interpreting proper orthogonal modes of randomly excited vibration systems proper orthogonal modes poms of displacements are interpreted for linear vibration systems under random excitation excitations are considered for which the fourier transform is convergent meaning that the input must have zero mean and no sustained sinusoidal component in such a case the poms in undamped discrete linear symmetric systems can represent linear natural modes if the mass distribution is known poms in one dimensional distributed parameter self adjoint systems can approximately represent the linear normal modes if the mass distribution is known simulation examples are presented simulations show that these ideas are also applicable under light modal damping

the discontinuous enrichment method for medium frequency helmholtz problems with a spatially variable wavenumber numerical dispersion or what is often referred to as the pollution effect presents a challenge to an efficient finite element discretization of the helmholtz equation in the medium frequency regime to alleviate this effect and improve the unsatisfactory pre asymptotic convergence of the classical galerkin finite element method based on piecewise polynomial basis functions several discretization methods based on plane wave bases have been proposed among them is the discontinuous enrichment method that has been shown to offer superior performance to the classical galerkin finite element method for a number of constant wavenumber helmholtz problems and has also outperformed two representative methods that use plane waves the partition of unity and the ultra weak variation formulation methods in this paper the discontinuous enrichment method is extended to the variable wavenumber helmholtz equation to this effect the concept of enrichment functions based on free space solutions of the homogeneous form of the governing differential equation is enlarged to include free space solutions of approximations of this equation obtained in this case by successive taylor series expansions of the wavenumber around a reference point this leads to plane wave enrichment functions based on the piece wise constant approximation of the wavenumber and to airy wave enrichment functions several elements based on these enrichment functions are constructed and evaluated on benchmark problems modeling sound hard scattering by a disk submerged in an acoustic fluid where the speed of sound varies in space all these elements are shown to outperform by a substantial margin their continuous polynomial counterparts

acoustic propagation in non uniform waveguides revisiting webster equation using evanescent boundary modes the scattering of an acoustic wave propagating in a non uniform waveguide is inspected by revisiting improved multimodal methods in which the introduction of additional modes so called boundary mo

a comparison of high order polynomial and wave based methods for helmholtz problems the application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation two common strategies to address this issue are to use high order polynomial shape functions e g hp fem or to use physics based or trefftz methods where the shape functions are local solutions of the problem typically plane waves both strategies have been actively developed over the past decades and both have demonstrated their benefits compared to conventional finite element methods but they have yet to be compared in this paper a high order polynomial method p fem with lobatto polynomials and the wave based discontinuous galerkin method are compared for two dimensional helmholtz problems a number of different benchmark problems are used to perform a detailed and systematic assessment of the relative merits of these two methods in terms of interpolation properties performance and conditioning it is generally assumed that a wave based method naturally provides better accuracy compared to polynomial methods since the plane waves or bessel functions used in these methods are exact solutions of the helmholtz equation results indicate that this expectation does not necessarily translate into a clear benefit and that the differences in performance accuracy and conditioning are more nuanced than generally assumed the high order polynomial method can in fact deliver comparable and in some cases superior performance compared to the wave based dgm in addition to benchmarking the intrinsic computational performance of these methods a number of practical issues associated with realistic applications are also discussed

an adaptive generalized finite element method applied to free vibration analysis of straight bars and trusses in this paper a first application of an adaptive generalized finite element method to free longitudinal vibration analysis of straight bars and trusses is presented the generalized finite element method is developed by enriching the standard finite element method space whose basis performs a partition of unity with knowledge about the differential equation being solved the enrichment functions used are dependent on the geometric and mechanical properties of the element the proposed approach converges very fast and is able to approximate the frequency related to any vibration mode the variational problem of free vibration is formulated and the main aspects of the adaptive generalized finite element method are presented and discussed the efficiency and convergence of the proposed method in vibration analysis of uniform and non uniform straight bars are checked the application of this technique in a truss is also presented the frequencies obtained by the adaptive generalized finite element method are compared with those obtained by the analytical solution the composite element method and the h version of finite element method

conservative integration of rigid body motion by quaternion parameters with implicit constraints summary an angular momentum and energy conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four component conjugate momentum vector via hamilton s equations the introduction of an extended mass matrix leads to a symmetric set of eight state space equations of motion the extra inertial parameter serves as a multiplier on the kinematic constraint and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments external loads enter these equations via the set of momentum equations initially the normalization of the quaternion array is introduced via a lagrange multiplier however this lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential and elimination of the lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient an algorithm is developed by forming a finite increment of the hamiltonian this procedure identifies the proper selection of increments and mean values and leads to an algorithm with conservation of momentum and energy implementation conservation properties and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top copyright 2012 john wiley sons ltd

multimodal admittance method in waveguides and singularity behavior at high frequencies this work presents a multimodal method for the propagation in a waveguide with varying height and its relation to trapped modes or quasi trapped modes the coupled mode equations are obtained by projecting the helmholtz equation on the local transverse modes to solve this problem we integrate the riccati equation governing the admittance matrix dirichlet to neumann operator for many propagating modes i e at high frequencies the numerical integration of the riccati equation shows that the rule is that this matrix has quasi singularities associated to quasi trapped modes

a study of wave propagation in varying cross section waveguides by modal decomposition part i theory and validation the propagation of acoustic waves in waveguides with variable cross section is considered using multimodal decomposition the approach adopted is to construct two infinite first order differential equations for the components of the pressure and the velocity projected over the normal modes from these an infinite matricial riccati equation is derived for the impedance matrix these equations are ordinary differential equations that can be integrated after truncation at a sufficient number of modes and take into account the coupling between modes the stiffness of the pressure velocity equations induced by the presence of evanescent modes is avoided by first calculating the impedance matrix along the guide the method is checked using different examples where the solutions of the plane wave approximation or the finite element method are known results show the method provides simple and accurate means to obtain the acoustic field with correct boundary conditions in a nonuniform guide with no restriction on the flare

an interpolation based parametric reduced order model combined with component mode synthesis abstract this paper presents an efficient interpolation based parametric reduced order model with component mode synthesis the off line sampling of a large scale structural dynamic system containing many parameters requires many computations to investigate the parameter dependency of a dynamical system therefore we introduce a dynamic substructuring scheme to execute off line sampling in the subdomain level in addition we suggest discriminating the interpolation of the subsystem depending on the characteristics of each substructural mode we synthesize the substructures and we then construct a two level semi parametrized rom by reducing the degrees of freedom of the interface in the on line stage we then demonstrate the efficiency and accuracy of the present method by computing the relative eigenvalue errors and the transient responses of the system with random values for the parameters and we conduct the design optimization of large scale systems under dynamic loading conditions a comparison of the present method with the full order model and a conventional reduced order model indicates that the present method is useful in carrying out an efficient analysis and design optimization for various large scale structural dynamic systems

a non intrusive reduced basis method for elastoplasticity problems in geotechnics abstract this work aims at investigating the use of reduced basis rb methods to diminish the cost of numerical simulation of elastoplasticity problems arising from geotechnics modeling and involving parameter dependent partial differential equations pdes computation times for large three dimensional analysis commonly take tens of hours making optimization procedures or sensitivity analysis relying on repeated simulations hardly feasible in many cases the geotechnical analysis requires very specific features such as highly non linear constitutive laws making the necessary modification of the fe calculation code for a standard rb method impossible an approach making it possible to use the reduced basis framework without having to modify the code gives the so called non intrusive reduced basis method a versatility of great practical interest our approach involves the computation of less expensive but less accurate fe approximation during the online stage and improvement of those solutions using a rb based rectification method

exact solutions for the longitudinal vibration of non uniform rods abstract the objective of this paper is to present exact analytical solutions for the longitudinal vibration of rods with non uniform cross section using appropriate transformation the equation of motion of axial vibration of a rod with varying cross section is reduced to analytically solvable standard differential equations whose form depends upon the specific area variation solutions are obtained for a rod with a polynomial area vibration and for a sinusoidal rod the solutions are obtained in terms of special functions such as bessel and neumann as well as trigonometric functions simple formulas to predict the natural frequencies of non uniform rods with various end conditions are presented the natural frequencies of non uniform rods for these end conditions are calculated and their dependence on taper is discussed the governing equation for the problem is the same as that of wave propagation through ducts with non uniform cross sections therefore solutions presented here can be used to investigate such problems

lamb wave propagation in inhomogeneous elastic waveguides the problem of lamb wave propagation in an axially multilayered waveguide is treated by a multimodal approach a general formalism is proposed that avoids the numerical divergence due to evanescent

on the partition function of the six vertex model with domain wall boundary conditions the six vertex model on an n n square lattice with domain wall boundary conditions is considered a fredholm determinant representation for the partition function of the model is given the kernel of the corresponding integral operator is of the so called integrable type and involves classical orthogonal polynomials from this representation a reconstruction formula is proposed which expresses the partition function as the trace of a suitably chosen quantum operator in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models

fast low rank modifications of the thin singular value decomposition this paper develops an identity for additive modivations of a singular value decomposition svd to reflect updates downdates shifts and edits of the data matrix this sets the stage for fast and ememory efficient sequential algorithms for tracking singular values and subspaces in conjunction with a fast solution for the pseudo inverse of a submatrix of an orthogonal matrix we develop a scheme for computing a thin svd of streaming data in a single pass with linear time complexity a rank r think svd of a p x q matrix can be computed in o pqr time for r less than or equal sqroot min p q linear algebra and its applications

nonlinear model order reduction based on local reduced order bases summary a new approach for the dimensional reduction via projection of nonlinear computational models based on the concept of local reduced order bases is presented it is particularly suited for problems characterized by different physical regimes parameter variations or moving features such as discontinuities and fronts instead of approximating the solution of interest in a fixed lower dimensional subspace of global basis vectors the proposed model order reduction method approximates this solution in a lower dimensional subspace generated by most appropriate local basis vectors to this effect the solution space is partitioned into subregions and a local reduced order basis is constructed and assigned to each subregion offline during the incremental solution online of the reduced problem a local basis is chosen according to the subregion of the solution space where the current high dimensional solution lies this is achievable in real time because the computational complexity of the selection algorithm scales with the dimension of the lower dimensional solution space because it is also applicable to the process of hyper reduction the proposed method for nonlinear model order reduction is computationally efficient its potential for achieving large speedups while maintaining good accuracy is demonstrated for two nonlinear computational fluid and fluid structure electric interaction problems copyright 2012 john wiley sons ltd

on the rational derivation of a hierarchy of dynamic equations for a homogeneous isotropic elastic plate abstract flexural equations of motion for a homogeneous isotropic elastic plate are derived by an antisymmetric expansion in the thickness coordinate of the displacement components all but the lowest order expansion functions are eliminated with the help of the three dimensional equations of motion and are plugged into the boundary conditions eliminating between these an equation is obtained for the mean plane vertical displacement which also includes arbitrary loading on the plate surface this equation can be truncated to any order in the thickness and it is in particular noted that the corresponding dispersion relation seems to correspond to a power series expansion of the exact rayleigh lamb dispersion relation to all orders various truncations of the equation are discussed and are compared numerically with each other the exact three dimensional solution and mindlin s plate equation both the dispersion relation and the corresponding displacement components as well as an excitation problem are used for the comparisons the theories are reasonably close to each other and in order to be on the safe side none of them should in fact be used for frequencies above the cutoff of the second flexural mode

input impedance of brass musical instruments comparison between experiment and numerical models we present a sequence of measured and calculated input impedance curves for tubes of increasing complexity culminating in curves for a trombone and for a tuba measurements were made by the indirect method in a manner which afforded unprecedented accuracy calculations were done with a one dimensional model including visco thermal losses by a method of discretization employing truncated cones the correspondence between theory and experiment is sufficiently good that one can use model calculations to predict with some confidence the changes in input impedance caused by modifications to real instruments

new enumeration formulas for alternating sign matrices and square ice partition functions abstract the refined enumeration of alternating sign matrices asms of given order having prescribed behavior near one or more of their boundary edges has been the subject of extensive study starting with the refined alternating sign matrix conjecture of mills robbins rumsey 1983 25 its proof by zeilberger 1996 31 and more recent work on doubly refined and triply refined enumerations by several authors in this paper we extend the previously known results on this problem by deriving explicit enumeration formulas for the top left bottom triply refined and top left bottom right quadruply refined enumerations the latter case solves the problem of computing the full boundary correlation function for asms the enumeration formulas are proved by deriving new representations which are of independent interest for the partition function of the square ice model with domain wall boundary conditions at the combinatorial point 2 3

arlequin framework for multi model multi time scale and heterogeneous time integrators for structural transient dynamics this paper presents a general framework for performing spatio temporal multi scale multi model coupling based on displacement continuity this approach guarantees a global energy balance of the system in the context of the arlequin method it introduces specific interpolation of the lagrange multipliers and parameters of time integration operators in the overlapping zone a dual schur implementation is then used to illustrate the efficiency and robustness of the method convergence analysis with numerical experiments is performed finally 1d 1d and 2d 1d coupling applications are presented in which each subdomain is integrated in time using different time steps and different schemes this approach gives the user more ways of controlling computational behaviour via newmark parameters time steps and sizing the overlapping zones

a parameterized non intrusive reduced order model and error analysis for general time dependent nonlinear partial differential equations and its applications abstract a novel parameterized non intrusive reduced order model p nirom based on proper orthogonal decomposition pod has been developed this p nirom is a generic and efficient approach for model reduction of parameterized partial differential equations p pdes over existing parameterized reduced order models p rom most of them are based on the reduced basis method it is non intrusive and independent on partial differential equations and computational codes during the training process the smolyak sparse grid method is used to select a set of parameters over a specific parameterized space p r p for each selected parameter the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model more generally the snapshots and basis function sets for any parameters over p can be obtained using an interpolation method the p nirom can then be constructed by using our recently developed technique xiao et al 2015 41 42 where either the smolyak or radial basis function rbf methods are used to generate a set of hyper surfaces representing the underlying dynamical system over the reduced space the new p nirom technique has been applied to parameterized navier stokes equations and implemented with an unstructured mesh finite element model the capability of this p nirom has been illustrated numerically by two test cases flow past a cylinder and lock exchange case the prediction capabilities of the p nirom have been evaluated by varying the viscosity initial and boundary conditions the results show that this p nirom has captured the quasi totality of the details of the flow with cpu speedup of three orders of magnitude an error analysis for the p nirom has been carried out

analysis of higher order mode effects in an expansion chamber using modal theory and equivalent electrical circuits abstract modal theory for propagation in an acoustic waveguide with variable cross section in particular in an expansion chamber can lead to several different numerical or approximate analytical solutions the advantages of these are compared and one of their number is shown to be a generalization of the plane piston inlet outlet approximation made by ih and lee the results are expressed in the form of two kinds of equivalent electrical circuits connecting the plane modes in the inlet and outlet in the first one the higher modes are described by two series inductances and one shunt inductance in the second one there is no shunt inductance and series inductances are replaced by complex series impedances the existence of resistive elements is discussed in terms of energy transport by higher modes especially at low frequencies when tunneling occurs approximate analytical numerical and experimental results for the elements of circuits are given with a study of coupling between the two step discontinuities

the latin multiscale computational method and the proper generalized decomposition this paper deals with the synergy between the latin multiscale method and what is called the proper generalized decomposition pgd which is the key of its performances

factorization of second order elliptic boundary value problems by dynamic programming we present a method to factorize a second order boundary value problem in a cylindrical domain in a system of uncoupled first order initial value problems together with a nonlinear riccati type equation for functional operators this uncoupling is obtained by a space invariant embedding technique along the axis of the cylinder this method can be viewed as an infinite dimensional generalization of the block gauss lu factorization

wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation we develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems bvps by considering the helmholtz equation at high wavenumber k as our model problem the key element in this theory is a novel k explicit regularity theory for helmholtz bvps that is based on decomposing the solution into two parts the first part has the sobolev regularity properties expected of second order elliptic pdes but features k independent regularity constants the second part is an analytic function for which k explicit bounds for all derivatives are given this decomposition is worked out in detail for several types of bvps namely the helmholtz equation in bounded smooth domains or convex polygonal domains with robin boundary conditions and in exterior domains with dirichlet boundary conditions we present an error analysis for the classical hp version of the finite element method hp fem where the dependence on the mesh width h the approximation order p and the wavenumber k is given explicitly in particular under the assumption that the solution operator for helmholtz problems is polynomially bounded in k it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

lamb wave propagation in elastic waveguides with variable thickness the problem of lamb wave propagation in waveguides with varying height is treated by a multimodal approach the technique is based on a rearrangement of the equations of elasticity that provides a new system of coupled mode equations preserving energy conservation these coupled mode equations avoid the usual problem at the cut offs with zero wavenumber thereafter we define an impedance matrix that is governed by a riccati equation yielding a stable numerical computation of the solution incidentally the versatility of the multimodal method is exemplified by treating analytically the case of slowly varying guide and by showing how to get easily the green tensor in any geometry the method is applied for a waveguide whose height is described by a gaussian function and the energy conservation in verified numerically we determine the green tensor in this geometry

boundary correlation functions of the six vertex model we consider the six vertex model on an n n square lattice with the domain wall boundary conditions boundary one point correlation functions of the model are expressed as determinants of n n matrices generalizing the known result for the partition function in the free fermion case explicit answers are obtained the introduced correlation functions are closely related to the problem of enumeration of alternating sign matrices and domino tilings

model order reduction and domain decomposition strategies for the solution of the dynamic elastic plastic structural problem abstract a new strategy for the efficient solution of highly nonlinear structural problems is proposed based on the combined use of domain decomposition dd and snapshots version of the proper orthogonal decomposition pod techniques the formulation here presented is tailored for applications in elastic plastic structural dynamics in this context the pod is applied to domains that remain elastic and a double strategy to update the reduced basis is adopted first the singular value decomposition svd proposed allows to update the reduced basis as soon as a new snapshot is stored secondly an online adaptation technique of the reduced space is performed through a plastic check during the reduced analysis the applications show that the computation time necessary for solving elastic plastic problems can be reduced of approximately 50 while keeping accuracy comparable to that obtained for the full model with a classical monolithic method

a pgd based homogenization technique for the resolution of nonlinear multiscale problems abstract this paper deals with the offline resolution of nonlinear multiscale problems leading to a pgd reduced model from which one can derive microinformation which is suitable for design here we focus on an improvement to the proper generalized decomposition pgd technique which is used for the calculation of the homogenized operator and which plays a central role in our multiscale computational strategy this homogenized behavior is calculated offline using the latin multiscale strategy and pgd leads to a drastic reduction in cpu cost indeed the multiscale aspect of the latin method is based on splitting the unknowns between a macroscale and a microcomplement this macroscale is used to capture the homogenized behavior of the structure and then to accelerate the convergence of the iterative algorithm the novelty of this work lies in the reduction of the computation cost of building the homogenized operator in order to do that the problems defined within each subdomain are solved using a new pgd representation of the unknowns in which the separation of the unknowns is performed not only in time and in space but also in terms of the interface macrodisplacements the numerical example presented shows that the convergence rate of the approach is not affected by this new representation and that a significant reduction in computation cost can be achieved particularly in the case of nonlinear behavior the technique used herein is then particularly important to enhance the performances of the latin strategy but is not limited to this method the more general path which is followed in this paper is to solve the microproblems arising in the homogenization for all the configurations that can be encountered in the calculation an additional and novel benefit of this approach is that it includes the calculation of the homogenized tangent operator which gives over the whole time interval and for any cell the relation between the perturbation in terms of macroforces and the perturbation in terms of macrodisplacements in linear viscoelasticity this operator represents the homogenized behavior of the structure exactly

a two point boundary value problem with a rapidly oscillating solution some numerical methods are developed for a two point boundary value problem with a rapidly oscillating solution the two point boundary value problem is chosen to model some of the difficulties that may be expected to occur in solving the reduced wave equation at moderately high frequencies

a wide class of analytical solutions of the webster equation abstract this paper aims at presenting closed form general analytical solutions of the webster equation describing plane elastic or acoustic waves the considered radius functions of non uniform cross sectioned rods or ducts are based on the triconfluent heun functions and contain some optional parameters enabling us to set various profiles of the radius functions in a relatively wide range while it is possible to employ the presented exact general analytical solution of the webster equation for all selected profiles if the radius functions are predetermined then the derived general analytical solution can also be employed for their triconfluent heun approximations including certain polynomial ones the applicability and correctness of the derived analytical solutions are demonstrated by calculations of natural frequencies and mode shapes for representative radius functions while the results based on approximate analytical solutions are verified numerically

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

vibration of non uniform rods and beams we show that for some non uniform rods and beams the equation of motion can be transformed into the equation of motion for a uniform rod or beam then when the ends are completely fixed the eigenvalues of the non uniform continuum are the same as those of uniform rods or beams for other end support conditions exact solutions are obtained an efficient procedure is used to analyze the free vibration of non uniform beams with general shape and arbitrary boundary conditions simple formulas are presented for predicting the fundamental natural frequency of non uniform beams with various end support conditions

approximation formulae for the acoustic radiation impedance of a cylindrical pipe useful approximation formulae for radiation impedance are given for the reflection coefficients of both infinitely flanged and unflanged rigid walled cylindrical ducts the expressions guarantee that simple but necessary physical and mathematical principles are met like hermitian symmetry for the reflection coefficient identical behaviour of positive and negative frequencies and causality for the impulse response a non causal but more accurate expression is also proposed that is suitable for frequency domain applications the formulae are obtained by analytical and numerical fitting to reference results from levine and schwinger for the unflanged case and extracted from the radiation impedance matrix given by zorumski for the infinite flanged case

unidimensional models of acoustic propagation in axisymmetric waveguides this paper presents a model of the linear acoustic propagation in axisymmetric waveguides the pressure depending on a single space variable the approach consists of writing the wave equation and the boundary conditions for a coordinate system rectifying the isobaric map at each time the two dimensional dependence of the problem is thus transferred from the pressure to the coefficients of the wave equation from this result an exclusively geometrical necessary condition is deduced for the admissibility of isobaric maps however the knowledge of the waveguide geometry is not sufficient to separate the pressure and the isobaric map solutions in order to develop a unidimensional wave equation a geometrical hypothesis is discussed for lossless and motionless rigid waveguides the deduced equation leads to exact results for tubes and cones it may be interpreted as a webster equation for a particular coordinate system so that the particular profiles for which analytical solutions of the pressure exist are redefined the wave equation is also established for pipes with visco thermal losses and more generally for mobile walls having a small admittance the compatibility of the geometrical hypothesis with the exact model is specified for this general case

multimodal method and conformal mapping for the scattering by a rough surface the scattering of a wave from a periodic rough surface in two dimensions is considered the proposed method is based on the use of a conformal mapping and of the multimodal admittance method the use of the conformal mappings induces a spatially varying refractive index and transforms the boundary condition on the rough surface into a boundary condition on a flat surface then the multimodal admittance method reduces this problem to a riccati equation for the modal admittance matrix which is solved numerically with a magnus mobius scheme the method is shown to converge exponentially with the number of fourier modes the method also allows to find geometries having trapped modes or quasi trapped modes at a given frequency by looking at singularities or quasi singularities of this riccati equation besides a simple perturbation expansion based on a small roughness approximation is developed

stabilization of projection based reduced order models summary a rigorous method for stabilizing projection based linear reduced order models without significantly affecting their accuracy is proposed unlike alternative approaches this method is computationally efficient it requires primarily the solution of a small scale convex optimization problem furthermore it is nonintrusive in the sense that it operates directly on readily available reduced order operators these can be precomputed using any data compression technique including balanced truncation balanced proper orthogonal decomposition proper orthogonal decomposition or moment matching the proposed method is illustrated with three applications the stabilization of the reduction of the computational fluid dynamics based model of a linearized unsteady supersonic flow the reduction of a computational structural dynamics system and the stabilization of the reduction of a coupled computational fluid dynamics computational structural dynamics model of a linearized aeroelastic system in the transonic flow regime copyright 2012 john wiley sons ltd

acoustic waves in slender axisymmetric tubes acoustic waves in a rigid axisymmetric tube with a variable cross section are considered the governing helmholtz equation is solved using neumann series expansions in bessel functions of various orders with a stretched radial coordinate leading to a hierarchy of one dimensional ordinary differential equations in the longitudinal direction the lowest approximation for axisymmetric motion includes webster s horn equation as a special case fourth order differential equations are obtained at the next level of approximation good agreement with existing asymptotic theories for waves in slender tubes is found

coarse grid computation of the one way propagation of coupled modes in a varying cross section waveguide a one way approximation is investigated for the computation of wave propagation in varying cross section waveguides the proposed method derives as a basic approximation of the extensively studied multimodal admittance method when integrated with a magnus scheme this matrix one way equation exhibits an unexpected behavior as the deviation from the exact solution is minimum when only two discretization points per wavelength are taken this peculiar property makes this method efficient to compute the wave propagation for a large variety of geometries beyond the initially stated framework of weakly non uniform waveguides

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

sound propagation in rigid bends a multimodal approach the sound propagation in a waveguide with bend of finite constant curvature is analyzed using multimodal decomposition two infinite first order differential equations are constructed for the pressure and velocity in the bend projected on the local transverse modes a riccati equation for the impedance matrix is then derived which can be numerically integrated after truncation at a sufficient number of modes an example of validation is considered and results show the accuracy of the method and its suitability for the formulation of radiation conditions reflection and transmission coefficients are also computed showing the importance of higher order mode generation at the junction between the bend and the straight ducts the case of varying cross section curved ducts is also considered using multimodal decomposition

passive time domain numerical models of viscothermal wave propagation in acoustic tubes of variable cross section numerical modeling of wave propagation in acoustic tubes is a subject of longstanding interest particularly for enclosures of varying cross section and especially when viscothermal losses due to boundary layer effects are taken into consideration though steady state or frequency domain methods are a common avenue of approach recursive time domain methods are an alternative allowing for the generation of wideband responses and offer a point of departure for more general modeling of nonlinear wave propagation the design of time domain methods is complicated by numerical stability considerations and to this end a passive representation is a useful design principle leading to simple stable and explicit numerical schemes particularly in the case of viscothermal loss modeling such schemes and the accompanying energy and stability analysis are presented here numerical examples are presented for a variety of duct profiles illustrating strict energy dissipation and for comparison of computed input impedances against frequency domain results

wave propagation in a waveguide containing restrictions with circular arc shape a multimodal method is used to analyze the wave propagation in waveguides containing restrictions or corrugations with circular arc shapes this is done using a geometrical transformation which transforms the waveguide with complex geometry in the real space to a straight waveguide in the transformed space or virtual space in this virtual space the helmholtz equation has a modified structure which encapsulates the complexity of the geometry it is solved using an improved modal method which was proposed in a paper by a maurel j f mercier and s felix proc r soc a 470 20130743 2014 that increases the accuracy and convergence of usual multimodal formulations results show the possibility to solve the wave propagation in a waveguide with a high density of circular arc shaped scatterers

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

determination of the complex acoustic scattering matrix of a right angled duct a method for determining the complete higher order scattering matrix of an acoustic discontinuity is developed the method is demonstrated for a right angled waveguide bend and the magnitude and phase of the reflection and transmission coefficients are extracted precisely the procedure is straightforward and based on the solutions to the helmholtz equation by the finite element method fem the consistency of the scattering coefficients found by this method is verified by their properties of symmetry and their accuracy is established by the conservation of energy the reliability of the new technique is further proved by means of an arbitrary sound source and by comparing the direct fem response to the reflection matrix calculation some features of the scattering matrix as a function of frequency are surprising such as the steps and reversion of the phase evolution or the complete loss of transmission of the incoming wave the methodology detailed in this paper can be extended to other multiport junctions such as t junctions or size discontinuities in ducts

improved multimodal method for the acoustic propagation in waveguides with a wall impedance and a uniform flow we present an efficient multimodal method to describe the acoustic propagation in the presence of a uniform flow in a waveguide with locally a wall impedance treatment the method relies on a variational formulation of the problem which allows to derive a multimodal formulation within a rigorous mathematical framework notably to properly account for the boundary conditions on the walls being locally the myers condition and the neumann condition otherwise also the method uses an enriched basis with respect to the usual cosine basis able to absorb the less converging part of the modal series and thus to improve the convergence of the method using the cosine basis the modal method has a low convergence 1 n with n the order of truncation using the enriched basis the improvement in the convergence is shown to depend on the mach number from 1 n5 to roughly 1 n1 5 for m 0 to m close to unity the case of a continuously varying wall impedance is considered and we discuss the limiting case of piecewise constant impedance which defines pressure edge conditions at the impedance discontinuities

model order reduction for dynamic simulation of a flexible multibody system via absolute nodal coordinate formulation abstract the absolute nodal coordinate formulation ancf can be used to model a flexible multibody system subject to both overall motions and large deformations with an increasing number of finite elements of ancf to mesh flexible bodies however the computation cost will become extremely high to improve the computational efficiency of ancf for a large scaled flexible multibody system a systematic method is proposed in this study for the model order reduction based on the proper orthogonal decomposition and the galerkin projection at first an approach for the selection of reduced constraint equations is proposed to deal with the singularity of the coefficient matrix of the reduced order model rom then the computation of the reduced stiffness matrix and generalized force vector of the rom are parallelized via the openmp directives afterwards two parametric approaches are presented to make the rom be adaptive to the change of system parameters one is the interpolation approach of reduced order basis vectors on a manifold to obtain the parametric rom and the other is the fast to slow rom approach via the reduced order basis vectors extracted from the fast dynamic responses to simulate slow dynamic responses finally four numerical examples are given to validate the efficacy of the proposed method for the dynamic simulations of both rigid and flexible multibody systems

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

boundary polarization in the six vertex model vertical arrow fluctuations near the boundaries in the six vertex model on the two dimensional n n square lattice with the domain wall boundary conditions are considered the one point correlation function boundary polarization is expressed via the partition function of the model on a sublattice the partition function is represented in terms of standard objects in the theory of orthogonal polynomials this representation is used to study the large n limit the presence of the boundary affects the macroscopic quantities of the model even in this limit the logarithmic terms obtained are compared with predictions from conformal field theory

aerodynamic characteristics of pretensioned elastic membrane rectangular sailwings presented in the paper are the details of a method for obtaining aerodynamic characteristics of pretensioned elastic membrane rectangular sailwings this is a nonlinear problem governed by the membrane equation for the inflated sail and the lifting surface theory integral equation for aerodynamic loads on the sail assuming an admissible mode shape for the inflated elastic sail an iterative procedure based on a doublet lattice method is employed to determine the inflated configuration as well as various aerodynamic characteristics application of the method is made to a typical nylon cotton sailwing of ar 6 0 and results are presented graphically to show the effect of various parameters the results are found to tend to plane wing values when the pretensions are large in magnitude

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

enriched finite elements for time harmonic webster s equation this work presents an enriched finite element method fem dedicated to the numerical resolution of webster s equation in the time harmonic regime which models many physical configurations e g wave propagation in acoustic waveguides or vibration of bars with varying cross section building on the wave based methods existing in the literature we present new enriched finite element bases that account for both the frequency of the problem and the heterogeneity of the coefficients of the equation the enriched method is compared to the classical fifth order polynomial fem and we show they share the same asymptotic convergence order present the same easiness of implementation and have similar computational costs the main improvement brought by these enriched bases relies on the convergence threshold the mesh size at which the convergence regime begins and the convergence multiplicative constant which are observed to be i better than the ones associated with polynomial bases and ii not only dependent on the resolution the number of elements per wavelength but also on the frequency for a fixed resolution making the method we propose well adapted to high frequency regimes moreover taking into account the heterogeneity of the coefficient in webster s equation by using an element dependent enrichment leads to a significant decrease of the approximation error on the considered examples compared to a uniform enrichment again with almost no additional cost several possible extensions of this work are finally discussed

some general properties of the exact acoustic fields in horns and baffles abstract the propagation of the fundamental longitudinal acoustic mode in a duct of variable cross section is considered and the webster wave equations for the sound pressure and velocity are used to establish some general properties of the exact acoustic fields the equipartition of kinetic and compression energies is shown section 2 1 to hold at all stations only for i a duct of constant cross section and ii an exponential horn these are the two cases for which the wave equations for the acoustic velocity and pressure coincide it is proved section 2 3 that there are only five duct shapes forming two dual families which have constant cut off frequency ies namely i the exponential duct which is self dual and is the only shape with constant and coincident cut offs both for the velocity and pressure ii the catenoidal horns of cross section s cosh 2 sinh 2 which with their duals iii the inverse catenoidal ducts s sech 2 csch 2 have one constant cut off frequency respectively for the acoustic pressure and velocity the existence of at least one constant cut off frequency implies that the corresponding wave equation can be transformed into one with constant coefficients and thus the acoustic fields calculated exactly in terms of elementary exponential circular and hyperbolic functions this property also applies to the imaginary transformations of the above shapes viz the sinusoidal s sin 2 and inverse sinusoidal s csc 2 ducts that have no cut off frequency i e are acoustically transparent it is shown that elementary exact solutions of the webster equation exist only section 3 1 for these seven shapes namely the exponential catenoidal sinusoidal and inverse ducts it is implied that for all other duct shapes the exact acoustic fields involve special functions in infinite or finite terms e g bessel and hermite functions respectively for power law and gaussian horns examples of the method of analysis are given by calculating in elementary form the exact acoustic fields in inverse catenoidal ducts for all cases of a propagating waves above b non oscillating modes below and c transition fields at the cut off frequency the inverse catenoidal ducts consist of a the horn of cross section s sech 2 ressembling the soliton of non linear water wave fame and b the baffle of cross section s csch 2 which also matches two exponentially converging ducts but has infinite instead of finite flare at the origin the geometrical and acoustic properties of these ducts are illustrated by sets of six plots in figure 1 a for the sech horn and in figure 1 b for the csch baffle the exact acoustic fields are described by amplitude and phase decompositions of the sound velocity and pressure plotted as functions of position along the duct for four frequencies ranging from the cut off condition to the ray limit or w k b j approximation

entropy functions for symmetric systems of conservation laws abstract using a simple symmetrizability criterion we show that symmetric systems of conservation laws are equipped with a one parameter family of entropy functions

trumpet with near perfect harmonicity design and acoustic results this paper presents a mathematical design methodology for determining the shape of a trumpet air column that has near perfect harmonicity whose components are discontinuity free and whose input impedance peak heights are balanced over the playing range the simulation model employed assumes linear wave propagation and uses cylindrical element discretization with a plane wave approximation acoustic measurements are made using a test set up with an estimated relative measurement error of 3 cents comparisons of measured results are given for the presented design macaluso trumpet and the same trumpet air column with the bell replaced by a commercially used generic trumpet bell of unknown shape generic trumpet for acoustic resonance modes 2 13 233 1515 hz the measured root mean square rms harmonicity deviation is 5 cents for the macaluso trumpet whereas it is 18 cents for the generic trumpet however considering the estimated measurement uncertainty each of those deviations is somewhat over s

multi time step explicit implicit method for non linear structural dynamics we present a method with domain decomposition to solve time dependent non linear problems this method enables arbitrary numeric schemes of the newmark family to be coupled with different time steps in each subdomain this coupling is achieved by prescribing continuity of velocities at the interface we are more specifically interested in the coupling of implicit explicit numeric schemes taking into account material and geometric non linearities the interfaces are modelled using a dual schur formulation where the lagrange multipliers represent the interfacial forces unlike the continuous formulation the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements velocities and accelerations at the interfaces we show that within the framework of the newmark family of numeric schemes continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged to prove this stability we use an energy method i e a global method over the whole time interval in order to verify the algorithms properties then we propose to extend this to non linear situations in the following cases implicit linear explicit non linear explicit non linear explicit non linear and implicit non linear explicit non linear finally we present some examples showing the feasibility of the method copyright 2001 john wiley sons ltd

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

determination of lamb mode eigenvalues an original method is presented to determine the complex lamb wave spectrum by using a numerical spectral method applied to the elasticity equations this method presents the advantage to directly determine complex wave numbers for a given frequency via a classical matricial eigenvalue problem and allows the wave numbers to be determined at relatively high frequencies i e corresponding to many propagating modes it does not need initial guess values for the wave numbers contrary to the usual method of root finding of the rayleigh lamb frequency equations dispersion relation in the complex plane results are presented and the method is discussed

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries roles of the inf sup stability constants in this paper we review and we extend the reduced basis approximation and a posteriori error estimation for steady stokes flows in affinely parametrized geometries focusing on the role played by the brezzi s and babuska s stability constants the crucial ingredients of the methodology are a galerkin projection onto a low dimensional space of basis functions properly selected an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure and a rigorous a posteriori error estimation on field variables the combinatiofn of these three factors yields substantial computational savings which are at the basis of an efficient model order reduction ideally suited for real time simulation and many query contexts e g optimization control or parameter identification in particular in this work we focus on i the stability of the reduced basis approximation based on the brezzi s saddle point theory and the introduction of a supremizer operator on the pressure terms ii a rigorous a posteriori error estimation procedure for velocity and pressure fields based on the babuska s inf sup constant including residuals calculations iii the computation of a lower bound of the stability constant and iv different options for the reduced basis spaces construction we present some illustrative results for both interior and external steady stokes flows in parametrized geometries representing two parametrized classical poiseuille and couette flows a channel contraction and a simple flow control problem around a curved obstacle

consistent coupled mode theory of sound propagation for a class of nonseparable problems this article examines the effects of boundary condition approximations that arise whenever the coupled mode theories of a d pierce j acoust soc am 37 19 27 1965 and d m milder j acoust soc am 46 1259 1263 1969 are applied to propagation problems involving range dependent boundaries this boundary condition approximation requires that the depth derivative rather than the normal derivative of the field be continuous across a sloping interface the approximation is necessary in order to carry out the partial separation of depth and range variables effected in the mathematical formulation of the theory this article will show that a consequence of this approximation is that conventional coupled mode theory applied to dissipationless media with nonhorizontal boundaries does not conserve energy it is shown that a correction to coupled mode theory can be derived such that the proper boundary conditions are satisfied and energy is conserved to first order in the local bottom slope moreover the corrections are not prohibitive in terms of added computational complexity numerical examples are presented which illustrate the nonconservation of energy effect and the corrections to the theory

improved precision in measurements of acoustic impedance spectra using resonance free calibration loads and controlled error distribution resonances and or singularities during measurement and calibration often limit the precision of acoustic impedance spectra this paper reviews and compares several established techniques and describes a technique that incorporates three features that considerably improve precision the first feature is to minimize problems due to resonances by calibrating the instrument using up to three different acoustic reference impedances that do not themselves exhibit resonances the second involves using multiple pressure transducers to reduce the effects of measurement singularities the third involves iteratively tailoring the spectrum of the stimulus signal to control the distribution of errors across the particular measured impedance spectrum examples are given of the performance of the technique on simple cylindrical waveguides

the acoustical klein gordon equation the wave mechanical step and barrier potential functions the transformed form of the webster equation is investigated usually described as analogous to the schrodinger equation of quantum mechanics it is noted that the second order time dependency defines a klein gordon problem this acoustical klein gordon equation is analyzed with particular reference to the acoustical properties of wave mechanical potential functions u x that give rise to geometry dependent dispersions at rapid variations in tract cross section such dispersions are not elucidated by other one dimensional cylindrical or conical duct models since sturm liouville analysis is not appropriate for inhomogeneous boundary conditions the exact solution of the klein gordon equation is achieved through a green s function methodology referring to the transfer matrix of an arbitrary string of square potential functions including a square barrier equivalent to a radiation impedance the general conclusion of the paper is that in the absence of precise knowledge of initial conditions on the area function any given potential function will map to a multiplicity of area functions of identical relative resonance characteristics since the potential function maps uniquely to the acoustical output it is suggested that the one dimensional wave physics is both most accurately and most compactly described within the klein gordon framework

localized and propagating modes in acoustical waveguides with variable cross section a method of determining the propagating and the localized modes in an acoustic waveguide with continuously varying cross section is studied for the rectangular and circular waveguides where the area of the cross section is dependent on the distance from the center of the waveguide the wave equation with dirichlet pressure release boundary condition can be transformed into an infinite set of coupled ordinary differential equations with variable coefficients this set of differential equations has a first integral which is the flux of the wave and is constant along the length of the waveguide if the cross section approaches a constant value at the two ends of the waveguide then there are a finite number of propagating modes plus a number of localized modes a technique for the numerical integration of the stiff ordinary differential equations resulting from the partial wave projection is discussed and applied to different examples

hyper reduction of mechanical models involving internal variables we propose to improve the efficiency of the computation of the reduced state variables related to a given reduced basis this basis is supposed to be built by using the snapshot proper orthogonal decomposition pod model reduction method in the framework of non linear mechanical problems involving internal variables the local integration of the constitutive laws can dramatically limit the computational savings provided by the reduction of the order of the model this drawback is due to the fact that using a galerkin formulation the size of the reduced basis has no effect on the complexity of the constitutive equations in this paper we show how a reduced basis approximation and a petrov galerkin formulation enable to reduce the computational effort related to the internal variables the key concept is a reduced integration domain where the integration of the constitutive equations is performed the local computations being not made over the entire domain we extrapolate the computed internal variable over the full domain using pod vectors related to the internal variables this paper shows the improvement of the computational saving obtained by the hyper reduction of the elasto plastic model of a simple structure

stochastic galerkin approximation of the reynolds equation with irregular film thickness abstract we consider the approximation of the reynolds equation with an uncertain film thickness the resulting stochastic partial differential equation is solved numerically by the stochastic galerkin finite element method with high order discretizations both in spatial and stochastic domains we compute the pressure field of a journal bearing in various numerical examples that demonstrate the effectiveness and versatility of the approach the results suggest that the stochastic galerkin method is capable of supporting design when manufacturing imperfections are the main sources of uncertainty

on webster s horn equation and some generalizations sound waves along a rigid axisymmetric tube with a variable cross section are considered the governing helmholtz equation is solved using power series expansions in a stretched radial coordinate leading to a hierarchy of one dimensional ordinary differential equations in the longitudinal direction the lowest approximation for axisymmetric motion turns out to be webster s horn equation fourth order differential equations are obtained at the next level of approximation comparisons with existing asymptotic theories for waves in slender tubes are made

thermodynamic limit of the six vertex model with domain wall boundary conditions we address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model here we compare the bulk free energy for periodic and domain wall boundary conditions using a determinant representation for the partition function with domain wall boundary conditions we derive toda differential equations and solve them asymptotically in order to extract the bulk free energy we find that it is different and bears no simple relation to the free energy for periodic boundary conditions the six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics alternating sign matrices this implies new results for the weighted counting for large size alternating sign matrices finally we comment on the interpretation of our results in particular in connection with domino tilings dimers on a square lattice

six vertex model with domain wall boundary conditions and one matrix model the partition function of the six vertex model on a square lattice with domain wall boundary conditions dwbc is rewritten as a hermitian one matrix model or a discretized version of it similar to sums over young diagrams depending on the phase the expression is exact for finite lattice size which is equal to the size of the corresponding matrix in the thermodynamic limit the matrix integral is computed using traditional matrix model techniques thus providing a complete treatment of the bulk free energy of the six vertex model with dwbc in the different phases in particular in the antiferroelectric phase the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions

physical interpretation of the proper orthogonal modes using the singular value decomposition abstract proper orthogonal decomposition is a statistical pattern analysis technique for finding the dominant structures called the proper orthogonal modes in an ensemble of spatially distributed data while the proper orthogonal modes are obtained through a statistical formulation they can be physically interpreted in the field of structural dynamics the purpose of this paper is thus to provide some insights into the physical interpretation of the proper orthogonal modes using the singular value decomposition

comparison between acoustic measurements of brass instruments and one dimensional models with curved wavefronts and transformed axial coordinates a progressive spherical or spheroidal wavefront approximation has previously been found to be a necessary step for a more accurate application of webster s wave equation to rapidly flaring horns this leads to a necessary transformation of the horn area function from the usual flat cross sectional area in terms of the axial coordinate into a curved cap like wavefront area as a function of either the axial coordinate the arc length coordinate along the horn profile the leading curved wavefront coordinate or still other possible longitudinal coordinates in this article horn functions and related frequency potential functions are calculated from the measured horn profiles of a trombone and a trumpet for several of the above parameterizations from them cutoff frequencies and effective lengths are determined a comparison is drawn between theoretical results using different parameterizations results calculated via transfer matrix models and experimental measurements of the acoustical input impedanc

numerical methods for higher order sturm liouville problems we review some numerical methods for self adjoint and non self adjoint boundary eigenvalue problems

determinant formula for the six vertex model the partition function of a six vertex model with domain wall boundary conditions is considered on the finite lattice the authors show that the partition function satisfies a recursive relation they solve the recursion relation by a determinant formula this gives a determinant representation for the partition function they use the quantum inverse scattering method qism