recent trends in ale formulation and its applications in solid mechanics this paper presents a complete derivation and implementation of the arbitrary lagrangian eulerian ale formulation for the simulation of nonlinear static and dynamic problems in solid mechanics the ale equations are presented using a fully coupled implicit approach full expression for the ale virtual work equations and finite element matrices are given the matrix equations are presented in a level of details that facilitates straightforward implementation time integration relations for the dynamic equations are also derived special techniques are presented to adapt the formulation for simulating crack propagation problems several applications in the areas of metal forming metal cutting and crack propagation are solved and presented

a penalty method for the vorticity velocity formulation we present a new vorticity velocity formulation and implementation for the unsteady three dimensional navier stokes equations based on a penalty method it relies on an equivalence theorem that employs exact boundary conditions and the vorticity definition on the domain boundary this approach is particularly attractive for high order methods for which the often used influence matrix method fails to converge for t 0 the accuracy and the robustness of the new method is demonstrated in the context of several spectral element simulations of unsteady two and three dimensional internal and external flows in particular the flow past a finite span cylinder attached to end plates is studied in some detail in order to evaluate the effects of the aspect ratio on the formation length

velocity vorticity helicity formulation and a solver for the navier stokes equations for the three dimensional incompressible navier stokes equations we present a formulation featuring velocity vorticity and helical density as independent variables we find the helical density can be observed as a lagrange multiplier corresponding to the divergence free constraint on the vorticity variable similar to the pressure in the case of the incompressibility condition for velocity as one possible practical application of this new formulation we consider a time splitting numerical scheme based on an alternating procedure between vorticity helical density and velocity bernoulli pressure systems of equations results of numerical experiments include a comparison with some well known schemes based on pressure velocity formulation and illustrate the competitiveness on the new scheme as well as the soundness of the new formulation

a constrained two dimensional triangulation and the solution of closest node problems in the presence of barriers a delaunay triangulation of a set of nodes is a collection of triangles whose vertices are at the nodes and whose union fills the convex hull of the set of nodes it also has several geometrical properties making it useful for solving closest point problems the generalization presented here allows the triangulation to cover nonconvex regions including those with holes although a variety of such generalizations are possible the one presented here is shown to retain important closest point characteristics thus it is useful for determining shortest paths within planar regions with polygonal boundaries

long time dynamics for nonlinear porous thermoelasticity with second sound and delay this paper is concerned with the long time behavior of a damped porous thermoelastic problem it has been studied by many authors and most of the known results are concerned with decay issues under different boundary conditions and damping terms there are just a few references on the long time dynamics of such systems the purpose of the present work is to complement and extend some early studies on porous thermoelastic systems by establishing new results on the existence of attractors and some of their properties motivated by this scenario we consider a one dimensional porous thermoelastic system with linear frictional damping nonlinear source terms and a time varying delay term in the internal feedback where the heat flux depends on cattaneo law under some suitable assumptions on the weights of feedback we establish the global well posedness of the system by using the c0 semigroup theory of linear operators then we show the existence of a global attractor for finite energy solutions and we prove its smoothness and finite fractal dimension furthermore the existence of a generalized fractal exponential attractor is also derived this paper is concerned with the long time behavior of a damped porous thermoelastic problem it has been studied by many authors and most of the known results are concerned with decay issues under different boundary conditions and damping terms there are just a few references on the long time dynamics of such systems the purpose of the present work is to complement and extend some early studies on porous thermoelastic systems by establishing new results on the existence of attractors and some of their properties motivated by this scenario we consider a one dimensional porous thermoelastic system with linear frictional damping nonlinear source terms and a time varying delay term in the internal feedback where the heat flux depends on cattaneo law under some suitable assumptions on the weights of feedback we establish the global well posedness of the system by using the c0 semigroup theory of linear operators then we show the existence of a global attractor for finite energy solutions and we pro

moving least square reproducing kernel method part ii fourier analysis in part i of this work the moving least square reproducing kernel mlsrk method is formulated and implemented based on its generic construction an m consistency structure is discovered and the convergence theorems are established in this part of the work a systematic fourier analysis is employed to evaluate and further establish the method the preliminary fourier analysis reveals that the mlsrk method is stable for sufficiently dense non degenerated particle distribution in the sense that the kernel function family satisfies the riesz bound one of the novelties of the current approach is to treat the mlsrk method as a variant of the standard finite element method and depart from there to make a connection with the multiresolution approximation in the spirits of multiresolution analysis we propose the following mlsrk transformation fm k hu i 1npu k ik h x xi x w1 the highlight of this paper is to embrace the mlsrk formulation with the notion of the controlled fp approximation based on its characterization the strang fix condition for example a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the mlsrk algorithm the main effort here is to obtain a constant correction function in the interior region of a general domain i e c h 1 this can create a leap in the approximation order of the mlsrk algorithm significantly if a highly smooth window function is embedded within the kernel one consequence of this development is the synchronized convergence phenomenon a unique convergence mechanism for the mlsrk method i e by properly tuning the dilation parameter the convergence rate of higher order error norms will approach the same order convergence rate of the l2 error norm they are synchronized

unconditional long time stability of a velocity vorticity method for the 2d navier stokes equations we prove unconditional long time stability for a particular velocity vorticity discretization of the 2d navier stokes equations the scheme begins with a formulation that uses the lamb vector to couple the usual velocity pressure system to the vorticity dynamics equation and then discretizes with the finite element method in space and implicit explicit bdf2 in time with the vorticity equation decoupling at each time step we prove the method s vorticity and velocity are both long time stable in the l 2 l2 and h 1 h1 norms without any timestep restriction moreover our analysis avoids the use of gronwall type estimates which leads us to stability bounds with only polynomial instead of exponential dependence on the reynolds number numerical experiments are given that demonstrate the effectiveness of the method

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

enforcement of essential boundary conditions in meshless approximations using finite elements abstract a technique for easily treating essential boundary conditions for approximations which are not interpolants such as moving least square approximations in the element free galerkin method is presented the technique employs a string of elements along the essential boundaries and combines the finite element shape functions with the approximation in the resulting approximation essential boundary conditions can be treated exactly as in finite elements it is shown that the resulting approximation can exactly reproduce linear polynomials so that it satisfies the patch test numerical studies show that the method retains the high rate of convergence associated with moving least square approximations

on the accuracy of the rotation form in simulations of the navier stokes equations the rotation form of the navier stokes equations nonlinearity is commonly used in high reynolds number flow simulations it was pointed out by a few authors and not widely known apparently that it can also lead to a less accurate approximate solution than the usual email protected u form we give a different explanation of this effect related to i resolution of the bernoulli pressure and ii the scaling of the coupling between velocity and pressure error with respect to the reynolds number we show analytically that i the difference between the two nonlinearities is governed by the difference in the resolution of the bernoulli and kinematic pressures and ii a simple linear grad div stabilization ameliorates much of the bad scaling of the velocity error with respect to re the rotation form does have superior conservation properties to the alternatives and it appears to be amenable to more efficient preconditioners thus the rotational form with grad div stabilization is a promising method we also give experiments that show bad velocity approximation is tied to poor pressure resolution in either form

constrained boundary recovery for three dimensional delaunay triangulations summary a new constrained boundary recovery method for three dimensional delaunay triangulations is presented it successfully resolves the difficulties related to the minimal addition of steiner points and their good placement applications to full mesh generation are discussed and numerical examples are provided to illustrate the effectiveness of guaranteed recovery procedure copyright 2004 john wiley sons ltd

finite element solution of the 3d compressible navier stokes equations by a velocity vorticity method this paper presents an alternative formulation to the primitive variable form of the navier stokes equations the approach is based on the use of the velocity and vorticity variables as a logical extension of the stream function vorticity method which is quite effective for two dimensional flows second order equations are obtained for the variables and discretization is through a weak galerkin finite element method followed by newton linearization the velocity and vorticity components are solved simultaneously by a direct solver the scheme is demonstrated for two dimensional and three dimensional incompressible and subsonic internal enclosed flow problems

a compact difference scheme for the navier stokes equations in vorticity velocity formulation this paper presents a new numerical method for solving the incompressible unsteady navier stokes equations in vorticity velocity formulation the method is applicable to spatial simulations of transitional and turbulent boundary layer flows it is based on a compact difference discretization of the streamwise and wall normal derivatives in cartesian coordinates a fourier collocation approach is used for the spanwise derivatives important new features of the numerical method are the use of nonequidistant differences in the wall normal direction the use of split compact differences in the streamwise direction a new fast iteration for a semi implicit time integration of the wall normal diffusion terms and an improvement of the buffer domain technique to prevent reflections of waves at the outflow boundary results of test calculations are presented to verify the improvements obtained by the use of these new techniques

on the numerical implementation of variational arbitrary lagrangian eulerian vale formulations this paper is concerned with the implementation of variational arbitrary lagrangian eulerian formulations also known as variational r adaption methods these methods seek to minimize the energy function with respect to the finite element mesh over the reference configuration of the body we propose a solution strategy based on a viscous regularization of the configurational forces this procedure eliminates the ill posedness of the problem without changing its solutions i e the minimizers of the regularized problems are also minimizers of the original functional we also develop strategies for optimizing the triangulation or mesh connectivity and for allowing nodes to migrate in and out of the boundary of the domain selected numerical examples demonstrate the robustness of the solution procedures and their ability to produce highly anisotropic mesh refinement in regions of high energy density

vorticity boundary conditions and boundary vorticity generation for two dimensional viscous incompressible flows abstract in this paper we present boundary conditions appropriate for the vorticity formulation of the two dimensional incompressible viscous navier stokes equations these boundary conditions are incorporated into a finite difference scheme and the resulting method is of the vorticity creation type i e vorticity is generated at the boundary to ensure that the tangential velocity boundary condition is satisfied the results of computations with this finite difference method are presented for flow past a circular cylinder a difference scheme and computational results for a model problem the prandtl boundary layer equations describing flow over a semi infinite flat plate are also presented

on error analysis for the 3d navier stokes equations in velocity vorticity helicity form we present a rigorous numerical analysis and computational tests for the galerkin finite element discretization of the velocity vorticity helicity formulation of the equilibrium navier stokes equations nses this formulation recently derived by the authors is the first nse formulation that directly solves for helicity and the first velocity vorticity formulation to naturally enforce incompressibility of the vorticity and preliminary computations confirm its potential we present a numerical scheme prove stability existence of solutions uniqueness under a small data condition and convergence and provide numerical experiments to confirm the theory and illustrate the effectiveness of the scheme on a benchmark problem

reproducing kernel element method part i theoretical formulation in this paper and its sequels we introduce and analyze a new class of methods collectively called the reproducing kernel element method rkem the central idea in the development of the new method is to combine the strengths of both finite element methods fem and meshfree methods two distinguished features of rkem are the arbitrarily high order smoothness and the interpolation property of the shape functions these properties are desirable especially in solving galerkin weak forms of higher order partial differential equations and in treating dirichlet boundary conditions so unlike the fem there is no need for special treatment with the rkem in solving high order equations compared to meshfree methods dirichlet boundary conditions do not present any difficulty in using the rkem a rigorous error analysis and convergence study of the method are presented the performance of the method is illustrated and assessed through some numerical examples

on the advantages of the vorticity velocity formulations of the equations of fluid dynamics the mathematical properties of the pressure velocity and vorticity velocity formulations of the equations of viscous flow are compared it is shown that a vorticity velocity formulation exists which has the interesting property that noninertial effects only enter into the problem through the implementation of initial and boundary conditions this valuable characteristic along with other advantages of the vorticity velocity approach are discussed in detail

l 2 estimates for the evolving surface finite element method in this paper we consider the evolving surface nite element method for the advection and di usion of a conserved scalar quantity on a moving surface in an earlier paper using a suitable variational formulation in time dependent sobolev space we proposed and analysed a nite element method using surface nite elements on evolving triangulated surfaces an optimal order h error bound was proved for linear nite elements in this work we prove the optimal error bound in l t uniformly in time

inclusion of no slip boundary conditions in the meevc scheme abstract this work presents three methods for enforcing tangential velocity boundary conditions for the meevc scheme which was shown to be mass enstrophy energy and vorticity conserving scheme in the case of inviscid flow 1 while the normal velocity component can be strongly imposed in a div conforming formulation for the velocity field inclusion of the tangential velocity needs to be set through an appropriate choice of vorticity boundary conditions three methods to impose the tangential velocity boundary condition will be discussed the kinematic dirichlet formulation the kinematic neumann formulation and the dynamic neumann formulation the conservation properties of each of the resulting schemes are analyzed and numerical results are shown for the taylor green vortex and for the dipole collision test cases these confirm that kinematic neumann vorticity boundary conditions perform best

longer time accuracy for incompressible navier stokes simulations with the emac formulation abstract in this paper we consider the recently introduced emac formulation for the incompressible navier stokes ns equations which is the only known ns formulation that conserves energy momentum and angular momentum when the divergence constraint is only weakly enforced since its introduction the emac formulation has been successfully used for a wide variety of fluid dynamics problems we prove that discretizations using the emac formulation are potentially better than those built on the commonly used skew symmetric formulation by deriving a better longer time error estimate for emac while the classical results for schemes using the skew symmetric formulation have gronwall constants dependent on exp c r e t with r e the reynolds number it turns out that the emac error estimate is free from this explicit exponential dependence on the reynolds number additionally it is demonstrated how emac admits smaller lower bounds on its velocity error since incorrect treatment of linear momentum angular momentum and energy induces lower bounds for l 2 velocity error and emac treats these quantities more accurately results of numerical tests for channel flow past a cylinder and 2d kelvin helmholtz instability are also given both of which show that the advantages of emac over the skew symmetric formulation increase as the reynolds number gets larger and for longer simulation times

dealing with extremely large deformation by nearest nodes fem with algorithm for updating element connectivity in the recently developed nearest nodes finite element method nn fem elements are mainly used for numerical integration while shape functions are constructed in a similar way as in meshless methods based on this strategy nn fem inherits major merits from both the classical finite element method and meshless methods one of them is that nn fem is nearly not affected by element distortion so nn fem is more efficient than the classical fem on dealing with large deformation problems nevertheless nn fem still has a requirement on finite element meshes that is elements in a mesh are required not to overlap or penetrate to each other to avoid difficulty in numerical integration to eliminate overlapped elements nn fem is supplemented with an algorithm for updating element connectivity with this supplement nn fem is able to deal with extremely large deformation in updating element connectivity element nodes are kept not changed and all information associated with nodes are not touched therefore there is no need to transfer solution data and error introduced by solution transfer is avoided

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

boundary conditions for viscous vortex methods this paper presents a neumann type vorticity boundary condition for the vorticity formulation of the navier stokes equations the vorticity creation process at the boundary due to the no slip condition is expressed in terms of a vorticity flux the scheme is incorporated then into a lagrangian vortex blob method that uses a particle strength exchange algorithm for viscous diffusion the no slip condition is not enforced by the generation of new vortices at the boundary but instead by modifying the strength of the vortices in the vicinity of the boundary

a new implementation of the element free galerkin method abstract element free galerkin methods efg are methods for solving partial differential equations with moving least squares interpolants efg methods require only nodal data no element connectivity is needed in a previous implementation of the efg method lagrange multipliers were used to enforce the essential boundary condition however the use of lagrange multipliers increases the cost of solving the linear algebraic equations a new implementation is developed based on a modified variational principle in which the lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded in addition weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes the latter makes it very advantageous for modelling progressive cracking

global well posedness of the velocity vorticity voigt model of the 3d navier stokes equations abstract the velocity vorticity formulation of the 3d navier stokes equations was recently found to give excellent numerical results for flows with strong rotation in this work we propose a new regularization of the 3d navier stokes equations which we call the 3d velocity vorticity voigt vvv model with a voigt regularization term added to momentum equation in velocity vorticity form but with no regularizing term in the vorticity equation we prove global well posedness and regularity of this model under periodic boundary conditions we prove convergence of the model s velocity and vorticity to their counterparts in the 3d navier stokes equations as the voigt modeling parameter tends to zero we prove that the curl of the model s velocity converges to the model vorticity which is solved for directly as the voigt modeling parameter tends to zero finally we provide a criterion for finite time blow up of the 3d navier stokes equations based on this inviscid regularization

arbitrary lagrangian eulerian finite element analysis of large deformation in contacting bodies the primary aim of this research is to implement the arbitrary lagrangian eulerian finite element description to model large deformation of interacting bodies a pure lagrangian description is often incapable of representing their contact with sufficient precision in this work the arbitrary lagrangian eulerian ale description has been extended to the surface nodes of interacting bodies to overcome these limitations a penalty formulation has been implemented to account for the impenetrability constraint the materials have been characterized as elastic viscoplastic implicit integration schemes have been used at the global and constitutive levels for improved stability

residual based vms subgrid modeling for vortex flows this paper presents a residual based subgrid modeling approach for large eddy simulations les based on the variational multiscale method as a cure for the problem of preservation of vortices in numerical flow simulation this approach combines a splitting of the non linear term in the navier stokes equations into strain and vorticity with a residual based modeling of the subgrid problems the benefit is that certain driving phenomena normally not present in subgrid modeling e g vortex stretching can be seen in the equations here we focus on two of the subgrid terms arising from the subgrid scale problem the effect of the two terms are illustrated in an les of a three dimensional flow around a wing where the main feature is the formation and preservation of a tip vortex an important phenomenon in many aerodynamic and hydrodynamical applications we see that the addition of the new subgrid terms correctly counteracts the dissipative effect arising from numerics and turbulence modeling on the vortex and thus strongly improves prediction of the tip vortex c 2009 elsevier b v all rights reserved

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

arbitrary lagrangian eulerian formulation for element free galerkin method abstract arbitrary lagrangian eulerian ale formulation of the element free galerkin efg method is presented efg is a meshless method for solving partial differentia equations in which the trial and test functions employed in the discretization process result from moving least square interpolants the most significant advantage of the method is that it requires only nodes and a description of internal and external boundaries and interfaces such as cracks of the model no element connectivity is needed however as for any discretization method acceptable solutions can only be obtained for a sufficiently refined discretization in dynamic fracture problems where the crack path can be arbitrary and is thus a priori unknown this necessitates a refined discretization in large parts of the computational domain which can lead to prohibitive computation costs ale formulation allows to continuously relocate nodes on the computational domain by combining efg with ale it is thus possible in a crack propagation problem to refine locally the spatial discretization in the neighborhood of a propagating crack tip results are presented for a wave propagation problem as well as for 2 d dynamic crack propagation problems

natural vorticity boundary conditions on solid walls abstract we derive a new kind of boundary conditions for the vorticity equation with solid wall boundaries for fluid flow problems the formulation uses a dirichlet condition for the normal component of vorticity and neumann type conditions for the tangential components in a galerkin integral formulation the tangential condition is natural i e it is enforced by a right hand side functional and does not impose a boundary constraint on trial and test spaces the functional involves the pressure variable and we discuss several velocity vorticity formulations where the proposed condition is appropriate several numerical experiments are given that illustrate the validity of the new approach

effective vorticity velocity formulations for three dimensional incompressible viscous flows the vorticity velocity formulation for incompressible viscous flows is studied the main concern is effective numerical solution of the kinematic cauchy riemann equations for velocity especially when the divergence free condition of vorticity is violated due to numerical error the mathematical formulation of a differential and an integral approaches are revisited a novel projection of vorticity onto the divergence free space and its application to the two approaches are studied as an illustration of the differential approach the 3d lid driven cavity problem is solved the projection scheme is justified through numerical tests by comparing it to a new fully divergence free scheme numerical tests indicate that keeping vorticity divergence free is important for obtaining correct and accurate solutions the theory and methods apply to other divergence free fields of physical interest as well such as that in electro magnetic dynamics

vorticity boundary condition and related issues for finite difference schemes this paper discusses three basic issues related to the design of finite difference schemes forunsteadyviscous incompressible flows using vorticity formulations the boundary condition for vorticity an efficient time stepping procedure and the relation between these schemes and the ones based on velocity pressure formulation we show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier we also show that if we couple a standard centered difference scheme with third or fourth order explicit runge kutta methods the resulting schemes havenocell reynolds number constraints for high reynolds number flows these schemes are stable under the cfl condition given by the convective terms finally we show that the classical mac scheme is the same as thom s formula coupled with second order centered differences in the interior in the sense that one can define discrete vorticity in a natural way for the mac scheme and get the same values as the ones computed from thom s formula we use this to derive an efficient fourth order runge kutta time discretization for the mac scheme from the one for thom s formula we present numerical results for driven cavity flow at high reynolds number 105

isogeometric divergence conforming b splines for the unsteady navier stokes equations divergence conforming b splines are developed for application to the incompressible navier stokes equations on geometrically mapped domains these enable smooth pointwise divergence free solutions and thus satisfy mass conservation in the strongest possible sense semi discrete methods based on divergence conforming b splines are shown to conserve linear and angular momentum and satisfy balance laws for energy vorticity enstrophy and helicity these are geometric structure preserving quantities and numerical simulations that are sensitive to them are shown to be qualitatively correct and quantitatively accurate the methods developed are anticipated to open new doors to the practical calculation of complex flows and to studies of their physical behavior