subdivision based mixed methods for isogeometric analysis of linear and nonlinear nearly incompressible materials this paper addresses the use of isogeometric analysis to solve solid mechanics problems involving nearly incompressible materials the present work is focused on extension of two field mixed variational formulations in both small and large strains to isogeometric analysis inf sup stable displacement pressure combinations for mixed formulations are developed based on the subdivision property of nurbs stability and convergence properties of the proposed displacement pressure combinations are illustrated by computing numerical inf sup constants and error norms the performance of the proposed formulations is assessed by studying several benchmark examples involving nearly incompressible and incompressible elastic and elasto plastic materials in both small and large strain regime

a robust nitsche s formulation for interface problems abstract in this work we propose a novel weighting for the interfacial consistency terms arising in a nitsche variational form we demonstrate through numerical analysis and extensive numerical evidence that the choice of the weighting parameter has a great bearing on the stability of the method consequently we propose a weighting that results in an estimate for the stabilization parameter such that the method remains well behaved in varied settings ranging from the configuration of embedded interfaces resulting in arbitrarily small elements to such cases where a large contrast in material properties exists an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of a elements with arbitrarily small volume fractions b large material heterogeneities and c both large heterogeneities as well as arbitrarily small elements we then highlight the accuracy and efficiency of the proposed formulation through numerical examples focusing particular attention on interfacial quantities of interest

c1 continuous stress recovery in finite element analysis abstract a penalized discrete least squares pdls variational principle is employed to recover c 1 continuous smooth stress fields from finite element stresses the error functional involves discrete least squares with a penalty constraint to enforce c 1 continuity of the recovered stresses the recovery procedure uses a smoothing element analysis sea with element based interpolation functions to minimize the error functional sea pdls recovers a superconvergent stress field of higher accuracy and continuity than the underlying consistent finite element stress field general and specialized formulations of the functional are given and an appropriate discretization strategy for the sea is discussed numerical results for both one and two dimensional stress fields are presented and compared with the corresponding solutions for the superconvergent patch recovery spr method of zienkiewicz and zhu the results demonstrate that in addition to achieving a higher degree of continuity sea pdls produces generally more accurate stresses than spr although an additional global analysis is required results indicate that two coarse mesh analyses finite element and smoothing produce solutions comparable to a single finite element analysis with a much greater degree of refinement and more computational effort the procedure is unique in that the recovered stress field is essentially c 1 continuous the results are shown to be insensitive to the value of the penalty parameter above a threshold and hence uncertainty in the choice of an appropriate penalty value is eliminated

immersed b spline i spline finite element method for geometrically complex domains a novel b spline based immersed finite element method is introduced for the computation of geometrically and topologically complex problems the geometry description and the finite element analysis rely on a block structured logically cartesian mesh which encloses the domain of interest a signed distance function is used for representing the domain on the cartesian mesh whereby the domain boundary is the zeroth level set of the signed distance function away from the domain boundaries the standard b spline basis functions are used for the finite element interpolation close to domain boundaries a new approach has been developed for modifying the b spline basis functions so that they locally interpolate the dirichlet boundary conditions the efficiency and robustness of the proposed approach is demonstrated with a number of one two and three dimensional linear boundary value problems

an analysis of the time integration algorithms for the finite element solutions of incompressible navier stokes equations based on a stabilised formulation abstract this work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the navier stokes equations the stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes particular attention is given to the generalised method and the linear discontinuous in time finite element scheme the time integration schemes are first applied to two model problems represented by a first order differential equation in time and the one dimensional advection diffusion equation and subjected to a detailed mathematical analysis based on the fourier series expansion in order to establish the accuracy and efficiency of the time integration schemes for the navier stokes equations a detailed computational study is performed of two standard numerical examples unsteady flow around a cylinder and flow across a backward facing step it is concluded that the semi discrete generalised method provides a viable alternative to the more sophisticated and expensive space time methods for simulations of unsteady flows of incompressible fluids governed by the navier stokes equations

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

isogeometric analysis of nearly incompressible solids this paper describes the use of isogeometric methods to solve problems in finite deformation solid mechanics in which compressible and nearly incompressible behavior may be encountered the work is based on a three field variational structures in which displacements mean stress and volume effects are independently approximated using this approach and tensor product nurbs approximations it is shown how the solution can be conducted in a standard approach where main variables are organized at nodes control points this leads to an efficient procedure to solve large problems of a general form without the need to develop special solution strategies the approach is demonstrated on some standard test problems copyright 2010 john wiley sons ltd

a higher order compact adi method with monotone iterative procedure for systems of reaction diffusion equations this paper is concerned with an existing compact finite difference adi method published in the paper by liao et al 2002 3 for solving systems of two dimensional reaction diffusion equations with nonlinear reaction terms this method has an accuracy of fourth order in space and second order in time the existence and uniqueness of its solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear reaction terms the convergence of the finite difference solution to the continuous solution is proved an efficient monotone iterative algorithm is presented for solving the resulting discrete system and some techniques for the construction of upper and lower solutions are discussed an application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method

on regulation under sampling the paper deals with linear and nonlinear regulation under sampling it is shown that digital solutions exist under assumptions which are closely related to the existence of robust solutions to the continuous problem approximated solutions are computed starting from the continuous ones

a stabilised immersed boundary method on hierarchical b spline grids for fluid rigid body interaction with solid solid contact abstract an accurate efficient and robust numerical scheme is presented for the simulation of the interaction between flexibly supported rigid bodies and incompressible fluid flow with topology changes and solid solid contact the solution of the incompressible navier stokes equations is approximated by employing a stabilised formulation on cartesian grids discretised with hierarchical b splines the solid is modelled as a rigid body and represented by linear segments along its boundary kinematic conditions along the fluid rigid body interface are enforced weakly using nitsche s method while ghost penalty operators are employed to avoid excessive ill conditioning of the system matrix arising from small cut cells a staggered scheme is used for resolving the coupled fluid rigid body interaction the contact between moving or moving and fixed solid bodies is modelled with lagrange multipliers the excellent performance and wide range of applicability of the proposed scheme are demonstrated in a number of benchmark tests as well as industrially relevant model problems the examples cover the galloping phenomena particulate flow hydraulic check valves and a model turbine

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

weighted extended b spline approximation of dirichlet problems we describe a new finite element method which uses weighted extended b splines on a regular grid as basis functions for solving dirichlet problems on bounded domains in arbitrary dimensions this web method does not require any grid generation and can be implemented very efficiently it yields smooth high order accurate approximations with relatively low dimensional subspaces

reduced integration post processing techniques for darcy s velocity recovery in bilinear quadrilaterals in this work we propose the use of a reduced integration technique for bilinear quadrilateral elements in the determination of the darcy s velocity via global and local post processing techniques this approach increases accuracy in distorted meshes and leads to a reduction in the computational effort for matrix coefficients and residuals evaluations numerical examples are shown to validate our approach accessing its efficiency and accuracy copyright 2007 john wiley sons ltd

analysis of material interface discontinuities and superconvergent fluxes in finite difference theory abstract an analysis of material interface discontinuities is developed and applied in finite difference theory to determine mathematically rigorous averaging techniques for material properties this result is compared with other averaging techniques particularly harmonic averaging which is often applied in practice we also develop a class of formulas of high accuracy for post processing the difference formula to compute derivatives fluxes stresses and conduct supporting numerical studies

a subdivision based implementation of the hierarchical b spline finite element method abstract a novel technique is presented to facilitate the implementation of hierarchical b splines and their interfacing with conventional finite element implementations the discrete interpretation of the two scale relation as common in subdivision schemes is used to establish algebraic relations between the basis functions and their coefficients on different levels of the hierarchical b spline basis the subdivision projection technique introduced allows us first to compute all element matrices and vectors using a fixed number of same level basis functions their subsequent multiplication with subdivision matrices projects them during the assembly stage to the correct levels of the hierarchical b spline basis the proposed technique is applied to convergence studies of linear and geometrically nonlinear problems in one two and three space dimensions

weakly enforced essential boundary conditions for nurbs embedded and trimmed nurbs geometries on the basis of the finite cell method summary enforcing essential boundary conditions plays a central role in immersed boundary methods nitsche s idea has proven to be a reliable concept to satisfy weakly boundary and interface constraints we formulate an extension of nitsche s method for elasticity problems in the framework of higher order and higher continuity approximation schemes such as the b spline and non uniform rational basis spline version of the finite cell method or isogeometric analysis on trimmed geometries furthermore we illustrate a significant improvement of the flexibility and applicability of this extension in the modeling process of complex 3d geometries with several benchmark problems we demonstrate the overall good convergence behavior of the proposed method and its good accuracy we provide extensive studies on the stability of the method its influence parameters and numerical properties and a rearrangement of the numerical integration concept that in many cases reduces the numerical effort by a factor two a newly composed boundary integration concept further enhances the modeling process and allows a flexible discretization independent introduction of boundary conditions finally we present our strategy in the framework of the modeling and isogeometric analysis process of trimmed non uniform rational basis spline geometries copyright 2013 john wiley sons ltd

an isogeometric design through analysis methodology based on adaptive hierarchical refinement of nurbs immersed boundary methods and t spline cad surfaces we explore hierarchical refinement of nurbs as a basis for adaptive isogeometric and immersed boundary analysis we use the principle of b spline subdivision to derive a local refinement procedure which combines full analysis suitability of the basis with straightforward implementation in tree data structures and simple generalization to higher dimensions we test hierarchical refinement of nurbs for some elementary fluid and structural analysis problems in two and three dimensions and attain good results in all cases using the b spline version of the finite cell method we illustrate the potential of immersed boundary methods as a seamless isogeometric design through analysis procedure for complex engineering parts defined by t spline cad surfaces specifically a ship propeller and an automobile wheel we show that hierarchical refinement considerably increases the flexibility of this approach by adaptively resolving local features

on uniform boundedness of parameterized discrete time systems with decaying inputs applications to cascades a framework for controller design of sampled data nonlinear systems via their approximate discrete time models has been established recently within this framework naturally arises the need to investigate stability properties of parameterized discrete time systems further results that guarantee appropriate stability of the parameterized family of discrete time systems that is used within this framework have been also established for systems with cascaded structure a fundamental condition that is required in this framework is uniform boundedness of solutions of the cascade however this is difficult to check in general in this paper we provide a range of sufficient conditions for uniform boundedness that are easier to check these results further contribute to the toolbox for controller design of sampled data nonlinear systems via their approximate discrete time models

guaranteed cost control of affine nonlinear systems via partition of unity method we consider the problem of guaranteed cost control gcc of affine nonlinear systems in this paper firstly the general affine nonlinear system with the origin being its equilibrium point is represented as a linear like structure with state dependent coefficient matrices secondly partition of unity method is used to approximate the coefficient matrices as a result of which the original affine nonlinear system is equivalently converted into a linear like system with modeling error a gcc law is then synthesized based on the equivalent model in the presence of modeling error under certain error condition the control law ensures that the system under control is asymptotically stable as well as that a given cost function is upper bounded a suboptimal gcc law can be obtained via solving an optimization problem in terms of linear matrix inequality lmi in stead of state dependent riccati equation sdre or hamilton jacobi equations that are usually required in solving nonlinear optimal control problems finally a numerical example is provided to illustrate the validity of the proposed method

a new method for stability analysis of nonlinear discrete time systems we address the problem of global lyapunov stability of discrete time systems with known coefficients we develop a method for reduction of dissipativity domain effectively testing if the system has a convex lyapunov function our implementation is immediately applicable to differentiable systems with bounded nonlinearities but the method proposed is more general and applicable to nondifferentiable systems with bounded right hand sides our main application emphasis is on stability analysis of recurrent neural networks we illustrate how to use our approach with examples

stabilized methods and post processing techniques for miscible displacements abstract stabilized finite element formulations and post processing techniques are presented for darcy flow and miscible displacement problems in these techniques all primary unknowns are approximated in space by standard equal order interpolations we presented a review of the main mathematical results including convergence estimates for global post processing to solve the pressure velocity problem and for the stabilized solution of the complete set of coupled partial differential equations in both space time and semi discrete frameworks demonstration problems are solved to show that we obtained numerically high rates of convergence than those predicted achieving good accuracy in the solution of monophasic flow problems in adverse conditions

high order linear and non linear residual distribution schemes for turbulent compressible flows a high order residual distribution scheme for the solution of the compressible rans equations is developed the one equation spalart allmaras turbulence model is solved in a fully coupled approach the mean flow equations as well as the turbulence equation are solved with high order of accuracy a continuous approximation of the solution is adopted and standard lagrangian basis functions are used to construct the discrete space since viscous terms involve the gradient of the numerical solution which has a discontinuous normal component across the faces of the elements a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same basis functions used for the solution the non linear system of equations resulting from the numerical discretization is solved with the non linear lu sgs method several numerical experiments are performed to verify the accuracy of the numerical method these are based on both subsonic and transonic flows in two and three spatial dimensions

a new family of projection schemes for the incompressible navier stokes equations with control of high frequency damping abstract a simple spatially discrete model problem consisting of mass points and dash pots is presented which allows for the assessment of the properties of different projection schemes for the solution of the incompressible navier stokes equations in particular the temporal accuracy the stability and the numerical damping are investigated the present study suggests that it is not possible to formulate a second order accurate projection pressure correction scheme which possesses any high frequency damping motivated by this observation two new families of projection schemes are proposed which are developed from the generalised midpoint rule and from the generalised method respectively and offer control over high frequency damping both schemes are investigated in detail on the basis of the model problem and subsequently implemented in the context of a finite element formulation for the incompressible navier stokes equations comprehensive numerical studies of the flow in a lid driven cavity and the flow around a cylinder are presented the observations made are in agreement with the conclusions drawn from the model problem

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

skeleton stabilized immersogeometric analysis for incompressible viscous flow problems a skeleton stabilized immersogeometric analysis technique is proposed for incompressible viscous flow problems with moderate reynolds numbers the proposed formulation fits within the framework of the finite cell method where essential boundary conditions are imposed weakly using a nitsche type method the key idea of the proposed formulation is to stabilize the jumps of high order derivatives of variables over the skeleton of the background mesh the formulation allows the use of identical finite dimensional spaces for the approximation of the pressure and velocity fields in immersed domains the stability issues observed for inf sup stable discretizations of immersed incompressible flow problems are avoided with this formulation for b spline basis functions of degree formula presented with highest regularity only the derivative of order formula presented has to be controlled which requires specification of only a single stabilization parameter for the pressure field the stokes and navier stokes equations are studied numerically in two and three dimensions using various immersed test cases oscillation free solutions and high order optimal convergence rates can be obtained the formulation is shown to be stable even in limit cases where almost every elements of the physical domain is cut and hence it does not require the existence of interior cells in terms of the sparsity pattern the algebraic system has a considerably smaller stencil than counterpart approaches based on lagrange basis functions this important property makes the proposed skeleton stabilized technique computationally practical to demonstrate the stability and robustness of the method we perform a simulation of fluid flow through a porous medium of which the geometry is directly extracted from 3d formula presented scan data

accuracy of mixed and control volume finite element approximations to darcy velocity and related quantities the accuracy of the darcy velocity flux and stream function computed from lowest order triangle based control volume and mixed finite element approximations to the two dimensional pressure equation is considered the control volume finite element method similar to integrated finite difference methods and analogous to the interpolation of galerkin finite element results over control volumes is shown to yield a conservative velocity field and smooth streamlines the streamlines and fluxes through the system computed with the control volume finite element approach are compared to those computed from the mixed finite element method which approximates the pressure and velocity variables separately it is shown that for systems with only moderate degrees of heterogeneity the control volume finite element method is the more computationally efficient alternative i e it provides more accurate flow results for a given number of unknowns for more variable or discontinuous permeability fields by contrast such as sand shale systems the mixed finite element method is shown to approximate flow variables more accurately and more realistically than the control volume method with the same number of unknowns

a high order immersed interface method for simulating unsteady incompressible flows on irregular domains immersed boundary methods and immersed interface methods are becoming increasingly popular for the computation of unsteady flows around complex geometries using a cartesian grid while good results both qualitative and quantitative have been obtained most of the methods rely on low order corrections to account for the immersed boundary the objective of the present work is to present as an alternative a high order modified immersed interface method for the 2d unsteady incompressible navier stokes equations in stream function vorticity formulation the method employs an explicit fourth order runge kutta time integration scheme fourth order compact finite differences for computation of spatial derivatives and a nine point fourth order compact discretization of the poisson equation for computation of the stream function corrections to the finite difference schemes are used to maintain high formal accuracy at the immersed boundary as confirmed by analytical tests to validate the method in its application to incompressible flows several physically relevant test cases are computed including uniform flow past a circular cylinder and tollmien schlichting waves in a boundary layer

sufficient conditions for stabilization of sampled data nonlinear systems via discrete time approximations given a parameterized by sampling period t family of approximate discrete time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all t sufficiently small we present conditions which guarantee that the same family of controllers semi globally practically stabilizes the exact discrete time model of the plant for sufficiently small sampling periods when the family of controllers is locally bounded uniformly in the sampling period the inter sample behavior can also be uniformly bounded so that the time varying sampled data model of the plant is uniformly semi globally practically stabilized the result justifies controller design for sampled data nonlinear systems based on the approximate discrete time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied our analysis is applicable to a wide range of time discretization schemes and general plant models

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

an unfitted hp adaptive finite element method based on hierarchical b splines for interface problems of complex geometry generating finite element discretizations with direct interface parameterizations constitutes a considerable computational expense in case of complex interface geometries the paper at hand introduces a b spline finite element method which circumvents parameterization of interfaces and offers fast and easy meshing irrespective of the geometric complexity involved its core idea is the adaptive approximation of discontinuities by hierarchical grid refinement which adds several levels of local basis functions in the close vicinity of interfaces but unfitted to their exact location so that a simple regular grid of knot span elements can be maintained numerical experiments show that an hp refinement strategy which simultaneously increases the polynomial degree of the b spline basis and the levels of refinement around interfaces achieves exponential rates of convergence despite the presence of discontinuities it is also demonstrated that the hierarchical b spline fem can be used to transfer the recently introduced finite cell concept to geometrically nonlinear problems its computational performance imposition of unfitted boundary conditions and fast hierarchical grid generation are illustrated for a set of benchmark problems in one two and three dimensions and the advantages of the regular grid approach for complex geometries are demonstrated by the geometrically nonlinear simulation of a voxel based foam composite

explicit strategies for incompressible fluid structure interaction problems nitsche type mortaring versus robin robin coupling we discuss explicit coupling schemes for fluid structure interaction problems where the added mass effect is important in this paper we show the close relation between coupling schemes using nitsche s method and a robin robin type coupling in the latter case the method may be implemented either using boundary integrals of the stresses or the more conventional discrete lifting operators recalling the explicit method proposed in comput methods appl mech engrg 198 5 8 766 784 2009 we make the observation that this scheme is stable under a hyperbolic type cfl condition but that optimal accuracy imposes a parabolic type cfl conditions due to the splitting error two strategies to enhance the accuracy of the coupling scheme under the hyperbolic cfl condition are suggested one using extrapolation and defect correction and one using a penalty free non symmetric nitsche method finally we illustrate the performance of the proposed schemes on some numerical examples in two and three space dimensions

approximate boundary flux calculations abstract a technique for determining derivatives fluxes or stresses from finite element solutions is developed the approach is a generalization to higher dimensions of a procedure known to give highly accurate results in one dimension numerical experiments demonstrate that certain difficulties are associated with corners in the higher dimensional extensions and two variants of the method are examined we consider both triangular and quadrilateral elements and observe some interesting differences in the numerical rates of convergence finally this post processing scheme is tested for nonlinear problems

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

lyapunov functions stability and input to state stability subtleties for discrete time discontinuous systems in this note we consider stability analysis of discrete time discontinuous systems using lyapunov functions we demonstrate via simple examples that the classical second method of lyapunov is precarious for discrete time discontinuous dynamics also we indicate that a particular type of lyapunov condition slightly stronger than the classical one is required to establish stability of discrete time discontinuous systems furthermore we examine the robustness of the stability property when it was attained via a discontinuous lyapunov function which is often the case for discrete time hybrid systems in contrast to existing results based on smooth lyapunov functions we develop several input to state stability tests that explicitly employ an available discontinuous lyapunov function

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

practical stability of approximating discrete time filters with respect to model mismatch this paper establishes practical stability results for an important range of approximate discrete time filtering problems involving mismatch between the true system and the approximating filter model practical stability is established in the sense of an asymptotic bound on the amount of bias introduced by the model approximation our analysis applies to a wide range of estimation problems and justifies the common practice of approximating intractable infinite dimensional nonlinear filters by simpler computationally tractable filters

analysis of the superconvergent patch recovery technique and a posteriori error estimator in the finite element method ii abstract this is the first in a series of two papers in which the patch recovery technique proposed by zienkiewicz and zhu is analyzed in this work it is proved that the recovered derivative by the least squares fitting is superconvergent for the two point boundary value problems further the a posteriori error estimator based on the recovery technique is shown to be asymptotically exact this confirms the computational results obtained in 1 2

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

subdivision stabilised immersed b spline finite elements for moving boundary flows an immersed finite element method is presented to compute flows with complex moving boundaries on a fixed cartesian grid the viscous incompressible fluid flow equations are discretized with b spline basis functions the two scale relation for b splines is used to implement an elegant and efficient technique to satisfy the lbb condition on non grid aligned fluid domains and at moving boundaries the boundary conditions are enforced with a consistent penalty method as originally proposed by nitsche in addition a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut cells the versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations

a fictitious domain distributed lagrange multiplier based fluid structure interaction scheme with hierarchical b spline grids abstract we present a numerical scheme for fluid structure interaction based on hierarchical b spline grids and fictitious domain distributed lagrange multipliers the incompressible navier stokes equations are solved over a cartesian grid discretised with b splines the fluid grid near the immersed solids is refined locally using hierarchical b splines the immersed solid is modelled as geometrically exact beam discretised with standard linear lagrange shape functions the kinematic constraint at the fluid solid interface is enforced with distributed lagrange multipliers the unconditionally stable and second order accurate generalised method is used for integration in time for both the fluid and solid domains a fully implicit and fully coupled solution scheme is developed by using the newton raphson method to solve the non linear system of equations obtained with galerkin weak formulation first the spatial and temporal convergence of the proposed scheme is assessed by studying steady and unsteady flow past a fixed cylinder then the scheme is applied to several benchmark problems to demonstrate the efficiency and robustness of the proposed scheme the results obtained with the present scheme are compared with the reference values

cutfem discretizing geometry and partial differential equations we discuss recent advances on robust unfitted finite element methods on cut meshes these methods are designed to facilitate computations on complex geometries obtained for example from computer

smooth lyapunov functions and robustness of stability for difference inclusions we demonstrate the equivalence of robust global asymptotic stability gas of the origin and the existence of a smooth lyapunov function for difference inclusions defined by upper semicontinuous set valued maps sufficient conditions for robust gas are given as an application of these results we give conditions for robust gas of difference equations defined by discontinuous right hand sides

derivative calculation from finite element solutions abstract a technique is considered whereby very accurate derivatives of a finite element solution can be calculated efficiently it is demonstrated here that most of the necessary quantities for this subsidiary computation are available as computed by products in the preceding finite element solution procedure the calculation is shown in this note to be a particular form of a procedure for which superconvergent theoretical error estimates have been proven elsewhere numerical experiments confirm the superior accuracy in the computed derivative stress or flux

multigrid methods for isogeometric discretization abstract we present geometric multigrid methods for isogeometric discretization of scalar second order elliptic problems the smoothing property of the relaxation method and the approximation property of the intergrid transfer operators are analyzed these properties when used in the framework of classical multigrid theory imply uniform convergence of two grid and multigrid methods supporting numerical results are provided for the smoothing property the approximation property convergence factor and iterations count for v w and f cycles and the linear dependence of v cycle convergence on the smoothing steps for two dimensions numerical results include the problems with variable coefficients simple multi patch geometry a quarter annulus and the dependence of convergence behavior on refinement levels l whereas for three dimensions only the constant coefficient problem in a unit cube is considered the numerical results are complete up to polynomial order p 4 and for c 0 and c p 1 smoothness

a robust nitsche s formulation for interface problems with spline based finite elements summary the extended finite element method x fem has proven to be an accurate robust method for solving embedded interface problems with a few exceptions the x fem has mostly been used in conjunction with piecewise linear shape functions and an associated piecewise linear geometrical representation of interfaces in the current work the use of spline based finite elements is examined along with a nitsche technique for enforcing constraints on an embedded interface to obtain optimal rates of convergence we employ a hierarchical local refinement approach to improve the geometrical representation of curved interfaces we further propose a novel weighting for the interfacial consistency terms arising in the nitsche variational form with b splines a qualitative dependence between the weights and the stabilization parameters is established with additional element level eigenvalue calculations an important consequence of this weighting is that the bulk as well as the interfacial fields remain well behaved in the presence of large heterogeneities as well as elements with arbitrarily small volume fractions we demonstrate the accuracy and robustness of the proposed method through several numerical examples copyright 2015 john wiley sons ltd

forward and converse lyapunov theorems for discrete dynamical systems this technical note addresses the stability analysis of nonlinear dynamic systems three main contributions are made first we show that the standard assumption of a continuous lyapunov function can be and in some cases must be relaxed we introduce the concept of the weak lyapunov function which requires that an annulus condition be satisfied we believe that this annulus condition is a more natural construct because it is precisely what is needed to make the forward lyapunov theorem true second we provide an example of a nonlinear system with stable equilibrium point that cannot be shown to be stable with a continuous lyapunov function finally we demonstrate a simpler and less restrictive proof of the converse lyapunov theorem

brief paper a discrete time observer design for spacecraft attitude determination using an orthogonality preserving algorithm this paper studies a nonlinear discrete time partial state observer design problem for rigid spacecraft systems particularly for spacecraft attitude estimation the construction is inspired by an orthogonality preserving numerical algorithm and it yields a semiglobal practical asymptotic observer we show that the separation principle holds and hence stabilization using output feedback is possible an example is provided to illustrate the usefulness of our results in a sampled data implementation of an attitude control for a rigid spacecraft using output feedback and to illustrate the advantages of the proposed observer design especially compared to a sample and hold version of a similar type observer designed in continuous time

a stabilised immersed boundary method on hierarchical b spline grids abstract in this work an immersed boundary finite element method is proposed which is based on a hierarchically refined cartesian b spline grid and employs the non symmetric and penalty free version of nitsche s method to enforce the boundary conditions the strategy allows for h and p refinement and employs a so called ghost penalty term to stabilise the cut cells an effective procedure based on hierarchical subdivision and sub cell merging which avoids excessive numbers of quadrature points is used for the integration of the cut cells a basic laplace problem is used to demonstrate the effectiveness of the cut cell stabilisation and of the penalty free nitsche method as well as their impact on accuracy the methodology is also applied to the incompressible navier stokes equations where the supg pspg stabilisation is employed simulations of the lid driven cavity flow and the flow around a cylinder at low reynolds number show the good performance of the methodology excessive ill conditioning of the system matrix is robustly avoided without jeopardising the accuracy at the immersed boundaries or in the field

formulas relating kl stability estimates of discrete time and sampled data nonlinear systems abstract we provide an explicit kl stability or input to state stability iss estimate for a sampled data nonlinear system in terms of the kl estimate for the corresponding discrete time system and a k function describing inter sample growth it is quite obvious that a uniform inter sample growth condition plus an iss property for the exact discrete time model of a closed loop system implies uniform iss of the sampled data nonlinear system our results serve to quantify these facts by means of comparison functions our results can be used as an alternative to prove and extend results in 1 or extend some results in 4 to a class of nonlinear systems finally the formulas we establish can be used as a tool for some other problems which we indicate

a penalty free nonsymmetric nitsche type method for the weak imposition of boundary conditions in this paper we show that the nonsymmetric version of nitsche s method for the weak imposition of boundary conditions is stable without penalty term for nonconforming elements we prove the same result for the symmetric formulation as well we prove optimal h 1 error estimates and l 2 error estimates that are suboptimal with half an order in h both the pure diffusion and the convection diffusion problems are discussed

the non symmetric nitsche method for the parameter free imposition of weak boundary and coupling conditions in immersed finite elements abstract we explore the use of the non symmetric nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh in contrast to symmetric nitsche methods it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters we first review the available mathematical background recollecting relevant aspects of the method from a numerical analysis viewpoint we then compare accuracy and convergence of symmetric and non symmetric nitsche methods for a laplace problem a kirchhoff plate and in 3d elasticity our numerical experiments confirm that the non symmetric method leads to reduced accuracy in the l 2 error but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux bending moments or stress based on our numerical evidence the non symmetric nitsche method is a viable alternative for problems with diffusion type operators in particular when the accuracy of derivative quantities is of primary interest

a variational method for finite element stress recovery and error estimation abstract a variational method for obtaining smoothed stresses from a finite element derived non smooth stress field is presented the method is based on minimizing a functional involving discrete least squares error plus a penalty constraint that ensures smoothness of the stress field an equivalent accuracy criterion is developed for the smoothing analysis which results in a c1 continuous smoothed stress field possessing the same order of accuracy as that found at the superconvergent optimal stress points of the original finite element analysis application of the smoothing analysis to residual error estimation is also demonstrated

high order compact scheme for the steady stream function vorticity equations a higher order compact scheme that is o h4 on the nine point 2 d stencil is formulated for the steady stream function vorticity form of the navier stokes equations the resulting stencil expressions are presented and hence this new scheme can be easily incorporated into existing industrial software we also show that special treatment of the wall boundary conditions is required the method is tested on representative model problems and compares very favourably with other schemes in the literature

system theoretic properties of sampled data representations of nonlinear systems obtained via taylor lie series it is possible to obtain a sampled data representation of a nonlinear dynamic system under zero order hold using taylor lie series within the context of taylor lie series theory explicit formulae for the resulting discrete time system are derived in terms of the system lie derivatives the effect of sampling on systemtheoretic properties such as equilibrium properties relative order stability zero dynamics and minimum phase characteristics is examined revealing the natural and transparent way in which the use of taylor lie series permeates the relevant theoretical aspects the results derived are also illustrated in two chemical engineering examples

mesh refinement strategies without mapping of nonlinear solutions for the generalized and standard fem analysis of 3 d cohesive fractures summary a robust and efficient strategy is proposed to simulate mechanical problems involving cohesive fractures this class of problems is characterized by a global structural behavior that is strongly affected by localized nonlinearities at relatively small sized critical regions the proposed approach is based on the division of a simulation into a suitable number of sub simulations where adaptive mesh refinement is performed only once based on refinement window s around crack front process zone s the initialization of newton raphson nonlinear iterations at the start of each sub simulation is accomplished by solving a linear problem based on a secant stiffness rather than a volume mapping of nonlinear solutions between meshes the secant stiffness is evaluated using material state information stored read on crack surface facets which are employed to explicitly represent the geometry of the discontinuity surface independently of the volume mesh within the generalized finite element method framework moreover a simplified version of the algorithm is proposed for its straightforward implementation into existing commercial software data transfer between sub simulations is not required in the simplified strategy the computational efficiency accuracy and robustness of the proposed strategies are demonstrated by an application to cohesive fracture simulations in 3 d copyright 2016 john wiley sons ltd

a new class of cn interpolations and its application to the finite element method a new class of cn continuous interpolations is presented these interpolations consist of two or more lagrangian interpolations blended by pseudo hermitian interpolation using this class of interpolations a new cn family of displacement type elements is developed the properties of this new family of elements such as their precision and their superconvergent points are discussed some solutions to the corner elements problem and implication of different boundary conditions are proposed using this new kind of element several examples of deflection of elastic beams and plates are treated the results of which are encouraging copyright 2001 john wiley sons ltd

input to state stabilization for nonlinear dual rate sampled data systems via approximate discrete time model the problem of state feedback stabilization of nonlinear sampled data systems is considered under the low measurement rate constraint a dual rate control scheme is proposed that utilizes a numerical integration scheme to approximately predict the current state given an approximate discrete time model of a sampled nonlinear plant and given a family of controllers that stabilizes the plant model in input to state sense we show that under some standard assumptions the closed loop dual rate sampled data system is input to state stable in the semiglobal practical sense

superconvergent derivatives a taylor series analysis an analysis based on the use of taylor series expansions is developed to determine accuracy estimates for derivatives in one and two dimensions computed by differentiation of a finite element interpolant or approximation the analysis clarifies some issues concerning special points at which the derivatives are believed to be exceptionally accurate with higher convergence rates superconvergence moreover it leads directly to a class of post processing strategies for the derivatives and offers a more direct constructive approach to the subject

global sampled data output feedback stabilisation for a class of nonlinear systems with unknown output function abstractthis paper discusses the problem of global sampled data output feedback stabilisation for a class of nonlinear systems whose output function is unknown a systematic design scheme is developed to construct a linear output feedback control law in sampled data form an explicit formula for the maximum allowable sampling period is computed to guarantee global stability of the uncertain nonlinear systems under the proposed controller with appropriate gains two examples are given to demonstrate the effectiveness of the proposed design procedure

asymptotic controllability and exponential stabilization of nonlinear control systems at singular points we discuss the relation between exponential stabilization and asymptotic controllability of nonlinear control systems with constrained control range at singular points using a discounted optimal control approach we construct discrete feedback laws minimizing the lyapunov exponent of the linearization thus we obtain an equivalence result between uniform exponential controllability and uniform exponential stabilizability by means of a discrete feedback law

a computational framework for fluid structure interaction finite element formulation and applications abstract this work is concerned with the modelling of the interaction of fluid flow with flexibly supported rigid bodies the fluid flow considered is governed by the incompressible navier stokes equations and modelled by employing stabilised low order velocity pressure finite elements the motion of the fluid domain is accounted for by an arbitrary lagrangian eulerian ale strategy the rigid body motion excited by the fluid flow is restricted by the elastic and damping properties of the supports for the temporal discretisation the discrete implicit generalised method is employed the resulting strongly coupled set of nonlinear equations is solved by means of a novel partitioned solution procedure which is based on the newton raphson methodology and incorporates full linearisation of the overall incremental problem the strong coupling is resolved and optimal convergence of the residuals is achieved several numerical examples are presented to demonstrate the robustness and efficiency of the methodology the examples clearly capture the phenomena of vortex induced oscillations and galloping

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd

recovery procedures in error estimation and adaptivity part i adaptivity in linear problems abstract the procedures of error estimation using stress or gradient recovery were introduced by zienkiewicz and zhu in 1987 3 and with the improvement of recovery procedures introduction of superconvergent patch recovery spr since that date the authors have succeeded in making these error estimators the most robust of those currently available very recently introduced methods of recovery such as rep recovery by equilibrium in patches allow general application in this paper the general idea of using recovery based error estimation in adaptive procedure of linear problems is explained through which the latest developments in recovery techniques is described the fundamental basis of a patch test introduced by babuska et al 11 14 is explained and applied to the recovery based error estimators using both spr and rep

superconvergence phenomena in the finite element method abstract we give a brief survey of superconvergence phenomena in finding a numerical solution of differential equations by finite elements several new results and open problems are introduced

controlling nonlinear time varying systems via euler approximations abstract a discrete time model reference control algorithm for continuous time systems represented by essentially unknown nonlinear time varying ordinary differential equations is presented and analysed provided the reference model and the inverse dynamics of the plant are bounded input bounded output stable it is shown that a discrete control using state measurements based on euler approximations for the systems response can achieve arbitrary tracking accuracy the effect of unmodelled dynamics and disturbances is analysed