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

continuous time covariance approaches for modal analysis proper orthogonal decomposition pod can be used to obtain complete information about the linear normal modes of structural systems only when special conditions are met principally related to the knowledge of mass distribution and to the absence of damping the main advantage of pod is the possibility to directly use covariance matrices estimated from time responses for the modal identification of the structure under investigation this paper proposes a method to extend this covariance based identification to general cases where significant damping is present and when no data is available on mass distribution a least square based decomposition is used to infer information about modal characteristics of the system this decomposition is applied to three different cases the first one uses records related to the free response of the system while the remaining two use records of the system under persistent white noise excitations in those latter cases the information on the excitation input time history can either be used or treated as unknown basic yet complete examples are presented to illustrate the properties of the proposed technique

on the physical interpretation of proper orthogonal modes in vibrations abstract the authors relate the proper orthogonal modes as applied in discrete vibration systems to normal modes of vibration in systems with a known mass matrix in the case of undamped free vibration the proper orthogonal modes converge to the linear normal modes as the amount of data increases this interpretation is also practical for lightly modally damped systems forced resonances lead to proper orthogonal modes which approximate vibration modes a particular case of non linear normal modes are looked at in which the motion of a single mode follows a curve in the co ordinate space in this case the proper orthogonal modes represent the principal axes of inertia formed by the distribution of data on the modal co ordinate curve more generally the proper orthogonal modes represent the principal axes of inertia of the data

proper orthogonal decomposition pod of a class of vibroimpact oscillations abstract this study employs the method of proper orthogonal decomposition pod also known as the karhunen loeve k l method to extract dominant coherent structures modes that approximate the system behavior from time series data these mode shapes can be used in a galerkin reconstruction process to obtain lower dimensional models for the structural systems under consideration the mode shapes and the energies obtained by this process can also be used to predict certain kinds of periodic or non periodic non linear motions the k l method has been applied successfully to fluid dynamical thermal processes and signal processing however only a handful of works exist in the area of vibration analysis and structural mechanics an extensive k l analysis is performed numerically for two vibroimpacting systems a beam and a rotor lower dimensional models are created and used to study non linear energy transmission from low to higher k l modes extensive reconstructions are also performed to prove the efficacy of the k l method to provide accurate low dimensional dynamical models in addition experimental investigations are presented for the case of the overhung impacting rotor and qualitative comparisons with the theoretical analysis are presented

poms analysis of randomly vibrating systems obtained from karhunen loeve expansion abstract the karhunen loeve kl decomposition establishes that a 2d random field can be expanded as a series involving a sequence of deterministic orthogonal functions with orthogonal random coefficients the proper orthogonal decomposition pod method consists in detecting spatially coherent modes in the dynamics of a spatio temporally varying system by diagonalizing the spatial correlation function given by an averaging operator the kl expansion is applied here to the responses of randomly excited vibrating systems with a view to performing a pod in separated variables time and space form discrete and continuous mechanical systems are considered in this study as well as stationary and transient non stationary responses an averaging operator involving time and ensemble averages is proposed to draw up the pod in separated variables form from the associated kl expansion the result obtained using this approach agrees with the classical pod in the case of deterministic or ergodic random signals the associated proper orthogonal modes are interpreted in case of linear and nonlinear vibrating systems subjected to white noise excitation in terms of normal modes

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

a locking free weak galerkin finite element method for elasticity problems in the primal formulation this paper presents an arbitrary order locking free numerical scheme for linear elasticity on general polygonal polyhedral partitions by using weak galerkin wg finite element methods like other wg methods the key idea for the linear elasticity is to introduce discrete weak strain and stress tensors which are defined and computed by solving inexpensive local problems on each element such local problems are derived from weak formulations of the corresponding differential operators through integration by parts locking free error estimates of optimal order are derived in a discrete h 1 norm and the usual l 2 norm for the approximate displacement when the exact solution is smooth numerical results are presented to demonstrate the efficiency accuracy and the locking free property of the weak galerkin finite element method

a weak galerkin finite element scheme with boundary continuity for second order elliptic problems abstract a new weak galerkin wg finite element method for solving the second order elliptic problems on polygonal meshes by using polynomials of boundary continuity is introduced and analyzed the wg method is utilizing weak functions and their weak derivatives which can be approximated by polynomials in different combination of polynomial spaces different combination gives rise to different weak galerkin finite element methods which makes wg methods highly flexible and efficient in practical computation this paper explores the possibility of certain combination of polynomial spaces that minimize the degree of freedom in the numerical scheme yet without losing the accuracy of the numerical approximation error estimates of optimal order are established for the corresponding wg approximations in both a discrete h 1 norm and the standard l 2 norm in addition the paper also presents some numerical experiments to demonstrate the power of the wg method the numerical results show that the wg method achieves the optimal convergence order with a relatively low computational cost which reveals a great promise of the flexibility and accuracy of the wg method

a nonsymmetric state variable decomposition for modal analysis a modal decomposition strategy based on state variable ensembles is formulated a nonsymmetric generalized eigenvalue problem is constructed the data based eigenvalue problem is related to the generalized eigenvalue problem associated with free vibration solutions of the state variable formulation of linear multi degree of freedom systems for linear free response data the inverse transpose of the eigenvector matrix converges to the state variable modal eigenvectors and the eigenvalues of the nonsymmetric eigenvalue problem approximate those of the state variable model as such the eigenvalues lead to estimates of frequencies and modal damping the interpretation holds for linear systems with multi modal free responses whether damping is small or somewhat large modal or nonmodal and without the need of input data

discrete maximum principle for the p 1 p 0 weak galerkin finite element approximations abstract this paper presents two discrete maximum principles dmp for the numerical solution of second order elliptic equations arising from the weak galerkin finite element method the results are established by assuming an h acute angle condition for the underlying finite element triangulations the mathematical theory is based on the well known de giorgi technique adapted in the finite element context some numerical results are reported to validate the theory of dmp

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

an optimal modal reduction of a system with frictional excitation abstract the proper orthogonal decomposition theorem is used in an optimal modal reduction of a frictionally excited system the chaotic dynamics are exploited in obtaining a basis that broadly represents the system s dynamics this basis is used in a galerkin s approximation to reduce the system order the reduced model is validated using qualitative and quantitative means and also from the occurrence of bifurcations the results confirm the validity of the method and expose the benefits of using proper orthogonal modes rather than the linear natural modes of the system

numerical solution of parabolic integro differential equations by the discontinuous galerkin method the numerical solution of a parabolic equation with memory is considered the equation is first discretized in time by means of the discontinuous galerkin method with piecewise constant or piecewise linear approximating functions the analysis presented allows variable time steps which as will be shown call then efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term or by nonsmooth initial data the combination with finite element discretizat in space is also studied

a stable numerical algorithm for the brinkman equations by weak galerkin finite element methods this paper presents a stable numerical algorithm for the brinkman equations by using weak galerkin wg finite element methods the brinkman equations can be viewed mathematically as a combination of the stokes and darcy equations which model fluid flow in a multi physics environment such as flow in complex porous media with a permeability coefficient highly varying in the simulation domain in such applications the flow is dominated by darcy in some regions and by stokes in others it is well known that the usual stokes stable elements do not work well for darcy flow and vice versa the challenge of this study is on the design of numerical schemes which are stable for both the stokes and the darcy equations this paper shows that the wg finite element method is capable of meeting this challenge by providing a numerical scheme that is stable and accurate for both darcy and the stokes dominated flows error estimates of optimal order are established for the corresponding wg finite element solutions the paper also presents some numerical experiments that demonstrate the robustness reliability flexibility and accuracy of the wg method for the brinkman equations

a symmetric weak galerkin method for solving non divergence form elliptic equations abstract we reformulate the second order non divergence form elliptic equations in a symmetric variation form i e a least squares form we design a weak galerkin finite element method for this high regularity formulation the optimal order of convergence is proved numerical results verify the theory in addition numerical results compared to the existing weak galerkin method for the non symmetric form and the new methods show advantage on the accuracy and the degree of freedoms

on the stability of the discontinuous galerkin method for the heat equation this paper analyzes stability properties of a class of discontinuous galerkin methods for the heat equation it is shown that the finite element projection associated with these methods is stable with respect to a mesh dependent norm a discrete analogue of the space time l 2 norm optimal order regularity error bounds in l 2 0 t l 2 omega are derived

a new error analysis of characteristics mixed fems for miscible displacement in porous media the method of characteristics type is especially effective for convection dominated diffusion problems due to the nature of characteristic temporal discretization the method allows one to use a large time step in many practical computations while all previous theoretical analyses always required certain restrictions on the time stepsize here we present a new analysis to establish unconditionally optimal error estimates for a modified method of characteristics with a mixed finite element approximation to the miscible displacement problem in mathbb r d d 2 3 for this purpose we introduce a new characteristic time discrete system we prove that the l 2 error bound the characteristic time discrete systemof the fully discrete method of characteristics to the time discrete system is tau independent and the numerical solution is bounded in w 1 infty norm unconditionally with the boundedness optimal error estimates are established in a traditional manner numerical results confirm our th

an effective implementation for stokes equation by the weak galerkin finite element method abstract in this paper we introduce and analyze the schur complement technique to a weak galerkin wg for short finite element method for solving stokes equation due to the special structure of the weak functions the main idea of schur decomposition method is to express the interior velocity functions defined inside partition elements in terms of the velocity functions defined on the partition boundary and the pressure functions this can largely reduce the degree of freedom of the derived discrete linear system we investigate the well posedness of the solution generated from schur complement and its coincidence with the standard wg method finally we exhibit some numerical experiments to illustrate the theoretical analysis

weak galerkin finite element method for viscoelastic wave equations abstract in this article we consider a weak galerkin finite element method wg fem for solving one type of viscoelastic wave equation a discrete weak gradient operator on discontinuous piecewise polynomials is used in the numerical scheme we introduce both the semi and fully discrete wg fems and establish the corresponding stability estimates optimal order error estimate in h 1 norm is derived numerical experiments are preformed to verify the theoretical results

reduced bases for nonlinear structural dynamic systems a comparative study abstract the presented work provides an overview of some commonly used approaches for generating reduced bases for discrete nonlinear dynamic systems it investigates the performance and the robustness of these bases if they are applied in a reduction by projection procedure on different test cases the bases are created from the linear normal modes the ritz vectors the proper and the smooth orthogonal decomposition method the a priori reduction the centroidal voronoi tessellation and the local equivalent linear stiffness method second order terms and an enhanced proper orthogonal decomposition formulation are included as variants the test cases are small dimensional locally or entirely nonlinear system subjected to a harmonic or an impulse force excitation the double objective of this numerical study is first to determine which bases are most adequate for a given combination of nonlinearity and excitation and second to which extend the bases exhibit an inherent robustness if the parameterisation of the excitation is changed a specific multicriteria decision analysis score is developed to assess the bases performance as a major result a strong dependence of the performance of the bases on the type of excitation is established and thus some bases become more adequate for a certain situation than others also a lack of robustness for all considered bases can be observed this situation improves in most cases if the basis is generated with the most critical values of the parameter

on the approximation of miscible displacement in porous media by a method of characteristics combined with a mixed method a miscible displacement of one incompressible fluid by another is modeled by a coupled system of partial differential equations the approximation of this system using a galerkin method that makes use of a modified method of characteristics for the concentration equation combined with a mixed finite element method for the pressure equation has been analyzed in comput math appl mech engrg 47 1984 pp 73 92 in this work we prove optimal error estimates in l 2 for this algorithm under milder restrictions in the time step than those required in the previous analysis in particular in the two dimensional case our estimates are valid when the time and space discretization parameters are of the same order which is a reasonable assumption in practice

legendre spectral collocation method for the solution of the model describing biological species living together in this article numerical techniques have been developed to solve a system of two nonlinear integro differential equations which arises in biological model this model has been solved by legendre spectral collocation method and bernstein polynomials the present methods reduce this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by newton s method also the results obtained by present methods have been compared with that of by some existing methods available in literature some illustrative examples have been discussed to show the applicability and accuracy of the present methods

a weak galerkin mixed finite element method for second order elliptic problems abstract a new weak galerkin wg method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations this method called wg mfem is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons polyhedra the wg mfem is capable of providing very accurate numerical approximations for both the primary and flux variables allowing the use of discontinuous approximating functions on arbitrary shape of polygons polyhedra makes the method highly flexible in practical computation optimal order error estimates in both discrete h and l norms are established for the corresponding weak galerkin mixed finite element solutions

a complex orthogonal decomposition for wave motion analysis a method is presented for decomposing wave motion into its principle components the basic idea is a complex generalization of proper orthogonal decomposition the method involves the representation of real oscillatory signals as complex analytic signals the relationship between complex modes and wave motion is explored from an ensemble of complex signals a complex correlation matrix is formed and its complex eigensolution is the basis of the decomposition like a complex singular value decomposition the complex eigenvectors contain standing and traveling characteristics a traveling index is proposed to quantify the relative degree of traveling and standing in a waveform a method of dissecting a wave mode into its traveling and standing parts is also proposed from the complex modes and modal coordinates frequencies wavelengths and characteristic wave speeds can be obtained the method is applied to traveling and standing wave examples

adaptive finite element methods for parabolic problems iv nonlinear problems we extend our program on adaptive finite element methods for parabolic problems to a class of nonlinear scalar problems we prove a posteriori error estimates design corresponding adaptive algorithms and present some numerical results

the weak galerkin method for solving the incompressible brinkman flow the brinkman equations are used to describe the dynamics of fluid flows in complex porous media with the high variability in the permeability coefficients which may take extremely large or small values this paper is devoted to the numerical analysis of a family of weak galerkin wg finite element methods for solving the time dependent brinkman problems this wg method is equipped with stable finite elements consisting of usual polynomials of degree k 1 for the velocity and polynomials of degree k 1 for the pressure the velocity element is enhanced by polynomials of degree k on the interface of the finite element partition all the finite element functions are discontinuous for which the usual gradient and divergence operators are implemented as distributions in properly defined spaces we further establish a priori error estimates in l 2 norm and h 1 norm and we provide a few numerical experiments to illustrate the behavior of the proposed scheme and confirm our theoretical findings regarding optimal convergence of the approximate solutions

unconditional convergence and optimal error estimates of a galerkin mixed fem for incompressible miscible flow in porous media in this paper we study the unconditional convergence and error estimates of a galerkin mixed fem with the linearized semi implicit euler scheme for the equations of incompressible miscible flow in porous media we prove that the optimal l 2 error estimates hold without any time step convergence conditions while all previous works require certain time step restrictions our theoretical results provide a new understanding on commonly used linearized schemes the proof is based on a splitting of the error into two parts the error from the time discretization of the pdes and the error from the finite element discretization of corresponding time discrete pdes the approach used in this paper can be applied to more general nonlinear parabolic systems and many other linearized semi implicit time discretizations

a weak galerkin finite element method for second order elliptic problems this paper introduces a finite element method by using a weakly defined gradient operator over generalized functions the use of weak gradients and their approximations results in a new concept called discrete weak gradients which is expected to play an important role in numerical methods for partial differential equations this article intends to provide a general framework for managing differential operators on generalized functions as a demonstrative example the discrete weak gradient operator is employed as a building block in the design of numerical schemes for a second order elliptic problem in which the classical gradient operator is replaced by the discrete weak gradient the resulting numerical scheme is called a weak galerkin wg finite element method it can be seen that the weak galerkin method allows the use of totally discontinuous functions in the finite element procedure for the second order elliptic problem an optimal order error estimate in both a discrete h 1 and l 2 norms are established for the corresponding weak galerkin finite element solutions a superconvergence is also observed for the weak galerkin approximation

modal parameter estimation for a wetted plate under flow excitation a challenging case in using pod abstract in this paper some issues related to the use of the proper orthogonal decomposition pod for the identification of the modal parameters of 2d structures are highlighted with reference to the experimental identification of the vibration modes of a plate excited on one side by a turbulent boundary layer the poor convergence of the pod modes to the linear normal modes is explained analytically and provides also the theoretical framework to avoid these issues by applying band pass filtering bp pod the application of the pod with data pre filtering has allowed for the evaluation of the modal damping from the proper orthogonal coordinates by using the random decrement technique to obtain the corresponding free decay function though the emphasis is about methodological aspects related to the pod as a byproduct of the performed analysis some more physical insights concerning the damping sensitivity and the hydrodynamic added mass are finally carried out

a weak galerkin finite element method for high dimensional time fractional diffusion equation abstract this paper is concerned with the application of a weak galerkin finite element method wg fem to the time fractional diffusion equation the wg fem with l1 formula and the fast evaluation scheme based on wg fem are designed the optimal convergence rates for both semi discrete and fully discrete wg fem schemes are obtained and the stability analysis for the semi discrete wg fem is derived numerical experiments are implemented to verify the theoretical results

smooth orthogonal decomposition for modal analysis of randomly excited systems modal parameter estimation in terms of natural frequencies and mode shapes is studied using smooth orthogonal decomposition for randomly excited vibration systems this work shows that under certain conditions the smooth orthogonal decomposition eigenvalue problem formulated from white noise induced response data can be tied to the unforced structural eigenvalue problem and thus can be used for modal parameter estimation using output response ensembles only the generalized eigenvalue problem is formed to estimate eigenfrequencies and modal vectors for an eight degree of freedom lightly damped vibratory system the estimated frequencies are compared against system frequencies obtained from structural eigenvalue problem and estimated modal vectors are checked using the modal assurance criterion for lightly damped simulations satisfactory results are obtained for estimating both system frequencies and modal vectors

an optimal order error estimate for a family of ellam mfem approximations to porous medium flow mathematical models used to describe porous medium flow lead to coupled systems of time dependent nonlinear partial differential equations which present serious mathematical and numerical difficulties standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid orientation effect the ellam mfem time stepping procedure in which an eulerian lagrangian localized adjoint method ellam is used to solve the transport equation and a mixed finite element method mfem is used for the pressure equation simulates porous medium flow accurately even if large spatial grids and time steps are used in this paper we prove an optimal order error estimate for a family of ellam mfem approximations

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

an optimal order l2 error estimate for nonsymmetric discontinuous galerkin methods for a parabolic equation in multiple space dimensions we analyze the nonsymmetric discontinuous galerkin methods nipg and iipg for linear elliptic and parabolic equations with a spatially varied coefficient in multiple spatial dimensions we consider d linear approximation spaces on a uniform rectangular mesh but our results can be extended to smoothly varying rectangular meshes using a blending or boolean interpolation we obtain a superconvergence error estimate in a discrete energy norm and an optimal order error estimate in a semi discrete norm for the parabolic equation the l 2 optimality for the elliptic problem follows directly from the parabolic estimates numerical results are provided to validate our theoretical estimates we also discuss the impact of penalty parameters on convergence behaviors of nipg

smooth karhunen loeve decomposition to analyze randomly vibrating systems abstract this paper deals with the application of smooth karhunen loeve decomposition skld procedure applied to random fields the skld is obtained by solving a generalized eigenproblem defined by combining the covariance matrix of the random field with that of the associated time derivative random field the main properties of the skld procedure thus obtained are described and compared with the classical karhunen loeve decomposition the skld is then applied to the responses of randomly excited vibrating systems in order to perform modal analysis the resulting skld characteristics are discussed in the case of linear vibrating systems subjected to white noise excitation in terms of normal modes

smooth orthogonal decomposition based vibration mode identification a new multivariate data analysis method called smooth orthogonal decomposition sod is proposed to extract linear normal modes and natural frequencies of multi degree of freedom and distributed parameter vibration systems it is demonstrated that for an undamped free vibration of a multi degree of freedom system the computed smooth orthogonal modes are in direct correspondence with the actual normal vibration modes and the smooth orthogonal values are related to the corresponding natural frequencies the same is also shown to be true for lightly damped free vibrations of both lumped and distributed parameter systems in contrast to the intrinsic limitations of the proper orthogonal decomposition pod analysis which requires the knowledge of system s mass matrix to extract normal modes and cannot uniquely identify modal subspaces that have similar proper orthogonal values the sod is shown to overcome both of these deficiencies numerical examples are provided to compare the performances of the pod and sod based modal identification in various types of vibration environment

an operational matrix based scheme for numerical solutions of nonlinear weakly singular partial integro differential equations abstract in this article we introduce an operational matrix scheme based on two dimensional wavelets for the volterra weakly singular nonlinear partial integro differential equations by implementing two dimensional wavelets approximations and its operational matrices of integration and differentiation along with collocation points the weakly singular partial integro differential equations are reduced into the system of nonlinear algebraic equations moreover bernoulli wavelet approximation and legendre wavelet approximation have been used for inspecting the errors and convergence analysis of the given problems some numerical examples are included to establish the accuracy of the proposed scheme via bernoulli wavelet approximation and legendre wavelet approximation respectively additionally comparisons of error values between the two wavelets have been presented