a dual mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions abstract in this paper we consider the hu washizu principle and propose a new dual mixed finite element method for nonlinear incompressible plane elasticity with mixed boundary conditions the approach extends a related previous work on the dirichlet problem and imposes the neumann essential boundary condition in a weak sense by means of an additional lagrange multiplier the resulting variational formulation becomes a twofold saddle point operator equation which for convenience of the subsequent analysis is shown to be equivalent to a nonlinear threefold saddle point problem in this way a slight generalization of the classical babuska brezzi theory is applied to show the well posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimates in particular the classical peers space is suitably enriched to define the associated galerkin scheme next we develop a local problems based a posteriori error analysis and derive an implicit reliable and quasi efficient estimate and a fully explicit reliable one finally several numerical results illustrating the good performance of the explicit a posteriori estimate for the adaptive computation of the discrete solutions are provided

accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

a mixed finite element method for nonlinear elasticity two fold saddle point approach and a posteriori error estimate we extend the applicability of stable mixed finite elements for linear plane elasticity such as peers to a mixed variational formulation of hyperelasticity the present approach is based on the introduction of the strain tensor as a further unknown which yields a two fold saddle point nonlinear operator equation for the corresponding weak formulation we provide the uniqueness of solution for the continuous and discrete schemes and derive the usual cea estimate for the associated error finally a reliable a posteriori error estimate based on the solution of local dirichlet problems and well suited for adaptive computations is also given

an hp adaptive mixed discontinuous galerkin fem for nearly incompressible linear elasticity we develop the a posteriori error estimation of mixed hp version discontinuous galerkin finite element methods for nearly incompressible elasticity problems in two space dimensions computable upper and lower bounds on the error measured in terms of a natural mesh dependent energy norm are derived the bounds are explicit in the local mesh sizes and approximation orders and are independent of the locking parameter a series of numerical experiments are presented which demonstrate the performance of the proposed error estimator within an automatic hp adaptive refinement procedure

a supra convergent finite difference scheme for the variable coefficient poisson equation on non graded grids we introduce a method for solving the variable coefficient poisson equation on non graded cartesian grids that yields second order accuracy for the solutions and their gradients we employ quadtree in 2d and octree in 3d data structures as an efficient means to represent the cartesian grid allowing for constraint free grid generation the schemes take advantage of sampling the solution at the nodes vertices of each cell in particular the discretization at one cell s node only uses nodes of two 2d or three 3d adjacent cells producing schemes that are straightforward to implement numerical results in two and three spatial dimensions demonstrate supra convergence in the l norm

strategies for computing goal oriented a posteriori error measures in non linear elasticity we investigate the characteristics and performance of goal oriented a posteriori error measures for a class of non linear elasticity models while restriction is made to small strain theory the chosen error measure of the displacement field can be global or local probing the chosen quantity in a specific spatial point the error is computable with the aid of the solution of a dual problem whose data depend on the error measure the main thrust of the paper is to evaluate the performance of a few different approximation strategies for computing the dual solution the chosen strategies are compared in terms of accuracy ease of implementation reliability and cost efficiency a well known numerical example the cook s membrane is used for the numerical evaluations

adjoint recovery of superconvergent functionals from pde approximations motivated by applications in computational fluid dynamics a method is presented for obtaining estimates of integral functionals such as lift or drag that have twice the order of accuracy of the computed flow solution on which they are based this is achieved through error analysis that uses an adjoint pde to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy numerical results are presented for the poisson equation in one and two dimensions and for the nonlinear quasi one dimensional euler equations the theory is equally applicable to nonlinear equations in complex multi dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

a second order accurate projection method for the incompressible navier stokes equations on non graded adaptive grids we present an unconditionally stable second order accurate projection method for the incompressible navier stokes equations on non graded adaptive cartesian grids we employ quadtree and octree data structures as an efficient means to represent the grid we use the supra convergent poisson solver of c h min f gibou h ceniceros a supra convergent finite difference scheme for the variable coefficient poisson equation on fully adaptive grids cam report 05 29 j comput phys in press a second order accurate semi lagrangian method to update the momentum equation an unconditionally stable backward difference scheme to treat the diffusion term and a new method that guarantees the stability of the projection step on highly non graded grids we sample all the variables at the grid nodes producing a scheme that is straightforward to implement we propose two and three dimensional examples to demonstrate second order accuracy for the velocity field and the divergence free condition in the l1 and l norms

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

smoothing for discrete valued time series we deal with smoothed estimators for conditional probability functions of discrete valued time series yt under two different settings when the conditional distribution of yt given its lagged values falls in a parametric family and depends on exogenous random variables a smoothed maximum partial likelihood estimator for the unknown parameter is proposed while there is no prior information on the distribution various nonparametric estimation methods have been compared and the adjusted nadaraya watson estimator stands out as it shares the advantages of both nadaraya watson and local linear regression estimators the asymptotic normality of the estimators proposed has been established in the manner of sparse asymptotics which shows that the smoothed methods proposed outperform their conventional unsmoothed parametric counterparts under very mild conditions simulation results lend further support to this assertion finally the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in hong kong in 1994 1995 an ad hoc model selection procedure based on a local akaike information criterion is proposed to select the significant pollutant indices

a fourth order accurate method for the incompressible navier stokes equations on overlapping grids a method is described to solve the time dependent incompressible navier stokes equations with finite differences on curvilinear overlapping grids in two or three space dimensions the scheme is fourth order accurate in space and uses the momentum equations for the velocity coupled to a poisson equation for the pressure the boundary condition for the pressure is taken as u 0 extra numerical boundary conditions are chosen to make the scheme accurate and stable the velocity is advanced explicitly in time any standard time stepping scheme such as runge kutta can be used the poisson equation is solved using direct or iterative sparse matrix solvers or by the multigrid algorithm computational results in two and three space dimensions are given

the post processing approach in the finite element method part 1 calculation of displacements stresses and other higher derivatives of the displacements this is the first in a series of three papers in which we discuss a method for post processing a finite element solution to obtain high accuracy approximations for displacements stresses stress intensity factors etc rather than take the values of these quantities directly from the finite element solution we evaluate certain weighted averages of the solution over the entire region these yield approximations are of the same order of accuracy as the strain energy we obtain error estimates and also present some numerical examples to illustrate the practical effectiveness of the technique in the third paper of this series we address the matters of adaptive mesh selection and a posteriori error estimation

tree methods for moving interfaces fast adaptive numerical methods for solving moving interface problems are presented the methods combine a level set approach with frequent redistancing and semi lagrangian time stepping schemes which are explicit yet unconditionally stable a quadtree mesh is used to concentrate computational effort on the interface so the methods move an interface withndegrees of freedom ino nlogn work per time step efficiency is increased by taking large time steps even for parabolic curvature flows the methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging anisotropy faceting and curvature

weighted matchings for preconditioning symmetric indefinite linear systems maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems especially in direct solvers in this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for incomplete mathrm ldl t factorizations the reorderings are constructed such that the matched entries form 1 times 1 or 2 times 2 diagonal blocks in order to increase the diagonal dominance of the system during the incomplete factorization only tridiagonal pivoting is used we report results for this approach and comparisons with other solution methods for a diverse set of symmetric indefinite matrices ranging from nonlinear elasticity to interior point optimization

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 adaptive level set approach for incompressible two phase flows we present a numerical method using the level set approach for solving incompressible two phase flow with surface tension in the level set approach the free surface is represented as the zero level set of a smooth function this has the effect of replacing the advection of density which has steep gradients at the free surface with the advection of the level set function which is smooth in addition the free surface can merge or break up with no special treatment we maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects in this work we couple the level set scheme to an adaptive projection method for the incompressible navier stokes equations in order to achieve higher resolution of the free surface with a minimum of additional expense we present two dimensional axisymmetric and fully three dimensional results of air bubble and water drop computations

a flux free a posteriori error estimator for the incompressible stokes problem using a mixed fe formulation in this contribution we present an a posteriori error estimator for the incompressible stokes problem valid for a conventional mixed fe formulation due to the saddle point property of the problem conventional error estimators developed for pure minimization problems cannot be utilized straight forwardly the new estimator is built up by two key ingredients at first a computed error approximation exactly fulfilling the continuity equation for the error is obtained via local dirichlet problems secondly we adopt the approach of solving local equilibrated flux free problems in order to bound the remaining incompressible error in this manner guaranteed upper and lower bounds of the velocity energy norm of the error as well as goal oriented linear output functionals with respect to a reference overkill mesh are obtained in particular it should be noted that this approach requires no computation of hybrid fluxes furthermore the estimator is applicable to mixed fe formulations using continuous pressure approximations such as the mini and taylor hood class of elements in conclusion a few simple numerical examples are presented illustrating the accuracy of the error bounds c 2010 elsevier b v all rights reserved

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

analysis finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity abstract in this paper we present constitutive models for nearly incompressible transversely isotropic materials in finite hyperelasticity particularly for reinforced rubber like materials which are of essential engineering interest the theory is developed using a convected curvilinear coordinate system based on a mixed two field displacement pressure energy functional furthermore an a posteriori error estimator without multiplicative constants is derived for non linear anisotropic problems which measures the discretization error in the first piola kirchhoff stresses in the l 2 norm by solving local neumann problems with equilibrated tractions illustrative numerical examples demonstrate the anisotropic material behaviour of reinforced materials and the efficiency of using adaptive finite element methods

locking effects in the finite element approximation of elasticity problems we consider the finite element approximation of the 2d elasticity problem when the poisson ratiov is close to 0 5 it is well known that the performance of certain commonly used finite elements deteriorates asv 0 a phenomenon calledlocking we analyze this phenomenon and characterize the strength of the locking androbustness of varioush version schemes using triangular and rectangular elements we prove that thep andh p versions are free of locking with respect to the error in the energy norm a generalization of our theory to the 3d problem is also discussed

locking free adaptive mixed finite element methods in linear elasticity mixed finite element methods such as peers or the bdms methods are designed to avoid locking for nearly incompressible materials in plane elasticity in this paper we establish a robust adaptive mesh refining algorithm that is rigorously based on a reliable and efficient a posteriori error estimate numerical evidence is provided for the independence of the constants in the a posteriori error bounds and for the efficiency of the adaptive mesh refining algorithm proposed

a posteriori error estimation and mesh adaptation for finite element models in elasto plasticity a new approach to a posteriori error estimation and adaptive mesh design based on techniques from optimal control is presented for primal mixed finite element models in elasto plasticity this method uses global duality arguments for deriving weighted a posteriori error bounds for arbitrary functionals of the error representing physical quantities of interest in these estimates local residuals of the computed solution are multiplied by certain weights which are obtained by solving a linearized global dual problem numerically the resulting local error indicators are used in a feed back process for generating economical meshes this approach is developed here for the hencky and prandtl reuss models in linear elastic perfect plasticity its performance is demonstrated for a plane strain benchmark problem

discontinuous galerkin finite element approximation of nonlinear second order elliptic and hyperbolic systems we develop the convergence analysis of discontinuous galerkin finite element approximations to symmetric second order quasi linear elliptic and hyperbolic systems of partial differential equations in divergence form in a bounded spatial domain in mathbb r d subject to mixed dirichlet neumann boundary conditions optimal order asymptotic bounds are derived on the discretization error in each case without requiring the global lipschitz continuity or uniform monotonicity of the stress tensor instead only local smoothness and a garding inequality are used in the analysis

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

a fast modular semi lagrangian method for moving interfaces a fast modular numerical method for solving general moving interface problems is presented it simplifies code development by providing a black box solver which moves a given interface one step with given normal velocity the method combines an efficiently redistanced level set approach a problem independent velocity extension and a second order semi lagrangian time stepping scheme which reduces numerical error by exact evaluation of the signed distance function adaptive quadtree meshes are used to concentrate computational effort on the interface so the method moves an n element interface in o n log n work per time step efficiency is increased by taking large time steps even for parabolic curvature flows numerical results show that the method computes accurate viscosity solutions to a wide variety of difficult geometric moving interface problems involving merging anisotropy faceting nonlocality and curvature

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

stability of approximate projection methods on cell centered grids projection methods are a popular class of methods for solving the incompressible navier stokes equations if a cell centered grid is chosen in order to use high resolution methods for the advection terms performing the projection exactly is problematic an attractive alternative is to use an approximate projection in which the velocity is required to be only approximately discretely divergence free the stability of the cell centered approximate projection is highly sensitive to the method used to update the pressure and compute the pressure gradient this is demonstrated by analyzing a model problem and conducting numerical simulations of the navier stokes equations

error estimation and adaptivity for incompressible hyperelasticity summary a galerkin fem is developed for nonlinear incompressible hyper elasticity that takes account of nonlinearities in both the strain tensor and the relationship between the strain tensor and the stress tensor by using suitably defined linearised dual problems with appropriate boundary conditions a posteriori error estimates are then derived for both linear functionals of the solution and linear functionals of the stress on a boundary where dirichlet boundary conditions are applied a second higher order method for calculating a linear functional of the stress on a dirichlet boundary is also presented together with an a posteriori error estimator for this approach an implementation for a 2d model problem with known solution where the entries of the strain tensor exhibit large rapid variations demonstrates the accuracy and sharpness of the error estimators finally using a selection of model problems the a posteriori error estimate is shown to provide a basis for effective mesh adaptivity copyright 2014 john wiley sons ltd

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

on the a priori and a posteriori error analysis of a two fold saddle point approach for nonlinear incompressible elasticity in this paper we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions the approach being based only on the fact that the resulting variational formulation becomes a two fold saddle point operator equation simplifies the analysis and improves the results provided recently in a previous work thus a well known generalization of the classical babuska brezzi theory is applied to show the well posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimate in particular enriched peers subspaces are required for the solvability and stability of the associated galerkin scheme in addition we use the ritz projection operator to obtain a new reliable and quasi efficient a posteriori error estimate finally several numerical results illustrating the good performance of the associated adaptive algorithm are presented copyright 2006 john wiley sons ltd