natural frequencies of structures with interval parameters abstract this paper deals with the evaluation of the lower and upper bounds of the natural frequencies of structures with uncertain but bounded parameters the solution of the generalized interval eigenvalue problem is pursued by taking into account the actual variability and dependencies of uncertain structural parameters affecting the mass and stiffness matrices to this aim interval uncertainties are handled by applying the improved interval analysis via extra unitary interval eui recently introduced by the first two authors by associating an eui to each uncertain but bounded parameter the cases of mass and stiffness matrices affected by fully disjoint completely or partially coincident uncertainties are considered then based on sensitivity analysis it is shown that the bounds of the interval eigenvalues can be evaluated as solution of two appropriate deterministic eigenvalue problems without requiring any combinatorial procedure if the eigenvalues are monotonic functions of the uncertain parameters then the exact bounds are obtained the accuracy of the proposed method is demonstrated by numerical results concerning truss and beam structures with material and or geometrical uncertainties

global existence and exponential stability in one dimensional nonlinear thermoelasticity with thermal memory in this paper we study the global existence uniqueness and asymptotic behaviour of solutions to the equations of one dimensional nonlinear thermoelasticity with thermal memory subject to dirichlet dirichlet boundary conditions for the temperature di5erence and displacement respectively the reference con6guration under consideration is the unit interval 0 1 the equations under consideration read as follows

propagation of singularities for semilinear hyperbolic initial boundary value problems in one space dimension on etudie la propagation et la reflexion de la regularite pour des systemes semi lineaires strictement hyperboliques a 1 variable d espace

maximum entropy power spectrum estimation with uncertainty in correlation measurements the purpose of this paper is to present a multidimensional mem algorithm valid for nonuniformly sampled arrays which satisfies a correlation approximating constraint to this end the correlation matching equality constraints of the usual mem are replaced by a single inequality constraint whose form is based on a measure of the noise in the given autocovariance function acf in this way one can incorporate into the model knowledge of the noisy nature of the given acf since the given acf is usually estimated from the samples of the wavefield specifically the covariance matrix of the correlation estimates is used in a quadratic form that weights the difference between the given acf and the one matched by the power spectrum the maximization of entropy under this inequality constraint leads ultimately to a steepest descent algorithm the algorithm has been tested with 1 d synthetic data representing multiple sinusoids buried in additive white noise the performance of this modified mem algorithm is compared to a traditional mem algorithm for extendible acf s and for different snr s examples of the mem spectrum are given for the case of nonextendible acf s

eigenvalue bounds of structures with uncertain but bounded parameters many analysis and design problems in engineering and science involve uncertainty to varying degrees this paper is concerned with the structural vibration problem involving uncertain material or geometric parameters specified as bounds on these parameters this produces interval stiffness and mass matrices and the problem is transformed into a generalized interval eigenvalue problem in interval mathematics however tighter bounds on the eigenvalues may be obtained by using the formulation of the structural dynamic problem often the stiffness and mass matrices can be formed as a non negative decomposition in the uncertain structural parameters in this case the eigenvalue bounds may be obtained from the parameter vertex solutions even more efficiently using interval extension from interval mathematics the generalized interval eigenvalue problem may be divided into two generalized eigenvalue problems for real symmetric matrix pairs the parameter vertex solution algorithm is compared with deif s solution the eigenvalue inclusion principle and the interval perturbation method in numerical examples

on a coupled nonlinear singular thermoelastic system abstract for a coupled nonlinear singular system of thermoelasticity with one space dimension we consider its initial boundary value problem on an interval for one of the unknowns a classical condition is replaced by a nonlocal constraint of integral type because of the presence of a memory term in one of the equations and the presence of a weighted boundary integral condition the solution requires a delicate set of techniques we first solve a particular case of the given nonlinear problem by using a functional analysis approach on the basis of the results obtained and an iteration method we establish the well posedness of solutions in weighted sobolev spaces

nonprobabilistic convex theoretic modeling of scatter in material properties nonprobabilistic convex modeling of uncertain material properties for viscoelastic structures is developed in this paper in particular the problem of forced vibrations of viscoelastic beams is studied first the analytic solution by inman is generalized for a deterministic set of variables describing material properties next these variables are treated as varying in a solid ball in the four dimensional space thus modeling the scatter in material properties the least favorable response needed for the design of the structure is determined

on some nonlinear hyperbolic systems with damping boundary conditions in recent years the effect of boundary damping on the solutions to interior initial boundary value problems for hyperbolic equations has been extensively studied in 111 quinn and russell considered the wave equation with damping boundary condition on a part of the boundary it was proved by control theoretic method that if the boundary where the boundary condition was posed was star shaped then the solution of the wave equation had polynomial decay rate later on chen in l got the exponential decay of solutions to the wave equation under the more restrictive assumptions on the boundary and also lagnese in 6 got the exponential decay of solutions to the three dimensional elastic wave equations with damping boundary condition on a part of the boundary provided the geometry of the boundary is suitably restricted as for the nonlinear problem greenberg and li in 131 considered the nonlinear conservation system in one space dimension with damping boundary condition and proved the global existence of smooth solution when the initial data are small shen and zheng 14 and nagasawa 9 proved the global existence of classical solutions to the system of one dimensional thermoelasticity and to the system of one dimensional nonlinear problem in higher space dimension respectively for the nonlinear problem in higher space dimension to our knowledge the works were done by qin lo and zajaczkowski 26 only for the nonlinear wave equation with damping boundary conditions based on chen s results in l in this paper our aim is to derive the decay rate of solutions to the linear hyperbolic system of second order with damping boundary condition without any assumptions on the geometry of the boundary by using the spectral analysis method cf section 2 below and then to prove the global existence of smooth solutions to the corresponding nonlinear system with applications to nonlinear elastodynamics and to nonlinear acoustic wave equations our approach to obtaining the rate of decay of solutions to linear problem is completely different from l 6 and lo

on a global existence theorem of small amplitude solutions for nonlinear wave equations in an exterior domain soit un domaine non borne dans un espace euclidien a n dimensions r n a frontiere compacte et c on considere le probleme mixte u u f t x d 1 d 1 u f t x dans 0 u 0 sur 0 u 0 x u 0 x t u 0 x u 1 x dans f est une fonction non lineaire a valeur reelle et d 1 d 1 u t j x u 1 j 2

microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems on developpe un calcul d operateurs pseudodifferentiels a coefficients non lisses afin d etudier la regularite des solutions des equations lineaires p x d u f

structured pseudospectra and random eigenvalues problems in vibrating systems this paper introduces the concept of pseudospectra as a generalized tool for uncertainty quantification and propagation in structural dynamics different types of pseudospectra of matrices and matrix polynomials are explained particular emphasis is given to structured pseudospectra for matrix polynomials which offer a deterministic way of dealing with uncertainties for structural dynamic systems the pseudospectra analysis is compared with the results from monte carlo simulations of uncertain discrete systems two illustrative example problems one with probabilistic uncertainty with various types of statistical distributions and the other with interval type of uncertainty are studied in detail excellent agreement is found between the pseudospectra results and monte carlo simulation results

stochastic interval analysis of natural frequency and mode shape of structures with uncertainties abstract in this paper natural frequencies and mode shapes of structures with mixed random and interval parameters are investigated by using a hybrid stochastic and interval approach expressions for the mean value and variance of natural frequencies and mode shapes are derived by using perturbation method and random interval moment method the bounds of these probabilistic characteristics are then determined by interval arithmetic two examples are given first to illustrate the feasibility of the presented method and the results are verified by monte carlo simulations the presented approach is also applicable to solve pure random and pure interval problems this capability is demonstrated in the third and fourth examples through the comparisons with the peer research outcomes

propagation of singularities in one dimensional thermoelasticity the propagation of singularities for the system of homogeneous thermoelasticity in one space dimension is studied linear and a class of semilinear cauchy problems are considered

global smooth solutions and asymptotic stability in one dimensional nonlinear thermoelasticity the global existence of smooth solutions to the equations of nonlinear thermoelasticity is shown for a one dimensional homogeneous reference configuration dirichlet boundary conditions are studied and the asymptotic behaviour of the solutions as t is described

a minimum free energy hybrid algorithm for 2 d spectral estimation a separable quasi 2 d spectral estimation algorithm is presented this hybrid algorithm combines the 1 d minimum free energy method with the 1 d periodogram the algorithm reduces to the corresponding maximum entropy hybrid spectral estimator in some cases the minimum free energy hybrid algorithm is preferable to the maximum entropy version when the estimated autocorrelation sequence is not extendable or when the signal to noise ratio is low a simulation helps illustrate this point

blow up of solutions to the cauchy problem in nonlinear one dimensional thermoelasticity this paper is concerned with the blow up phenomena of solutions to the cauchy problem in non autonomous nonlinear one dimensional thermoelastic models obeying both fourier s law of heat flux and the theory due to gurtin and pipkin moreover some previously related results have been extended

two dimensional ar spectral estimation using a two dimensional minimum free energy method we present a 2 d extension of the minimum free energy mfe autoregressive parameter and spectral estimation technique mfe can outperform the multidimensional levinson algorithm producing superior spectral estimation of 2 d sinusoids in white gaussian noise mfe spectral estimates are found to be comparable to modified covariance method estimates mfe also outperforms the conventional fourier method for closely spaced sinusoids at low signal to noise ratio and low data set size

conormal and piecewise smooth solutions to quasilinear wave equations in this paper we show first that if a solution u of the equation p2 t x u du d u f t x u du where p2 t x u du d is a second order strictly hyperbolic quasilinear operator is conormal with respect to a single characteristic hypersurface i of p2 in the past and i is smooth in the past then i is smooth and u is conormal with respect to i for all time second let z0 and z1 be characteristic hypersurfaces of p2 which intersect transversally and let r 0 n ii if 4o and ii are smooth in the past and u is conormal with repect to o 11 in the past then r is smooth and u is conormal with respect to 1o ii locally in time outside of r even though z0 and ii are no longer necessarily smooth across r finally we show that if u o x and tu o x are in an appropriate sobolev space and are piecewise smooth outside of r then u is piecewise smooth locally in time outside of iouz1

an interval algorithm for uncertain dynamic stability analysis abstract analysis on the stability of dynamic systems is a very important field in structural elasticity theory for possible engineering situations where the system parameters are uncertain but bounded the propagation from the initial uncertainties of parameters to the consequent uncertainty of dynamic stability of structure is elaborately investigated by interval analysis for the first time the interval dynamic stability issue is studied through the column structure as a generalized interval eigenvalue problem and an analytic theorem is rigorously derived and numerically verified for solving the generalized interval eigenvalue problem moreover to avoid the exaggeration on actual bounds of solution an effective approach is established to limit the dependency phenomenon in the interval dynamic stability analysis the interval relationships of the excitation parameter and the critical load frequency with five types of system parameters are further bridged according to the interval arithmetic to propose a three stage interval scheme for evaluating the effects of interval system parameters on the dynamic stability of structures numerical studies demonstrate that the uncertainty of the constant part of load influences the boundaries of principal instability region much more than that of the variable part of load within all structural parameters the uncertainty of column length has the most effect on the boundaries which could make the critical frequency of load be magnified about ten times the uncertainty of parameter in particular if all the system parameters are interval with an identical uncertainty degree the uncertainty of parameter would be propagated in the system of dynamic stability and enlarged as high as 20 times the uncertainty of parameter consequently the impacts of parameter uncertainties on the dynamic stability of structures are fairly significant

the time response of structures with bounded parameters and interval initial conditions uncertainty plays an important role in the performance of structures in this paper we focus on the dynamic response of structures with bounded parameters and interval initial conditions and present a new method to determine the supremum and infimum of the time response the method is based on the vertex solution theorem for the first order deviation of the dynamic response from its central value and avoids interval extension problems present in current methods where the length of the interval increases significantly due to the intermediate calculations the method is more accurate than existing perturbation methods and provides tighter bounds on the response the approach neglects the second order terms in the equation of motion and care should be exercised when the parameter variations are large the other advantage of this method is its ability to solve problems with uncertainties in the initial conditions

propagation of singularities in cauchy problems for quasilinear thermoelastic systems in three space variables by using microlocal analysis the propagation of weak singularities in cauchy problems for quasilinear thermoelastic systems in three space variables are investigated first paradifferential operators are employed to decouple the quasilinear thermoelastic systems second by investigating the decoupled hyperbolic parabolic systems and using the classical bootstrap argument the property of finite propagation speeds of singularities in cauchy problems for the quasilinear thermoelastic systems is obtained finally it is shown that the microlocal weak singularities for cauchy problems of the thermoelastic systems are propagated along the null bicharacteristics of the hyperbolic operators

singular solutions for semilinear hyperbolic equations i on construit des solutions avec singularites de type choc pour des equations semilineaires strictement hyperboliques

global existence and asymptotic behavior for one dimensional thermoelasticity of integral type with hyperbolic heat conduction abstract we consider an initial boundary value problem for thermoelasticity of integral type with hyperbolic heat conduction we show that for small initial data in h 1 norm there exist globally defined solutions which are exponentially stable in bounded spatial domains

singularities of solutions to semilinear boundary value problems soit p un operateur differentiel d ordre 2 strictement hyperbolique sur une region ouverte cr n a frontiere lisse non caracteristique etant donnee une solution u h s loc s n 1 2 de pu f x u on etudie la propagation des singularites microlocales h r pour s r 2n n 2 1 2 dans la situation generale ou le champ hamiltonien de p hp peut etre tangent a t o a un ordre infini ou fini arbitrairement eleve

interaction of nonlinear progressing waves for semilinear wave equations on etudie l interaction transversale de 3 ondes planes progressives pour une equation d onde semi lineaire a 3 dimensions

remarks on propagation of singularities in thermoelasticity the propagation of high order weak singularities for the system of homogeneous thermoelasticity in one space variable is studied by using paralinearization and a new decoupling technique introduced by the author microlocal analysis in nonlinear thermoelasticity to appear for the linear system one shows that the nonsmooth initial data for the parabolic part lead to singularities in the hyperbolic part of solutions even when the initial data for that part are identically zero both the cauchy problem and the problem inside of a domain for the semilinear system are considered it is shown that the propagation of high order singularities is essentially dominated by the hyperbolic operator in the system of thermoelasticity

linear minimum free energy estimation a computationally efficient noise suppression spectral estimation algorithm simplified linear versions of the nonlinear minimum free energy mfe noise suppression algorithms are introduced the lmfe algorithms result from the addition of a smoothness penalty function to the linear prediction cost function the autoregressive ar parameters are chosen commensurate with a global minimum of the modified cost function it is shown that this constrained optimization procedure reduces to a form of matrix regularization the lmfe algorithms are calculable in real time as they require a negligible increase in complexity over the conventional autoregressive algorithms including fast computational versions thereof linear mfe extensions are applicable to all conventional ar algorithms and should in each case substantially increase the useful snr range of these algorithms simulation results illustrating the single snapshot performance of these algorithms are given for both narrowband sources and combinations of narrowband and broadband sources subjected to various levels of gaussian white noise

high dimensional model representation method for fuzzy structural dynamics abstract uncertainty propagation in multi parameter complex structures possess significant computational challenges this paper investigates the possibility of using the high dimensional model representation hdmr approach when uncertain system parameters are modeled using fuzzy variables in particular the application of hdmr is proposed for fuzzy finite element analysis of linear dynamical systems the hdmr expansion is an efficient formulation for high dimensional mapping in complex systems if the higher order variable correlations are weak thereby permitting the input output relationship behavior to be captured by the terms of low order the computational effort to determine the expansion functions using the cut method scales polynomically with the number of variables rather than exponentially this logic is based on the fundamental assumption underlying the hdmr representation that only low order correlations among the input variables are likely to have significant impacts upon the outputs for most high dimensional complex systems the proposed method is first illustrated for multi parameter nonlinear mathematical test functions with fuzzy variables the method is then integrated with a commercial finite element software adina modal analysis of a simplified aircraft wing with fuzzy parameters has been used to illustrate the generality of the proposed approach in the numerical examples triangular membership functions have been used and the results have been validated against direct monte carlo simulations it is shown that using the proposed hdmr approach the number of finite element function calls can be reduced without significantly compromising the accuracy

line array beamforming using linear prediction for aperture interpolation and extrapolation it is demonstrated that linear prediction can be successfully utilized in conjunction with otherwise conventional narrow band line array beamforming for aperture extrapolation apex where there are significant snr signal to noise ratio and resolution advantages compared to the conventional approach aperture interpolation where faulty sensor data are replaced and a combined technique alpinex the performance of all three methods is investigated in connection with both simulated and real data the apex technique is shown to be amenable to a simple theoretical treatment further the approach is demonstrated to offer a practical performance comparable to the well known high resolution technique music when the observation time is short

global existence of small solutions to the initial value problem for nonlinear thermoelasticity abstract we prove the existence of global small solutions to the initial value problem in three dimensional nonlinear thermoelasticity with quadratic completely coupled nonlinear terms the asymptotic behaviour of the solution as t is also described

propagation of mild singularities in higher dimensional thermoelasticity the propagation of mild singularities for the semilinear model of three dimensional thermoelasticity is studied it is shown that the propagation picture of such singularities of the solution to the semilinear model coincides with one of the solutions to the corresponding linear model as a simple consequence of our method a similar result for the full semilinear cauchy problem of one dimensional thermoelasticity is also presented

computation of bounds for eigenvalues of structures with interval parameters in this paper the computation of eigenvalue bounds for generalized interval eigenvalue problem is considered two algorithms based on the properties of continuous functions are developed for evaluating upper and lower eigenvalue bounds of structures with interval parameters the method can provide the tightest bounds within a given precision numerical examples illustrate the effectiveness of the proposed method

a spectral approach for fuzzy uncertainty propagation in finite element analysis abstract uncertainty propagation in complex engineering systems with fuzzy variables constitutes a significant challenge this paper proposes a polynomial chaos type spectral approach based on orthogonal function expansion a fuzzy variable is represented as a set of interval variables via the membership function the interval variables are further transformed into the standard interval 1 1 smooth nonlinear functions of standard interval variables are projected in the basis of legendre polynomials by exploiting its orthogonal properties over the interval 1 1 the coefficients associated with the basis functions are obtained by a galerkin type of error minimisation the method is first illustrated using scalar functions of multiple fuzzy variables later the method is proposed for elliptic type finite element problems where the technique is extended to vector valued functions with multiple fuzzy variables the response of such systems can be expressed in the complete basis of multivariate legendre polynomials the coefficients obtained by galerkin type of error minimisation can be calculated from the solution of an extended set of linear algebraic equations an eigenfunction based model reduction technique is proposed to obtain the coefficient vectors in an efficient way a numerical example of axial deformation of a rod with fuzzy axial stiffness is considered to illustrate the proposed methods linear and nonlinear membership functions are used and the results are compared with direct numerical simulation results

exponential energy decay in a linear thermoelastic rod abstract we examine the stability of solutions to a pair of coupled linear partial differential equations which describe the temperature distribution and displacement within a one dimensional thermoelastic rod for particular sets of natural boundary conditions the eigenfunctions are shown to form a riesz basis on the hilbert space of finite energy states for these cases asymptotic eigenvalue estimates show that the spectrum is to the left of some vertical line re contained in the open left half of the complex plane an energy decay estimate of the form e t me t e 0 is then obtained

propagation of singularities of solutions to semilinear boundary value problems soit p un operateur differentiel d ordre 2 strictement hyperbolique sur une region ouverte r n n 3 a frontiere non caracteristique lisse etant donnee une solution u h loc s s n 1 2 a pu f x u on discute de la propagation des singularites h r microlocales dans le domaine s r 2n n 2 dans le cas general ou le champ de hamilton de p peut etre tangent a t 0 a un ordre arbitrairement eleve fini ou infini

microlocal analysis in nonlinear thermoelasticity abstract this paper is devoted to the study of the propagation of singularities for semilinear hyperbolic parabolic coupled systems in thermoelasticity first a linear transformation involving pseudodifferential operators for unknowns is introduced to decouple the hyperbolic and the parabolic operators by using the decoupled system we obtain that the semilinear system of thermoelasticity has finite speeds for the propagation of singularities furthermore it is proved that the microlocal singularities of solutions to this semilinear hyperbolic parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of nonsmooth pseudodifferential operators finally the cauchy problem for the semilinear system of thermoelasticity is investigated when the initial data have singularities

a deterministic method to predict the effect of unknown but bounded elastic moduli on the buckling of composite structures abstract a new non probabilistic method is developed in this study to predict the variability in buckling loads of composite plates and shells stemming from the unavoidable scatter in elastic moduli the available measurements of elastic moduli are fitted by the four dimensional uncertainty box and appropriate ellipsoid for the specific cases considered the maximum buckling load of the composite plate exceeds by 11 the value of the buckling load based on the calculation using the nominal values of elastic moduli in the case of the cylindrical shell an appropriate exceedance constitutes 9

safety margin analysis of buckling for structures with unknown but bounded uncertainties abstract a safety margin of buckling for structures with unknown but bounded uncertainties is proposed in this paper the perturbation matrixes consisting of the elastic stiffness matrix and the geometric stiffness matrix are introduced to derive the concept of safety margin the concept of safety margin of buckling is first proposed to quantify the influence of uncertain parameters on structural buckling based on the spectrum norm and the singular value analysis the mathematical basis of the safety margin is expounded furthermore to obtain a more accurate estimation of safety margin the correction factor for the safety margin is introduced through the interval extension of interval analysis two engineering examples compared with traditional interval analysis method are utilized to demonstrate the validity of the proposed safety margin and numerical results indicate that the safety margin can provide a relatively accurate index to indicate the allowable perturbation range of the perturbation matrix of a structure under given loads

natural frequency intervals for vibrating systems with polytopic uncertainty we consider the problem of locating the natural frequencies of uncertain systems whose describing matrices are functions of an unknown parameter vector which is included in an assigned bounding set we face what we call the weak frequency interval detection problem of determining the smallest interval which includes all possible frequencies we show that if the system matrices depend affinely on the parameter vector whose bounding set is a compact polyhedron then this problem requires the solution of a finite number of eigenvalue problems associated with the vertices of such a polyhedron unfortunately detecting the intervals associated with all the natural frequencies strong frequency interval detection problem cannot rely on this property so that one must resort to monte carlo methods or numerical optimization to find them we show that the strong version is solvable exploring the vertices only under some stronger assumptions in the case in which the uncertainty bounding set is not defined by linear inequalities not even the extremal frequencies can be associated with the vertices of the admissibility domain then again numerical approach is necessary unless we accept to merge the original system in a larger one of an affine nature finally we present as an application the study of structures with uncertain mass distribution