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

almost sure convergence of extremum seeking algorithm using stochastic perturbation abstract this paper presents an extremum seeking es algorithm where the perturbation signal is a martingale difference sequence m d s with a vanishing variance the measurement noise at the plant output is modeled by a superposition of deterministic component and a non stationary colored noise signal the optimizing set point of the uncertain reference to output equilibrium map is estimated by a stochastic approximation sa type algorithm the algorithm has a vanishing gain sequence dependent on the set point estimates by utilizing powerful tools of the martingale convergence theory it is proved that with probability one the set point estimates converge to the optimizing equilibrium point in spite of the presence of a measurement noise this result is derived without requiring boundedness or any prior condition on the set point estimates

brief stability of extremum seeking feedback for general nonlinear dynamic systems while the mainstream methods of adaptive control both linear and nonlinear deal only with regulation to known set points or reference trajectories in many applications the set point should be selected to achieve a maximum of an uncertain reference to output equilibrium map the techniques of the so called extremum control or self optimizing control developed for this problem in the 1950 1960s have long gone out of fashion in the theoretical control literature because of the difficulties that arise in a rigorous analytical treatment in this paper we provide the first proof of stability of an extremum seeking feedback scheme by employing the tools of averaging and singular perturbation analysis our scheme is much more general that the existing extremum control results which represent the plant as a static nonlinear map possibly cascaded with a linear dynamic block we allow the plant to be a general nonlinear dynamic system possibly non affine in control and open loop unstable whose reference to output equilibrium map has a maximum and whose equilibria are locally exponentially stabilizable

an assumed covariant strain based 9 node shell element a new shell element which is free from serious locking problems and which does not possess hourglass modes is proposed solutions obtained with this element exhibit good convergence and are satisfactorily insensitive to mesh distortion the element also exhibits good convergence for plates with variable thickness this element is based on the use of assumed covariant strains which are obtained from the covariant strain field defined with respect to the element natural co ordinate system only the linear version for thin shell cases is considered the element performance is tested by application to several standard plate and shell problems a problem involving variable thickness is also presented

on stress interpolation for hybrid models a technique is proposed for the selection of stress interpolations for hybrid models the present paper applies this approach to plane problems the stiffness matrix is derived using the hellinger reissner variational principle this formulation uses infinitesimal equilibrium relationships and divides the assumed stress into its lower order and higher order parts the patch test can be passed and the resulting elements are generally invariant a plane four node quadrilateral element is described and compared with existing elements numerical studies show that the accuracy of the element is generally good

assumed strain stabilization procedure for the 9 node lagrange shell element an assumed strain strain interpolation method is used to construct a stabilization matrix for the 9 node shell element the stabilization procedure can be justified based on the hellinger reissner variational method it involves a projection vector which is orthogonal to both linear and quadratic fields in the local co ordinate system of each quadrature point all terms in the development involve 2 2 quadrature in the 9 node element example problems show good accuracy and an almost optimal rate of convergence

multivariable extremum seeking for pde dynamic systems in this article we address the design and analysis of multivariable newton based extremum seeking es for static locally quadratic maps subject to actuation dynamics governed by diffusion partial differential equations pdes multi input systems with distinct diffusion coefficients in each individual input channel are dealt with the phase compensation of the dither signals is handled as a trajectory generation problem and the inclusion of a multivariable diffusion feedback controller with a perturbation based averaging based estimate of the hessian s inverse allow to obtain local exponential convergence results to a small neighborhood of the optimal point the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the diffusion pdes in addition the generalization of the results for different classes of parabolic reaction advection diffusion pdes wave equations and or first order hyperbolic transport dominated pdes is also discussed as for ordinary differential equations case the proposed newton approach removes the dependence of the algorithm s convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable unlike the gradient algorithm a numerical example illustrates the performance of our multivariable es for compensating pde based systems

the development of hybrid axisymmetric elements based on the hellinger reissner variational principle we present a general procedure for the development of hybrid axisymmetric elements based on the hellinger reissner principle within the context of linear elasticity similar to planar elements the stress interpolation is obtained by an identification of the zero energy modes we illustrate our procedure by designing a lower order four node and a higher order nine node element both elements are of correct rank and moreover use the minimum number of stress parameters namely seven and 17 several examples are presented to show the excellent performance of both elements under various demanding situations such as when the material is almost incompressible when the thickness to radius ratio is very small etc when the variation of the field variables is along the radial direction alone when the mesh is uniform and the loading is of pressure type the developed elements are superconvergent i e they yield the exact nodal displacement values

a nested matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems a new infinitesimal sufficient condition is given for uniform global asymptotic stability ugas for time varying nonlinear systems it is used to show that a certain relaxed persistency of excitation condition called uniform spl delta persistency of excitation u spl delta pe is sufficient for uniform global asymptotic stability in certain situations u spl delta pe of the right hand side of a time varying differential equation is also shown to be necessary under a uniform lipschitz condition the infinitesimal sufficient condition for ugas involves the inner products of the flow field with the gradients of a finite number of possibly sign indefinite locally lipschitz lyapunov like functions these inner products are supposed to be bounded by functions that have a certain nested or triangular negative semidefinite structure this idea is reminiscent of a previous idea of matrosov who supplemented a lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the lyapunov function is zero for this reason we call the main result a nested matrosov theorem the utility of our results on stability analysis is illustrated through the well known case study of the nonholonomic integrator

a level set method for the evolution of faceted crystals a level set formulation for the motion of faceted interfaces is presented the evolving surface of a crystal is represented as the zero level of a phase function the crystal is identified by its orientation and facet speeds accuracy is tested on a single crystal by comparison with the exact evolution the method is extended to study the evolution of a polycrystal numerical examples in two and three dimensions are presented

higher order shell elements based on a cosserat model and their use in the topology design of structures for the purpose of carrying out topology optimization of shell structures we show that it is advantageous to use shell elements based on a classical shell theory such as the cosserat theory rather than elements based on the degenerated solid approach since the shell thickness appears explicitly in the formulation thereby greatly simplifying the sensitivity analysis one of the well known shell elements based on the cosserat shell theory is the four node element presented by simo et al comput methods appl mech engrg 72 1989 267 73 1989 53 although one could use this element the use of lower order elements often results in instabilities such as the checkerboard instability in the resulting topologies this provides the motivation for developing higher order shell elements based on a classical shell theory in this work we present the formulation and implementation details for six node and seven node triangular and nine node quadrilateral shell elements which are based on the variational formulation of simo et al and show that good accuracy is obtained even with coarse meshes in fairly demanding problems we also present the topology optimization formulation and some examples of optimal topologies that are obtained using this formulation

observer based perturbation extremum seeking control with input constraints for direct contact membrane distillation process abstractan observer based perturbation extremum seeking control is proposed for a direct contact membrane distillation dcmd process the process is described with a dynamic model that is based on a 2d advection diffusion equation model which has pump flow rates as process inputs the objective of the controller is to optimise the trade off between the permeate mass flux and the energy consumption by the pumps inside the process cases of single and multiple control inputs are considered through the use of only the feed pump flow rate or both the feed and the permeate pump flow rates a nonlinear lyapunov based observer is designed to provide an estimation for the temperature distribution all over the designated domain of the dcmd process moreover control inputs are constrained with an anti windup technique to be within feasible and physical ranges performance of the proposed structure is analysed and simulations based on real dcmd process parameters for each control input are provided

a hybrid mixed model for non linear shell analysis and its applications to large rotation problems adopting an updated lagrange approach the general framework for the fully non linear analysis of curved shells is developed using a simple quadrilateral c0 model hmsh5 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified hellinger reissner type with independent assumptions for displacement and strain fields emphasis is placed on devising effective solution procedures to deal with large rotations in space finite stretches and generalized rate type material models in particular a geometrically exact scheme for configuration update is developed by making use of the so called exponential mapping algorithm and the resulting element was shown to exhibit a quadratic rate of asymptotic convergence in solving practical shell problems with newton raphson type iterative schemes for the purpose of updating the spatial stress field of the element an objective generalized midpoint integration rule is utilized which relies crucially on the concept of polar decomposition for the deformation gradient and is in keeping with the underlying mixed method finally the effectiveness and practical usefulness of the hmsh5 element are demonstrated through a number of test cases involving beams plates and shells undergoing very large displacements and rotations

a new efficient approach to the formulation of mixed finite element models for structural analysis abstract a new finite element formulation is developed based on the hellinger reissner principle with independent strain by dividing the assumed strain into the lower order part and the higher order part the new formulation can be made much more efficient than the standard mixed formulation in addition the present new approach provides a rational way of introducing stabilization matrix to suppress undesirable kinematic modes author

batch to batch finite horizon lq control for unknown discrete time linear systems via stochastic extremum seeking we employ our recent discrete time stochastic averaging theorems and stochastic extremum seeking to iteratively batch to batch optimize open loop control sequences for unknown but reachable discrete time linear systems with a scalar input and without known system dimension for a cost that is quadratic in the measurable output and the input first for a multivariable gradient based stochastic extremum seeking algorithm we prove local exponential convergence to the optimal open loop control sequence second to remove the convergence rate s dependence on the hessian matrix of the cost function which is unknown since the system s model the system matrices a b c is unknown we develop a multivariable discrete time newton based stochastic extremum seeking method design the newton based algorithm for the iteration of the input sequence and prove local exponential convergence to the optimal input sequence finally two simulation examples are given to illustrate the effectiveness of the two methods

a new efficient mixed formulation for thin shell finite element models abstract a nine node shell element is developed by a new and more efficient mixed formulation the new shell element formulation is based on the hellinger reissner principle with independent strain and the concept of degenerate solid shell the new formulation is made more efficient in terms of computing time than the conventional mixed formulation by dividing the assumed strain fields into a lower part and a higher order part numerical results demonstrate that the present nine node element is free of locking even for very thin plates and shells and is also kinematically stable in fact the stiffness matrix associated with the higher order assumed strain plays the role of a stabilization matrix

a unified framework for input to state stability in systems with two time scales this paper develops a unified framework for studying robustness of the input to state stability iss property and presents new results on robustness of iss to slowly varying parameters to rapidly varying signals and to generalized singular perturbations the common feature in these problems is a time scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory to address various robustness problems simultaneously the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables the main results establish that if the boundary layer and averaged systems are iss then the iss bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time scales the generality of the framework is illustrated by making connection to various classical two time scale problems and suggesting extensions

extremum seeking for static maps with delays in this paper we address the design and analysis of multi variable extremum seeking for static maps subject to arbitrarily long time delays both gradient and newton based methods are considered multi input systems with different time delays in each individual input channel as well as output delays are dealt with the phase compensation of the dither signals and the inclusion of predictor feedback with a perturbation based averaging based estimate of the hessian allow to obtain local exponential convergence results to a small neighborhood of the optimal point even in the presence of delays the stability analysis is carried out using backstepping transformation and averaging in infinite dimensions capturing the infinite dimensional state due the time delay in particular a new backstepping like transformation is introduced to design the predictor for the gradient based extremum seeking scheme with multiple and distinct input delays the proposed newton based extremum seeking approach removes the dependence of the convergence rate on the unknown hessian of the nonlinear map to be optimized being user assignable as in the literature free of delays a source seeking example illustrates the performance of the proposed delay compensated extremum seeking schemes

a variational approach to modeling and simulation of grain growth most technologically useful materials arise as polycrystalline microstructures composed of a myriad of small crystallites called grains separated by their interfaces called grain boundaries the orientations and arrangements of the grains and their network of boundaries are implicated in many properties across wide scales for example functional properties like conductivity in microprocessors and lifetime properties like fracture toughness in structures simulation is becoming an important tool for understanding both materials properties and their processing requirements here we offer a consistent variational approach to the mesoscale simulation of these systems subject to the mullins equation of curvature driven growth in a two dimensional setting the main objective is to provide a calibration for future two dimensional and three dimensional efforts we discuss several novel features of our approach which we anticipate will render it a flexible scalable and robust tool to aid in microstructural prediction simulation results offer compelling evidence of the predictability and robustness of statistical properties of large systems such as grain size distribution and texture that are of immediate interest in materials science

a finite element formulation with stabilization matrix for geometrically non linear shells an assumed strain finite element formulation with a stabilization matrix is developed for analysis of geometrically non linear problems of isotropic and laminated composite shells the present formulation utilizes the degenerate solid shell concept and assumes an independent strain as well as displacement the assumed independent strain field is divided into a lower order part and a higher order part subsequently the lower order part is set equal to the displacement dependent strain evaluated at the lower order integration points and the remaining higher order part leads to a stabilization matrix the strains and the determinant of the jacobian matrix are assumed to vary linearly in the thickness direction this assumption allows analytical integration through thickness independent of the number of plies a nine node element with a judiciously chosen set of higher order assumed strain field is developed numerical tests involving isotropic and composite shells undergoing large deflections demonstrate the validity of the present formulation

brief adaptive extremum seeking control of nonlinear dynamic systems with parametric uncertainties we pose and solve an extremum seeking control problem for a class of nonlinear systems with unknown parameters extremum seeking controllers are developed to drive system states to the desired set points that extremize the value of an objective function the proposed adaptive extremum seeking controller is inverse optimal in the sense that it minimizes a meaningful cost function that incorporates penalty on both the performance error and control action simulation studies are provided to verify the effectiveness of the proposed approach

two higher order shell finite elements with stabilization matrix a 32 node three dimensional solid element and a 16 node degenerate solid element that are applicable to analysis of thin shell structural problems are presented in this paper both elements are formulated by using the hellinger reissner principle with independent strain the assumed independent strain field is divided into higherand lower order polynomial terms the element stiffness matrix associated with the higher order assumed strain plays the role of the stabilization matrix various types of the higher order assumed strain are tested for the present elements the resulting finite element models require much less time to compute the element stiffness matrix than the corresponding conventional mixed models or assumed displacement models with a full gaussian quadrature rule numerical results demonstrate that with a properly chosen set of higher order assumed strain the present elements produce reliable and very accurate solutions even for very thin plates and thin shells with deeply curved geometry

on non local stability properties of extremum seeking control in this paper we consider several extremum seeking schemes and show under appropriate conditions that these schemes achieve extremum seeking from an arbitrarily large domain of initial conditions if the parameters in the controller are appropriately adjusted this non local stability result is proved by showing semi global practical stability of the closed loop system with respect to the design parameters we show that reducing the size of the parameters typically slows down the convergence rate of the extremum seeking controllers and enlarges the domain of the attraction our results provide guidelines on how to tune the controller parameters in order to achieve extremum seeking simulation examples illustrate our results

model free plant tuning given a static plant described by a differentiable input output function which is completely unknown but whose jacobian takes values in a known polytope in the matrix space this paper considers the problem of tuning i e driving to a desired value the output by suitably choosing the input it is shown that if the polytope is robustly nonsingular or has full rank in the nonsquare case then a suitable tuning scheme drives the output to the desired point the proof exploits a lyapunov like function and applies a well known game theoretic result concerning the existence of a saddle point for a min max zero sum game when the plant output is represented in an implicit form it is shown that the same result can be obtained resorting to a different lyapunov like function the case in which proper input or output constraints must be enforced during the transient is considered as well some application examples are proposed to show the effectiveness of the approach

a duality based method for generating geometric representations of polycrystals we present a method which we have termed relaxed dual complex rdc for generating geometric representations and computational models of polycrystals of arbitrary shape the rdc method combines a first topological step which defines an initial unrelaxed polycrystal geometry as the barycentric dual of an input triangulation of the solid and a second relaxation step in which the grain boundaries are relaxed by means of a gradient flow driven by grain boundary energy the rdc method applies to arbitrary solids defined by means of a triangulation and in this manner it couples seamlessly to standard solid modelling engines an additional appealing feature of the rdc method is that it generates a conforming tetrahedral mesh of the polycrystal that can be used as a basis for subsequent simulations the rdc method also affords some control over the statistical properties of the polycrystal including grain size which provides a convenient device for matching experimental statistical data the range versatility and performance of the rdc method have been demonstrated by means of selected examples

on the stable equilibrium points of gradient systems this paper studies the relations between the local minima of a cost function f and the stable equilibria of the gradient descent flow of f in particular it is shown that under the assumption that f is real analytic local minimality is necessary and sufficient for stability under the weaker assumption that f is indefinitely continuously differentiable local minimality is neither necessary nor sufficient for stability

brief paper parameter convergence in adaptive extremum seeking control this paper addresses the problem of parameter convergence in adaptive extremum seeking control design an alternate version of the popular persistence of excitation condition is proposed for a class of nonlinear systems with parametric uncertainties the condition is translated to an asymptotic sufficient richness condition on the reference set point since the desired optimal set point is not known a priori in this type of problem the proposed method includes a technique for generating perturbation signal that satisfies this condition in closed loop this demonstrates its superiority in terms of parameter convergence the method guarantees parameter convergence with minimal but sufficient level of perturbation the effectiveness of the proposed method is illustrated with a simulation example

a uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

unified frameworks for sampled data extremum seeking control global optimisation and multi unit systems two frameworks are proposed for extremum seeking of general nonlinear plants based on a sampled data control law within which a broad class of nonlinear programming methods is accommodated it is established that under some generic assumptions semi global practical convergence to a global extremum can be achieved in the case where the extremum seeking algorithm satisfies a stronger asymptotic stability property the converging sequence is also shown to be stable using a trajectory based proof as opposed to a lyapunov function type approach the former is more straightforward and insightful this allows for more general optimisation algorithms than considered in existing literature such as those which do not admit a state update realisation and or lyapunov functions lying at the heart of the analysis throughout is robustness of the optimisation algorithms to additive perturbations of the objective function multi unit extremum seeking is also investigated with the objective of accelerating the speed of convergence

fast extremum seeking on hammerstein plants partial plant knowledge may be used to develop model based extremum seekers however existing results rely on a type of time scale separation which leads to slow optimization relative to the plant dynamics in this work a fast model based extremum seeking scheme is proposed for a hammerstein plant and semi global stability results are provided the structure of a hammerstein plant is used to advantage in designing filters that enable the extremum seeker to act on a faster time scale than the plant dynamics this leads to fast convergence while maintaining semi global stability

a framework for extremum seeking control of systems with parameter uncertainties traditionally the design of extremum seeking algorithm treats the system as essentially a black box which for many applications means disregarding known information about the model structure in contrast to this approach there have been recent examples where a known plant structure with uncertain parameters has been used in the online optimization of plant operation however the results for these approaches have been restricted to specific classes of plants and optimization algorithms this paper seeks to provide general results and a framework for the design of extremum seekers applied to systems with parameter uncertainties general conditions for an optimization method and a parameter estimator are presented so that their combination guarantees convergence of the extremum seeker for both static and dynamic plants tuning guidelines for the closed loop scheme are also presented the generality and flexibility of the proposed framework is demonstrated through a number of parameter estimators and optimization algorithms that can be combined to obtain extremum seeking examples of anti lock braking and model reference adaptive control are used to illustrate the effectiveness of the proposed framework

a framework for a class of hybrid extremum seeking controllers with dynamic inclusions abstract this paper presents a prescriptive framework for the design and analysis of a class of deterministic extremum seeking controllers escs based on hybrid dynamic inclusions this type of escs combines continuous and discrete time dynamics during the seeking process and its evolution in time is characterized by differential and difference inclusions rather than standard difference and differential equations examples of hybrid escs include but are not limited to purely continuous esc with optimizers described by set valued mappings esc with arbitrarily fast and slow switching modes esc with weakly jumping parameters as well as distributed escs for multi agent systems with time varying graphs since solutions of systems described by set valued mappings are usually not unique we do not insist on this property but rather we characterize the behavior of all possible solutions generated by the closed loop system making use of recent results in singular perturbations averaging and limit sets of sets for hybrid dynamical systems we establish a practical asymptotic stability result some examples of common classes of seeking dynamics described by hybrid systems are presented as well as some numerical simulations illustrating their application in standard maximization problems and nash seeking problems in game theoretical scenarios

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

control of spurious modes in the nine node quadrilateral element the nine node quadrilateral is much more accurate than its eight node relative if elements are non rectangular however the nine node element may display spurious low energy modes under 2 2 gauss quadrature these modes can be controlled by using the 3 3 quadrature rule but at added cost this paper investigates two inexpensive control devices that can be used with 2 2 gauss quadrature

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

mixed models and reduced selective integration displacement models for nonlinear analysis of curved beams simple mixed models are developed for the geometrically nonlinear analysis of deep arches a total lagrangian description of the arch deformation is used and the analytical formulation is based on a form of the nonlinear deep arch theory with the effects of transverse shear deformation included the fundamental unknowns consist of the six internal forces and generalized displacements of the arch and the element characteristic arrays are obtained by using hellinger reissner mixed variational principle the polynomial interpolation functions used in approximating the forces are one degree lower than those used for approximating the displacements and the forces are discontinuous at the interelement boundaries the equivalence between the mixed models developed herein and displacement models based on reduced integration of both the transverse shear and extensional energy terms is discussed the advantages of mixed models over equivalent displacement models are outlined numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed and compare their performance with other mixed models reported in the literature

brief paper on global extremum seeking in the presence of local extrema we propose a global extremum seeking scheme which can seek the global optimal value in the presence of local extrema it is shown that the proposed global extremum seeking scheme can converge to an arbitrarily small neighborhood of the global extremum from an arbitrarily large set of initial conditions if sufficient conditions are satisfied a simple example illustrates the effectiveness of the proposed scheme