spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

limit angular velocities of variable thickness rotating disks elastic and plastic limit angular velocities are calculated for rotating disks of variable thickness in power function form analytical solution is obtained and used to calculate elastic limit angular velocities of variable thickness rotating annular disks and annular disks with rigid inclusion an efficient numerical solution procedure is designed and used to obtain the elastic limit angular velocities of variable thickness rotating solid disks von mises yield criterion and its flow rule is used to estimate plastic limit angular velocities both linear and nonlinear hardening material behaviors are treated numerically the results are verified by comparing with those of uniform thickness rotating solid disks available in the literature elastic and plastic limit angular velocities are found to increase with the reduction of the disk thickness at the edge as well as the reduction in the disk mass due to the shape of the profile

non linear finite element formulation of kinematic limit analysis the objective of the research presented in this paper was to develop a general computational method for kinematic limit analysis problems that involve the determination of an optimal kinematically admissible velocity field of the studied structure under external loads with the failure velocity field available we can obtain a numerical estimate of the limit load exerted to the structure the general kinematic limit analysis is characterized by the minimization of the plastic potential power dissipation functional of the mechanical system this minimization problem was approached previously by linear and non linear programming schemes in this paper a new non linear solution scheme is described which is based on previous research on the regularized method the regularized functional is a convex and non linear functional whose minimum represents a viscoplastic potential power dissipation the velocity which minimizes this functional was proved to be kinematically admissible and can be obtained by the augmented lagrangian method this paper demonstrates that the non linear programming scheme is applicable to direct limit analysis the basic idea of using this optimization method is to tranform the functional with a non linear term of the first order derivatives of the velocity to a functional in which this term is uncoupled with the strain rates and in consequence the augmented lagrangian transformed functional can be solved more easily some special problems related to incompressibility and discontinuity are discussed a simple and accurate scheme is proposed to deal with the incompressibility problem and the problem of the linear mohr coulomb yield surface in the principal stress space examples of plane stress and plane strain problems are given for von mises and mohr coulomb materials numerical results are provided in graphical form which serves to illustrate the kinematic admissibility the limit loads obtained agreed well with the analytical results which demonstrates the efficiency and accuracy of the non linear computational method presented

a general algorithm for limit solutions of plane stress problems a computational approach to limit solutions is considered most challenging for two major reasons a limit solution is likely to be non smooth such that certain non differentiable functions are perfectly admissible and make physical and mathematical sense moreover the possibility of non unique solutions makes it difficult to analyze the convergence of an iterative algorithm or even to define a criterion of convergence in this paper we use two mathematical tools to resolve these difficulties a duality theorem defines convergence from above and from below the exact solution a combined smoothing and successive approximation applied to the upper bound formulation perturbs the original problem into a smooth one by a small parameter e as e 0 the solution of the original problem is recovered this general computational algorithm is robust such that from any initial trial solution the first iteration falls into a convex hull that contains the exact solution s of the problem unlike an incremental method thut invariably renders the limit problem ill conditioned the algorithm is numerically stable limit analysis itself is a highly efficient concept which bypasses the tedium of the intermediate elastic plastic deformation and seeks the most important information directly with the said algorithm we have produced many limit solutions of plane stress problems certain non smooth characters of the limit solutions are shown in the examples presented two well known as well as one parametric family of yield functions are used to allow comparison with some classical solutions

least squares methods for stokes equations based on a discrete minus one inner product abstract the purpose of this paper is to develop and analyze least squares approximations for stokes and elasticity problems the major advantage of the least squares formulation is that it does not require that the classical ladyzhenskaya bab ska brezzi lbb condition be satisfied we provide two methods the first is posed in terms of the velocity pressure pair without the introduction of additional variables the second adds a vorticity variable in both cases we employ least squares functionals which involve a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of such inner products applied to second order problems was proposed in an earlier paper by bramble lazarov and pasciak 1994

accelerated convergence in newton s method newton s method is based on a linear approximation of the function whose roots are to be determined taken at the current point and the resulting algorithm is known to converge quadratically in a procedure to increase the rate of convergence the author modifies the target function in such a way that newton s method applied to the modified function will yield a faster rate of convergence

computation of limit loads for computing the collapse state of a perfect plastic continuum we suggest a mixed finite element approximation to the infinite dimensional mathematical programming problem of limit analysis the convergence of the limit load to the exact solution is discussed as well as solution methods for the discrete problem finally the method is applied to a classical problem in plane strain

indefinite mean field stochastic linear quadratic optimal control this paper is concerned with the discrete time indefinite mean field linear quadratic optimal control problem the so called mean field type stochastic control problems refer to the problem of incorporating the means of the state variables into the state equations and cost functionals such as the mean variance portfolio selection problems a dynamic optimization problem is called to be nonseparable in the sense of dynamic programming if it is not decomposable by a stage wise backward recursion the classical dynamic programming based optimal stochastic control methods would fail in such nonseparable situations as the principle of optimality no longer applies in this paper we show that both the well posedness and the solvability of the indefinite mean field linear quadratic problem are equivalent to the solvability of two coupled constrained generalized difference riccati equations and a constrained linear recursive equation we characterize the optimal control set completely and obtain a set of necessary and sufficient conditions on the mean variance portfolio selection problem the results established in this paper offer a more accurate solution scheme in tackling directly the issue of nonseparability and deriving the optimal policies analytically for the mean variance type portfolio selection problems

solvability of indefinite stochastic riccati equations and linear quadratic optimal control problems abstract a new approach to study the indefinite stochastic linear quadratic lq optimal control problems which we called the equivalent cost functional method is introduced by yu 2013 in the setup of hamiltonian system on the other hand another important issue along this research direction is the possible state feedback representation of optimal control and the solvability of associated indefinite stochastic riccati equations as the response this paper continues to develop the equivalent cost functional method by extending it to the riccati equation setup our analysis is featured by its introduction of some equivalent cost functionals which enable us to have the bridge between the indefinite and positive definite stochastic lq problems with such bridge some solvability relation between the indefinite and positive definite riccati equations is further characterized it is remarkable the solvability of the former is rather complicated than the latter hence our relation provides some alternative but useful viewpoint consequently the corresponding indefinite linear quadratic problem is discussed for which the unique optimal control is derived in terms of state feedback via the solution of the riccati equation in addition some example is studied using our theoretical results

penalty finite element analysis of 3 d navier stokes equations abstract the steady state navier stokes equations in three dimensions are solved by a penalty finite element method for the problems of wall driven cavity flow in a cubical box and natural convection in a cubical cavity subjected to differential side heating the present solutions are compared with recent finite difference solutions of the wall driven cavity problem for reynolds numbersre 100 and 400 the agreement is found to be very good

a general algorithm for plastic flow simulation by finite element limit analysis abstract limit analysis has been rendered versatile in many structural and metal forming problems in metal forming analysis the slip line method and the upper bound method have filled the role of limit analysis as a breakthrough of the previous work a computational approach to limit solutions is considered as the most challenging area in the present work a general algorithm for limit solutions of plastic flow is developed with the use of finite element limit analysis the algorithm deals with a generalized holder inequality a duality theorem and combined smoothing and successive approximation in addition to a general procedure for finite element analysis the algorithm is robust such that from any initial trial solution the first iteration falls into a convex set which contains the exact solution s of the problem the idea of the algorithm for limit solutions is extended from rigid perfectly plastic materials to work hardening materials by the nature of the limit formulation which is also robust with numerically stable convergence and highly efficient computing time

a variational principle and an algorithm for limit analysis of beams and plates abstract a variational principle for limit analysis of beams and plates is developed from a yield function based on the frobenius matrix norm the formulation produced a pair of maximization and minimization problems with a duality relation between them exact solutions of two simple problems are presented as verification to the validity of the new variational principle an iterative algorithm is constructed to solve the minimization problem the algorithm tested successfully on the two example problems is intended for beams and plates with general loading and boundary conditions and shapes

linear quadratic optimal control problems for mean field stochastic differential equations linear quadratic optimal control problems are considered for mean field stochastic differential equations with deterministic coefficients by a variational method the optimality system is derived which is a linear mean field forward backward stochastic differential equation using a decoupling technique two riccati differential equations are obtained which are uniquely solvable under certain conditions then a feedback representation is obtained for the optimal control

a closed loop saddle point for zero sum linear quadratic stochastic differential games with mean field type abstract this paper is concerned with a linear quadratic zero sum differential game with mean field type the notions of explicit and implicit strategy laws are proposed based on them a closed loop formulation for saddle points in the mixed strategy law form is established in order to construct the saddle point the classical entire completion of square argument is developed to a four step completion of square argument and a pair of optimality conditions is proposed

analysis of least squares finite element methods for the navier stokes equations in this paper we study finite element methods of least squares type for the stationary incompressible navier stokes equations in two and three dimensions we consider methods based on velocity vorticity pressure form of the navier stokes equations augmented with several nonstandard boundary conditions least squares minimization principles for these boundary value problems are developed with the aid of the agmon douglis nirenberg adn elliptic theory among the main results of this paper are optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions results of several computational experiments with least squares methods which illustrate among other things the optimal convergence rates are also reported

space time least squares spectral elements for convection dominated unsteady flows legendre polynomials are employed in a space time least squares spectral element formulation applied to linear and nonlinear hyperbolic scalar equations no stabilization techniques are required to render a stable high order accurate scheme in parts of the domain where the underlying exact solution is smooth the scheme exhibits exponential convergence with polynomial enrichment whereas in parts of the domain where the underlying exact solution contains discontinuities the solution displays a gibbs like behavior numerical results will be given in which the capabilities of the space time formulation to capture discontinuities will be demonstrated

least squares finite element formulations for viscous incompressible and compressible fluid flows abstract we present least squares based finite element formulations for the numerical solution of viscous fluid flows governed by the navier stokes equations as an alternate approach to the well known weak form galerkin finite element formulations the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise and the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix we address the issue of norm equivalence of the least squares functional and its implications on the resulting finite element model in particular we develop an understanding of the compromise that must exist between the optimality and practicality of the finite element model the latter measured in terms of c k regularity across inter element boundaries we show through numerical examples that for the traditional c 0 basis such a compromise is possible when high p levels are used to span the finite element spaces when a low p level solution is desired guidelines are presented to obtain a reliable least squares collocation solution numerical examples are presented to demonstrate the high and low p level approaches these include incompressible flow past two circular cylinders in a side by side arrangement for gap sizes s d 2 0 and 0 85 incompressible flow past a square cylinder and subsonic transonic and supersonic compressible flow past a circular cylinder in addition we present a discontinuous least squares formulation where c k regularity across inter element boundaries is enforced in a weak sense through the least squares functional allowing for h and p type non conformities in the computational domain

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

a least squares finite element formulation for unsteady incompressible flows with improved velocity pressure coupling in the weak form galerkin formulation for incompressible flows the pressure has a well understood role at all times it may be interpreted as a lagrange multiplier that enforces the divergence free constraint on the velocity field this is not the case in least squares formulations for incompressible flows where the divergence free constraint is enforced in a least squares sense in a variational setting of residual minimization thus the role of the pressure in a least squares formulation is rather vague we find that this lack of velocity pressure coupling in least squares formulations may induce spurious temporal pressure oscillations when using the non stationary form of the equations we present a least squares formulation with improved velocity pressure coupling based on the use of a regularized divergence free constraint a first order system least squares fosls approach based on velocity pressure and vorticity is used to allow the use of practical c0 element expansions in the finite element model we use high order spectral element expansions in space and second and third order time stepping schemes excellent conservation of mass and accuracy of computed pressure metrics are demonstrated in the numerical results

one kind of linear quadratic zero sum stochastic differential game with jumps this paper is concerned with one kind of two person zero sum linear quadratic stochastic differential game with jumps a closed loop formulation for the saddle points of the game is established in

upper bound limit analysis using discontinuous velocity fields abstract a new method for computing rigorous upper bounds under plane strain conditions is described it is based on a linear three noded triangular element which has six unknown nodal velocities and a fixed number of unknown plastic multiplier rates and uses the kinematic theorem to define a kinematically admissible velocity field as the solution of a linear programming problem unlike existing formulations which permit only a limited number of velocity discontinuities whose directions of shearing must be specified a priori the new formulation permits velocity discontinuities at all edges shared by adjacent triangles and the directions of shearing are found automatically the variation of the velocity jump along each discontinuity is described by an additional set of four unknowns all of the unknowns are subject to the constraints imposed by an associated flow rule and the velocity boundary conditions the objective function corresponds to the dissipated power or some related load parameter of interest and is minimised to yield the desired upper bound since plastic deformation may occur not only in the discontinuities but also throughout the triangular elements as well the method is capable of modelling complex velocity fields accurately and typically produces tight upper bounds on the true limit load the formulation is applicable to materials whose strength is cohesive frictional purely cohesive and uniform or purely cohesive and linearly varying and thus quite general the new procedure is very efficient and always requires fewer elements than existing methods to obtain useful upper bound solutions moreover because of the extra degrees of freedom introduced by the discontinuities the linear elements no longer need to be arranged in a special pattern to model incompressible behaviour accurately

two dimensional elastoplasticity approximation by mixed finite elements two dimensional elastoplastic problems are used to assess the performance of a family of mixed finite elements in the non linear regime in particular four node bilinear quadrilateral elements that exhibit high accuracy in coarse meshes are used to simulate plane stress plane strain axisymmetric and shear deformable plate bending problems the excellent performance of these elements is demonstrated with a number of discriminating examples selected from the literature in addition agreement is observed in the limit load predicted by the axisymmetric solid and plate bending elements

least squares finite element method and preconditioned conjugate gradient solution a least squares variational procedure for first order systems of differential equations and an approximate formulation based on finite elements are developed error estimates a condition number bound and analysis of weighting factors are given steepest descent and conjugate gradient solution procedures are examined and an appropriate preconditioner constructed which is demonstrated to yield rapid convergence and to be insensitive to problem size numerical studies of rates of convergence for a test problem are given

stochastic differential games and viscosity solutions of hamilton jacobi bellman isaacs equations in this paper we study zero sum two player stochastic differential games with the help of the theory of backward stochastic differential equations bsdes more precisely we generalize the results of the pioneering work of fleming and souganidis indiana univ math j 38 1989 pp 293 314 by considering cost functionals defined by controlled bsdes and by allowing the admissible control processes to depend on events occurring before the beginning of the game this extension of the class of admissible control processes has the consequence that the cost functionals become random variables however by making use of a girsanov transformation argument which is new in this context we prove that the upper and the lower value functions of the game remain deterministic apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information its combination with bsde methods in particular that of the notion of stochastic backward semigroups introduced by peng bsde and stochastic optimizations in topics in stochastic analysis science press beijing 1997 allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way the upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower hamilton jacobi bellman isaacs equations respectively for this peng s bsde method is extended from the framework of stochastic control theory into that of stochastic differential games

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

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

least squares spectral elements applied to the stokes problem least squares spectral element methods are based on two important and successful numerical methods spectral hp element methods and least squares finite element methods in this respect least squares spectral element methods seem very powerful since they combine the generality of finite element methods with the accuracy of the spectral methods and also have the theoretical and computational advantages of the least squares methods these features make the proposed method a competitive candidate for the solution of large scale problems arising in scientific computing in order to demonstrate its competitiveness the method has been applied to an analytical problem and the theoretical optimal and suboptimal a priori estimates have been confirmed for various boundary conditions moreover the exponential convergence rates typical for a spectral element discretization have also been confirmed the comparison with the classical galerkin spectral element method revealed that the least squares spectral element method is as accurate as the galerkin method for the smooth model problem

analytical and numerical investigation of strain hardening viscoplastic thick walled cylinders under internal pressure by using sequential limit analysis plastic limit load of strain hardening viscoplastic thick walled cylinders subjected to internal pressure is investigated numerically and analytically in the paper the paper applies sequential limit analysis to deal with the quasi static problem involving hardening material properties and weakening behavior corresponding to the strain rate sensitivity and widening deformation by sequential limit analysis the paper treats the plasticity problems as a sequence of limit analysis problems stated in the upper bound formulation rigorous upper bounds are acquired iteratively through a computational optimization procedure with the internal pressure factor as the objective function especially rigorous validation was conducted by numerical and analytical studies of thick walled cylinders in terms of the plastic limit load as well as the onset of instability it is found that the computed limit loads are rigorous upper bounds and agree very well with the analytical solutions 2007 elsevier b v all rights reserved

analysis of velocity flux least squares principles for the navier stokes equations part ii this paper continues the development of the least squares methodology for the solution of the incompressible navier stokes equations started in part i here we again use a velocity flux first order navier stokes system but our focus now is on a practical algorithm based on a discrete negative norm

investigation of rotating hollow cylinders of strain hardening viscoplastic materials by sequential limit analysis the paper employs sequential limit analysis to investigate rotating hollow cylinders of nonlinear isotropic strain hardening viscoplastic materials a computational optimization procedure was developed to account for hardening material properties and weakening behavior corresponding to the strain rate sensitivity and widening deformation a sequence of limit analysis problems was then conducted to seek the corresponding plastic limit angular velocities sequentially by a general algorithm incorporated with an inner and outer iterative sequence particularly analytical solutions of plastic limit angular velocities and the onset of instability were derived for rigorous comparisons the corresponding stability condition was also obtained explicitly specifically the implicit form of the onset of instability was solved by the fixed point iteration it is found that the computed limit angular velocities are rigorous upper bounds and match well with analytical solutions

a space time coupled p version least squares finite element formulation for unsteady two dimensional navier stokes equations this paper presents a p version least squares finite element formulation for two dimensional unsteady fluid flow described by navier stokes equations where the effects of space and time are coupled the dimensionless form of the navier stokes equations are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c0 element approximation the element properties are derived by utilizing the p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure the application of least squares minimization to the set of coupled first order partial differential equations results in finding a solution vector which makes gradient of error functional with respect to a null vector this is accomplished by using newton s method with a line search a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach the generality success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements

a space time coupled p version least squares finite element formulation for unsteady fluid dynamics problems this paper presents a p version least squares finite element formulation for unsteady fluid dynamics problems where the effects of space and time are coupled the dimensionless form of the differential equations describing the problem are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c element approximation the element properties are derived by utilizing p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach for non linear problems the non linear algebraic equations resulting from the least squares process are solved using newton s method with a line search this procedure results in a symmetric hessian matrix equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the generality success and superiority of the present formulation procedure is demonstrated by presenting specific formulations and examples for the advection diffusion and burgers equations the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements the work presented in this paper provides the basis for the extension of the space time coupled least squares minimization concept to two and three dimensional unsteady fluid flow

forward backward stochastic differential equations and controlled mckean vlasov dynamics the purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of the mckean vlasov type motivated by the recent interest in mean field games we highlight the connection and the differences between the two sets of problems we prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control we also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied finally we show that our solution to the control problem provides approximate equilibria for large stochastic games with mean field interactions

convergence analysis and validation of sequential limit analysis of plane strain problems of the von mises model with non linear isotropic hardening the paper presents sequential limit analysis of plane strain problems of the von mises model with non linear isotropic hardening by using a general algorithm the general algorithm is a combined smoothing and successive approximation cssa method in the paper emphasis is placed on its convergence analysis and validation applied to sequential limit analysis involving materials with isotropic hardening by sequential limit analysis the paper treats deforming problems as a sequence of limit analysis problems stated in the upper bound formulation especially the cssa algorithm was proved to be unconditionally convergent by utilizing the cauchy schwarz inequality finally rigorous validation was conducted by numerical and analytical studies of a thick walled cylinder under pressure it is found that the computed limit loads are rigorous upper bounds and agree very well with the analytical solutions copyright 2005 john wiley sons ltd

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

analysis of velocity flux first order system least squares principles for the navier stokes equations part i this paper develops a least squares approach to the solution of the incompressible navier stokes equations in primitive variables as with our earlier work on stokes equations we recast the navier stokes equations as a first order system by introducing a velocity flux variable and associated curl and trace equations we show that a least squares principle based on l2 norms applied to this system yields optimal discretization error estimates in the h1 norm in each variable including the velocity flux an analogous principle based on the use of an h 1 norm for the reduced system with no curl or trace constraints is shown to yield similar estimates but now in the l2 norm for velocity flux and pressure although the h 1 least squares principle does not allow practical implementation these results are critical to the analysis of a practical least squares method for the reduced system based on a discrete equivalent of the negative norm a practical method of this type is the subject of a companion paper finally we establish optimal multigrid convergence estimates for the algebraic system resulting from the l2 norm approach

least squares finite element method with high continuity nurbs basis for incompressible navier stokes equations modern least squares finite element method lsfem has advantage over mixed finite element method for non self adjoint problem like navier stokes equations but has problem to be norm equivalent and mass conservative when using c 0 type basis in this paper lsfem with non uniform b splines nurbs is proposed for navier stokes equations high order continuity nurbs is used to construct the finite dimensional spaces for both velocity and pressure variational form is derived from the governing equations with primitive variables and the dofs due to additional variables are not necessary there is a novel k refinement which has spectral convergence of least squares functional the method also has same advantages as isogeometric analysis like automatic mesh generation and exact geometry representation several benchmark problems are solved using the proposed method the results agree well with the benchmark solutions available in literature the results also show good performance in mass conservation

on generalized ho der inequality inequality may apply to general paired non euclidean norms we restrict the discussion to finite dimensional spaces a theorem is established for such generalized holder inequality together with its sharpness condition the new sharpness condition involves the gradient of the norm function the previous inequalities of this type and their associated sharpness conditions become special cases of this new inequality which is a fundamental primal dual relation of two spaces defined by paired norms on their respective elements given an arbitrarily defined primal norm in one space a method is presented for constructing the dual norm in the other the generalized holder inequality is the basis for many known minimax duality theorems in applied fields particular examples of this duality are found in mathematical models of plastic behavior inequalities appear frequently in algebra geometry and analysis this form of mathematical statements is so prevalent that it is difficult to state their relevance and applications exhaustively inequalities are used to demarcate numbers vectors matrices and functions e g to be positive definite to define sets norm measures convexity of sets and functions to compare and bound functions with another function just to name a few there are books devoted exclusively to inequalities 11 a class of inequalities concerning inner products of vectors and functions can be grouped into two frequently encountered ones in literature although one is a special case of the other the schwarz inequality applies to the euclidean and hilbert spaces 2 the holder inequality extends the concept to certain non euclidean and non hilbert spaces associated with a specific family of norm measures defined by minkowski 3 we shall restrict our discussion to finite dimensional spaces a further extension of the holder inequality is presented in this paper for general paired non euclidean spaces the familiar schwarz inequality for the euclidean space r can be stated for the inner product

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

a collocation penalty least squares finite element formulation for incompressible flows abstract we present a penalty least squares finite element formulation for incompressible flows the pressure degree of freedom is eliminated using the classical penalty approach and the least squares model is formed in terms of velocity vorticity and dilatation an iterative penalization is implemented which allows the use of low penalty parameters and retains a manageable conditioning number of the global coefficient matrix the h convergence is verified using the exact solution of kovasznay s flow numerical results are presented for a number of benchmark problems e g steady flow past a large circular cylinder in a channel transient flow over a backward facing step unsteady flow past a circular cylinder and buoyancy driven natural convection in a square cavity for all numerical examples the effect of penalty parameter on the accuracy is investigated thoroughly and it is concluded that the present model produces accurate results for low penalty parameters 10 100 these problems are also solved on coarse meshes to show that good mass conservation is achieved

spectral hp penalty least squares finite element formulation for the steady incompressible navier stokes equations in this paper we present spectral hp penalty least squares finite element formulation for the numerical solution of steady incompressible navier stokes equations the continuity equation is treated as a constraint on velocity field and the constraint is enforced using the penalty method which is equivalent to replacing the pressure variable p with p n p n email protected u where u is the velocity field c is the penalty parameter and superscript n is non linear iteration number the modified navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity dilatation or stresses as additional independent variables and the least squares method is used to develop the finite element model high order basis functions are used to construct the discrete model unlike other penalty finite element formulations equal order integration is used for all terms of the coefficient matrix schur complement method is used and all interior element degrees of freedom are condensed out and interface degrees of freedom are solved either by direct solver banded cholesky factorization or iterative solver preconditioned conjugate gradient present formulation produces very accurate results for even very low penalty parameters 10 40 in this way present formulation and implementation fixes perennial problem of bad conditioning of coefficient matrix associated with penalty formulations proposed formulation is verified using the exact solution of kovasznay flow numerical results are also presented for a number of two dimensional benchmark problems e g 2d lid driven cavity flow flow over a backward facing step steady flow past a circular cylinder and 2d flow past a large circular cylinder in a channel finally the proposed formulation is extended to coupled velocity temperature problems and buoyant flow inside a square enclosure is solved to test the penalty least squares finite element model developed herein

large deformation of structures by sequential limit analysis although most structures are designed to function under small deformation their large deformation behavior can be used to estimate reliability and safety for survivorship from an accident or a natural disaster structures such as buildings bridges ships vehicles and machinery are designed with a safety factor to protect certain assets from the unexpected and unknot elements the large deformation analysis provides the rational basis for a safety factor in this paper the method of sequential limit analysis is used to compute large deformation solutions of truss and frame problems differing from the incremental method of plasticity the limit analysis method is numerically stable more effecient and requires simpler input data a duality theorem serves as the foundation of an algorithm for computing the complete static and kinematic solutions sim uhaneously and for establishing their accuracy in each step of a deformation sequence the phenom ena encountered in large deformation such as loading unloading under monotone deformation bifurcation more generally loss of uniqueness of solutions and internal contact of structural members are revealed by the sequential limit analysis presented in this paper

an mpec approach for the critical post collapse behavior of rigid plastic structures this paper presents a numerical method to identify and trace the critical post collapse response of rigid perfectly plastic structures to account for the possibility of multiple equilibrium paths the critical one is directly identified using the minimum 2nd order work criterion our proposed enhanced sequential limit analysis is formulated as an instance of the challenging class of optimization problems known as a mathematical program with equilibrium constraints mpec this mpec formulation minimizes the 2nd order work expression subject to the set of constraints describing the complete complementarity system in mixed static kinematic variables governing simultaneously the two adjacent equilibrium configurations namely the current one and its neighboring state we use a nonlinear programming based algorithm involving relaxation of the complementarity terms to solve the mpec four numerical examples are provided to illustrate application of the proposed scheme

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

an optimal feedback control strategy pair for zero sum linear quadratic stochastic differential game the riccati equation approach in this paper we study a two person zero sum linear quadratic stochastic differential game problem from a new viewpoint we construct an optimal feedback control strategy pair for the game in a closed loop form based on the solution of a riccati equation a key part of our analysis involves proving the global solvability of this riccati equation which is interesting in its own right moreover we demonstrate an indefinite phenomenon arising from the linear quadratic game

an open loop stackelberg strategy for the linear quadratic mean field stochastic differential game this paper is concerned with the open loop linear quadratic lq stackelberg game of the mean field stochastic systems in finite horizon by means of two generalized differential riccati equations the follower first solves a mean field stochastic lq optimal control problem then the leader turns to solve an optimization problem for a linear mean field forward backward stochastic differential equation by introducing new state and costate variables we present a sufficient condition for the existence and uniqueness of the stackelberg strategy in terms of the solvability of some riccati equations and a convexity condition furthermore it is shown that the open loop stackelberg equilibrium admits a feedback representation involving the new state and its mean finally two examples are given to show the effectiveness of the proposed results

on the least squares method abstract it is shown that the theoretical basis of the general least squares method is the bounded inverse theorem this explains why the least squares finite element method lsfem within one mathematical computational framework without any special treatment can provide numerical solutions for all types of partial differential equations error estimates of lsfem for general and elliptic first order systems of partial differential equations are established as an example the incompressible stokes equations are considered the principal part of the stokes equations in the first order velocity pressure vorticity formulation consists of two div curl systems this knowledge is employed to derive all permissible combinations of non standard boundary conditions for the stokes equations and to show that under these conditions the stokes operator is bounded below in the full h1 sense and hence the corresponding lsfem is optimal the numerical results are also given to support this conclusion

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100