discontinuous galerkin methods for incompressible and nearly incompressible elasticity by nitsche s method we propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes we show optimal error estimates that are uniform with respect to poisson s ratio the method is thus locking free we also introduce an equivalent mixed formulation allowing for completely incompressible elasticity problems numerical results are presented

a posteriori error estimates for continuous discontinuous galerkin approximations of the kirchhoff love plate we present energy norm a posteriori error estimates for continuous discontinuous galerkin c dg approximations of the kirchhoff love plate problem the method is based on a continuous displacement field inserted into a symmetric discontinuous galerkin formulation of the fourth order partial differential equation governing the deflection of a thin plate we also give explicit formulas for the penalty parameter involved in the formulation

energy estimates and model order reduction for stochastic bilinear systems in this paper we investigate a large scale stochastic system with bilinear drift and linear diffusion term such high dimensional systems appear for example when discretizing a stochastic partial

c0 interior penalty methods for a dynamic nonlinear beam model abstract in this work we aim to develop efficient numerical schemes for a nonlinear fourth order partial differential equation arising from the so called dynamic gao beam model we use c 0 interior penalty finite element methods over the spatial domain to set up the semi discrete formulations convergence results for the semi discrete case are shown based on a truncated variational formulation and its equivalent abstract formulations we combine time discretizations to derive fully discrete numerical formulations newton s method is applied to compute one time step numerical solutions of a nonlinear system two numerical examples are provided one supports our theoretical results and the other presents a buckling state of the gao beams

a discontinuous galerkin method for nonlinear shear flexible shells abstract in this paper a discontinuous galerkin method for a nonlinear shear flexible shell theory is proposed that is suitable for both thick and thin shell analysis the proposed method extends recent work on reissner mindlin plates to avoid locking without the use of projection operators such as mixed methods or reduced integration techniques instead the flexibility inherent to discontinuous galerkin methods in the choice of approximation spaces is exploited to satisfy the thin plate compatibility conditions a priori a benefit of this approach is that only generalized displacements appear as unknowns we take advantage of this to craft the method in terms of a discrete energy minimization principle thereby restoring the rayleigh ritz approach in addition to providing a straightforward and elegant derivation of the discrete equilibrium equations the variational character of the method could afford numerous advantages in terms of mesh adaptation and available solution techniques the proposed method is exercised on a set of benchmarks and example problems to assess its performance numerically and to test for shear and membrane locking

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

c0 discontinuous galerkin methods for a wind driven ocean circulation model two grid algorithm abstract this paper presents a nonconforming finite element method for a streamfunction formulation of the stationary quasi geostrophic equations which describe the large scale wind driven ocean circulation the streamfunction formulation is a fourth order nonlinear pde and the nonconforming method is based on c 0 elements instead of c 1 elements existence and uniqueness of the approximation are proved and optimal error estimates in several norms of interest are demonstrated under a small data assumption two grid algorithms based on picard and newton type linearizations are then presented to efficiently resolve nonlinearities and computational results are given to demonstrate the efficiency of the algorithm the mediterranean sea example is tested with real world coastline data which illustrates the effectiveness of the two grid approach in the wind driven ocean circulation simulation

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

adaptive rational krylov subspaces for large scale dynamical systems the rational krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems however its success has been hindered by the lack of a parameter free procedure which would effectively generate the sequence of shifts used to build the space in this paper we propose an adaptive computation of these shifts the whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix while further information is automatically generated during the process the approach is a full generalization to the nonsymmetric case of the idea first proposed in druskin et al 2010 18 and it is used for two important problems in control the approximation of the transfer function and the numerical solution of large lyapunov equations the procedure can be naturally extended to other related problems such as the solution of the sylvester equation and parametric or higher order systems several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors

a locking free weak galerkin finite element method for reissner mindlin plate on polygonal meshes abstract a new weak galerkin finite element method is introduced and analyzed for the reissner mindlin plate model in the primary form without introducing shear strain as an extra unknown which results in a linear system with symmetric positive definite stiffness matrix the proposed method achieves uniform convergence with respect to plate thickness the so called locking free without introducing any projection reduced integration etc in addition the new method can be applied to general polygonal meshes in particular we implement pentagonal and hexagonal meshes in our numerical tests the numerical study confirms our theory

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

on mixed finite element methods for the reissner mindlin plate model in this paper we analyze the convergence of mixed finite element approximations to the solution of the reissner mindlin plate problem we show that several known elements fall into our analysis thus providing a unified approach we also introduce a low order triangular element which is optimalorder convergent uniformly in the plate thickness

finite time balanced truncation for linear systems via shifted legendre polynomials abstract in this paper we present a finite time model order reduction method for linear systems via shifted legendre polynomials the main idea of the approach is to use finite time empirical gramians which are constructed from impulse responses by solving block tridiagonal linear systems to generate approximate balanced system for the large scale system the balancing transformation is directly computed from the expansion coefficients of impulse responses in the space spanned by shifted legendre polynomials without individual reduction of the gramians and a separate eigenvector solve then the reduced order model is constructed by truncating the states corresponding to the small approximate hankel singular values hsvs the stability preservation of the reduced model is briefly discussed and in combination with the dominant subspace projection method we modify the reduction procedure to alleviate the shortcomings of the above method which may unexpectedly lead to unstable systems even though the original one is stable furthermore the properties of the resulting reduced models are considered finally numerical experiments are given to demonstrate the effectiveness of the proposed methods

a uniformly accurate finite element method for the reissner mindlin plate a simple finite element method for the reissner mindlin plate model in the prim itive variables is presented and analyzed the method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging it is proved that the method converges with optimal order uniformly with respect to thickness

the tdnns method for reissner mindlin plates a new family of locking free finite elements for shear deformable reissner mindlin plates is presented the elements are based on the tangential displacement normal normal stress formulation of elasticity in this formulation the bending moments are treated as separate unknowns the degrees of freedom for the plate element are the nodal values of the deflection tangential components of the rotations and normal normal components of the bending strain contrary to other plate bending elements no special treatment for the shear term such as reduced integration is necessary the elements attain an optimal order of convergence

a c0 discontinuous galerkin method for the stationary quasi geostrophic equations of the ocean abstract this work concerns the development of a finite element algorithm for the stationary quasi geostrophic equations to treat the large scale wind driven ocean circulation the algorithm is developed based on the streamfunction formulation involving fourth order gradients of the streamfunction here we examine the adaptation of a relatively inexpensive nonconforming method based on c 0 basis functions we develop the variational form of the method and establish consistency the method weakly enforces continuity of the normal flux across interelement boundaries and stabilization is achieved via nitsche s method explicit expression for the choice of the stabilization parameter is derived moreover the theoretical error estimate for the linear stommel munk model is performed numerical results from several benchmark problems on rectangular and curved domains are provided to demonstrate the accuracy and robustness of the method we also provide the mediterranean sea example to demonstrate the capability of the approach to the wind driven ocean circulation simulation

locking free quadrilateral continuous discontinuous finite element methods for the reissner mindlin plate we develop a finite element method with continuous displacements and discontinuous rotations for the reissner mindlin plate model on quadrilateral elements to avoid shear locking the rotations mu

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

an output error bound for time limited balanced truncation abstract model order reduction mor is often applied to spatially discretized partial differential equations to reduce their order and hence decrease computational complexity a reduced system can be obtained e g by time limited balanced truncation a method that aims to construct an accurate reduced order model on a given finite time interval 0 t this particular balancing related mor technique is studied in this paper an l t 2 error bound based on the truncated time limited singular values is proved and is the main result of this paper the derived error bound converges as t to the well known h error bound of unrestricted balanced truncation a scheme that is used to construct a good reduced system on the entire time line the techniques within the proofs of this paper can also be applied to unrestricted balanced truncation so that a relatively short time domain proof of the h error bound is found here

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

dual pairs of generalized lyapunov inequalities and balanced truncation of stochastic linear systems we consider two approaches to balanced truncation of stochastic linear systems which follow from different generalizations of the reachability gramian of deterministic systems both preserve mean square asymptotic stability but only the second leads to a stochastic h type bound for the approximation error of the truncated system

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

weakly over penalized discontinuous galerkin schemes for reissner mindlin plates without the shear variable this paper introduces a new locking free formulation that combines the discontinuous galerkin methods with weakly over penalized techniques for reissner mindlin plates we derive optimal a priori error estimates in both the energy norm and l 2 l2 norm for polynomials of degree k 2 k 2 and we extend the results concerning the energy norm to higher order polynomial degrees numerical tests confirm our theoretical predictions

a finite element method with discontinuous rotations for the mindlin reissner plate model we present a continuous discontinuous finite element method for the mindlin reissner plate model based on continuous polynomials of degree k 2 for the transverse displacements and discontinuous polynomials of degree k 1 for the rotations we prove a priori convergence estimates uniformly in the thickness of the plate and thus show that locking is avoided we also derive a posteriori error estimates based on duality together with corresponding adaptive procedures for controlling linear functionals of the error finally we present some numerical results

a four node plate bending element based on mindlin reissner plate theory and a mixed interpolation this communication discusses a 4 node plate bending element for linear elastic analysis which is obtained as a special case from a general nonlinear continuum mechanics based 4 node shell element formulation the formulation of the plate element is presented and the results of various example solutions are given that yield insight into the predictive capability of the plate and shell element

analysis of some low order finite element schemes for mindlin reissner and kirchhoff plates we set up a framework for analyzing mixed finite element methods for the plate problem using a mesh dependent energy norm which applies both to the kirchhoff and to the mindlin reissner formulation of the problem the analysis techniques are applied to some low order finite element schemes where three degrees of freedom are associated to each vertex of a triangulation of the domain the schemes proceed from the mindlin reissner formulation with modified shear energy

an h2 type error bound for balancing related model order reduction of linear systems with levy noise abstract to solve a stochastic linear evolution equation numerically finite dimensional approximations are commonly used for a good approximation one might end up with a sequence of ordinary stochastic linear differential equations of high order to reduce the high dimension for practical computations model order reduction is frequently used balanced truncation bt is a well known technique from deterministic control theory and it was already extended for controlled linear systems with levy noise recently a new ansatz was investigated which provides an alternative way to generalize bt for stochastic systems there the question of the existence of an h 2 error bound was asked which we answer in this paper we discuss how this bound can be computed practically and how to use it to find a suitable reduced order dimension

time complexity of radio broadcasting adaptiveness vs obliviousness and randomization vs determinism we consider the time of broadcasting in ad hoc radio networks modeled as undirected graphs in such networks every node knows only its own label and a linear bound on the number of nodes but is unaware of the topology of the network or even of its own neighborhood our aim is to study to what extent the availability of two important characteristics of a broadcasting algorithm influences optimal broadcasting time these characteristics are adaptiveness and randomization our contribution is establishing upper and lower bounds on optimal broadcasting time for three classes of algorithms adaptive deterministic oblivious randomized and oblivious deterministic in two cases we present tight bounds and in one case a small gap remains we show that for deterministic adaptive algorithms time n is required even for n node networks of constant diameter this lower bound is strongest possible since linear time algorithms are known and hence establishes optimal time n for this class for oblivious randomized algorithms we show an upper bound o n min d log n and a lower bound n on optimal expected broadcasting time in n node networks of diameter d finally for oblivious deterministic algorithms we show matching upper and lower bounds n min d n on optimal broadcasting time our results imply that enforcing obliviousness has at least as strong negative impact on broadcasting time as enforcing determinism and that algorithms having both these features are strictly less efficient than those having only one of them