an isogeometric method for the reissner mindlin plate bending problem we present a new isogeometric method for the discretization of the reissner mindlin plate bending problem the proposed scheme follows a recent theoretical framework that makes possible the construction of a space of smooth discrete deflections wh and a space of smooth discrete rotations h such that the kirchhoff constraint is exactly satisfied at the limit therefore we obtain a formulation which is natural from the theoretical mechanical viewpoint and locking free by construction we prove that the method is uniformly stable and satisfies optimal convergence estimates finally the theoretical results are fully supported by numerical tests

generalized mixed variational principles and solutions of ill conditioned problems in computational mechanics part ii shear locking abstract although the finite element method fem has been extensively applied to various areas of engineering the ill conditioned problems occurring in many situations are still thorny to deal with this study attempts to provide a high performing and simple approach to the solutions of ill conditioned problems the theoretical foundation of it is the parametrized variational principles called the generalized mixed variational principles gmvps initiated by rong in 1981 gmvps can solve many kinds of ill conditioned problems in computational mechanics among them four cases are investigated in detail the volumetric locking the shear locking the inhomogeneousness and the membrane locking problems composing four parts of the study part i part iv respectively this paper is part ii wherein a gmvp specially suited to the reissner plate theory and timoshenko beam theory is constructed providing a mathematical foundation for establishing fem formulations which can automatically unlock the shear locking and produce no spurious zero energy modes

current state opacity formulations in probabilistic finite automata a system is said to be current state opaque if the entrance of the system state to a set of secret states remains opaque uncertain to an intruder at least until the system leaves the set of secret states this notion of opacity has been studied in nondeterministic finite automata settings where the intruder observes a subset of events for example via some natural projection mapping and has been shown to be useful in characterizing security requirements in many applications including encryption using pseudorandom generators and coverage properties in sensor networks one limitation of the majority of existing analysis is that it fails to provide a quantifiable measure of opacity for a given system instead it simply provides a binary characterization of the system being opaque or not opaque in this paper we address this limitation by extending current state opacity formulations to systems that can be modeled as probabilistic finite automata under partial observation we introduce three notions of opacity namely 1 step based almost current state opacity 2 almost current state opacity and 3 probabilistic current state opacity all of which can be used to provide a measure of a given system s opacity we also propose verification methods for these probabilistic notions of opacity and characterize their corresponding computational complexities

analysis of kinematic linked interpolation methods for reissner mindlin plate problems the approximation to the solution of reissner mindlin plate problem is considered in the framework of finite element techniques a general strategy involving a linking operator between rotations and vertical displacements is analyzed an abstract convergence result is provided examples of elements falling into this framework are presented and shown to be stable and locking free numerical tests detail the performances of the elements email lovadina ing unitn it

a new class of mixed finite element methods for reissner mindlin plates a new class of finite elements for the reissner mindlin plate problem is presented the family is based on a modified mixed formulation recently introduced by arnold and brezzi boundary value problems for partial differential equations and applications j l lions and c baiocchi eds masson paris 1993 a result of stability and convergence uniformly in the thickness is provided

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

analysis of some stabilized low order mixed finite element methods for reissner mindlin plates abstract new low order mixed finite element methods are established for the reissner mindlin plate problem under a clamped boundary condition employing equal order bi linear interpolants for the rotation and the lateral displacement and discontinuous constant or constant locally enriched by a special function or linear interpolant for the shear stress where two stabilization terms are introduced one of which is used to control the jump of the normal trace of the shear stress across the interelement boundaries and the other is in essence used to control the rotation by modifying the thickness of the plate when using macro element for the displacement the stabilization term pertaining to the jump of the normal trace of the shear stress can be dropped since the jump can be automatically controlled it is shown that all these methods are stable and optimal convergent including in the l 2 norm for the rotation and the displacement uniform in the thickness of the plate

a posteriori error analysis of the linked interpolation technique for plate bending problems we develop a posteriori error estimates for the so called linked interpolation technique to approximate the solution of plate bending problems we show that the proposed residual based estimator is both reliable and efficient

wavelet based ilu preconditioners for the numerical solution by pufem of high frequency elastic wave scattering daubechies family of wavelets combined to the incomplete lower upper ilu factorization are considered as preconditioners for a block sparse linear system arising from the approximation of the time harmonic elastic wave equations by the partition of unity finite element method pufem after applying the discrete wavelet transform dwt to each dense block in the final matrix and the known right hand side due to the local enrichment by pressure p and shear s plane waves the resulting linear system is solved by the restarted generalized minimum residual method gmres with ilu preconditioners allowing fill in elements in the l and u matrix factors a reordering algorithm of the vertices based on gibbs method is also introduced it leads to a significant reduction of the bandwidth of the wave based finite element fe matrix and enables the ilu preconditioners to be more effective to study the performance of the proposed preconditioners a problem of a horizontal s plane wave scattered by a rigid circular body in an infinite elastic medium is considered the numerical tests show the good performance of the dwt based ilu preconditioners in improving the rate of convergence of gmres for high numbers of approximating p and s plane waves on coarse mesh grids containing multi wavelength sized elements moreover the haar dwt enhances the scalability with respect to the problem size when the number of the nodal points increases of the ilu preconditioner which uses the threshold strategy in the control of the fill in elements despite the high level of the conditioning a good accuracy may be achieved for a discretization level of around 1 9 degrees of freedom per s wavelength which is far below the rule of thumb of 10 nodal points per wavelength adopted for the conventional fe

a stable bilinear element for the reissner mindlin plate model abstract in this paper we introduce a new quadrilateral element using isoparametric bilinear basis functions for both components of the rotation vector and the deflection the element is a stable modification of the mitc4 element we report on calculations with this new element the original mitc4 and also the bilinear element with selective reduced integration the numerical results are in accordance with the results of the numerical analysis and they show that i the method with reduced integration is highly unreliable and cannot be recommended ii the mitc4 performs rather well but its instability can lead to a decrease in accuracy for the deflection and especially to an inaccurate and oscillating shear force iii the drawbacks of the mitc4 are not present in the modified method

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

finite dimensional approximation of bifurcation problems in presence of symmetries we give here a brief summary of the results of 4 14 the aim of our work was not only the finite dimensional approximation of the specific problem studied in 3 but we also wanted to show how to adapt to discrete problems and how to use in this case some classical mathematical tools e g the splitting lemma of gromoll meyer magnus the singularity theory etc such considerations are developed in 5 in particular we prove in 5 that in most cases e g in the examples given in 6 7 9 10 the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of 12 another way to use universal unfoldings in the approxima tion of bifurcation problems is suggested by 2

opacity of nondeterministic transition systems a bi simulation relation approach in this paper we propose several opacity preserving bi simulation relations for nondeterministic transition systems ntss in terms of initial state opacity current state opacity k step opacity and infinite step opacity we also show how one can leverage quotient constructions to compute such relations as a result although the opacity verification problem for infinite ntss is generally undecidable if one can find such an opacity preserving relation from an infinite nts to a finite one the lack of opacity of the infinite nts can be easily verified over the finite one which is decidable

verification of state based opacity using petri nets a system is said to be opaque if a given secret behavior remains opaque uncertain to an intruder who can partially observe system activities this work addresses the verification of state based opacity in systems modeled with petri nets the secret behavior of a system is defined as a set of states more precisely two state based opacity properties are considered current state opacity and initial state opacity we show that both current state and initial state opacity problems in bounded petri nets can be efficiently solved by using a compact representation of the reachability graph called basis reachability graph brg this approach is practically efficient since the exhaustive enumeration of the reachability space can be avoided

linked interpolation for reissner mindlin plate elements part ii a simple triangle the formulation and shape functions given in part i are extended to develop a simple plate bending triangle with good performance in both thin and thick situations indeed its performance is better than that of other nine dof elements and its computer implementation simpler when used with selective reduced integration the element produces identical results as that of xu 1 2 to save space details given in part i are not repeated and some results are also presented in the figures of part i

numerical methods for treating problems of viscoelastic isotropic solid deformation mathematical models for treating problems of linear viscoelasticity involving hereditary constitutive relations for compressible solids are discussed and their discretization using finite element methods in space together with quadrature rules in time to treat the hereditary integrals is described the range of applicability of this type of formulation is reviewed in the context of geometric and constitutive linearity nonlinearity and the limitations imposed by the availability of physical data are discussed one of the above models is a volterra integral equation of the second kind in this when the kernel is separable an established technique due to goursat 1933 can be exploited to reformulate the problem as a system of ordinary differential equations this approach will be described for the special case of a linear viscoelastic isotropic homogeneous synchronous constant poisson s ratio solid this method results in a complete decoupling of the space and time dependence in this case the problem can be solved at each time level by solving first a problem of linear elasticity and then a system of ordinary differential equations for each point in the spatial mesh at which the viscoelastic displacements are required the advantages disadvantages and limitations offered by this and various other schemes for solving problems of viscoelasticity as outlined below are discussed

opacity of discrete event systems and its applications in this paper we investigate opacity of discrete event systems we define two types of opacities strong opacity and weak opacity given a general observation mapping a language is strongly opaque if all strings in the language are confused with some strings in another language and it is weakly opaque if some strings in the language are confused with some strings in another language we show that security and privacy of computer systems and communication protocols can be investigated in terms of opacity in particular two important properties in security and privacy namely anonymity and secrecy can be studied as special cases of opacity we also show that by properly specifying the languages and the observation mapping three important properties of discrete event systems namely observability diagnosability and detectability can all be reformulated as opacity thus opacity has a wide range of applications also in this paper we provide algorithms for checking strong opacity and weak opacity for systems described by regular languages and having a generalized projection as the observation mapping

an edge based smoothed finite element method es fem with stabilized discrete shear gap technique for analysis of reissner mindlin plates abstract an edge based smoothed finite element method es fem for static free vibration and buckling analyses of reissner mindlin plates using 3 node triangular elements is studied in this paper the calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements in order to avoid the transverse shear locking and to improve the accuracy of the present formulation the es fem is incorporated with the discrete shear gap dsg method together with a stabilization technique to give a so called edge based smoothed stabilized discrete shear gap method es dsg the numerical examples demonstrated that the present es dsg method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature

analysis of a mixed finite element method for the reissner mindlin plate problems abstract an analysis of a triangular mixed finite element method proposed by taylor and auricchio cf 13 is presented the method is based on a linked interpolation between deflections and rotations in order to avoid the locking phenomenon cf 15 the analysis shows that the approximated deflections and rotations are first order convergent to the exact solution uniformly in the thickness

critical friction for wedged configurations abstract a wedged configuration with coulomb friction is a nontrivial equilibrium state of a linear elastic body in a frictional unilateral contact with a rigid body under vanishing external loads we analyze here the relation between the geometry of the elastic body and the friction coefficient for which wedged configurations exist in a 3 d context the critical friction coefficient w is defined as the infimum of a supremal functional defined on the set of admissible normal displacement and tangential stresses for friction coefficients with w the wedged problem has at least a solution and for the wedged problem is stated in a discrete framework using a mixed finite element approach and the discrete critical friction coefficient h w is introduced as the solution of a global minimization problem involving a non differentiable and non convex functional the existence of h the displacement field of a critical wedged state is proved and a specific numerical method based on a genetic algorithm was developed to compute the critical wedged configurations some techniques to handle the discontinuities of the normal vector on the contact surface are presented and the analysis is illustrated with three numerical simulations

an isostatic assumed stress triangular element for the reissner mindlin plate bending problem the paper describes a new assumed stress triangular element for reissner mindlin plates called tip3 with three nodes and three degrees of freedom per node the kinematics is constructed by means of the so called linked interpolation ruled by technically significant degrees of freedom i e one transversal displacement and two rotations per node without using additional bubble modes the static representation starts from a moment shear uncoupled polynomial approximation and is constrained to satisfy some equilibrium conditions in order to reduce the stress parameters to a minimum number the resulting element is locking free presents correct rank and passes the bending patch test even for very thin plates the good performances of the element are demonstrated by several comparisons with other triangular plate elements available in the literature copyright 2007 john wiley sons ltd

on some recent adventures into improved finite element cfd methods for convective transport abstract the quest continues for computational fluid dynamics cfd algorithms that are accurate and efficient for convection dominated applications including shocks travelling fronts and wall layers the boundary value optimal galerkin weak statement invariably requires manipulation either in the test space or in an augmented form for the conservation law system to handle the disruptive character introduced by the discretized first order convection term an incredible variety of methodologies have been derived and examined to address this issue in particular seeking achievement of accurate and monotone discrete approximate solutions in an efficient implementation the utk cfd research group continues its search on examining the breadth of approaches leading to development of a consistent encompassing theoretical statement exhibiting quality performance included herein are adventures into generalized taylor series lax wendroff methods characteristic euler flux resolutions sub grid embedded high degree lagrange bases with static condensation and assembled stencil fourier analysis optimization for finite element weak statement implementations for appropriate model problems including steady convection diffusion and pure unsteady convection and benchmark navier stokes definitions recent advances have lead to candidate accurate monotone methods with linear basis efficiency this contribution highlights the theoretical developments and presents quantitative documentation of achievable high quality solutions

a posteriori error analysis of finite element methods for reissner mindlin plates this paper develops an a posteriori error control theory of finite element methods for reissner mindlin plates which states that one can derive a uniformly reliable and efficient a posteriori error estimate for a given scheme by only 1 checking three conditions 2 designing three functions and one parameter 3 bounding the last three terms of the abstract estimator we apply this theory to two classes of methods and achieve robust a posteriori error controls for them

operator calculus for information field theory signal inference problems with non gaussian posteriors can be hard to tackle through using the concept of gibbs free energy these posteriors are rephrased as gaussian posteriors for the price of computing various expectation values with respect to a gaussian distribution we present a way of translating these expectation values to a language of operators which is similar to that in quantum mechanics this simplifies many calculations for instance such as those involving log normal priors the operator calculus is illustrated by deriving a self calibrating algorithm which is tested with mock data

linked interpolation for reissner mindlin plate elements part i a simple quadrilateral in most plate elements using the reissner mindlin assumptions the interpolations used for the lateral displacements w and the rotation involve the independent representation of each variable by its nodal values usually with identical interpolations to ensure a higher order of expansion for displacement w its representation is linked in the present paper with both sets of nodal variables conditions necessary for the use of such expansions are established here and the paper shows the development of a linear quadrilateral element q4bl whose performance and robustness are good although it possesses one singularity if only three degrees of freedom are prescribed in part ii we apply the identical formulation to develop a triangular element t3bl which performs equally well and is fully robust

a new approach for the verification of infinite step and k step opacity using two way observers abstract in the context of security analysis for information flow properties where a potentially malicious observer intruder tracks the observed behavior of a given system infinite step opacity respectively k step opacity holds if the intruder can never determine for sure that the system was in a secret state for any instant within infinite steps respectively k steps prior to that particular instant we present new algorithms for the verification of the properties of infinite step opacity and k step opacity for partially observed discrete event systems modeled as finite state automata our new algorithms are based on a novel separation principle for state estimates that characterizes the information dependence in opacity verification problems and they have lower computational complexity than previously proposed ones in the literature specifically we propose a new information structure called the two way observer that is used for the verification of infinite step and k step opacity based on the two way observer a new upper bound for the delay in k step opacity is derived which also improves previously known results

mixed and nonconforming finite element approximations of reissner mindlin plates abstract we consider a mixed finite element approximation of the reissner mindlin plate model which plays an important role in the simulation of plates and shells of small to moderate thickness on triangles and quadrilaterals low order finite elements are proposed the rotation is approximated by conforming isoparametric bi linear elements and the transverse displacement by constant elements and the shear stress by the lowest order raviart thomas elements we prove that the new method satisfies the k ellipticity and the inf sup condition in the abstract framework of the babuska brezzi theory for mixed problems consequently the method is uniformly stable and uniformly optimally convergent independent of the thickness of the plate moreover a local postprocessing approach and the implementation by either the augmented lagrangian algorithm or the hybridization method are discussed including equivalence to some nonconforming methods in particular a new nonconforming method is deduced which can accommodate arbitrary regular quadrilaterals

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

a priori and a posteriori error analysis for a large scale ocean circulation finite element model we consider the finite element solution of the stream function vorticity formulation for a large scale ocean circulation model first we study existence and uniqueness of solution for the continuous and discrete problems under appropriate regularity assumptions we prove that the stream function can be computed with an error of order h in h1 seminorm second we introduce and analyze an h adaptive mesh refinement strategy to reduce the spurious oscillations and poor resolution which arise when convective terms are dominant we propose an a posteriori anisotropic error indicator based on the recovery of the hessian from the finite element solution which allows us to obtain well adapted meshes the numerical experiments show an optimal order of convergence of the adaptive scheme furthermore this strategy is efficient to eliminate the oscillations around the boundary layer

robust bddc preconditioners for reissner mindlin plate bending problems and mitc elements a balancing domain decomposition method by constraints bddc is constructed and analyzed for the reissner mindlin plate bending problem discretized with mixed interpolation of tensorial components mitc finite elements this bddc algorithm is based on selecting the plate rotations and deflection degrees of freedom at the subdomain vertices as primal continuity constraints after the implicit elimination of the interior degrees of freedom in each subdomain the resulting plate schur complement is solved by the preconditioned conjugate gradient method the preconditioner is based on the solution of local reissner mindlin plate problems on each subdomain with clamping conditions at the primal degrees of freedom and on the solution of a coarse reissner mindlin plate problem for the primal degrees of freedom the main results of the paper are the proof and numerical verification that the proposed bddc plate algorithm is scalable quasi optimal and most important robust with respect to the plate thickness while this result is due to an underlying mixed formulation of the problem both the interface plate problem and the preconditioner are positive definite the numerical results also show that the proposed algorithm is robust with respect to discontinuities of the material properties

an improved reissner mindlin triangular element abstract a modified version of the low order mixed finite element method proposed by onate et al internat j numer method engng 37 1994 2569 for the reissner mindlin plate model is analyzed it is proved that the new method is optimal convergent uniform in the thickness of the plate

parallel domain decomposition procedures of improved d d type for parabolic problems two parallel domain decomposition procedures for solving initial boundary value problems of parabolic partial differential equations are proposed one is the extended d d type algorithm which extends the explicit implicit conservative galerkin domain decomposition procedures given in 5 from a rectangle domain and its decomposition that consisted of a stripe of sub rectangles into a general domain and its general decomposition with a net like structure an almost optimal error estimate without the factor h 1 2 given in dawson dupont s error estimate is proved another is the parallel domain decomposition algorithm of improved d d type in which an additional term is introduced to produce an approximation of an optimal error accuracy in l 2 norm

domain decomposition based mathcal h lu preconditioning hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an mathcal h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approximation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based mathcal h matrices this new approach yields mathcal h lu factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an mathcal h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decomposition based mathcal h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

a low order nonconforming finite element for reissner mindlin plates we propose a locking free element for plate bending problems based on the use of nonconforming piecewise linear functions for both rotations and deflections we prove optimal error estimates with respect to both the meshsize and the analytical solution regularity

a locking free reissner mindlin quadrilateral element on arbitrary regular quadrilaterals a new finite element method for the reissner mindlin plate is proposed where both transverse displacement and rotation are approximated by isoparametric bilinear elements with local bubbles enriching rotation and a local reduction operator is applied to the shear energy term this new method gives optimal error bounds uniform in the thickness of the plate for both transverse displacement and rotation with respect to h1 and l2 norms

signal inference with unknown response calibration uncertainty renormalized estimator the calibration of a measurement device is crucial for every scientific experiment where a signal has to be inferred from data we present cure the calibration uncertainty renormalized estimator to reconstruct a signal and simultaneously the instrument s calibration from the same data without knowing the exact calibration but its covariance structure the idea of the cure method developed in the framework of information field theory is to start with an assumed calibration to successively include more and more portions of calibration uncertainty into the signal inference equations and to absorb the resulting corrections into renormalized signal and calibration solutions thereby the signal inference and calibration problem turns into a problem of solving a single system of ordinary differential equations and can be identified with common resummation techniques used in field theories we verify the cure method by applying it to a simplistic toy example and compare it against existent self calibration schemes wiener filter solutions and markov chain monte carlo sampling we conclude that the method is able to keep up in accuracy with the best self calibration methods and serves as a noniterative alternative to them

improving self calibration response calibration is the process of inferring how much the measured data depend on the signal one is interested in it is essential for any quantitative signal estimation on the basis of the data here we investigate self calibration methods for linear signal measurements and linear dependence of the response on the calibration parameters the common practice is to augment an external calibration solution using a known reference signal with an internal calibration on the unknown measurement signal itself contemporary self calibration schemes try to find a self consistent solution for signal and calibration by exploiting redundancies in the measurements this can be understood in terms of maximizing the joint probability of signal and calibration however the full uncertainty structure of this joint probability around its maximum is thereby not taken into account by these schemes therefore better schemes in sense of minimal square error can be designed by accounting for asymmetries in the uncertainty of signal and calibration we argue that at least a systematic correction of the common self calibration scheme should be applied in many measurement situations in order to properly treat uncertainties of the signal on which one calibrates otherwise the calibration solutions suffer from a systematic bias which consequently distorts the signal reconstruction furthermore we argue that nonparametric signal to noise filtered calibration should provide more accurate reconstructions than the common bin averages and provide a new improved self calibration scheme we illustrate our findings with a simplistic numerical example

numerical approximation of mindlin reissner plates we consider a finite element approximation of the so called mindlin reissner formulation for moderately thick elastic plates we show that stability and optimal error bounds hold independently of the value of the thickness 1 introduction the so called mindlin reissner model for moderately thin plates is often used by engineers in connection with plate and shell problems it is well known that many numerical schemes for this model are satisfactory only when the thickness parameter t is not too small for a very small t some bad behavior such as the locking phenomenon might occur here we present a method which is uniformly good as t goes to zero and we prove optimal error estimates for transversal displacement rotations and shear stresses with constants independent of t

verification of initial state opacity in security applications of discrete event systems in this paper we formulate and analyze methodologies for verifying the notion of initial state opacity in discrete event systems that are modeled as non deterministic finite automata with partial observation on their transitions a system is initial state opaque if the membership of its true initial state to a set of secret states remains opaque i e uncertain to an intruder who observes system activity through some projection map initial state opacity can be used to characterize security requirements in a variety of applications including tracking problems in sensor networks in order to model and analyze the intruder capabilities regarding initial state opacity we first address the initial state estimation problem in a non deterministic finite automaton via the construction of an initial state estimator we analyze the properties and complexity of the initial state estimator and show how the complexity of the verification method can be greatly reduced in the special case when the set of secret states is invariant we also establish that the verification of initial state opacity is a pspace complete problem

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

opacity enforcing supervisory strategies via state estimator constructions state based notions of opacity such as initial state opacity and infinite step opacity emerge as key properties in numerous security applications of discrete event systems we consider systems that are modeled as partially observed nondeterministic finite automata and tackle the problem of constructing a minimally restrictive opacity enforcing supervisor moes which limits the system s behavior within some prespecified legal behavior while enforcing initial state opacity or infinite step opacity requirements we characterize the solution to moes under some mild assumptions in terms of the supremal element of certain controllable normal and opaque languages we also show that this supremal element always exists and that it can be implemented using state estimators the result is a supervisor that achieves conformance to the pre specified legal behavior while enforcing initial state opacity by disabling at any given time a subset of the controllable system events in a way that minimally restricts the range of allowable system behavior although infinite step opacity cannot be easily translated to language based opacity we show that by using a finite bank of supervisors the aforementioned approach can be extended to enforce infinite step opacity in a minimally restrictive way