a new higher order finite volume method based on moving least squares for the resolution of the incompressible navier stokes equations on unstructured grids in this work a new higher order 2 accurate finite volume method for the resolution of the incompressible navier stokes equations on unstructured grids is presented the formulation is based on the use of moving least squares mls approximants third and fourth order accurate discretizations of the convective and viscous fluxes are obtained on collocated meshes in addition mls is employed to design a new momentum interpolation method that allows interpolations better than linear on any kind of mesh the accuracy and performance of the proposed method is demonstrated by solving different steady and unsteady benchmark problems on unstructured grids

finite surface discretization for incompressible navier stokes equations and coupled conservation laws abstract we present a new finite surface discretization fsd aiming at the incompressible navier stokes equations nse and other coupled conservation laws this discretization defines the velocities as surfaced averaged values living on the faces of the pressure volumes in which the mass is set to be conserved consequently the calculation of the mass balance on these control volumes is exact which allows more accurate information to be kept in the velocity field and produces a very accurate prediction of the pressure in the next time step the proposed discretization reduces the stencil size of the poisson equation in the projection method compared to the finite volume and finite difference discretizations due to highly accurate mass conservation the compact sixth order approximation of fsd can be used with an explicit fourth order pressure treatment this property greatly reduces the cost and complexity of the implementation we present the discrete evolution equation of the surface averaged velocities together with the enforcement of mass conservation and the solution procedure for the pressure the approximation of the nse under this new discretization uses a combination of finite difference and finite volume methods the proposed method is validated using standard laminar test cases we identify the conditions under which a fourth order pressure treatment can support the sixth order and eighth order approximations of the convection term using fourier analysis the performance of the method is evaluated on turbulent channel flows up to friction reynolds number of 950 the quantitative relationships between the accuracy of the solution and grid size are identified we present two performance indices for comparison with other methods at the error level of 1 per mille the proposed method is 28 times faster than the classic second order scheme

accuracy analysis of explicit runge kutta methods applied to the incompressible navier stokes equations this paper investigates the temporal accuracy of the velocity and pressure when explicit runge kutta methods are applied to the incompressible navier stokes equations it is shown that at least up to and including fourth order the velocity attains the classical order of accuracy without further constraints however in case of a time dependent gradient operator which can appear in case of time varying meshes additional order conditions need to be satisfied to ensure the correct order of accuracy furthermore the pressure is only first order accurate unless additional order conditions are satisfied two new methods that lead to a second order accurate pressure are proposed which are applicable to a certain class of three and four stage methods a special case appears when the boundary conditions for the continuity equation are independent of time since in that case the pressure can be computed to the same accuracy as the velocity field without additional cost relevant computations of decaying vortices and of an actuator disk in a time dependent inflow support the analysis and the proposed methods

regular article on a class of pade finite volume methods a class of pade finite volume methods providing an improved spectral resolution is presented and compared with well known methods the formulation is based on the sliding averages of the variables and allows the computation of derivatives of all orders using the fourier analysis these methods are examined with respect to i order of accuracy ii spectral resolution iii boundary conditions and iv stability

energy conserving runge kutta methods for the incompressible navier stokes equations energy conserving methods have recently gained popularity for the spatial discretization of the incompressible navier stokes equations in this paper implicit runge kutta methods are investigated which keep this property when integrating in time firstly a number of energy conserving runge kutta methods based on gauss radau and lobatto quadrature are constructed these methods are suitable for convection dominated problems such as turbulent flows because they do not introduce artificial diffusion and are stable for any time step secondly to obtain robust time integration methods that work also for stiff problems the energy conserving methods are extended to a new class of additive runge kutta methods which combine energy conservation with l stability in this class the radau iia b method has the best properties results for a number of test cases on two stage methods indicate that for pure convection problems the additive radau iia b method is competitive with the gauss methods however for stiff problems such as convection dominated flows with thin boundary layers both the higher order gauss and radau iia b method suffer from order reduction overall the gauss methods are the preferred method for energy conserving time integration of the incompressible navier stokes equations

a semi smooth newton method for elasto plastic contact problems in this paper we reformulate the frictional contact problem for elasto plastic bodies as a set of unconstrained non smooth equations the equations are semi smooth so that pang s newton method for b differentiable equations can be applied an algorithm based on this method is described in detail an example demonstrating the efficiency of the algorithm is presented

optimization of time integration schemes coupled to spatial discretization for use in caa applications in this paper errors arising from spatial and temporal discretization are discussed a new formulation to optimize time integration schemes is proposed the new optimization also takes the errors coming from the spatial discretization into account which leads to a minimization of the total errors it is applicable to central and upwind type spatial discretizations a six stage runge kutta scheme is optimized for central and upwind spatial discretizations its efficiency is demonstrated on a one dimensional convection equation and on a linearized euler problem the results indicate an important improvement

an optimal adaptive finite element method for elastoplasticity an adaptive finite element algorithm for problems in elastoplasticity with hardening is proven to be of optimal convergence with respect to the notion of approximation classes the results rely on the equivalence of the errors of the stresses and energies resulting from jensen s inequality numerical experiments study the influence of the hardening and bulk parameters to the convergence behavior of the afem algorithm this is the first optimal adaptive fem for a variational inequality

on the implementation of inelastic constitutive equations with special reference to large deformation problems abstract methods for finite element analysis of inelastic problems are reviewed problems with currently popular methods based on rate principles of virtual work are pointed out a formulation is presented based on the exact incremental virtual work equation the procedure for integration of the constitutive rate equations to obtain incremental constitutive equations is based on the assumption of a constant strain rate or straight strain path during the increment full newton raphson iteration is used to solve the resulting set of nonlinear equations the treatment of the constitutive equations in this newton raphson scheme leads to a linearized form which is substantially different from the classical rate equations the procedure is implemented in a finite element program and its effectiveness is shown in a number of examples

continuation newton methods severely nonlinear problems can only be solved by some homotopy continuation method an example of a homotopy method is the continuous newton method which however must be discretized which leads to the damped step version of newton s method the standard newton iteration method for solving systems of nonlinear equations f u 0 must be modified in order to get global convergence i e convergence from any initial point the control of steplengths in the damped step newton method can lead to many small steps and slow convergence furthermore the applicability of the method is restricted in as much as it assumes a nonsingular and everywhere differentiable mapping f classical continuation methods are surveyed then a new method in the form of a coupled newton and load increment method is presented and shown to have a global convergence already from the start and second order of accuracy with respect to the load increment step and with less restrictive regularity assumptions than for the standard newton method the method is applied for an elastoplastic problem with hardening

the existence of patterns on surfaces of revolution without boundary abstract we give a sufficient condition for the existence of patterns on surfaces of revolution of r 3 without boundary such a condition involves the gauss curvature of the surface and the geodesic curvature of parallels an analogous result for surfaces of revolution with boundary is established in bandle et al 2012 4

a nonsmooth newton method for elastoplastic problems in this work we reformulate the incremental small strain j2 plasticity problem with linear kinematic and nonlinear isotropic hardening as a set of unconstrained nonsmooth equations the reformul

a high order compact difference algorithm for half staggered grids for laminar and turbulent incompressible flows abstract the paper presents a novel efficient and accurate algorithm for laminar and turbulent flow simulations the spatial discretisation is performed with help of the compact difference schemes up to 10th order for collocated and half staggered grid arrangements the time integration is performed by a predictor corrector approach combined with the projection method for pressure velocity coupling at this stage a low order discretisation is introduced which considerably decreases the computational costs it is demonstrated that such approach does not deteriorate the solution accuracy significantly following boersma b j 13 the interpolation formulas developed for staggered uniform meshes are used also in the computations with a non uniform strongly varying nodes distribution in the proposed formulation of the projection method such interpolation is performed twice it is shown that it acts implicitly as a high order low pass filter and therefore the resulting algorithm is very robust its accuracy is first demonstrated based on simple 2d and 3d problems an inviscid vortex advection a decay of taylor green vortices a modified lid driven cavity flow and a dipole wall interaction in periodic flow problems the first two cases the solution accuracy exhibits the 10th order behaviour in the latter cases the 3rd and the 4th order is obtained robustness of the proposed method in the computations of turbulent flows is demonstrated for two classical cases a periodic channel with re 395 and re 590 and a round jet with re 21 000 the solutions are obtained without any turbulence model and also without any explicit techniques aiming to stabilise the solution the results are in a very good agreement with literature dns and les data both the mean and r m s values are predicted correctly

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

algorithms for gradient damage models based on a semi smooth newton method in this paper two different algorithms are applied to a model of brittle damage including the gradient of the damage variable both algorithms are based on a modified newton method in the first algorithm the newton method is applied directly to the whole system of equations while in the second algorithm the equilibrium equations and the damage evolution problem are solved uncoupled from each other in a gauss seidel scheme the algorithms are applied to two different formulations of the problem in the first formulation the thermodynamic force work conjugate to the damage variable is kept as an auxiliary variable while in the second formulation this variable is eliminated it turns out that the formulation using the extra variable is more robust in the sense that the algorithms converge even for very large load increments on the other hand the formulation where the extra variable is eliminated is more efficient for small time steps since a smaller system of equations is solved for each increment furthermore it is found that the direct approach is more efficient than the decoupled approach for this problem

high order immersed boundary method for incompressible flows in this work a high order numerical approach based on the immersed boundary method to simulate viscous incompressible flows involving arbitrary boundaries is described the idea is to combine compact finite difference schemes and a high order direct forcing approach of the sharp interface immersed boundary method in this approach a particular challenge is to solve the pressure poisson equation using the compact stencil this issue is addressed by introducing an efficient approach based on high order corrections to the right hand side of the poisson equation that is the source term while retaining the second order short stencil for the laplace operator the two dimensional implementation is described for which third order global convergence is achieved when an immersed boundary is present

a semi smooth newton method for orthotropic plasticity and frictional contact at finite strains abstract a new approach for the unified treatment of frictional contact and orthotropic plasticity at finite strains using semi smooth newton methods is presented the contact discretization is based on the well known mortar finite element method using dual lagrange multipliers to facilitate the handling of the additional lagrange multiplier degrees of freedom exploiting the similarity of the typical inequality constraints of plasticity and friction all involved discrete inequalities are reformulated as nonlinear non smooth equations using complementarity functions the resulting set of discrete semi smooth equations can be solved efficiently by a variant of newton s method where all additionally introduced variables are condensed from the global system so that a linear system only consisting of the displacement degrees of freedom has to be solved in each iteration step in contrast to classical radial return mapping methods for computational plasticity the plastic constraints are not required to hold at every iterate in the nonlinear solution procedure but only at convergence this relaxation in the pre asymptotic behavior results in an increased flexibility regarding algorithm design and a potentially higher robustness compared to radial return mapping algorithms the presented elasto plasticity algorithm includes arbitrary isotropic hyperelasticity an anisotropic hill type yield function with isotropic and kinematic hardening plastic spin and appropriate finite element technology for nearly incompressible materials therefore it is well suited for the modeling of sheet metal forming and similar processes several numerical examples underline the robustness of the proposed plasticity algorithm and the efficient treatment of elasto plastic contact problems

high accuracy solutions of incompressible navier stokes equations abstract in recent years we have developed high accuracy finite difference approximations for partial differential equations of elliptic type with particular emphasis on the convection diffusion equation these approximations are of compact type have a local truncation error of fourth order and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations in this paper we extend these high accuracy approximations to the solution of navier stokes equations solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors it is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows

on performance of methods with third and fifth order compact upwind differencing the difference schemes for fluid dynamics type of equations based on third and fifth order compact upwind differencing cud are considered to validate their properties following from a linear analysis calculations were carried out using the inviscid and viscous burgers equation as well as the compressible navier stokes equation written in the conservative form for curvilinear coordinates in the latter case transonic cascade flow was chosen as a representative example the performance of the cud methods was estimated by investigating mesh convergence of the solutions and comparing with the results of second order schemes it is demonstrated that the oscillation free steep gradients solutions obtained without using smoothing techniques can provide considerable increase of accuracy even when exploiting coarse meshes

the stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

optimized compact difference based finite volume schemes for linear wave phenomena high order compact difference based finite volume schemes are developed analyzed and implemented for linear wave propagation phenomena with particular emphasis on computational electromagnetics in the time domain the formulation combines the primitive function approach with five point spatially sixth and fourth order methods optimization in the semi discrete case is achieved by minimizing dispersion and isotropy error the fully discrete scheme is examined by adopting the classical fourth order runge kutta technique stability bounds are established and the coefficients for the spatial discretization are readjusted for optimum performance the scheme is then formally extended to multiple dimensions consistent boundary conditions are presented for the reconstruction operator as well as flux specification several calculations in one and three dimensions confirm the properties of the method

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

a family of high order finite difference schemes with good spectral resolution this paper presents a family of finite difference schemes for the first and second derivatives of smooth functions the schemes are hermitian and symmetric and may be considered a more general version of the standard compact pade schemes discussed by lele they are different from the standard pade schemes in that the first and second derivatives are evaluated simultaneously for the same stencil width the proposed schemes are two orders higher in accuracy and have significantly better spectral representation eigenvalue analysis and numerical solutions of the one dimensional advection equation are used to demonstrate the numerical stability of the schemes the computational cost of computing both derivatives is assessed and shown to be essentially the same as the standard pade schemes the proposed schemes appear to be attractive alternatives to the standard pade schemes for computations of the navier stokes equations

a tangent secant approach to rate independent elastoplasticity formulations and computational issues a general and robust solution procedure for nonlinear finite element equations in small strain elastoplastic structural problems is presented its peculiar feature lies in the choice of the most suitable constitutive operator to be adopted at each iteration of a generic load step in order to ensure the utmost stability and convergence rate namely the consistent tangent operator is replaced by a secant one or vice versa whether the adopted norm of the residual does not or does conveniently decrease at the current iteration the secant operator is defined as to recover the finite step increment of the plastically admissible stress from the total not iterative strain increment the original formulation of the solution procedure consisting of alternate tangent and secant iterations is then extended to achieve an effective coupling with line searches the excellent performances of the two procedures are illustrated by numerical examples carried out for typical benchmark problems in plane strain and three dimensional cases

family of runge kutta based algorithms for unsteady incompressible flows a family of algorithms based on implicit runge kutta time integration suitable for unsteady incompressible flow is developed spatially discretized momentum equations are integrated in time using i

a computational stream function method for two dimensional incompressible viscous flows this work concerns the development of a numerical method based on the stream function formulation of the navier stokes equations to simulate two dimensional plane or axisymmetric viscous flows the main features of the proposed method are the use of the high order finite difference compact method for the discretization of the stream function equation the implicit pseudo transient newton krylov multigrid matrix free method for the stationary stream function equation and the fourth order runge kutta method for the integration of non stationary flows copyright 2005 john wiley sons ltd

compact finite volume schemes on boundary fitted grids the paper focuses on the development of a framework for high order compact finite volume discretization of the three dimensional scalar advection diffusion equation in order to deal with irregular domains a coordinate transformation is applied between a curvilinear non orthogonal grid in the physical space and the computational space advective fluxes are computed by the fifth order upwind scheme introduced by pirozzoli s pirozzoli conservative hybrid compact weno schemes for shock turbulence interaction j comp phys 178 2002 81 while the coupled derivative scheme m h kobayashi on a class of pade finite volume methods j comp phys 156 1999 137 is used for the discretization of the diffusive fluxes numerical tests include unsteady diffusion over a distorted grid linear free surface gravity waves in a irregular domain and the advection of a scalar field the proposed methodology attains high order formal accuracy and shows very favorable resolution characteristics for the simulation of problems with a wide range of length scales

compact fourth order finite volume method for numerical solutions of navier stokes equations on staggered grids the development of a compact fourth order finite volume method for solutions of the navier stokes equations on staggered grids is presented a special attention is given to the conservation laws on momentum control volumes a higher order divergence free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes three forms of the nonlinear correction for staggered grids are proposed and studied the accuracy of each approximation is assessed comparatively in fourier space the importance of higher order approximations of pressure is discussed and numerically demonstrated fourth order accuracy of the complete scheme is illustrated by the doubly periodic shear layer and the instability of plane channel flow the efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations the proposed scheme is highly accurate and efficient at the same level of accuracy the fourth order scheme can be ten times faster than the second order counterpart this gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost

equilibration techniques for solving contact problems with coulomb friction abstract in this paper we consider residual and equilibrated error indicators for contact problems with coulomb friction the contact problem is handled within the abstract framework of saddle point problems more precisely the non penetration constraint and the friction law is realized as a variationally consistent weak formulation in terms of a localized dual lagrange multiplier space thus from the displacement we can easily compute in a local post process the lagrange multiplier which acts as a neumann condition on the possible contact zone having computed the discrete lagrange multiplier we can apply standard error estimators by replacing the unknown neumann data by its approximation as it is shown in 1 this results in an error estimator for a one sided contact problem without friction here we consider more general situations and discuss two additional contact terms which measure the non conformity of the discrete lagrange multiplier numerical results in two and three dimensions illustrate the flexibility of the approach and show the influence of the material parameters on the adaptive refinement process

convergence analysis of some algorithms for solving nonsmooth equations this paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally lipschitzian functions for the directional derivative based and the generalized jacobian based newton methods both the iterates and the corresponding function values are locally superlinearly convergent globally a limiting point of the iterate sequence generated by the damped directional derivative based newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one provided that the system is strongly bd regular and semismooth at this point in this case the convergence is superlinear a general attraction theorem is presented which can be applied to two algorithms proposed by han pang and rangaraj a hybrid method which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent is also presented

an implicit compact scheme solver with application to chemically reacting flows a novel stable implicit compact scheme solver that is higher order in space suitable for modeling steady state and time dependent phenomena on nonuniform grids for one dimensional configurations is presented several properties of compact scheme discretizations are introduced to develop efficient algorithms for jacobian matrix generation and jacobian vector multiplication using a new component form for jacobian operations composite nonuniform grids are introduced that enable the implicit compact scheme solver to achieve sixth order accuracy a robust newton s method is employed with explicit generation of jacobian matrices superior resolution characteristics of the implicit compact scheme solver are demonstrated with several steady state and time dependent problems for the burgers equation the example of the solution of stiff flame problem is given an analysis of spectral properties of jacobian matrices is presented which shows that the condition number and the eigenvalue distributions behave similarly to those found in jacobians associated with low order discretizations two sparsification strategies are developed for the systematic approximation of a dense jacobian aimed at the practical implementation of linear system preconditioning through partial jacobians

a finite volume formulation of compact central schemes on arbitrary structured grids in the present paper a formulation allowing the use of compact schemes in the finite volume context on arbitrary meshes is presented the proposed formulation is based on the use of an implicit formula to evaluate the fluxes on the cell faces a series of numerical experiments for a 2d model convection equation a flat plate a subsonic vortical problem as well as the les simulation of channel flow has been carried out the results indicate an important improvement in obtained accuracy compared to a standard central finite volume formulation

a fourth order accurate finite volume compact method for the incompressible navier stokes solutions abstract this paper presents a finite volume fourth order accurate compact scheme for discretization of the incompressible navier stokes equations in primitive variable formulation the numerical method of integrating the navier stokes equations comprises a compact finite volume formulation of the average convective and diffusive fluxes the pressure velocity coupling is achieved via the coupled solution of the resulting system of equations the solution of the coupled set of equations is performed with an implicit newton krylov matrix free method for stationary problems for simulation of unsteady flows a standard fourth order runge kutta method was used for temporal discretization and the velocity pressure coupling was ensured at each stage also using the matrix free method several incompressible viscous steady and unsteady flow problems have been computed to assess the robustness and accuracy of the proposed method

explicit and implicit multidimensional compact high resolution shock capturing methods two families of explicit and implicit compact high resolution shock capturing methods for the multidimensional compressible euler equations for fluid dynamics are constructed some of these schemes can be fourth and sixth order accurate away from discontinuities for the semi discrete case their shock capturing properties are of the total variation diminishing tvd total variation bounded tvb total variation diminishing in the mean tvdm essentially nonoscillatory eno or positive type of scheme for 1d scalar hyperbolic conservation laws and are positive schemes in more than one dimension these higher order compact schemes require the same grid stencil per spatial direction as their second order noncompact cousins the added terms over the second order noncompact cousins involve extra vector additions but no added flux evaluations due to the construction these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on 1d extensions the extension of these families of compact schemes to coupled nonlinear systems can be accomplished using the roe approximate riemann solver the generalized steger and warming flux vector splitting or the van leer type flux vector splitting modification to existing high resolution second or third order noncompact shock capturing computer codes is minimal high resolution shock capturing properties can also be achieved via a variant of the second order lax friedrichs numerical flux without the use of riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock capturing counterparts an efficient and compatible high resolution shock capturing filter for spatially fourth and sixth order classical compact and noncompact schemes is discussed the simplest extension to viscous flows can be achieved by using the standard fourth order compact or noncompact formula for the viscous terms

formulation and comparison of algorithms for frictional contact problems this paper presents two algorithms for solving the discrete quasi static small displacement linear elastic contact problem with coulomb friction the algorithms are adoptions of a newton method for solving b differentiable equations and an interior point method for solving smooth constrained equations for the application of the former method the contact problem is formulated as a system of b differentiable equations involving the projection operator onto sets with simple structure for the application of the latter method the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non negativity of variables treated as constraints the two algorithms are numerically tested for two dimensional problems containing up to 100 contact nodes and up to 100 time increments results show that at the present stage of development the newton method is superior both in robustness and speed additional comparison is made with a commercial finite element code

return mapping for nonsmooth and multiscale elastoplasticity we present a semi implicit return mapping algorithm for integrating generic nonsmooth elastoplastic models the semi implicit nature of the algorithm stems from freezing the plastic internal variables at their previous state followed by implicitly integrating the stresses and plastic multiplier the plastic internal variables are incrementally updated once convergence is achieved a posteriori locally the algorithm behaves as a classic return mapping for perfect plasticity and hence inherits the stability of implicit integrators however it differs from purely implicit integrators by keeping the plastic internal variables locally constant this feature affords the method the ability to integrate nonsmooth c0 evolution laws that may not be integrable using implicit methods as a result we propose and use the algorithm as the backbone of a semi concurrent multiscale framework in which nonsmooth constitutive relationships can be directly extracted from the underlying micromechanical processes and faithfully incorporated into elastoplastic continuum models though accuracy of the proposed algorithm is step size dependent its simplicity and its remarkable ability to handle nonsmooth relations make the method promising and computationally appealing

convergence of newton type methods in incremental return mapping analysis of elasto plastic problems abstract the incremental finite element algorithm with return mapping stress computation is considered for the solution of problems of elasto plasticity this algorithm performed in load steps leads to the necessity of solving large scale nonlinear systems the properties of these systems are investigated in the presented paper together with two iterative techniques for their solution the main results of the paper are rigorous proofs of the convergence of the inexact initial stiffness method and the local quadratic convergence of the inexact newton method with the consistent tangent operator

conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics the proposed scheme which is a conservative form of the interpolated differential operator scheme ido cf can provide high accurate solutions for both compressible and incompressible fluid equations spatial discretizations with fourth order accuracy are derived from interpolation functions locally constructed by both cell integrated values and point values these values are coupled and time integrated by solving fluid equations in the flux forms for the cell integrated values and in the derivative forms for the point values the ido cf scheme exactly conserves mass momentum and energy retaining the high resolution more than the non conservative form of the ido scheme a direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods benchmark tests of riemann problems and lid driven cavity flows show that the ido cf scheme is immensely promising in compressible and incompressible fluid dynamics studies

very high order compact finite difference schemes on non uniform grids for incompressible navier stokes equations this article presents a family of very high order non uniform grid compact finite difference schemes with spatial orders of accuracy ranging from 4th to 20th for the incompressible navier stokes equations the high order compact schemes on non uniform grids developed in shukla and zhong r k shukla x zhong derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation j comput phys 204 2005 404 for linear model equations are extended to the full navier stokes equations in the vorticity and streamfunction formulation two methods for the solution of helmholtz and poisson equations using high order compact schemes on non uniform grids are developed the schemes are constructed so that they maintain a high order of accuracy not only in the interior but also at the boundary second order semi implicit temporal discretization is achieved through an implicit backward differentiation scheme for the linear viscous terms and an explicit adam bashforth scheme for the non linear convective terms the boundary values of vorticity are determined using an influence matrix technique the resulting discretized system with boundary closures of the same high order as the interior is shown to be stable when applied to the two dimensional incompressible navier stokes equations provided enough grid points are clustered at the boundary the resolution characteristics of the high order compact finite difference schemes are illustrated through their application to the one dimensional linear wave equation and the two dimensional driven cavity flow comparisons with the benchmark solutions for the two dimensional driven cavity flow thermal convection in a square box and flow past an impulsively started cylinder show that the high order compact schemes are stable and produce extremely accurate results on a stretched grid with more points clustered at the boundary

fully implicit integration and consistent tangent modulus in elasto plasticity this paper deals with the numerical integration of a class of rate independent elasto plastic models the backward euler scheme is used to integrate the rate constitutive relations the non linear equations obtained are solved by the newton method the consistent tangent modulus is obtained by exact linearization of the algorithm in the case of j 2 elastoplasticity with non linear isotropic hardening and non linear kinematic hardening chaboche marquis model explicit formulas are derived without any approximations

a primal dual finite element approximation for a nonlocal model in plasticity we study the numerical approximation of a static infinitesimal plasticity model of kinematic hardening with a nonlocal extension here the free energy to be minimized is a combination of the elastic energy and an additional term depending on the curl of the plastic variable first we introduce the stress as dual variable and provide an equivalent primal dual formulation resulting in a local flow rule the discretization is based on curl conforming nedelec elements to obtain optimal a priori estimates the finite element spaces have to satisfy a uniform inf sup condition this can be guaranteed by adding locally defined face and element bubbles second the discrete variational inequality system is reformulated as a nonlinear equality we show that the classical radial return algorithm applied to the mixed inequality formulation is equivalent to a semismooth newton method for the nonlinear system of equations numerical results illustrate the convergence of the applied discretization and the solver

continuity constrained least squares interpolation for sfo suppression in immersed boundary methods a new immersed boundary interpolation for discrete forcing methods is presented the method decreases spurious oscillations in the pressure field and consequently in the body force calculations which are a common issue in several immersed boundary methods the method applies a continuity constraint in the least squares interpolation and guarantees that adjacent interpolation polynomials are continuous between each other this approach strictly enforces continuity in the flow reconstruction domain reducing time discontinuities caused by the boundary conditions applied at the immersed boundary several moving body problems test cases are simulated including a three dimensional one to demonstrate the new method s capability of computing stable pressure fields even for coarse grids and small cfl numbers which are known to increase the pressure oscillations

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

finite volume compact schemes on staggered grids compact finite difference schemes have been recently used in several direct numerical simulations of turbulent flows since they can achieve high order accuracy and high resolution without exceedingly increasing the size of the computational stencil the development of compact finite volume schemes is more revolved due to the appearance of surface and volume integrals while pereira et al j comput phys 167 2001 and smirnov et al aiaa paper 2546 2001 focused on collocated grids in this paper we use the staggered grid arrangement compact schemes can be tuned to achieve very high resolution for a given formal order of accuracy we develop and test high resolution schemes by following a procedure proposed by lele j comput phys 103 1992 which to the best of our knowledge has not yet been applied to compact finite volume methods on staggered grids results from several one and two dimensional simulations for the scalar transport and navier stokes equations are presented showing that the proposed method is capable to accurately reproduce complex steady and unsteady flows

derivation of high order compact finite difference schemes for non uniform grid using polynomial interpolation in this paper simple polynomial interpolation is used to derive arbitrarily high order compact schemes for the first derivative and tridiagonal compact schemes for the second derivative consisting of three second derivative nodes in the interior and two on the boundary on non uniform grids boundary and near boundary schemes of the same order as the interior are also developed using polynomial interpolation and for a general compact scheme on a non uniform grid it is shown that polynomial interpolation is more efficient than the conventional method of undetermined coefficients for finding coefficients of the scheme the high order non uniform schemes along with boundary closure of up to 14th order thus obtained are shown to be stable on a non uniform grid with appropriate stretching so that more grid points are clustered near the boundary the stability and resolution properties of the high order non uniform grid schemes are studied and the results of three numerical tests on stability and accuracy properties are also presented

a direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures this paper presents a direct complementarity approach for carrying out the elastoplastic analysis of plane stress and plane strain structures founded on a traditional finite step formulation our approach however avoids the typically cumbersome implementation of iterative predictorcorrector procedures associated with the ubiquitous return mapping algorithm instead at each predefined step the governing formulationcast in its most natural mathematical programming format known as a mixed complementarity problemis directly solved by using a complementarity solver run from within a mathematical modeling system we have chosen the industry standard general algebraic modeling system path mixed complementarity problem solver that is called from within the general algebraic modeling system environment we consider both von mises and tresca materials with perfect or hardening kinematic and isotropic behaviors our numerical tests five benchmark examples of which are presented in this paper have been carried out using models constructed from the mixed finite element of capsoni and corradi comput methods appl mech eng 1997 141 6793 which beneficially offers a locking free behavior and coarse mesh accuracy the results indicate in addition to an isochoric locking free behavior good accuracy and the ability to circumvent the difficult singularity problem associated with the corners of tresca yield surfaces

a new flux vector splitting compact finite volume scheme a new high resolution finite volume method is proposed here that uses flux vector splitting as the building block to represent the physical processes the high accuracy projection in space is obtained here inter alia by using some compact schemes of sengupta et al 1 and lele 2 that have extremely high accuracy in spectral plane this produces a new class of finite volume fv scheme for spatial discretization that has been analyzed for spectral accuracy numerical stability and dispersion relation preservation drp property following the full domain analysis method of 1 using two different time integration strategies these combination of spatial and temporal methods have been tested for two linear wave problems reported in 1 3 and the solution of burgers equation is compared with the analytical results presented in adams and shariff 4 the shock capturing method is based on the proposed technique in 5 that does not require the usage of any nonlinear differencing techniques or limiters to show that the proposed method can also handle irregular grids the burgers equation is solved using non uniform grids also to demonstrate the utility of the proposed scheme to solve practical problems the benchmark problem of nonlinear wave propagation in a one dimensional shock tube has been solved with the initial data as given in 23 by solving the euler equation the analysis and comparison of this flux vector splitting scheme with other well known methods clearly demonstrate the superior scale resolution of the proposed method at the same time providing a neutrally stable scheme when used with four stage runge kutta scheme also the same scheme displays extended ranges of wave numbers and circular frequencies over which drp property is valid the computed results of three test cases and their comparison with analytical results clearly reveal that the presented method can be used for practical problems

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

nonlinear complementarity functions for plasticity problems with frictional contact in this paper a unifying framework for the numerical treatment of frictional contact in combination with elastoplasticity with hardening is presented both kinds of problems lead to the mathematical structure of variational inequalities which are discretized by introducing additional variables for the contact stresses or the deviatoric stresses leading to a mixed formulation by defining a family of nonlinear complementarity functions which can be applied to both frictional and plastic yield conditions these inequality constraints are reformulated as equalities combined with the equilibrium equations a system of nonlinear equations is obtained that is solved in terms of a semismooth newton method to obtain a larger domain of convergence and better stability properties the introduction of additional scaling parameters is motivated and their benefit to the convergence of the scheme is demonstrated due to the locality of the contact and plastic constraints the resulting method can be interpreted as a primal dual active set strategy furthermore for linear hardening laws our method is a generalization of the well known radial return algorithm which is obtained for a special suboptimal choice of the plastic scaling parameters the efficiency and the generality of the proposed quasi newton scheme is demonstrated by several numerical examples including the simulation of a three dimensional contact problem with several elastoplastic bodies exponential isotropic hardening and coulomb friction

curvilinear finite volume schemes using high order compact interpolation during the last years the need of high fidelity simulations on complex geometries for aeroacoustics predictions has grown most of high fidelity numerical schemes in terms of low dissipative and low dispersive effects lie on finite difference fd approach but for industrial applications fd schemes are less robust compared to finite volume fv ones thus the present study focuses on the development of a new compact fv scheme for two and three dimensional applications the proposed schemes are formulated in the physical space and not in the computational space as it is the case in most of the known works therefore they are more appropriate for general grids they are based on compact interpolation to approximate interface averaged field values using known cell averaged values for each interface the interpolation coefficients are determined by matching taylor series expansions around the interface center two types of schemes can be distinguished the first one uses only the curvilinear abscissa along a mesh line to derive a sixth order compact interpolation formulae while the second more general uses coordinates in a spatial three dimensional frame well chosen this latter is formally sixth order accurate in a preferred direction almost orthogonal to the interface and at most fourth order accurate in transversal directions for non linear problems different approaches can be used to keep the high order scheme however in the present paper a muscl like formulation was sufficient to address the presented test cases all schemes have been modified to treat multiblock and periodic interfaces in such a way that high order accuracy stability good spectral resolution conservativeness and low computational costs are guaranteed this is a first step to insure good scalability of the schemes although parallel performances issues are not addressed as high frequency waves badly resolved could be amplified and then destabilize the scheme compact filtering operators have been used numerous test cases as the linear convection of a gaussian wave the convection of a lamb oseen vortex and the diffraction of an acoustic wave on a plane have been realized to validate the schemes the most efficient schemes are shown to be at least fifth order accurate on linear and non linear convection problems they are also less dissipative and less dispersive on non uniform curvilinear grids than schemes using implicit interpolation with constant coefficients of the same order on uniform cartesian grids

a 2d compact fourth order projection decomposition method a 2d fourth order compact direct scheme projection decomposition method for solving incompressible viscous flows in multi connected rectangular domains is devised in each subdomain the governing navier stokes equations are discretized by using fourth order compact schemes in space and second order scheme in time the coupling between subdomains is based on a direct non overlapping multidomain method it allows to solve each helmholtz poisson problem resulting of a projection method in complex geometries the major difficulty of the poisson neumann problem solvability is addressed and correctly treated the present numerical method is checked through some classical numerical experiments first the second order accuracy in time and the fourth order accuracy in space are shown by matching with the analytical solution of the taylor problem the method is also tested by simulating the flow in a 2d lid driven cavity the utility of the compact scheme projection decomposition method approach is further illustrated by two other benchmark problems viz the flow over a backward facing step and the laminar flow past a square prism the present results are in good agreement with the experimental data and other numerical solutions available in the literature

a collocated method for the incompressible navier stokes equations inspired by the box scheme we present a new finite difference numerical method to solve the incompressible navier stokes equations using a collocated discretization in space on a logically cartesian grid the method shares some common aspects with and it was inspired by the box scheme it uses centered second order accurate finite difference approximations for the spatial derivatives combined with semi implicit time integration the proposed method is constructed to ensure discrete conservation of mass and momentum by discretizing the primitive velocity pressure form of the equations the continuity equation is enforced exactly to machine accuracy at the collocated locations whereas the momentum equations are evaluated in a staggered manner this formulation preempts the appearance of spurious pressure modes in the embedded elliptic problem associated with the pressure the method shows uniform order of accuracy both in space and time for velocity and pressure in addition the skew symmetric form of the non linear advection term of the navier stokes equations improves discrete conservation of kinetic energy in the inviscid limit to within the order of the truncation error of the time integrator the method has been formulated to accommodate different types of boundary conditions fully periodic periodic channel inflow outflow and lid driven cavity always ensuring global mass conservation a novel aspect of this finite difference formulation is the derivation of the discretization near boundaries using the weak form of the equations as in the finite element method the method of manufactured solutions is utilized to perform accuracy analysis and verification of the solver to assess the applicability of the new method presented in this paper four realistic flow problems have been simulated and results are compared with those in the literature these cases include a lid driven cavity backward facing step kovasznay flow and fully developed turbulent channel

semismooth newton and augmented lagrangian methods for a simplified friction problem in this paper a simplified friction problem and iterative second order algorithms for its solution are analyzed in infinite dimensional function spaces motivated from the dual formulation a primal dual active set strategy and a semismooth newton method for a regularized problem as well as an augmented lagrangian method for the original problem are presented and their close relation is analyzed local as well as global convergence results are given by means of numerical tests we discuss among others convergence properties the dependence on the mesh and the role of the regularization and illustrate the efficiency of the proposed methodologies