optimal matrix pencil approximation problem in structural dynamic model updating abstract the problem of finding the least change adjustment to a given matrix pencil is considered in this paper desired matrix properties including satisfaction of characteristic equation symmetry positive semidefiniteness and sparsity are imposed as side constraints to form the optimal matrix pencil approximation problem such a problem is related to the frequently encountered engineering problem of a structural modification on the dynamic behavior of a structure alternating direction method is applied to this constrained minimization problem numerical results are included to illustrate the performance and application of the proposed method

a symmetric generalized inverse eigenvalue problem in structural dynamics model updating abstract updating an existing but inaccurate structural dynamics model with measured data can be mathematically reduced to the problem of the best approximation to a given matrix pencil in the frobenius norm under a given spectral constraint and a submatrix pencil constraint in this paper a direct method and the associated mathematical theories for solving this problem are proposed it is shown that the best approximation solution is unique and an explicit expression of the solution is derived numerical examples show that the proposed method is quite accurate and efficient

discontinuous finite elements for a hyperbolic problem with singular coefficient a convergence theory for one dimensional spherical transport a theory of convergence is presented for the discontinuous galerkin finite element method of solving the nonscattering spherically symmetric boltzmann transport equation using piecewise constant test and trial functions results are then extended to higher order polynomial spaces comparisons of numerical properties were presented in earlier work

the nearness problems for symmetric matrix with a submatrix constraint in this paper we first give the representation of the general solution of the following least squares problem lsp given a matrix email protected r n x p and symmetric matrices email protected r p x p a email protected r r x r find an nxn symmetric matrix a such that x email protected min s t a 1 r a 0 where a 1 r is the rxr leading principal submatrix of the matrix a we then consider a best approximation problem given an nxn symmetric matrix email protected with email protected 1 r a 0 find email protected s e such that email protected email protected min a s email protected email protected email protected where s e is the solution set of lsp we show that the best approximation solution email protected is unique and derive an explicit formula for it

direct update of dynamic mathematical models from modal test data two methods for the direct updating of mathematical models based on modal test data are described a set of minimum required constraints derived from eigendynamic and force equilibrium conditions are presented both methods generate mass and stiffness matrices fitted exactly to the given modal test data one method changes all coefficients of the mass respective to the stiffness matrix and lessens the solution effort by the algebraical elimination of the lagrangian multipliers the other method changes only those selected matrix coefficients requiring additional solutions for the numerical calculation of the lagrangian multipliers examples and comments on the applicability of the methods are given

propagation of uncertainty in test analysis correlation of substructured spacecraft determining a finite element model s fem fidelity to data or the process of test analysis correlation is a vital component of the overall verification and validation procedure that is applied to analytical models of spacecraft prior to flight the goal of this work is to develop a methodology for studying the effects of uncertainty on metrics used for test analysis correlation of complex spacecraft that are validated on a substructure by substructure basis uncertainty will be propagated from substructures to system for low frequency applications

an analysis of some high accuracy finite element methods for hyperbolic problems some high accuracy finite element methods for hyperbolic problems are studied in this paper it is proven that over a finite element mesh the convergence order of linear finite element solutions for both linear and nonlinear equations can be higher than one even if the exact solutions are discontinuous the theoretical tools for the convergence analysis are some superclose error estimates that are also developed in this paper for nonsmooth solutions

superconvergence and a posteriori error estimates of the dg method for scalar hyperbolic problems on cartesian grids in this paper we analyze the discontinuous galerkin dg finite element method for the steady two dimensional transport reaction equation on cartesian grids we prove the l2 stability and optimal l2 error estimates for the dg scheme we identify a special numerical flux for which the l2 norm of the solution is of order p 1 when tensor product polynomials of degree at most p are used we further prove superconvergence towards a particular projection of the directional derivative the order of superconvergence is proved to be p 1 2 we also provide a very simple derivative recovery formula which is o h p 1 superconvergent approximation to the directional derivative moreover we establish an o h 2 p 1 global superconvergence for the solution flux at the outflow boundary of the domain these results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element finally we prove that the proposed a posteriori dg error estimates converge to the true errors in the l2 norm at o h p 1 rate and that the global effectivity index converges to unity at o h rate our results are valid without the flow condition restrictions we perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal

constructively well posed approximation methods with unity inf sup and continuity constants for partial differential equations starting from the generalized lax milgram theorem and from the fact that the approximation error is minimized when the continuity and inf sup constants are unity we develop a theory that provably delivers well posed approximation methods with unity continuity and inf sup constants for numerical solution of linear partial differential equations we demonstrate our single framework theory on scalar hyperbolic equations to constructively derive two different hp finite element methods the first one coincides with a least squares discontinuous galerkin method and the other appears to be new both methods are proven to be trivially well posed with optimal hpconvergence rates the numerical results show that our new discontinuous finite element method namely a discontinuous petrov galerkin method is more accurate has optimal convergence rate and does not seem to have unphysical diffusion compared to the upwind discontinuous galerkin method

correcting finite element models using a symmetric eigenstructure assignment technique improvement of structural models by incorporating measured structural modal parameters is approached from a controls aspect the approach is developed for linear structures that exhibit nonproportional damping residual damping and stiffness matrices are determined such that the improved analytical model eigenstructure matches more closely that obtained experimentally

optimum approximation for residual stiffness in linear system identification a new method of system identification uti 1 i zes projector m atrix theory and the moorepenrose generalized inverse to derive an analytical stiffness matrix which when combined with the analytical mass matrix will more closely p redict modal test results weighting matrices a re used to enforce connectivity and make weighted corrections to the original analytical stiffness matrix a simple and straightforward mathematical formulation is obtained the method is compared and contrasted with other methods found in the literature and a simple numerical example is presented

an adaptive discontinuous finite element method for the transport equation abstract in this paper we introduce a discontinuous finite element method in our approach it is possible to combine the advantages of finite element and finite difference methods the main ingredients are numerical flux approximation and local orthogonal basis functions the scheme is defined on arbitrary triangulations and two different error indicators are derived especially the second one is closely connected to our approach and able to handle arbitrary varying flow directions numerical resuls are given for boundary value problems in two dimensions they demonstrate the performance of the scheme combined with the two error indicators

crosswind smear and pointwise errors in streamline diffusion finite element methods for a model convection dominated singularly perturbed convection diffusion prob lem it is shown that crosswind smear in the numerical streamline diffusion finite element method is minimized by introducing a judicious amount of artificial crosswind diffusion the ensuing method with piecewise linear elements converges with a pointwise accuracy of almost h 5 4 under local smoothness assumptions 1 introduction the streamline diffusion method is a finite element method for convection dominated convection diffusion problems which combines formal high accuracy with decent stability properties the method was introduced in the case of stationary problems by hughes and brooks 7 cf raithby and torrance 14 and wahlbin 15 for earlier thoughts in this direction the mathematical analysis of the method was started in johnson and navert 8 and continued with extensions to e g time dependent problems in navert 12 johnson navert and pitkaranta 9 and johnson and saranen 10 in these papers local error estimates in l2 of order o h k 1 2 in regions of smoothness with piecewise polynomial finite elements of degree k were derived together with estimates stating as a typical example that in the zero diffusion limit a sharp discontinuity in the exact solution across a streamline will be captured in a numerical crosswind layer of width 0 h1 2 essentially the purpose of the present paper is first to improve the result just mentioned on numerical crosswind smear to 0 h3 4 the improvement from 0 h1 2 to 0 h3 4 is obtained by adding a small amount 0 h3 2 of artificial crosswind diffusion to the method in the piecewise linear case k 1 this does not destroy the known o h3 2 accuracy in l2 in smooth regions using our first result we then obtain our second main result localized pointwise error estimates of order 0 h5 4 in regions of smoothness the previously known best pointwise error estimate in the piecewise linear situation is 0 h1 2 another consequence is a global l estimate of order 0 h 2 in the presence of typical crosswind and downwind singularities we shall consider the model problem of finding u u x y such that

superconvergence of discontinuous galerkin and local discontinuous galerkin methods eigen structure analysis based on fourier approach various superconvergence properties of discontinuous galerkin dg and local dg ldg methods for linear hyperbolic and parabolic equations have been investigated in the past due to these superconvergence properties dg and ldg methods have been known to provide good wave resolution properties especially for long time integrations zhong and shu 2011 26 in this paper under the assumption of uniform mesh and via fourier approach we observe that the error of the dg or ldg solution can be decomposed into three parts 1 dissipation and dispersion errors of the physically relevant eigenvalue this part of error will grow linearly in time and is of order 2 k 1 for dg method and 2 k 2 for ldg method 2 projection error there exists a special projection of the exact solution such that the numerical solution is much closer to this special projection than the exact solution itself this part of error will not grow in time 3 the dissipation of non physically relevant eigenvectors this part of error will be damped exponentially fast with respect to the spatial mesh size x along this line we analyze the error for a fully discrete runge kutta rk dg scheme a collection of numerical examples for linear equations are presented to verify our observations above we also provide numerical examples based on non uniform mesh nonlinear burgers equation and high dimensional maxwell equations to explore superconvergence properties of dg methods in a more general setting

a note on the convergence of the discontinuous galerkin method for a scalar hyperbolic equation a numerical example is presented which shows that the known l 2 error estimate left u u h right leqq ch k 1 2 left u right k 1 for the discontinuous galerkin method cannot be improved within the class of quasi uniform meshes not even for smooth exact solutions u

rigorous computation of smooth branches of equilibria for the three dimensional cahn hilliard equation in this paper we propose a new general method to compute rigorously global smooth branches of equilibria of higher dimensional partial differential equations the theoretical framework is based on a combination of the theory introduced in global smooth solution curves using rigorous branch following van den berg et al math comput 79 271 1565 1584 2010 and in analytic estimates and rigorous continuation for equilibria of higher dimensional pdes gameiro and lessard j diff equ 249 9 2237 2268 2010 using this method one can obtain proofs of existence of global smooth solution curves of equilibria for large continuous parameter ranges and about local uniqueness of the solutions on the curve as an application we compute several smooth branches of equilibria for the three dimensional cahn hilliard equation

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

a unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty we discuss stabilized galerkin approximations in a new framework widening the scope from the usual dichotomy of the discontinuous galerkin method on the one hand and petrov galerkin methods such as the supg method on the other the idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation thus circumventing the need of a petrov galerkin type choice of spaces this is made possible by adding a higher order penalty term giving l2 control of the jumps in the gradients between adjacent elements we consider convection diffusion reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces the standard h1 conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides the crouzeix raviart element moreover we show how the formulation extends to discontinuous galerkin interior penalty methods in a natural way by domain decomposition using nitsche s method

positivity preserving discontinuous galerkin schemes for linear vlasov boltzmann transport equations we develop a high order positivity preserving discontinuous galerkin dg scheme for linear vlasov boltzmann transport equations vlasov bte under the action of quadratically confined electrostatic potentials the solutions of such btes are positive probability distribution functions and it is very challenging to have a mass conservative high order accurate scheme that preserves positivity of the numerical solutions in high dimensions our work extends the maximum principle satisfying scheme for scalar conservation laws in a recent work by x zhang and c w shu to include the linear boltzmann collision term the dg schemes we developed conserve mass and preserve the positivity of the solution without sacrificing accuracy a discussion of the standard semi discrete dg schemes for the bte are included as a foundation for the stability and error estimates for this new scheme numerical results of the relaxation models are provided to validate the method

pointwise error estimates for a streamline diffusion finite element scheme pointwise error estimates for a streamline diffusion scheme for solving a model convection dominated singularly perturbed convection diffusion problem are given these estimates improve pointwise error estimates obtained by johnson et al 5

an optimal order error estimate for the discontinuous galerkin method in this paper a new approach is developed for analyzing the discontinuous galerkin method for hyperbolic equations for a model problem in r2 the method is shown to converge at a rate 0 hn l when applied with nth degree polynomial approxi mations over a semiuniform triangulation assuming sufficient regularity in the solution

model updating via weighted reference basis with connectivity constraints abstract the paper considers the problem of updating an analytical model from experimental data using the reference basis approach in the general framework of the reference basis method certain quantities e g natural frequencies or modeshapes are considered to be completely accurate and the others are updated by solving a constrained optimization problem however the underlying structure known as connectivity existing in the model is not preserved and the method is not suited for parametric updating in this paper a method for introducing connectivity constraints into reference basis while maintaining its advantages is presented it brings the reference basis method closer to a broad class of updating methods that use parametric updating the notions of connectivity cost and parameterization cost are defined and used to obtain the best model for a given parameterization and to compare the outcomes of different parameterizations

optimal error estimates of the direct discontinuous galerkin method for convection diffusion equations abstract in this paper we present the optimal l2 error estimate ofo hk 1 for polynomial elements of degree k of the semidiscrete direct discontinuous galerkin method for convection diffusion equations the main technical difficulty lies in the control of the inter element jump terms which arise because of the convection and the discontinuous nature of numerical solutions the main idea is to use some global projections satisfying interface conditions dictated by the choice of numerical fluxes so that trouble terms at the cell interfaces are eliminated or controlled in multi dimensional case the orders of k 1 hinge on a superconvergence estimate when tensor product polynomials of degree k are used on cartesian grids a collection of projection errors in both oneand multi dimensional cases is established

error estimates for discontinuous galerkin approximations of implicit parabolic equations we analyze the classical discontinuous galerkin method for implicit parabolic equations symmetric error estimates for schemes of arbitrary order are presented the ideas developed allow certain assumptions frequently required in previous work to be relaxed for example different discrete spaces are allowed at each time step and the spatial operator is not required to be self adjoint or independent of time error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular

error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation this paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions we introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in mathbf r nd and every nonvanishing vector mathbf a of mathbf r nd first we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the finite volume scheme occurs afterwards we prove that this norm is indeed bounded by the mesh size in several cases including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases such as the one with independent refined meshes

stiffness matrix adjustment using mode data a procedure is introduced that uses in addition to mode data structural connectivity information to optimally adjust deficient stiffness matrices the adjustments performed are such that the percentage change to each stiffness coefficient is minimized the physical configuration of the analytical model is preserved and the adjusted model will exactly reproduce the modes used in the identification the theoretical development is presented and the procedure is demonstrated by numerical simulation of a test problem

the weak galerkin finite element method for the transport reaction equation abstract we present and analyze a weak galerkin finite element method for solving the transport reaction equation in d space dimensions this method is highly flexible by allowing the use of discontinuous finite element on general meshes consisting of arbitrary polygon polyhedra we derive the l 2 error estimate of o h k 1 2 order for the discrete solution when the kth order polynomials are used for k 0 moreover for a special class of meshes we also obtain the optimal error estimate of o h k 1 order in the l 2 norm a derivative recovery formula is presented to approximate the convection directional derivative and the corresponding superconvergence estimate is given numerical examples on compatible and non compatible meshes are provided to show the effectiveness of this weak galerkin method

an adaptive higher order method for solving the radiation transport equation on unstructured grids a new class of solution methods based on a short characteristics approach embedded in the finite element framework for the solution of the radiation transport equation is developed and tested the order of convergence and the error to runtime ratio are determined in numerical tests furthermore the efficiency of the scheme is demonstrated in the case of a challenging problem from the field of solar physics a quantitative comparison with other methods based either on the characteristics or on the finite element approach is carried out this emphasizes the efficiency and accuracy of our method two different adaptation mechanisms are studied grid adaptation and local variation of the order of the scheme

analysis of a combined cg1 dg2 method for the transport equation in this paper we introduce a reduced discontinuous galerkin method in which the space of continuous piecewise linear functions cg1 is enriched with discontinuous piecewise quadratics dg2 the resultant finite element approximation is continuous at the vertices of the mesh and discontinuous across edges faces we analyze the properties of the cg1 dg2 discretization in the context of a steady linear transport equation the presented a priori error estimate shows that the discontinuous enrichment stabilizes the continuous coarse scale component and delivers optimal convergence rates numerical studies for steady and unsteady convection problems confirm this result

a generalized inverse eigenvalue problem in structural dynamic model updating this paper is concerned with the problem of the best approximation for a given matrix pencil under a given spectral constraint and a submatrix pencil constraint such a problem arises in structural dynamic model updating by using the moore penrose generalized inverse and the singular value decomposition svd matrices the solvability condition and the expression for the solution of the problem are presented a numerical algorithm for solving the problem is developed

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

discontinuous galerkin methods for advection diffusion reaction problems we apply the weighted residual approach recently introduced in f brezzi et al comput methods appl mech engrg 195 2006 pp 3293 3310 to derive discontinuous galerkin formulations for advection diffusion reaction problems we devise the basic ingredients to ensure stability and optimal error estimates in suitable norms and propose two new methods

an explicit a priori estimate for a finite volume approximation of linear advection on non cartesian grids we propose an elementary proof of strong convergence for a finite volume approximation of nonstationary linear advection on arbitrary grids in two space dimensions this proof is elementary in the sense that the basic a priori estimate uses only some discrete integrations by parts and the cauchy schwarz inequality numerical results show that the estimate is probably nonoptimal

a posteriori error analysis of the discontinuous finite element methods for first order hyperbolic problems abstract in this paper we consider the a posteriori error analysis of discontinuous galerkin finite element methods for the steady and nonsteady first order hyperbolic problems with inflow boundary conditions we establish several residual based a posteriori error estimators which provide global upper bounds and a local lower bound on the error further for nonsteady problem we construct a fully discrete discontinuous finite element scheme and derive the a posteriori error estimators which yield global upper bound on the error in time and space our a posteriori error analysis is based on the mesh dependent a priori estimates for the first order hyperbolic problems these a posteriori error analysis results can be applied to develop the adaptive discontinuous finite element methods

error estimates to smooth solutions of runge kutta discontinuous galerkin methods for scalar conservation laws in this paper we study the error estimates to sufficiently smooth solutions of scalar conservation laws for runge kutta discontinuous galerkin rkdg methods where the time discretization is the second order explicit total variation diminishing tvd runge kutta method error estimates for the mathbb p 1 piecewise linear elements are obtained under the usual cfl condition tau leq gamma h for general nonlinear conservation laws in one dimension and for linear conservation laws in multiple space dimensions where h and tau are the maximum element lengths and time steps respectively and the positive constant gamma is independent of h and tau however error estimates for higher order mathbb p k k geq 2 elements need a more restrictive time step tau leq gamma h 4 3 we remark that this stronger condition is indeed necessary as the method is linearly unstable under the usual cfl condition tau leq gamma h for the mathbb p k elements of degree k geq 2 error estimates of o h k 1 2 tau 2 are obtained for general monotone numerical fluxes and optimal error estimates of o h k 1 tau 2 are obtained for upwind numerical fluxes

a vertex centered dual discontinuous galerkin method this note introduces a new version of the discontinuous galerkin method for discretizing first order hyperbolic partial differential equations the method uses piecewise polynomials that are continuous on a macroelement surrounding the nodes in the unstructured mesh but discontinuous between the macroelements at lowest order the method reduces to a vertex centered finite volume method with control volumes based on a dual mesh and the method can be implemented using an edge based data structure the method provides therefore a strategy to extend existing vertex centered finite volume codes to higher order using the discontinuous galerkin method preliminary tests on a model linear hyperbolic equation in two dimensional indicate a favorable qualitative behavior for nonsmooth solutions and optimal convergence rates for smooth solutions

convergence of an explicit finite volume scheme for first order symmetric systems this paper is devoted to the derivation of a o h1 2 error estimate for the classical upwind explicit in time finite volume scheme for linear first order symmetric systems such a result already existed for the corresponding implicit in time finite volume scheme since it can be interpreted as a particular case of the space time discontinuous galerkin method but the technique of proof used in that case does not extend to explicit schemes the general framework recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws can not be used either because it is not adapted for systems even linear in this article we propose a new technique which takes advantage of the linearity of the problem the first step consists in controlling the approximation error u u h l2 by an expression of the form 2 where u is the exact solution g is a particular smooth function and h h are some linear forms depending on the approximate solution u h the second step consists in carefully estimating the error terms and by using uniform stability results for the discrete problem and regularity properties of the continuous solution

least squares solutions of quadratic inverse eigenvalue problem with partially bisymmetric matrices under prescribed submatrix constraints abstract the inverse eigenvalue problems play an important role in broad application areas such as system identification hopfield neural networks control design mass spring system and molecular spectroscopy this paper proposes an algorithm that yields a new method to efficiently and accurately compute the partially bisymmetric solutions m c k under prescribed submatrix constraints of the quadratic inverse eigenvalue problem m x 2 c x k x 0 the algorithm is developed based on the conjugate gradient normal equations residual minimizing cgnr method we discuss the convergence properties of the algorithm finally the performance of the algorithm is tested on two numerical examples and compared to the previous algorithm

an efficient algorithm based on lanczos type of bcr to solve constrained quadratic inverse eigenvalue problems abstract this paper proposes the lanczos type of biconjugate residual bcr algorithm for solving the quadratic inverse eigenvalue problem l 2 x 2 l 1 x l 0 x 0 where l 2 l 1 and l 0 should be partially bisymmetric under a prescribed submatrix constraint an analysis reveals that the algorithm obtains the solutions of the constrained quadratic inverse eigenvalue problem in finitely many steps in the absence of round off errors finally numerical results are performed to confirm the analysis and to illustrate the efficiency of the proposed algorithm

model updating using an incomplete set of experimental modes new approaches are presented that use the measured natural frequencies and mode shapes to update the analytical mass and stiffness matrices of a structure by adding known masses to the structure and measuring its new modes of vibration we can utilize this additional information to correct the mass matrix of the system after which the stiffness matrix can be updated by requiring it to satisfy the generalized eigenvalue problem associated with the structure manipulating the unknown system matrices into vector forms the connectivity information can be easily implemented to preserve the physical configuration of the structure and to reduce the computational efforts required to correct the system matrices a comparison is made between the proposed updating schemes introduced in this paper and other updating algorithms found in the literature and drastic improvements are observed

secant method adjustment for structural models on orbit testing will be required for final tuning and validation of any mathematical model of large space structures identification methods using limited response data to produce optimally adjusted property matrices seem ideal for this purpose but difficulties exist in the application of previously published methods to large space truss structures this article presents new stiffness matrix adjustment methods that generalize optimal update secant methods found in quasi newton approaches for nonlinear optimization many aspects of previously published methods of stiffness matrix adjustment may be better understood within this new framework of secant methods one of the new methods preserves realistic structural connectivity with minimal storage requirements and computational effort a method for systematic compensation for errors in measured data is introduced that also preserves structural connectivity two demonstrations are presented to compare the new methods results to those of previously published techniques

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

taming the cfl number for discontinuous galerkin methods on structured meshes the upwind discontinuous galerkin method is an attractive method for solving time dependent hyperbolic conservation laws it is possible to use high order explicit time stepping methods and high order spatial approximations without incurring heavy numerical linear algebra overheads however the courant friedrichs lewy cfl condition for these methods depends on the polynomial order used and there is a somewhat excessive cost for using very high order spatial approximation we discuss the impact of a covolume mesh based filter on the cfl number for these methods and present an algorithm which has a cfl number independent of the spatial order of approximation we present computational results for the advection equation and the wave equation on one dimensional meshes using up to tenth order in space and time

analysis of the cell vertex finite volume method for hyperbolic problems with variable coefficients an analysis of the cell vertex finite volume method is presented for a scalar linear hyperbolic equation with variable coefficients the cell vertex approximation is shown to be second order convergent in the l2 norm and first order convergent in the h1 norm provided the exact solution belongs to h3 and the characteristic curves of the differential equation are transversal to mesh lines on the closure of the domain of integration when characteristics are parallel to mesh lines on a subset of positive measure the scheme is only first order convergent in the l2 norm however if transversality fails on part of the outflow boundary only then the cell vertex scheme is second order convergent in a weighted l2 norm

how accurate is the streamline diffusion finite element method we investigate the optimal accuracy of the streamline diffusion finite element method applied to convection dominated problems for linear bilinear elements the theoretical order of convergence given in the literature is either o h 3 2 for quasi uniform meshes or o h 2 for some uniform meshes the determination of the optimal order in general was an open problem by studying a special type of meshes it is shown that the streamline diffusion method may actually converge with any order within this range depending on the characterization of the meshes

adaptive least squares finite element methods for linear transport equations based on an h div flux reformulation abstract in this paper we study the least squares finite element methods lsfem for the linear hyperbolic transport equations the linear transport equation naturally allows discontinuous solutions and discontinuous inflow conditions while the normal component of the flux across the mesh faces needs to be continuous traditional lsfems using continuous finite element approximations will introduce unnecessary extra error for discontinuous solutions and boundary conditions in order to separate the continuity requirements a new flux variable is introduced with this reformulation the continuities of the flux and the solution can be handled separately in natural h div l 2 conforming finite element spaces several variants of the methods are developed to handle the inflow boundary condition strongly or weakly with the reformulation the new lsfems can handle discontinuous solutions and boundary conditions much better than the traditional lsfems with continuous polynomial approximations with least squares functionals as a posteriori error estimators the adaptive methods can naturally identify error sources including singularity and non matching discontinuity for discontinuity aligned mesh no extra error is introduced if an r t 0 p 0 pair is used to approximate the flux and the solution the new adaptive lsfems can approximate discontinuous solutions with almost no overshooting even when the mesh is not aligned with discontinuity existence and uniqueness of the solutions and a priori and a posteriori error estimates are established for the proposed methods extensive numerical tests are performed to show the effectiveness of the methods developed in the paper

optimal convergence of the original dg method for the transport reaction equation on special meshes we show that the approximation given by the original discontinuous galerkin method for the transport reaction equation in d space dimensions is optimal provided the meshes are suitably chosen the l 2 norm of the error is of order k 1 when the method uses polynomials of degree k these meshes are not necessarily conforming and do not satisfy any uniformity condition they are required only to be made of simplexes each of which has a unique outflow face we also find a new element by element postprocessing of the derivative in the direction of the flow which superconverges with order k 1

superconvergence of discontinuous galerkin methods for hyperbolic systems in this paper the discontinuous galerkin method for the positive and symmetric linear hyperbolic systems is constructed and analyzed by using bilinear finite elements on a rectangular domain and an o h 2 order superconvergence error estimate is established under the conditions of almost uniform partition and the h 3 regularity for the exact solutions the convergence analysis is based on some superclose estimates derived in this paper finally as an application the numerical treatment of maxwell equation is discussed and computational results are presented

a hybridized discontinuous petrov galerkin scheme for scalar conservation laws summary we present a hybridized discontinuous petrov galerkin hdpg method for the numerical solution of steady and time dependent scalar conservation laws the method combines a hybridization technique with a local petrov galerkin approach in which the test functions are computed to maximize the inf sup condition since the petrov galerkin approach does not guarantee a conservative solution we propose to enforce this explicitly by introducing a constraint into the local petrov galerkin problem when the resulting nonlinear system is solved using the newton raphson procedure the solution inside each element can be locally condensed to yield a global linear system involving only the degrees of freedom of the numerical trace this results in a significant reduction in memory storage and computation time for the solution of the matrix system albeit at the cost of solving the local petrov galerkin problems however these local problems are independent of each other and thus perfectly scalable we present several numerical examples to assess the performance of the proposed method the results show that the hdpg method outperforms the hybridizable discontinuous galerkin method for problems involving discontinuities moreover for the test case proposed by peterson the hdpg method provides optimal convergence of order k 1 copyright 2012 john wiley sons ltd

on the order of convergence of the discontinuous galerkin method for hyperbolic equations the basic error estimate for the discontinuous galerkin method for hyperbolic equations indicates an o h n convergence rate for nth degree polynomial approximation over a triangular mesh of size h however the optimal o h 1 rate is frequently seen in practice here we extend the class of meshes for which sharpness of the o h estimate can be demonstrated using as an example a problem with a nonaligned mesh in which all triangle sides are bounded away from the characteristic direction the key to realizing h n convergence is a mesh which to the extent possible directs the error to lower frequency modes which are approximated not damped as h 0

superconvergence of discontinuous galerkin methods for scalar nonlinear conservation laws in one space dimension in this paper an analysis of the superconvergence property of the semidiscrete discontinuous galerkin dg method applied to one dimensional time dependent nonlinear scalar conservation laws is carried out we prove that the error between the dg solution and a particular projection of the exact solution achieves left k frac32 right th order superconvergence when upwind fluxes are used the results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree k k ge 1 under the condition that f u possesses a uniform positive lower bound numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions indicating that the restriction on f u is artificial

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

direct least squares formulation of a stiffness adjustment method a direct least squares formulation of the stiffness matrix adjustment method known as the kma method kabe a m stiffness matrix adjustment using mode data aiaa journal vol 23 no 9 1985 pp 1431 1436 is presented the kma method belongs to a class of procedures that refine dynamic models using test measured modes and structural connectivity viewed as a constrained minimization problem most of these methods have applied lagrange multiplier techniques in their development it is shown that these two approaches result in linear systems of equations that are algebraically equivalent and therefore have identical solutions by virtue of this equivalence the direct least squares formulation of the kma method is shown to be equivalent to its original lagrange multiplier formulation the direct least squares versions of other model refinement methods that preserve structural connectivity are also discussed in this respect this approach provides a rigorous framework for unifying these optimal update procedures and indicates how their solutions should compare numerical results are presented that illustrate the equivalence between the two formulations and also the conditions under which the various methods yield identical solutions the results also indicate that because of improved numerical conditioning the direct least squares approach yields more accurate computational solutions and generally requires less computer storage than methods that were developed using lagrange multipliers

improvement of a large analytical model using test data a method has been developed which uses measured normal modes and natural frequencies to improve an analytical mass and stiffness matrix model of a structure the method directly identifies without iteration a set of minimum changes in the analytical matrices which force the eigensolutions to agree with the test measurements an application is presented in which the analytical model had 508 degrees of freedom and 19 modes were measured at 101 locations on the structure the resulting changes in the model are judged to be small compared to expectations of error in the analysis thus the improved model is accepted as a reasonable model of the structure with improved dynamic response characteristics in addition it is shown that the procedure may be a useful tool in identifying apparent measured modes which are not true normal modes of the structure nomenclature analytical matrix matrix of changes identity matrix full improved stiffness and mass matrices n x n full analytical k m matrices n x n partitions of ka ma corresponding to test coordinates partitions of ka ma corresponding to coupling elements partitions of ka ma corresponding to unmea sured coordinates number of measured modes number of degrees of freedom in the model measures of changes eqs 15 17 matrix norm sum of the squares of all the elements rectangular modal matrix normalized n x m th mode th column of measured and unmeasured partitions of diagonal matrix of measured natural frequencies m xm natural frequency of th mode 12 17 sum of the squares of all elements of matrix

structural damage identification of the highway bridge z24 by fe model updating abstract the development of a methodology for accurate and reliable condition assessment of civil structures has become very important the finite element fe model updating method provides an efficient non destructive global damage identification technique which is based on the fact that the modal parameters eigenfrequencies and mode shapes of the structure are affected by structural damage in the fe model the damage is represented by a reduction of the stiffness properties of the elements and can be identified by tuning the fe model to the measured modal parameters this paper describes an iterative sensitivity based fe model updating method in which the discrepancies in both the eigenfrequencies and unscaled mode shape data obtained from ambient tests are minimized furthermore the paper proposes the use of damage functions to approximate the stiffness distribution as an efficient approach to reduce the number of unknowns additionally the optimization process is made more robust by using the trust region strategy in the implementation of the gauss newton method which is another original contribution of this work the combination of the damage function approach with the trust region strategy is a practical alternative to the pure mathematical regularization techniques such as tikhonov approach afterwards the updating procedure is validated with a real application to a prestressed concrete bridge the damage in the highway bridge is identified by updating the young s and the shear modulus whose distribution over the fe model are approximated by piecewise linear functions

convergence of a fully discrete scheme for two dimensional neutron transport we prove an error estimate for a fully discrete method for the numerical solution of a two dimensional model problem in neutron transport theory based on using the discrete ordinates method for the

an upwind like discontinuous galerkin method for hyperbolic systems we investigate an upwind like dg method for solving first order hyperbolic problems written as the friedrichs systems under certain condition this dg scheme may be semi explicit such that the discrete equations can be solved layer by layer we give the stability analysis and error estimate of order k 1 2 in the dg norm in particular for some hyperbolic systems we show that the convergence rate is of order k 1 in the l 2 norm if the q k elements are used on rectangular meshes finally we provide some numerical experiments to illustrate the theoretical analysis

optimal convergence of the original dg method on special meshes for variable transport velocity we prove optimal convergence rates for the approximation provided by the original discontinuous galerkin method for the transport reaction problem this is achieved in any dimension on meshes related in a suitable way to the possibly variable velocity carrying out the transport thus if the method uses polynomials of degree k the l 2 norm of the error is of order k 1 moreover we also show that by means of an element by element postprocessing a new approximate flux can be obtained which superconverges with order k 1

implicit runge kutta methods and discontinuous galerkin discretizations for linear maxwell s equations in this paper we consider implicit runge kutta methods for the time integration of linear maxwell s equations we first present error bounds for the abstract cauchy problem which respect the unboundedness of the differential operators using energy techniques the error bounds hold for algebraically stable and coercive methods such as gauss and radau collocation methods the results for the abstract evolution equation are then combined with a discontinuous galerkin discretization in space using upwind fluxes for the case that permeability and permittivity are piecewise constant functions we show error bounds for the full discretization where the constants do not deteriorate if the spatial mesh width tends to zero

optimal error estimate and superconvergence of the dg method for first order hyperbolic problems we consider the original discontinuous galerkin method for the first order hyperbolic problems in d dimensional space we show that when the method uses polynomials of degree k the l 2 error estimate is of order k 1 provided the triangulation is made of rectangular elements satisfying certain conditions further we show the o h 2 k 1 order superconvergence for the error on average on some suitably chosen subdomains including the whole domain and their outflow faces moreover we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k 1

structural damage identification using assigned partial eigenstructure a method of identifying damaged structural elements from measured modal data using an incomplete measurement set is presented the method uses a mathematical optimization strategy to minimize deviations between measured and analytical modal frequencies and partial mode shapes damage is identified by determining the stiffness change to a finite element model required to match the measured data of the damaged structure damaged elements are obtained directly from the results of the cost function minimization because the allowed structural changes are consistent with the original finite element formulation the cost function minimization is based on an assigned partial eigenstructure algorithm in which the physical properties of the structural elements are treated as control variables which are chosen to achieve the measured partial eigenstructure an iterative solution is used to solve the nonlinear optimization problem experimental results are reported for a cantilevered eight bay truss assembly consisting of 104 elements