dimensional hyper reduction of nonlinear finite element models via empirical cubature we present a general framework for the dimensional reduction in terms of number of degrees of freedom as well as number of integration points hyper reduction of nonlinear parameterized finite element fe models the reduction process is divided into two sequential stages the first stage consists in a common galerkin projection onto a reduced order space as well as in the condensation of boundary conditions and external forces for the second stage reduction in number of integration points we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized the distinguishing features of the proposed method are 1 the minimization problem is posed in terms of orthogonal basis vector obtained via a partitioned singular value decomposition rather that in terms of snapshots of the integrand 2 the volume of the domain is exactly integrated 3 the selection algorithm need not solve in all iterations a nonnegative least squares problem to force the positiveness of the weights furthermore we show that the proposed method converges to the absolute minimum zero integration error when the number of selected points is equal to the number of internal force modes included in the objective function we illustrate this model reduction methodology by two nonlinear structural examples quasi static bending and resonant vibration of elastoplastic composite plates in both examples the number of integration points is reduced three order of magnitudes with respect to fe analyses without significantly sacrificing accuracy

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

finite slicing in superreflexive banach spaces abstract for every superreflexive banach space x there exists a supermultiplicative function which is the supremum in a very natural ordering of the set of all the moduli of convexity of equivalent norms if this supremum is actually a maximum achieved under some equivalent renorming of x then its modulus of convexity is the best possible in asymptotic sense otherwise we can give an almost optimal uniformly convex renorming of x beyond the classical power type bound obtained by pisier 21

travelling wave solutions of reduced super kdv equation a perspective from lame equation abstract reduced supersymmetric korteweg de vries super kdv equation is investigated in this paper with the help of an auxiliary lame equation and jacobi elliptic function travelling wave solutions of reduced super kdv equation are obtained in the form of jacobi elliptic periodic function meanwhile trigonal solutions and hyperbolic solutions are derived

multidimensional a priori hyper reduction of mechanical models involving internal variables this paper concerns the adaptation of reduced order models during simulations of series of elastoviscoplastic problems in continuation with previous works this paper aimed at extending the a priori hyper reduction method aphr method for nonlinear thermal problems to nonlinear mechanical problems involving internal variables this method is an a priori approach because full incremental responses of detailed models are not forecasted in order to build reduced order models the recent extension of the hyper reduction method to reduction of mechanical models involving internal variables makes possible the reduction of degrees of freedom and the reduction of integration points a multi level formulation is introduced to focus on the capability of the method to perform efficient parallel computations to adapt reduced order models

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

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

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

nonpropagating solitons of the variable coefficient and nonisospectral korteweg de vries equation the method of inverse scattering is extended to solve the initial value problem of a variable coefficient and nonisospectral korteweg de vries equation with time varying boundary condition one and two soliton solutions are examined in detail by an appropriate decomposition soliton interactions and asymptotic behaviors are investigated oscillating and asymptotically standing two soliton solutions are discussed

investigation on a nonisospectral fifth order korteweg de vries equation generalized from fluids in this paper a nonisospectral fifth order korteweg de vries equation generalized from fluids is investigated with symbolic computation such equation is transformed into its bilinear form through a proposed dependent variable transformation with one more parameter than those in the existing literature n soliton solutions backlund transformation and lax pair in the explicit forms are constructed based on the above results the characteristic line method is applied to discuss the features of the solitons for the nonisospectral problem i e the controllable solitonic velocities and widths four types of solitonic structures with the different solitonic velocities widths amplitudes and backgrounds are also illustrated

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

are the snapshot difference quotients needed in the proper orthogonal decomposition this paper presents a theoretical and numerical investigation of the following practical question should the time difference quotients dqs of the snapshots be used to generate the proper orthogonal decomposition pod basis functions the answer to this question is important since some published numerical studies use the time dqs whereas other numerical studies do not the criterion used in this paper to answer this question is the optimality of the convergence rate of the error of the reduced order model with respect to the number of pod basis functions since to the best of our knowledge a definition of the optimality of the convergence rate is not available we propose one in definition 3 1 in this paper two cases are considered the no dq case in which the snapshot dqs are not used and the dq case in which the snapshot dqs are used for each case two types of pod bases are used the l 2 pod basis in which the basis is generated in the l 2 norm and the h 1 pod basis in which the basi

analytical solutions of kdv equation with relaxation effect of inhomogeneous medium kdv korteweg de vries equation with relaxation effect of inhomogeneous medium with time changing can be employed in many different physical fields in this paper some new analytical solutions of the equation are obtained which may be very useful in numerical simulation by using of the truncated expansion and jacobi elliptic function expansion methods

an adaptive scheme for a class of interpolatory model reduction methods for frequency response problems frequency sweep problems arise in many structural dynamic acoustic and structural acoustic applications in each case they incur the evaluation of a frequency response function for a typically large number of frequencies because each function evaluation requires the solution of an often large scale system of equations frequency sweep problems are computationally intensive interpolatory model order reduction is a powerful tool for reducing their cost however the performance of this tool depends on the location and number of the interpolation frequency points it also depends on the number of consecutive frequency derivatives of the response function that are matched at each frequency point so far these two choices have been made in the literature in a heuristic manner in contrast this paper proposes an automatic adaptive strategy based on monitoring the euclidean norm of the relative residual associated with the function to be evaluated over the frequency band of interest more specifically the number of interpolation points and the number of matched frequency derivatives are adaptively increased until the global euclidean norm of the relative residual is reduced below a user specified tolerance the robustness accuracy and computational efficiency of this adaptive strategy are highlighted with the solution of several frequency sweep problems associated with large scale structural dynamic acoustic and structural acoustic finite element models copyright 2012 john wiley sons ltd

symmetries conservation laws and hamiltonian structures of the non isospectral and variable coefficient kdv and mkdv equations an infinite number of form invariant symmetries is obtained and a one to one correspondence between symmetries and conservation laws is established for the non isospectral and variable coefficient generalizations of both the kdv and the mkdv equations two families of symmetries and their lie algebraic structures are constructed some interesting facts about their hamiltonian structures are presented

sequential karhunen loeve basis extraction and its application to images the karhunen loeve kl transform is an optimal method for approximating a set of vectors or images which was used in image processing and computer vision for several tasks such as face and object recognition its computational demands and its batch calculation nature have limited its application here we present a new sequential algorithm for calculating the kl basis which is faster in typical applications and is especially advantageous for image sequences the kl basis calculation is done with much lower delay and allows for dynamic updating of image databases systematic tests of the implemented algorithm show that these advantages are indeed obtained with the same accuracy available from batch kl algorithms

szlenk indices and uniform homeomorphisms we prove some rather precise renorming theorems for banach spaces with szlenk index wo we use these theorems to show the invariance of certain quantitative szlenk type indices under uniform homeomorphisms

compact spaces of szlenk index ω abstract the szlenk index has found many applications in the isomorphic theory of banach spaces its definition is based in some kind of interplay between a weak topology and the norm metric with not much care on the linear structure there is no obstacle to consider the notion of szlenk index in more general settings in this paper we study the compact spaces of szlenk index at most with respect to an associated metric we include new applications to banach spaces of the these methods where the estimations of the growth speed of the finite szlenk indices play a fundamental role

symmetries and a hierarchy of the general modified kdv equation two groups of symmetries and their lie algebra properties for the modified kdv equation are extended to the general modified kdv equation and the miura transformation between the general kdv equation and the general modified equation is also established

symmetries and a hierarchy of the general kdv equation two groups old and new of symmetries and their lie algebra properties for the kdv and cylindrical kdv equations are unified and extended to the general kdv equation