shape optimization using adaptive shape refinement a large part of the computational effort in shape optimization problems is expended in the numerical computation of the gradients for sensitivity information this effort increases dramatically with an increase in the number of variables used to represent the shape an adaptation of the gradient projection algorithm for shape optimization problems is described here along with a method to reduce the intermediate size of the optimization problem by allowing adaptive refinement of the shape the method is demonstrated with a simple representative test case

optimization methods and stability of inclusions in banach spaces our paper deals with the interrelation of optimization methods and lipschitz stability of multifunctions in arbitrary banach spaces roughly speaking we show that linear convergence of several first order methods and lipschitz stability mean the same particularly we characterize calmness and the aubin property by uniformly with respect to certain starting points linear convergence of descent methods and approximate projection methods so we obtain e g solution methods for solving equations or variational problems which require calmness only the relations of these methods to several known basic algorithms are discussed and errors in the subroutines as well as deformations of the given mappings are permitted we also recall how such deformations are related to standard algorithms like barrier penalty or regularization methods in optimization

on projection algorithms for solving convex feasibility problems due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences most notably computerized tomography algorithms for solving convex feasibility problems continue to receive great attention to unify generalize and review some of these algorithms a very broad and flexible framework is investigated several crucial new concepts which allow a systematic discussion of questions on behaviour in general hilbert spaces and on the quality of convergence are brought out numerous examples are given

a conservative three dimensional eulerian method for coupled solid fluid shock capturing a new method is presented for the explicit eulerian finite difference computation of shock capturing problems involving multiple resolved material phases in three dimensions we solve separately for each phase the equations of fluid dynamics or solid mechanics using as interface boundary conditions artificially extended representations of the individual phases for fluids we use a new 3d spatially unsplit implementation of the piecewise parabolic ppm method of colella and woodward for solids we use the 3d spatially unsplit eulerian solid mechanics method of miller and colella vacuum and perfectly incompressible obstacles may also be employed as phases a separate problem is the time evolution of material interfaces which are represented by planar segments constructed with a volume of fluid method the volume fractions are advanced in time using a second order 3d spatially unsplit advection routine with a velocity field determined by solution of interface normal two phase riemann problems from the riemann problem solutions we also determine cross interface momentum and energy fluxes the volume fractions in mixed cells may be arbitrarily small which would ordinarily make the courant friedrichs lewy time step stability limit arbitrarily small as well we overcome this limitation using the mass redistribution formalism to conservatively redistribute generalized mass in the neighborhood of the split cells we present an application of this method to an explosion contained in a metal can a reactive fluid approximating pbx 9404 is encased within an elastic plastic solid approximating copper surrounded by vacuum our implementation is in parallel and with adaptive mesh refinement

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

adaptive mesh generation for fluid mechanics problems in this study we give an in depth investigation of the adaptive procedures for compressible and incompressible flow problems many tests have been carried out with different error indicators for different problems these refinement procedures show that the interpolation error indicators are adequate to predict an accurate solution comparison between the gradient and curvature based error indicators show that the former yields better performance in most situations however we found that the latter indicators predict better motion in the recirculatory and unstable regions than the gradient based method many test cases have been presented for both compressible and incompressible flow problems in the later part of the paper we suggest a simple way of combining the curvature gradient indicators of different key variables and present an appropriate problem where it is applicable a wide coverage of relevant literature has also been presented in this paper copyright 2000 john wiley sons ltd

convergence rate analysis for averaged fixed point iterations in common fixed point problems in this paper we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a hilbert space our results are achieved under a bounded holder regularity

a 3d adaptive mesh refinement algorithm for multimaterial gas dynamics abstract adaptive mesh refinement amr in conjunction with high order upwind finite difference methods has been used effectively on a variety of problems in this paper we discuss an implementation of an amr finite difference method that solves the equations of gas dynamics with two material species in three dimensions an equation for the evolution of volume fractions augments the gas dynamics system the material interface is preserved and tracked from the volume fractions using a piecewise linear reconstruction technique

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

necessary conditions for linear convergence of iterated expansive set valued mappings we present necessary conditions for monotonicity of fixed point iterations of mappings that may violate the usual nonexpansive property notions of linear type monotonicity of fixed point sequences weaker than fejer monotonicity are shown to imply metric subregularity this together with the almost averaging property recently introduced by luke et al math oper res 2018 https doi org 10 1287 moor 2017 0898 guarantees linear convergence of the sequence to a fixed point we specialize these results to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called subtransversality subtransversality is shown to be necessary for linear convergence of alternating projections for consistent feasibility

minimum weight design of trusses by an optimality criteria method this work presents an optimality criteria method which accurately and efficiently solves the structural optimization problem of trusses subject to deflection and stress constraints the method is based upon the newton raphson method and a first order approximation both for deflection and stress constraints using the method of virtual work it solves the problem rigorously and no serious difficulties have yet been encountered in spite of the drawbacks inherent in the newton raphson method worked examples show its superiority to other methods and its ability to cope with trusses with many members and subject to multiple constraints

p version least squares finite element formulation for convection diffusion problems this paper presents a p version least squares finite element formulation of the convection diffusion equation the second order differential equation describing convection diffusion is reduced to a series of equivalent first order differential equations for which the least squares formulation is constructed using the same order of approximation for each of the dependent variables the hierarchical approximation functions and the nodal variable operators are established by first constructing the one dimensional hierarchical approximation functions of orders p and p and the corresponding nodal variable operators in and direction and then taking their products numerical results are presented and compared with analytical and numerical solutions for a two dimensional test problem to demonstrate the accuracy and the convergence characteristics of the present formulation the gaussian quadrature rule used to calculate the numerical values of the element matrices vectors as well as the error functional i e is established based on the highest degree of the polynomial in the integrands it is demonstrated that this quadrature rule with the present p version formulation produces excellent results for very low as well as extremely high peclet numbers 10 106 and furthermore the error functional i sum of the integrals of e2 is a monotonically decreasing function of the number of degrees of freedom as the p levels are increased for a fixed mesh it is shown that exact integration with the h version linear and parabolic elements produces inaccurate solutions at high peclet numbers results are also presented using reduced integration it is shown that the reduced integration with p version produces accurate values of the primary variable even for relatively low p levels but the error functional i when calculated using the proper integration rule has a much higher value due to errors in the derivatives of the primary variable and is a non monotonic function of the degrees of freedom as p levels are increased for a fixed mesh similar behaviour of the error functional i is also observed for the h models using linear elements when reduced integration is used although the h models using parabolic elements produce monotonic error functional behaviour as the number of degrees of freedom are increased the error values are inferior to the p version results using exact integration

adaptive moment of fluid method a novel adaptive mesh refinement amr strategy based on the moment of fluid mof method for volume tracking of evolving interfaces is presented moment of fluid method is a new interface reconstruction and volume advection method using volume fractions as well as material centroids the mesh refinement criterion is based on the deviation of the actual centroid obtained by interface reconstruction from the reference centroid given by moment advection process the centroid error indicator detects not only high curvature regions but also regions with complicated subcell structures like filaments a new lagrange remap scheme is presented for advecting moments which includes lagrangian backtracking polygon intersection based remapping and forward tracking to define the material centroid the effectiveness and efficiency of amr mof method is demonstrated with classical test problems such as zalesak s disk and reversible vortex problem the comparison with previously published results for these problems shows the superior accuracy of the amr mof method over other methods in addition two new test cases with severe deformation rates are introduced namely droplet deformation and s shape deformation problems for further demonstration of the capabilities of the amr mof method

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

backward forward algorithms for structured monotone inclusions in hilbert spaces abstract in this paper we study the backward forward algorithm as a splitting method to solve structured monotone inclusions and convex minimization problems in hilbert spaces it has a natural link with the forward backward algorithm and has the same computational complexity since it involves the same basic blocks but organized differently surprisingly enough this kind of iteration arises when studying the time discretization of the regularized newton method for maximally monotone operators first we show that these two methods enjoy remarkable involutive relations which go far beyond the evident inversion of the order in which the forward and backward steps are applied next we establish several convergence properties for both methods some of which were unknown even for the forward backward algorithm this brings further insight into this well known scheme finally we specialize our results to structured convex minimization problems the gradient projection algorithms and give a numerical illustration of theoretical interest

uniformity and inexact version of a proximal method for metrically regular mappings abstract we study stability properties of a proximal point algorithm for solving the inclusion 0 t x when t is a set valued mapping that is not necessarily monotone more precisely we show that the convergence of our algorithm is uniform in the sense that it is stable under small perturbations whenever the set valued mapping t is metrically regular at a given solution we present also an inexact proximal point method for strongly metrically subregular mappings and show that it is super linearly convergent to a solution to the inclusion 0 t x

new approach to the η proximal point algorithm and nonlinear variational inclusion problems abstract a new approach using the over relaxed proximal point algorithm in the context of solving a class of inclusion problems based on the notion of maximal monotonicity is developed and examined convergence analysis seems to be reasonable and finally some specializations are included furthermore the model developed in this communication seems to be appropriate to the yosida approximation in the sense that it can be applied to first order evolution equations inclusions as well

convergence rates with inexact non expansive operators in this paper we present a convergence rate analysis for the inexact krasnosel ski mann iteration built from non expansive operators the presented results include two main parts we first establish the global pointwise and ergodic iteration complexity bounds then under a metric sub regularity assumption we establish a local linear convergence for the distance of the iterates to the set of fixed points the obtained results can be applied to analyze the convergence rate of various monotone operator splitting methods in the literature including the forward backward splitting the generalized forward backward the douglas rachford splitting alternating direction method of multipliers and primal dual splitting methods for these methods we also develop easily verifiable termination criteria for finding an approximate solution which can be seen as a generalization of the termination criterion for the classical gradient descent method we finally develop a parallel analysis for the non stationary krasnosel ski mann iteration

second order accurate volume of fluid algorithms for tracking material interfaces we introduce two new volume of fluid interface reconstruction algorithms and compare the accuracy of these algorithms to four other widely used volume of fluid interface reconstruction algorithms we find that when the interface is smooth e g continuous with two continuous derivatives the new methods are second order accurate and the other algorithms are first order accurate we propose a design criteria for a volume of fluid interface reconstruction algorithm to be second order accurate namely that it reproduce lines in two space dimensions or planes in three space dimensions exactly we also introduce a second order unsplit volume of fluid advection algorithm that is based on a second order finite difference method for scalar conservation laws due to bell dawson and shubin we test this advection algorithm by modeling several different interface shapes propagating in two simple incompressible flows and compare the results with the standard second order operator split advection algorithm although both methods are second order accurate when the interface is smooth we find that the unsplit algorithm exhibits noticeably better resolution in regions where the interface has discontinuous derivatives such as at corners

a proximal method for composite minimization we consider minimization of functions that are compositions of convex or prox regular functions possibly extended valued with smooth vector functions a wide variety of important optimization problems fall into this framework we describe an algorithmic framework based on a subproblem constructed from a linearized approximation to the objective and a regularization term properties of local solutions of this subproblem underlie both a global convergence result and an identification property of the active manifold containing the solution of the original problem preliminary computational results on both convex and nonconvex examples are promising

douglas rachford splitting for nonconvex optimization with application to nonconvex feasibility problems we adapt the douglas rachford dr splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem while the convergence properties of the method for convex problems have been well studied far less is known in the nonconvex setting in this paper for the direct adaptation of the method to minimize the sum of a proper closed function g and a smooth function f with a lipschitz continuous gradient we show that if the step size parameter is smaller than a computable threshold and the sequence generated has a cluster point then it gives a stationary point of the optimization problem convergence of the whole sequence and a local convergence rate are also established under the additional assumption that f and g are semi algebraic we also give simple sufficient conditions guaranteeing the boundedness of the sequence generated we then apply our nonconvex dr splitting method to finding a point in the intersection of a closed convex set c and a general closed set d by minimizing the squared distance to c subject to d we show that if either set is bounded and the step size parameter is smaller than a computable threshold then the sequence generated from the dr splitting method is actually bounded consequently the sequence generated will have cluster points that are stationary for an optimization problem and the whole sequence is convergent under an additional assumption that c and d are semi algebraic we achieve these results based on a new merit function constructed particularly for the dr splitting method our preliminary numerical results indicate that our dr splitting method usually outperforms the alternating projection method in finding a sparse solution of a linear system in terms of both the solution quality and the number of iterations taken

issues related to least squares finite element methods for the stokes equations least squares finite element methods have become increasingly popular for the approximate solution of first order systems of partial differential equations here after a brief review of some existing theories a number of issues connected with the use of such methods for the velocity vorticity pressure formulation of the stokes equations in two dimensions in realistic settings are studied through a series of computational experiments finite element spaces that are not covered by existing theories are considered included in these are piecewise linear approximations for the velocity mixed boundary conditions which are also not covered by existing theories are also considered as is enhancing mass conservation next problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved a conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh dependent weights in the least squares functional almost always improves the accuracy of the approximations concluding remarks concerning three dimensional problems the nonlinear navier stokes equations and the conditioning of the discrete systems are provided

computational procedures for topological shape design abstract this research describes a method for the design of general shapes that is able to change the shape topology during the design process and can handle geometric and behavioral constraints expressed as algebraic or partial differential equations the proposed method uses an iterative process that alternates between formulating and solving a sequence of numerical optimization problems at each iteration a design parameterization is generated and a corresponding domain discretization is then derived automatically from this design parameterization a numerical optimization problem is then formulated by expressing the objective and the problem constraints in terms of the design variables the results of this optimization update the shape and the resulting shape then initiates a new design iteration the process continues until convergence the parameterization proposed is based on a geometric abstraction called the skeleton and leads to a relatively small number of design variables the parameterization is adaptive to shape evolutions guarantees shape integrity during the design and allows the redefinition of the objective and constraints in terms of adaptively generated design variables the skeleton also forms the basis for automatic domain discretization parameterized by the design variables this parameterized discretization provides efficient means for evaluating pde constraints and computing their gradients the method overcomes some of the problems associated with current shape design procedures including boundary variation methods and homogeniz ation methods computational examples with geometric and pde constraints are presented to demonstrate the capabilities of the proposed shape design method in generating general geometries and topologies

a lattice boltzmann approach for free surface flow simulations on non uniform block structured grids in this paper we present a hybrid volume of fluid based algorithm for the simulation of free surface flow problems for the solution of the flow field the lattice boltzmann method is used the additional advection equation for the volume of fluid vof fill level is discretized with a classical finite volume method for the interface reconstruction a piecewise linear interface reconstruction in 3d has been implemented the free surface tracking algorithm is embedded into the 3d non uniform lattice boltzmann based solver virtualfluids freudiger et al 2009 1 freudiger 2009 2 the advection algorithm is verified and validated with well known advection test cases for the validation of the free surface algorithm we present simulations of a breaking dam benchmark

least squares finite element methods for the elasticity problem abstract a new first order system formulation for the linear elasticity problem in displacement stress form is proposed the formulation is derived by introducing additional variables of derivatives of the displacements whose combinations represent the usual stresses standard and weighted least squares finite element methods are then applied to this extended system these methods offer certain advantages such as that they need not satisfy the inf sup condition which is required in the mixed finite element formulation that a single continuous piecewise polynomial space can be used for the approximation of all the unknowns that the resulting algebraic systems are symmetric and positive definite and that accurate approximations of the displacements and the stresses can be obtained simultaneously with displacement boundary conditions it is shown that both methods achieve optimal rates of convergence in the h 1 norm and in the l 2 norm for all the unknowns numerical experiments with various poisson ratios are given to demonstrate the theoretical error estimates

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

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

p version least squares finite element formulation for axisymmetric incompressible non newtonian fluid flow abstract the paper presents a p version least squares finite element formulation lsfef for axisymmetric incompressible non newtonian fluid flow the dimensionless form of the equations describing the fluid motion are cast into a set of first order coupled partial differential equations involving non newtonian stresses as auxiliary variables the pressure velocities primary variables and the stresses auxiliary variables are interpolated using equal order c0 continuity p version hierarchical approximation functions the least squares functional or error functional is constructed using the system of coupled first order non linear partial differential equations integrated sum of squares of the errors resulting from the individual equations for the entire discretization without linearization approximations or assumptions the minimization of this least squares error functional results in finding a solution vector for which the partial derivatives of the error functional with respect to the nodal degrees of freedom becomes zero this is accomplished by using newton s method with a line search the paper presents an implementation of the power law model for non newtonian viscosity numerical examples are presented to demonstrate the convergence characteristics and the accuracy of the proposed formulation

nonconvex notions of regularity and convergence of fundamental algorithms for feasibility problems we consider projection algorithms for solving nonconvex feasibility problems in euclidean spaces of special interest are the method of alternating projections ap and the douglas rachford algorithm dr in the case of convex feasibility firm nonexpansiveness of projection mappings is a global property that yields global convergence of ap and for consistent problems dr a notion of local subfirm nonexpansiveness with respect to the intersection is introduced for consistent feasibility problems this together with a coercivity condition that relates to the regularity of the collection of sets at points in the intersection yields local linear convergence of ap for a wide class of nonconvex problems and even local linear convergence of nonconvex instances of the dr algorithm

an ellipsoidal branch and bound algorithm for global optimization a branch and bound algorithm is developed for global optimization branching in the algorithm is accomplished by subdividing the feasible set using ellipses lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate a ball approximation algorithm obtained by generalizing of a scheme of lin and han is used to solve the convex relaxation of the original problem the ball approximation algorithm is compared to sedumi as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal constraints

smooth methods of multipliers for complementarity problems this paper describes several methods for solving nonlinear complementarity problems a general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem obtaining primal dual and primal dual formulations we derive bregman function based generalized proximal algorithms for each of these formulations generating three classes of complementarity algorithms the primal class is well known the dual class is new and constitutes a general collection of methods of multipliers or augmented lagrangian methods for complementarity problems in a special case it corresponds to a class of variational inequality algorithms proposed by gabay by appropriate choice of bregman function the augmented lagrangian subproblem in these methods can be made continuously differentiable the primal dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems

an element by element strategy for non linear shape optimization a strategy for the efficient solution of non linear shape optimization problems is developed this strategy employs an integrated element by element approach to the solution of the governing partial differential equations and more particularly to the computation of the necessary gradients of the objective function and constraints using an adjoint formulation this proves to be a very efficient strategy and also is relatively easy to implement because the local effect of design changes can be exploited the method is tested with an application involving the design of the shape of electromagnet poles in order to obtain a desired field in the interpolar region

multiscale analytical sensitivity analysis for composite materials we describe a methodology aimed at determining the sensitivity of the global structural behaviour such deformation or vibration modes with respect to the local characteristics such as material constants of micro constituents an analytical gradient computation which involves the direct differentiation of the multiple scale strong forms with respect to the design parameters is developed comparison of the multiple scale sensitivity analysis masa to the central finite difference cfd approximation in terms of accuracy and computation efficiency is carried out we demonstrate the robustness of the masa approach compared to the cfd approximation which has been found to be highly sensitive to the choice of the step size whose optimal value is problem dependent copyright 2001 john wiley sons ltd

on mass conservation in least squares methods we compare three least squares finite element reformulations of the stokes equations paying particular attention to mass conservation the first problem we approximate has a simple analytical solution over a convex region even for this simple problem without special treatment of the conservation of mass term very poor numerical solutions may result sufficiently weighting this term leads to a dramatic improvement in the results over a range of test problems

alternating projections and douglas rachford for sparse affine feasibility the problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an np complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations in this work we consider elementary methods based on projections for solving a sparse feasibility problem without employing convex heuristics it has been shown recently that locally the fundamental method of alternating projections must converge linearly to a solution to the sparse feasibility problem with an affine constraint in this paper we apply different analytical tools that allow us to show global linear convergence of alternating projections under familiar constraint qualifications these analytical tools can also be applied to other algorithms this is demonstrated with the prominent douglas rachford algorithm where we establish local linear convergence of this method applied to the sparse affine feasibility problem

numerical methods for tracking interfaces an overview is given of special numerical methods for tracking discontinuous fronts and interfaces these methods include surface tracking methods based on connected marker points along the interface volume tracking methods that track the volume occupied by the solution regions bounded by the interfaces and moving mesh methods where the underlying mesh is aligned and moved with the interface the pros and cons of the current methods are discussed and a new method is proposed that overcomes some of the difficulties encountered in approximating equations with multiple interacting interfaces

space time discontinuous galerkin method for nonlinear water waves a space time discontinuous galerkin dg finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented the space time dg method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolution the algorithm is higher order accurate both in space and time and closely related to an arbitrary lagrangian eulerian ale approach a detailed derivation of the numerical algorithm is given including an efficient procedure to solve the nonlinear algebraic equations resulting from the space time discretization numerical examples are shown on a series of model problems to demonstrate the accuracy and capabilities of the method

a cartesian grid projection method for the incompressible euler equations in complex geometries many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow here we present a method for calculating time dependent incompressible inviscid flow which combines a projection method with a cartesian grid approach for representing geometry in this approach the body is represented as an interface embedded in a regular cartesian mesh the advection step is based on a cartesian grid algorithm for compressible flow in which the discretization of the body near the flow uses a volume of fluid representation a redistribution procedure is used to eliminate time step restrictions due to small cells where the boundary intersects the mesh the projection step uses an approximate projection based on a cartesian grid method for potential flow the method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells convergence results are given for the projection itself and for the time dependent algorithm in two dimensions the method is also demonstrated on flow past a half cylinder with vortex shedding

quasi least squares finite element method for steady flow and heat transfer with system rotation two quasi least squares finite element schemes based on l2 inner product are proposed to solve a steady navier stokes equations coupled to the energy equation by the boussinesq approximation and augmented by a coriolis forcing term to account for system rotation the resulting nonlinear systems are linearized around a characteristic state resulting in linearized least squares models that yield algebraic systems with symmetric positive definite coefficient matrices existence of solutions are investigated and a priori error estimates are obtained the performance of the formulation is illustrated by using a direct iteration procedure to treat the nonlinearities and shown theoretical convergent rate for general initial guess

space time discontinuous galerkin finite element method for two fluid flows a novel numerical method for two fluid flow computations is presented which combines the space time discontinuous galerkin finite element discretization with the level set method and cut cell based interface tracking the space time discontinuous galerkin stdg finite element method offers high accuracy an inherent ability to handle discontinuities and a very local stencil making it relatively easy to combine with local hp refinement the front tracking is incorporated via cut cell mesh refinement to ensure a sharp interface between the fluids to compute the interface dynamics the level set method lsm is used because of its ability to deal with merging and breakup also the lsm is easy to extend to higher dimensions small cells arising from the cut cell refinement are merged to improve the stability and performance the interface conditions are incorporated in the numerical flux at the interface and the stdg discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative the numerical method is applied to one and two dimensional two fluid test problems using the euler equations

local error estimation and adaptive remeshing scheme for least squares mixed finite elements following the theme of our previous work on least squares finite elements 10 28 we describe an adaptive remeshing scheme using local residuals as the error indicator this choice of indicator is natural and exact at the element level in the norm associated with the corresponding least squares statement the remeshing strategy applied here involves mesh enrichment by point insertion in a delaunay scheme several refined grids and error plots are included for a representative model elliptic boundary value problem

linear convergence and metric selection for douglas rachford splitting and admm recently several convergence rate results for douglas rachford splitting and the alternating direction method of multipliers admm have been presented in the literature in this paper we show global linear convergence rate bounds for douglas rachford splitting and admm under strong convexity and smoothness assumptions we further show that the rate bounds are tight for the class of problems under consideration for all feasible algorithm parameters for problems that satisfy the assumptions we show how to select step size and metric for the algorithm that optimize the derived convergence rate bounds for problems with a similar structure that do not satisfy the assumptions we present heuristic step size and metric selection methods

p version least squares finite element formulation of burgers equation a p version least squares finite element formulation for non linear problems is presented and applied to the steady state one dimensional burgers equation the second order equation is recast as a set of first order equations which permit the use of c0 elements the primary and auxiliary variables are approximated using equal order p version hierarchical approximation functions the system of non linear simultaneous algebraic equations resulting from the least squares process is solved using newton s method with a line search the use of exact and reduced quadrature rules is investigated and the results are compared the formulation is found to produce excellent results when the exact integration rule is used the combination of least squares finite element formulation and p version works extremely well for burgers equation and appears to have great potential in fluid dynamics problems

computational method for optimization of structural shapes in the direct method of shape optimum design mesh regridding is a necessary step in avoiding the distortion of finite element grids the effect of internal nodal movements on the shape design sensitivity associated with the regridding process has been carefully examined in this paper it shows that to maintain computational accuracy the effects of internal nodal movement should be considered in shape design sensitivity analysis to take the internal nodal movement into account the numerical accuracy of the speed method is investigated a new approach which constructs the velocity field over the element domain is presented although it takes more computational time the new approach does improve the accuracy of shape design sensitivity

inexact variants of the proximal point algorithm without monotonicity this paper studies convergence properties of inexact variants of the proximal point algorithm when applied to a certain class of nonmonotone mappings the presented algorithms allow for constant relative errors in the line of the recently proposed hybrid proximal extragradient algorithm the main convergence result extends a recent work of the second author where exact solutions for the proximal subproblems were required we also show that the linear convergence property is preserved in the case when the inverse of the operator is locally lipschitz continuous near the origin as an application we give a convergence analysis for an inexact version of the proximal method of multipliers for a rather general family of problems which includes variational inequalities and constrained optimization problems

a physics based parameterization method for shape optimization a physical approach is presented to provide shape modification of nurbs curve the movement of the control polygon of nurbs is simulated by a physical frame structure and the curve deformation is then carried out through changes of the mode coordinates which are derived from the eigenvectors of the finite element equations of the physical frame system the shape of the design boundaries is described by a linear combination of the mode shape vectors based on the presented design variable representation and p version finite element analysis an integrated approach to structural shape optimization is discussed a method is developed that enables the description of the shape to be adaptively refined during the shape optimization process numerical examples are used to demonstrate the applicability of this approach to structural shape optimization

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

proximal methods for cohypomonotone operators conditions are given for the viability and the weak convergence of an inexact relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a hilbert space in turn new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

set regularities and feasibility problems we synthesize and unify notions of regularity both of individual sets and of collections of sets as they appear in the convergence theory of projection methods for consistent feasibility problems several new characterizations of regularities are presented which shed light on the relations between seemingly different ideas and point to possible necessary conditions for local linear convergence of fundamental algorithms

analysis of a least squares finite element method for the circular arch problem1this work was supported in part by nsc grant 88 2119 m 214 001 taiwan 1 the stability and convergence of a least squares finite element method for the circular arch problem with shear deformation in a first order system formulation are investigated it is shown that the least squares finite element approximations are stable and convergent in a natural energy norm associated with the least squares functional for the shallow arch case the optimal order of convergence in the h 1 norm for all the unknowns can be achieved uniformly with respect to the small thickness parameter and thus the locking phenomenon does not occur in this case a simple and sharp a posteriori error estimator is also addressed

proximal methods in reflexive banach spaces without monotonicity abstract we introduce the concept of hypomonotone point to set operators in banach spaces with respect to a regularizing function this notion coincides with the one given by rockafellar and wets in hilbertian spaces when the regularizing function is the square of the norm we study the associated proximal mapping which leads to a hybrid proximal extragradient and proximal projection methods for nonmonotone operators in reflexive banach spaces these methods allow for inexact solution of the proximal subproblems with relative error criteria we then consider the notion of local hypomonotonicity and propose localized versions of the algorithms which are locally convergent

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

analysis of the l2 least squares finite element method for the velocity vorticity pressure stokes equations with velocity boundary conditions a theoretical analysis of the l 2 least squares finite element method lsfem for solving the stokes equations in the velocity vorticity pressure vvp first order system formulation with the dirichlet velocity boundary condition is given the least squares energy functional is defined to be the sum of the squared l 2 norms of the residuals in the partial differential equations weighted appropriately by the viscosity constant n it is shown that with many advantages the method is stable and convergent without requiring extra smoothness of the exact solution and the piecewise linear finite elements can be used to approximate all the unknowns furthermore with respect to the order of approximation for smooth exact solutions the method exhibits an optimal rate of convergence in the h 1 norm for velocity and a suboptimal rate of convergence in the l 2 norm for vorticity and pressure some numerical experiments in two and three dimensions are given which confirm the a priori error estimates since the boundary of the bounded domain under consideration is polygonal in r 2 or polyhedral in r 3 instead of c 1 smooth the authors adopt the more direct technique of bramble pasciak and cai manteuffel mccormick which departs from the agmon douglis nirenberg theory in showing the coercivity bound of the least squares functional

least squares finite element approximations to the timoshenko beam problem in this paper a least squares finite element method for the timoshenko beam problem is proposed and analyzed the method is shown to be convergent and stable without requiring extra smoothness of the exact solutions for sufficiently regular exact solutions the method achieves optimal order of convergence in the h 1 norm for all the unknowns displacement rotation shear moment uniformly in the small parameter which is generally proportional to the ratio of thickness to length thus the locking phenomenon disappears as the parameter tends to zero a sharp a posteriori error estimator which is exact in the energy norm and equivalent in the h 1 norm is also briefly discussed

finding best approximation pairs relative to a convex and prox regular set in a hilbert space we study the convergence of an iterative projection reflection algorithm originally proposed for solving what are known as phase retrieval problems in optics there are two features that frustrate any analysis of iterative methods for solving the phase retrieval problem nonconvexity and infeasibility the algorithm that we developed called relaxed averaged alternating reflections raar was designed primarily to address infeasibility though our strategy has advantages for nonconvex problems as well in the present work we investigate the asymptotic behavior of the raar algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a hilbert space with empty intersection and for the problem of finding points that achieve a local minimum distance between one closed convex set and a closed prox regular set also possibly nonintersecting the nonconvex theory includes and expands prior results limited to convex sets with nonempty intersection to place the raar algorithm in context we develop parallel statements about the standard alternating projections algorithm and gradient descent all of the various algorithms are unified as instances of iterated averaged alternating proximal reflectors applied to a sum of regularized maximal monotone mappings

a perspective on adaptive modeling and meshing am m the purpose of this study is to develop an integrated approach to adaptive modeling in conjunction with adaptive meshing this involves consideration of the modeling error from various sources and the discretization error in the numerical scheme adapting the model to control error due to geometry boundary data constitutive relations and model parameters can be integrated with adaptive mesh control to improve numerical reliability other factors such as stabilization by artificial dissipation and regularization are considered a class of hybrid algorithms are developed to combine these approaches

exact a posteriori error analysis of the least squares finite element method1this work was supported by nsc under grant nsc87 2115 m 009 005 taiwan 1 a residual type a posteriori error estimator is presented for the least squares finite element method the estimator is proved to equal the exact error in a norm induced by some least squares functional the error indicator of each element is equal to the exact error norm restricted to the element as well in other words the estimator is perfectly effective and reliable for error control and for adaptive mesh refinement the exactness property requires virtually no assumptions on the regularity of the solution and on the finite element order in the approximation or in the estimation the least squares method is in a very general setting that applies to various linear boundary value problems such as the elliptic systems of first order and of even order and the mixed type partial differential equations numerical results are given to demonstrate the exactness

on the douglas rachford algorithm the douglas rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators the behaviour of the algorithm remains mysterious in the general inconsistent case i e when the sum problem has no zeros however more than a decade ago it was shown that in the possibly inconsistent convex feasibility setting the shadow sequence remains bounded and its weak cluster points solve a best approximation problem in this paper we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting in fact a more general sufficient condition for the weak convergence in the general case is presented our proof relies on a new convergence principle for fejer monotone sequences numerous examples illustrate our results

generalized eckstein bertsekas proximal point algorithm based on a maximal monotonicity design a general design for the eckstein bertsekas proximal point algorithm using the notion of the a maximal monotonicity is developed convergence analysis for the generalized eckstein bertsekas proximal point algorithm in the context of solving a class of nonlinear inclusion problems is explored some auxiliary results of interest involving a maximal monotone mappings are also included the obtained results generalize investigations on general maximal monotonicity and beyond

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

least squares spectral element solution of incompressible navier stokes equations with adaptive refinement least squares spectral element solution of steady two dimensional incompressible flows are obtained by approximating velocity pressure and vorticity variable set on gauss lobatto legendre nodes constrained approximation method is used for h and p type nonconforming interfaces of quadrilateral elements adaptive solutions are obtained using a posteriori error estimates based on least squares functional and spectral coefficient effective use of p refinement to overcome poor mass conservation drawback of least squares formulation and successful use of h and p refinement together to solve problems with geometric singularities are demonstrated capabilities and limitations of the developed code are presented using kovasznay flow flow past a circular cylinder in a channel and backward facing step flow

isoparametric hermite elements isoparametric hermite elements are created using bogner fox schmit rectangles on a reference domain and mapping these numerically onto the computational domain the difficulties involved in devising explicit c1 shape functions for isoparametric elements are thus avoided and the resulting elements have all the benefits of full c1 continuity the simplicity of the bogner fox schmit element and the geometrical flexibility expected from higher order isoparametric elements the numerical mapping consists in the finite element solution of a linear boundary value problem which is inexpensive and is carried out as a preprocessing operation the required derivatives of the mapping then being supplied to the main analysis as data some care is required in defining the differential boundary conditions and guidance on this is provided examples are given showing the success of the mapping procedure and the use of the resulting elements in the solution of some boundary value problems the numerical results confirm a convergence analysis provided for the new isoparametric hermite element

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

adaptive refinement for least squares finite elements with element by element conjugate gradient solution an error analysis and an adaptive refinement scheme are developed for use with a class of least squares finite element methods the error indicators for the refinement procedure are based on element residuals which are calculated as part of the least squares method the solution scheme used in the supporting numerical studies employs an elment by element conjugate gradient algorithm consequently storage and computation are not influenced by nodal numbering

regular article a computational model for suspended large rigid bodies in 3d unsteady viscous flows a 3d numerical model for computing large rigid objects suspended in fluid flow has been developed rather than calculating the surface pressure upon the solid body we evaluate the net force and torque based on a volume force formulation the total effective force is obtained by summing up the forces at the eulerian grids occupied by the rigid body the effects of the moving bodies are coupled to the fluid flow by imposing the velocity field of the bodies to the fluid a poisson equation is used to compute the pressure over the whole domain the objects are identified by color functions and calculated by the ppm scheme and a tangent function transformation which scales the transition region of the computed interface to a compact thickness the model is then implemented on a parallel computer of distributed memory and validated with stokes and low reynolds number flows

perturbation of error bounds our aim in the current article is to extend the developments in kruger et al siam j optim 20 6 3280 3296 2010 doi 10 1137 100782206 and more precisely to characterize in the banach space setting the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations we propose new concepts of arbitrary convex and linear perturbations of the given function defining the system under consideration which turn out to be a useful tool in our analysis the characterizations of error bounds for families of perturbations can be interpreted as estimates of the radius of error bounds the definitions and characterizations are illustrated by examples