duct acoustics a numerical technique for the higher order mode solution of three dimensional problems with rigid walls and no flow abstract a three dimensional time dependent finite difference technique is developed for the steady state solution of the acoustic wave equation in the absence of flow the acoustic characteristics of a mitred bend are used to illustrate the use of the technique and numerical results are corroborated by experimental data

a three dimensional boundary element scheme for acoustic radiation in low mach number flows a boundary element scheme is proposed for acoustical radiation in moving flows the scheme relies upon a transformation valid at low mach numbers which permits the case with flow to be restated as an equivalent problem in the absence of flow but with modified boundary conditions this restatement of the original problem is then amenable to boundary integral representation and boundary element solution results obtained by using this approach are presented for the test case of a pulsating or juddering sphere of finite dimensions in a uniform low mach number flow a fully three dimensional boundary element model is used the validity of the results is established by comparison with an analytic perturbation solution where applicable and by comparison with results from an alternative axisymmetric numerical scheme

approximation of a laminated microstructure for a rotationally invariant double well energy density we give error estimates for the approximation of a laminated microstructure which minimizes the energy int omega phi nabla v x dx for a rotationally invariant double well energy density phi a we present error estimates for the convergence of the deformation in l 2 the convergence of directional derivatives of the deformation in the twin planes the weak convergence of the deformation gradient the convergence of the microstructure or young measure of the deformation gradients and the convergence of nonlinear integrals of the deformation gradient

adaptive nonlinear system identification in the short time fourier transform domain in this paper we introduce an adaptive algorithm for nonlinear system identification in the short time fourier transform stft domain the adaptive scheme consists of a parallel combination of a linear component represented by crossband filters between subbands and a quadratic component which is modeled by multiplicative cross terms we adaptively update the model parameters using the least mean square lms algorithm and derive explicit expressions for the transient and steady state mean square error mse in frequency bins for white gaussian inputs we show that estimation of the nonlinear component improves the mse performance only when the power ratio of nonlinear to linear components is relatively high furthermore as the number of crossband filters increases a lower steady state mse may be obtained at the expense of slower convergence experimental results support the theoretical derivations

numerical approximation of the solution of variational problem with a double well potential variational problems with a double well potential are not lower semicontinuous and can fail to attain a minimum value rather the gradients of minimizing sequences do not converge pointwise and can have oscillations however the gradients do converge in the weak topology i e their local spatial averages converge such functionals arise in the description of equilibria of crystals or other ordered materials stable configurations for solid crystals which have symmetry related martensitic energy wells have a fine scale microstructure which can be related to the oscillations that energy minimizing sequences for the bulk energy exhibit an analysis is given of approximation methods for variational problems with a double well potential to give a rigorous justification for the use of such numerical methods to model the behavior of this class of solid crystals

optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem nonconvex variational problems arise in models for the equilibria of crystals and other ordered materials the solution of these variational prob lems must be described in terms of a microstructure rather than in terms of a deformation moreover the numerical approximation of the deformation gra dient often does not converge strongly as the mesh is refined nevertheless the probability distribution of the deformation gradients near each material point does converge recently we introduced a metric to analyze this convergence in this paper we give an optimal order error estimate for the convergence of the deformation gradient in a norm which is stronger than the metric used earlier

the simply laminated microstructure in martensitic crystals that undergo a cubic to orthorhombic phase transformation we study simply laminated microstructures of a martensitic crystal capable of undergoing a cubic to orthorhombic transformation of type mathcal p 432 to mathcal p 222 the free energy density modeling such a crystal is minimized on six energy wells that are pairwise rank one connected we consider the energy minimization problem with dirichlet boundary data compatible with an arbitrary but fixed simple laminate we first show that for all but a few isolated values of transformation strains this problem has a unique young measure solution solely characterized by the boundary data that represents the simply laminated microstructure we then present a theory of stability for such a microstructure and apply it to the conforming finite element approximation to obtain the corresponding error estimates for the finite element energy minimizers

modeling and identification of nonlinear systems in the short time fourier transform domain in this paper we introduce a novel approach for improved nonlinear system identification in the short time fourier transform stft domain we first derive explicit representations of discrete time volterra filters in the stft domain based on these representations approximate nonlinear stft models which consist of parallel combinations of linear and nonlinear components are developed the linear components are represented by cross band filters between subbands while the nonlinear components are modeled by multiplicative cross terms we consider the identification of quadratically nonlinear systems and show that a significant reduction in computational cost as well as substantial improvement in estimation accuracy can be achieved over a time domain volterra model particularly when long memory systems are considered experimental results validate the theoretical derivations and demonstrate the effectiveness of the proposed approach

estimates for numerical approximations of rank one convex envelopes summary we present a convergence analysis of an algorithm for the numerical computation of the rank one convex envelope of a function f mathrm m m times n rightarrow mathbb r a rate of convergence for the scheme is established and numerical experiments are presented to illustrate the analytical results and applications of the algorithm

duct acoustics a time dependent difference approach for steady state solutions abstract a time dependent numerical technique is developed to study the propagation of sound in ducts in the absence of flow the method is shown to be stable and convergent for propagating and decaying modes sample calculations are compared with analytical solutions in straight ducts and with numerical and experimental results in a mitred bend

on the computation of the rank one convex hull of a function we report on several numerical experiments where the rank one convexification of an energy density is computed the explicit examples cover a whole spectrum of typical situations one may encounter one of those is especially relevant for the computation of microstructures in crystalline solids

constitutive theory for some constrained elastic crystals abstract some of the observations of a 15 superconductors near cubic tetragonal phase transformations suggest treating them as thermoelastic bodies subject to certain material constraints here we begin to develop a theory of this kind

on the numerical analysis of non convex variational problems we discuss a numerical method for finding young measure valued minimizers of non convex variational problems to have any hope of a convergence theorem one must work in a setting where the minimizer is unique and minimizing sequences converge strongly this paper has two main goals i we specify a method for producing strongly convergent minimizing sequences despite the failure of strict convexity and ii we show how uniqueness of the young measure can be parlayed into a numerical convergence theorem the treatment of ii is done in the setting of two model problems one involving scalar valued functions and a multiwell energy the other from micromagnetics

numerical analysis of oscillations in multiple well problems variational problems which fail to be convex occur often in the study of ordered materials such as crystals in these problems the energy density for the material has multiple potential wells in this paper we study multiple well problems by first determining the analytic properties of energy minimizing sequences and then by estimating the continuous problem by an approximation using piecewise linear finite elements we show that even when there is no minimizer of the energy the approximations still take on a predictable structure

on algorithms for nonconvex optimization in the calculus of variations a simple algorithm for the computation of local minima of nonconvex problems in one dimension is proposed this algorithm avoids certain well known local minima that are distant from the global minimum and can be used to improve the minima obtained using classical descent methods the calculated minima have mesh scale oscillations typical of weakly converging sequences it is shown that these local minima converge weakly to the correct limit for certain simple problems however numerical evidence suggests that the gradients may not converge to the correct weak limits and this is verified using asymptotic expansions extensions to multiple dimensions and some numerical examples in two dimensions are considered

numerical approximations in variational problems with potential wells in this paper some numerical aspects of variational problems which fail to be convex are studied it is well known that for such a problem in general the infimum of the energy the functional that has to be minimized fails to be attained instead minimizing sequences develop oscillations which allow them to decrease the energy it is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed it is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant an estimate of the approximate deformation as it approximates a measure and some numerical results are also given

strong convergence of numerical solutions to degenerate variational problems numerical approximations of strongly degenerate variational problems of the form j u 0 1 f u u f 2 are considered where f is assumed convex but may have intervals where f 0 it is shown that in spite of the degeneracy natural numerical approximations still converge in w 1 p rates in weaker norms and the connection with nonconvex variational problems are also considered

application of the time dependent finite difference theory to the study of sound and vibration interactions in ducts abstract the time dependent finite difference theory is extended to the solution of the acoustic wave equation in rectangular ducts when acoustic structural interactions are allowed at a duct wall the treatment of the boundary condition which describes the coupling is examined and the stability of the procedure is studied and found to depend on the nature of this coupling the convergence of solutions is discussed as a function of the discretization of the solution domain particularly at frequencies approaching resonance

approximated convex envelope of a function the goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope such a problem arises in various physical situations where the function considered is some energy that has to be minimized this study is a first step toward understanding how to approximate the quasi convex envelope of a function the importance of this issue is due to the various applications that are encountered in particular in the field of material science 1994 copyright society for industrial and applied mathematics

a stochastic analysis of a modified gain extended kalman filter with applications to estimation with bearings only measurements a new globally convergent nonlinear observer called the modified gain extended kalman observer mgeko is developed for a special class of systems this observer structure forms the basis of a new stochastic filter mechanization called the modified gain extended kalman filter mgekf a sufficient condition for the estimation errors of the mgekf to be exponentially bounded in the mean square is obtained finally the mgeko and the mgekf are applied to the three dimensional bearings only measurement problem where the extended kalman filter often shows erratic behavior

numerical analysis of compatible phase transitions in elastic solids the variational model of phase transitions for elastic materials based on linearized elasticity leads to a nonconvex minimization problem p in which a minimum need not be attained in the design of advanced materials the main interest is in reliable numerical predictions of certain macroscopic quantities such as the global deformation and the stress field determined in a relaxed problem qp an explicit formula of the quasi convexified energy density in qp due to r v kohn provides us with a well posed numerical problem first a mathematical a priori and a posteriori error analysis is established for the finite element approximation of the stress variable then the residual based error indicator is implemented within an adaptive mesh refinement algorithm numerical examples illustrate that the macroscopic properties of the materials are computed efficiently with appropriate error control

numerical solution of the scalar double well problem allowing microstructure the direct numerical solution of a non convex variational problem p typically faces the difficulty of the finite element approximation of rapid oscillations although the oscillatory discrete minimisers are properly related to corresponding young measures and describe real physical phenomena they are costly and difficult to compute in this work we treat the scalar double well problem by numerical solution of the relaxed problem rp leading to a degenerate convex minimisation problem the problem rp has a minimiser u and a related stress field dw u which is known to coincide with the stress field obtained by solving p in a generalised sense involving young measures if u h is a finite element solution h dw u h is the related discrete stress field we prove a priori and a posteriori estimates for h in l 4 3 and weaker weighted estimates for u u h the a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments

a partitioned adaptive approach to nonlinear channel equalization the problem of identifying digital transmitted symbols over nonlinear communication channels is addressed the equalization scenario is considered from the decision point of view and constitutes a joint identification and estimation situation due to incomplete knowledge of the system model a new class of multilinearization algorithms for nonlinear systems is derived according to partitioning theory concepts the procedure targets on adaptively selecting the best reference points for linearization from an ensemble of generated trajectories that span the whole state space of the desired signal in the various simulations examined the partitioned based equalizer is found superior to the classical extended kalman filter

nonconforming finite element approximation of crystalline microstructure we consider a class of nonconforming nite element approxima tions of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can un dergo either an orthorhombic to monoclinic double well or a cubic to tetrag onal triple well transformation we rst establish a series of error bounds in terms of elastic energies for the l 2 approximation of derivatives of the de formation in the direction tangential to parallel layers of the laminate for the l 2 approximation of the deformation for the weak approximation of the de formation gradient for the approximation of volume fractions of deformation gradients and for the approximation of nonlinear integrals of the deformation gradient we then use these bounds to give corresponding convergence rates for quasi optimal nite element approximations

numerical analysis of a nonconvex variational problem related to solid solid phase transitions the description of equilibria of shape memory alloys or other ordered materials gives rise to nonconvex variational problems in this paper a two dimensional model of such materials is studied due to the fact that the corresponding functional has two symmetry related martensitic energy wells the numerical approximation of the deformation gradient does not converge but tends to oscillate between the two wells as the size of the mesh is refined these oscillations may be interpreted in terms of microstructures using a nonconforming mathcal p 1 finite element an estimate is given for the rate of convergence of the probability for the approximated deformation to have its gradient near one of the two martensitic wells

the time dependent finite difference procedure for propagation of sound in a non uniform lined duct a comparison with experiments abstract a time dependent finite difference procedure is developed for the calculation of acoustic fields which incorporate finite impedance and rigid wall boundaries a numerical mapping technique is evoked to transform the solution domain into a rectangular geometry and the procedure is used to solve for frequencies where cross modes can propagate in the straight sections of duct numerical results are corroborated with experiments

on longitudinal acoustic propagation in convergent and divergent nozzle flows abstract the longitudinal propagation of sound in quasi one dimensional low mach number nozzle flow is considered in section 1 the solution in the ray approximation section 2 1 is used to transform the wave equation into a schrodinger form which is studied for the family of power law ducts including as a particular case the conical nozzle it is shown that the coincidence of flow sources sinks with sound sources can lead to appearance of essential singularities section 2 2 which can be removed by using a riccati transformation section 2 4 the exact solutions of the acoustic equations for the parabolic figure 1 and hyperbolic figure 2 nozzles are obtained in terms of bessel functions section 2 3 respectively of complex order and argument the general formulas together with limiting forms in the compactness ray and asymptotic approximations section 3 1 are used to establish properties of the acoustic velocity and pressure section 3 2 kinetic and compression energies and energy flux and wave action section 3 3 for example it is shown that the equipartition of energies for moderate variations in cross section gives way table 2 to a predominance of kinetic compression energies respectively near blockages openings the effects of non uniform mean flow table 1 are discussed by comparing horns with nozzles section 3 4 e g it is shown that the duality principle in three alternative forms does not extend from horn to nozzles and the acoustic equations have no elementary solutions for the latter in contrast with the former

analysis of the finite element approximation of microstructure in micromagnetics the solution to nonconvex variational problems is often characterized by microstructure the variational problem for the magnetization field in micromagnetics has a nonconvex constraint and energy minimizing sequences of magnetization fields can have oscillations whose scale converges to zero but whose amplitude remains finite it is shown that the finite element approximation of the magnetization field for this variational problem does not converge pointwise as the mesh is refined but that nonlinear functions of the approximate magnetization fields converge weakly or equivalently all local spatial averages of nonlinear functions of the approximate magnetization field converge as the mesh is refined which implies the convergence as the mesh is refined of the probability distribution of the approximate magnetization fields in local spatial domains a norm is given to measure the convergence of this microstructure and we prove a rate of convergence in this norm

undermodeling error quantification for quadratically nonlinear system identification in the short time fourier transform domain in this paper we introduce an estimation error analysis for quadratically nonlinear system identification in the short time fourier transform stft domain the identification scheme consists of a parallel connection of a linear component represented by crossband filters between subbands and a quadratic component which is modeled by multiplicative cross terms we mainly concentrate on two types of undermodeling errors the first is caused by employing a purely linear model in the estimation process i e nonlinear undermodeling and the second is a consequence of restricting the number of estimated crossband filters in the linear component we derive analytical relations between the noise level nonlinearity strength and the obtainable mean square error mse in subbands we show that for low signal to noise ratio snr conditions a lower mse is achieved by allowing for nonlinear undermodeling and utilizing a purely linear model however as the snr increases the performance can be generally improved by incorporating a nonlinear component into the model we further show that as the snr increases a larger number of crossband filters should be estimated to attain a lower mse whether a linear or nonlinear model is employed experimental results support the theoretical derivations

on the convergence of finite element approximations of a relaxed variational problem the accuracy of finite element approximations to a convex but not strictly convex variational problem is considered convergence is proved for a finite element approximation of a particular vector field related to the solution in a special one dimensional case 0 h convergence is shown for a piecewise linear approximation of the derivative h denotes the size of each element domain numerical results are also presented for this one dimensional case

transmission and reflection of higher order acoustic modes in a mitred duct bend a two dimensional finite element technique was used to investigate the characteristics of a 90 mitred bend for higher order modes experiments were used to corroborate the numerical results and good agreement was obtained between the two approaches for values of the wave number parameter extending to a little above the cut on value of the second cross mode the results are presented as modal contributions of the reflected and transmitted sounds when a propagating mode is incident on the bend the magnitude of the energy transfer is also examined for the case where an evanescent mode is incident on the bend

finite element analysis of microstructure for the cubic to tetragonal transformation martensitic crystals which can undergo a cubic to tetragonal phase transformation have a nonconvex energy density with three symmetry related rotationally invariant energy wells we give estimates for the numerical approximation of a first order laminate for such martensitic crystals we give bounds for the l2 convergence of directional derivatives in the twin plane for the l2 convergence of the deformation for the weak convergence of the deformation gradient for the convergence of the microstructure and for the convergence of nonlinear integrals of the deformation gradient

numerical analysis of oscillations in nonconvex problems we study numerically the pattern of the minimizing sequences of nonconvex problems which do not admit a minimizer