form the method should be viewed as a cheap and automatic way of completing a linear buckling analysis it is illustrated in two examples a square plate under in plane compression and a cylindrical shell under pressure

the trajectory tracking problem for an unmanned four rotor system flatness based approach in this article we present a flatness based controller that solves the trajectory tracking problem for an unmanned four rotor aircraft a simple version of the dynamical system obtained from a suitable coordinate transformation of the euler lagrange model of the system is used this model combined with a suitable input saturation nonlinearity also avoids the inconvenient system singularities the corresponding tracking error is shown to be globally asymptotically stable but only locally exponentially stable while the controller maintains the four rotor system motions inside a feasible region numerical simulations are presented to assesses the effectiveness of the proposed control strategy

computing finite rotations of shells by an asymptotic numerical method abstract we present an asymptotic numerical algorithm for the computation of elastic shells with large rotations the theoretical formulation involves a three field hu washizu functional which allows us to put the problem into a quadratic framework the spatial discretization is based on geometrically exact element recently presented by buchter et al buchter et al three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept int j numer methods engrg 37 1994 2551 1568 several classical benchmarks are discussed to define the best strategy and to assess the validity and the efficiency of the present method as compared to more classical iterative algorithms

nonlinear finite element analysis of elastic systems under nonconservative loading natural formulation part ii dynamic problems abstract the paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces attention is focused on the stability behaviour of such systems this leads mathematically to non self adjoint boundary value problems which are of great theoretical and practical interest in particular in connection with alternative modes of instability like divergence of flutter only quasistatic effects are however considered in the present part the methodology of our theory is general but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement dependent nonconservative follower forces in a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives in general rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix as a result the total tangent stiffness matrix becomes unsymmetric an indication of the non self adjoint character of the problem our theory is based on the natural mode technique 1 2 3 and permits i a a simple but elegant derivation of the load correction matrix the application of the general theory as evolved in this paper is demonstrated on the beam element in space a number of numerical examples are considered including divergence and flutter types of instability for which exact analytic solutions are known the problems demonstrate the efficiency of the present finite element formulation the paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in 2 3

tracing post limit point paths with reduced basis technique abstract a reduced basis technique and a problem adaptive computational algorithm are presented for predicting the post limit point paths of structures in the proposed approach the structure is discretized by using displacement finite element models the nodal displacement vector is expressed as a linear combination of a small number of vectors and a rayleigh ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations to circumvent the difficulties associated with the singularity of the stiffness matrix at limit points a constraint equation defining a generalized arc length in the solution space is added to the system of nonlinear algebraic equations and the rayleigh ritz approximation functions or basis vectors are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc length the potential of the proposed approach and its advantages over the reduced basis load control technique are outlined the effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap through and snap back phenomena

a high order purely frequency based harmonic balance formulation for continuation of periodic solutions combining the harmonic balance method hbm and a continuation method is a well known technique to follow the periodic solutions of dynamical systems when a control parameter is varied however since deriving the algebraic system containing the fourier coefficients can be a highly cumbersome procedure the classical hbm is often limited to polynomial quadratic and cubic nonlinearities and or a few harmonics several variations on the classical hbm such as the incremental hbm or the alternating frequency time domain hbm have been presented in the literature to overcome this shortcoming here we present an alternative approach that can be applied to a very large class of dynamical systems autonomous or forced with smooth equations the main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the hbm once the equations have been rendered quadratic it becomes obvious to derive the algebraic system and solve it by the so called asymptotic numerical method anm continuation technique several classical examples are presented to illustrate the use of this numerical approach

finite rotation elements for the non linear analysis of thin shell structures abstract for the numerical analysis of shells undergoing finite rotations doubly curved finite shell elements are developed via the displacement formulation the derivation starts from a consistent finite rotation shell theory which is transformed by a variational procedure into an incremental formulation thus the non linearity can be treated by an incremental iterative technique the non linear element matrices are obtained by a tensor oriented procedure permitting a direct transformation of the initial equations into efficient numerical models unlike in the usual procedure the kirchhoff love assumption is treated as a subsidiary condition at the element level this computer oriented approach permits the elimination of the dependent rotational degrees of freedom without loss of accuracy finally some examples are presented to demonstrate the ability of the resulting finite elements to deal with finite rotation problems

a parallel computer implementation of the asymptotic numerical method to study thermal convection instabilities we have developed a numerical model to efficiently compute steady state combined buoyancy and thermocapillary convection solutions it features a parallel computer implementation of an asymptotic numerical method to perform steady state path following and locate bifurcation points in problems involving large size algebraic systems up to few million degrees of freedom the model has been first validated on a problem for which a reference solution exists and then is used to analyse the influence of the container size and shape on rayleigh benard marangoni convection and its related cellular pattern

frequency discrimination of stylized synthetic vowels with two formants just noticeable differences jnd s in the formant frequencies of synthetic two formant vowels were measured for normal hearing subjects the jnd s were examined for a change in only the first or the second formant and for a combined change of both formants for the combined change two quantitative relations between the formant frequencies were used one with equal relative changes in both formants and one with a double relative change for the first formant formant frequencies were 500 550 and 600 hz in the first and 2000 2050 and 2100 hz in the second formant region both formants had either shallow or steep slopes for the fundamental frequency of the complexes we used 100 and 200 hz for the single formant changes a natural and a random phase relation were used between the individual components of the complexes these results were compared to jnd s for a gaussian noise that was filtered with the same spectral envelopes as the harmonic complexes for the combined formant changes only the natural phase relation was used a three interval three alternative forced choice task was used all measurements were performed with roving stimulus level for the single formant changes the phase relations had no effect on the results for the harmonic stimuli jnd s were mostly smaller for the formants between two harmonics than for those at a harmonic the results for the harmonic stimuli as well as the noise bands could be described by a model using a spectral profile comparison for the combined formant changes smaller jnd s were found than for the single changes these jnd s could be explained by combining measures of the perceived differences from the two separately changed formants in this combination these measures were summed as independent variables

fundamental frequency effects on thresholds for vowel formant discrimination in a recent report thresholds for discrimination of vowel formants are described as a piecewise linear function that is flat in the f1 region and linearly increasing in the f2 region kewley port and watson j acoust soc am 95 485 496 1994 results also suggested a trend for higher fundamental frequency to produce increased formant thresholds for these synthetic female vowels further study of the effect of various stimulus parameters on formant thresholds is needed to describe discrimination using excitation pattern models the present experiment investigated the effect of glottal source on formant thresholds in synthesized male vowels thresholds for formant frequency discrimination were obtained for six vowels with two fundamental frequencies high 126 hz and low 101 hz four well trained subjects performed an adaptive tracking task under minimal uncertainty conditions discrimination thresholds for vowels with high f0 were significantly greater than those with low f0 consistent with resu

an excursion into large rotations the present discourse develops an enlarged exploration of the matrix formulation of finite rotations in space initiated in 1 it is shown how a consistent but subtle matrix calculus inevitably leads to a number of elegant expressions for the transformation or rotation matrix t appertaining to a rotation about an arbitrary axis also analysed is the case of multiple rotations about fixed or follower axes particular attention is paid to an explicit derivation of a single compound rotation vector equivalent to two consecutive arbitrary rotations this theme is discussed in some detail for a number of cases semitangential rotations for which commutativity holds first proposed in 2 3 are also considered furthermore an elementary geometrical analysis of large rotations is also given finally we deduce in an appendix using a judicious reformulation of quarternions the compound pseudovector representing the combined effect of n rotations in the author s opinion the present approach appears preferable to a pure vectorial scheme and even more so to an indicial formulation and is computationally more convenient

asymptotic numerical method for problems coupling several nonlinearities abstract the objective of this paper is to assess the efficiency of the asymptotic numerical method to solve problems coupling various nonlinearities the 3d hemispherical stretching of a circular sheet that involves geometrical material and red unilateral contact nonlinearities is chosen as an example an elastoplastic model based on the plasticity deformation theory is adopted the structural discretization is performed by a shell finite element well adapted for problems involving large displacements and large rotations the unilateral contact problem is identified to boundary conditions which are replaced by force displacement relations and solved using a special algorithm comparisons with results obtained by the help of an industrial code establish the interest and the performance of the present method

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects