achieving minimum length scale in topology optimization using nodal design variables and projection functions a methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology the projection is made via mesh independent functions that are based upon the minimum length scale a simple linear projection scheme and a non linear scheme using a regularized heaviside step function to achieve nearly 0 1 solutions are examined the new approach is demonstrated on the minimum compliance problem and the popular simp method is used to penalize the stiffness of intermediate volume fraction elements solutions are shown to meet user defined length scale criterion without additional constraints penalty functions or sensitivity filters no instances of mesh dependence or checkerboard patterns have been observed copyright 2004 john wiley sons ltd

automated derivation of the adjoint of high level transient finite element programs in this paper we demonstrate a new technique for deriving discrete adjoint and tangent linear models of a finite element model the technique is significantly more efficient and automatic than standard algorithmic differentiation techniques the approach relies on a high level symbolic representation of the forward problem in contrast to developing a model directly in fortran or c high level systems allow the developer to express the variational problems to be solved in near mathematical notation as such these systems have a key advantage since the mathematical structure of the problem is preserved they are more amenable to automated analysis and manipulation the framework introduced here is implemented in a freely available software package named dolfin adjoint based on the fenics project our approach to automated adjoint derivation relies on run time annotation of the temporal structure of the model and employs the fenics finite element form compiler to automatically generate the low level co

an infeasible bundle method for nonsmooth convex constrained optimization without a penalty function or a filter global convergence in constrained optimization algorithms has traditionally been enforced by the use of parametrized penalty functions recently the filter strategy has been introduced as an alternative at least part of the motivation for using filter methods consists of avoiding the need for estimating a suitable penalty parameter which is often a delicate task in this paper we demonstrate that the use of a parametrized penalty function in nonsmooth convex optimization can be avoided without using the relatively complex filter methods we propose an approach which appears to be more direct and easier to implement in the sense that it is closer in spirit and structure to the well developed unconstrained bundle methods preliminary computational results are also reported

general topology optimization method with continuous and discrete orientation design using isoparametric projection a general topology optimization method which is capable of simultaneous design of density and orientation of anisotropic material is proposed by introducing orientation design variables in addition to the density design variable in this work the cartesian components of the orientation vector are utilized as the orientation design variables the proposed method supports continuous orientation design which is out of the scope of discrete material optimization approaches as well as design using discrete angle sets the advantage of this approach is that vector element representation is less likely to fail into local optima because it depends less on designs of former steps especially compared with using the angle as a design variable continuous fiber angle optimization by providing a flexible path from one angle to another with relaxation of orientation design space an additional advantage is that it is compatible with various projection or filtering methods such as sensitivity filters and density filters because it is free from unphysical bound or discontinuity such as the one at theta 2 pi and theta 0 seen with direct angle representation one complication of cartesian component representation is the point wise quadratic bound of the design variables that is each pair of element values has to reside in a given circular bound to overcome this issue we propose an isoparametric projection method which transforms box bounds into circular bounds by a coordinate transformation with isoparametric shape functions without having the singular point that is seen at the origin with polar coordinate representation a new topology optimization method is built by taking advantage of the aforementioned features and modern topology optimization techniques several numerical examples are provided to demonstrate its capability copyright c 2014 john wiley sons ltd

box constrained quadratic programming with proportioning and projections two new closely related concepts are introduced that depend on a positive constant gamma an iteration is proportional if the norm of violation of the kuhn tucker conditions at active variables does not excessively exceed the norm of the part of the gradient that corresponds to free variables while a progressive direction determines a descent direction that enables the released variables to move far enough from the boundary in a step called proportioning an algorithm that uses the conjugate gradient method to explore the face of the region defined by the current iterate until a disproportional iteration is generated is proposed it then changes the face by means of the progressive direction it is proved that for strictly convex problems the proportioning is a spacer iteration so that the algorithm converges to the solution if the solution is nondegenerate then the algorithm finds the solution in a finite number of steps moreover a simple lower bound on gamma is given to ensure finite termination even for problems with degenerate solutions the theory covers a class of algorithms allowing many constraints to be added or dropped at a time and accepting approximate solutions of auxiliary problems preliminary numerical results are promising

augmented lagrangians with adaptive precision control for quadratic programming with simple bounds and equality constraints in this paper we discuss a specialization of the augmented lagrangian type algorithm of conn gould and toint to the solution of strictly convex quadratic programming problems with simple bounds and equality constraints the new feature of the presented algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm which yields a rate of convergence that does not have any term that accounts for inexact solution of auxiliary problems moreover boundedness of the penalty parameter is achieved for the precision control used numerical experiments illustrate the efficiency of the presented algorithm and encourage its usage

a reflective newton method for minimizing a quadratic function subject to bounds on some of the variables we propose a new algorithm a reflective newton method for the minimization of a quadratic function of many variables subject to upper and lower bounds on some of the variables this method applies to a general indefinite quadratic function for which a local minimizer subject to bounds is required and is particularly suitable for the large scale problem our new method exhibits strong convergence properties global and quadratic convergence and appears to have significant practical potential strictly feasible points are generated experimental results on moderately large and sparse problems support the claim of practicality for large scale problems

relaxation methods for monotropic programs we propose a dual descent method for the problem of minimizing a convex possibly nondifferentiable separable cost subject to linear constraints the method has properties reminiscent of the gauss seidel method in numerical analysis and uses the e complementary slackness mechanism introduced in bertsekas hosein and tseng 1987 to ensure finite convergence to near optimality as special cases we obtain the methods in bertsekas hosein and tseng 1987 for network flow programs and the methods in tseng and bertsekas 1987 for linear programs

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

stationarity and superlinear convergence of an algorithm for univariate locally lipschitz constrained minimization this paper introduces an algorithm for minimizing a single variable locally lipschitz function subject to a like function being nonpositive the method combines polyhedral and quadratic approximation a new type of penalty technique and a safeguard in such a way as to give convergence to a stationary point the convergence is shown to be superlinear under somewhat stronger assumptions that allow both nonsmooth and nonconvex cases the algorithm can be an effective subroutine for solving line search subproblems called for by multivariable optimization algorithms

relaxation methods for problems with strictly convex costs and linear constraints consider the problem of minimizing a strictly convex possibly nondifferentiable and nonseparable cost subject to linear constraints we propose a dual coordinate ascent method for this problem that uses inexact line search and either essentially cyclic or gauss southwell order of coordinate relaxation we show under very weak conditions that this method generates a sequence of primal vectors converging to the optimal primal solution under an additional regularity assumption and assuming that the effective domain of the cost function is polyhedral we show that a related sequence of dual vectors converges in cost to the optimal cost if the constraint set has an interior point in the effective domain of the cost function then this sequence of dual vectors is bounded and each of its limit points is an optimal dual solution when the cost function is strongly convex we show that the order of coordinate relaxation can become progressively more chaotic these results significantly improve upon those obtained previously

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

optimal design for non newtonian flows using a topology optimization approach we study non newtonian effects on the layout and geometry of flow channels using a material distribution based topology optimization approach the flow is modeled with the single relaxation hydrodynamic lattice boltzmann method and the shear dependence of viscosity is included through the carreau yasuda model for non newtonian fluids to represent the viscosity of blood in this model we use non newtonian similarity further we introduce a scaling to decrease the effects of the non newtonian model in porous regions in order to stabilize the coupling of the lbm porosity and non newtonian flow models for the resulting flow model we derive the non newtonian sensitivity analysis for steady state conditions and illustrate the non newtonian effect on channel layouts for a 2d dual pipe design problem at different reynolds numbers

a high level programming language implementation of topology optimization applied to steady state navier stokes flow we present a versatile high level programming language implementation of non linear topology optimization our implementation is based on the commercial software package femlab and it allows a wide range of optimization objectives to be dealt with easily we exemplify our method by studies of steady state navier stokes flow problems thus extending the work by borrvall and petersson on topology optimization of fluids in stokes flow int j num meth fluids 2003 41 77 107 we analyse the physical aspects of the solutions and how they are affected by different parameters of the optimization algorithm a complete example of our implementation is included as femlab code in an appendix copyright 2005 john wiley sons ltd

separating plane algorithms for convex optimization the equivalent formulation of a convex optimization problem is the computation of a value of a conjugate function at the origin the latter can be achieved by approximation of the epigraph of the conjugate function around the origin and gradual refinement of the approximation this yields a generic algorithm of convex optimization which transforms into some well known techniques when certain strategies of approximation are employed it also suggests new algorithmic approaches with promising computational experience and provides a uniform treatment of constrained and unconstrained optimization

a bracketing technique to ensure desirable convergence in univariate minimization this paper gives a general safeguarded bracketing technique for minimizing a function of a single variable in certain cases the technique guarantees convergence to a stationary point and when combined with sequential polynomial and or polyhedral fitting algorithms preserves rapid convergence each bracket has an interior point whose function value does not exceed those of the two bracket endpoints the safeguarding technique consists of replacing the fitting algorithm s iterate candidate by a close point whose distance from the three bracket points exceeds a positive multiple of the square of the bracket length it is shown that a given safeguarded quadratic fitting algorithm converges in a certain better than linear manner with respect to the bracket endpoints for a strongly convex twice continuously differentiable function

relaxation methods for problems with strictly convex separable costs and linear constraints we consider the minimization problem with strictly convex possibly nondifferentiable separable cost and linear constraints the dual of this problem is an unconstrained minimization problem with differentiable cost which is well suited for solution by parallel methods based on gauss seidel relaxation we show that these methods yield the optimal primal solution and under additional assumptions an optimal dual solution to do this it is necessary to extend the classical gauss seidel convergence results because the dual cost may not be strictly convex and may have unbounded level sets

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

a superlinearly convergent algorithm for one dimensional constrained minimization problems with convex functions this paper presents a globally and superlinearly convergent algorithm for solving one dimensional constrained minimization problems involving not necessarily smooth convex functions the constraint is handled by what can be interpreted as a new type of penalty method the algorithm does not require the objective function to be evaluated at infeasible points and it does not use constraint function values at feasible points the penalty parameter is automatically generated by the algorithm via linear approximation of the constraint function as in the unconstrained case developed by lemarechal and the author the algorithm uses a step that is the shorter of a quadratic approximation step and a polyhedral approximation step here the latter is actually a penalized polyhedral step whose computation is well conditioned if the constraint satisfies a nondegeneracy assumption

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

an implementation of an algorithm for univariate minimization and an application to nested optimization this paper provides an efficient numerical algorithm for solving quite general single variable constrained minimization problems also given are a listing of the corresponding fortran subroutine pq1 an explanation of its general use and a description of its particular use in solving a practical resource allocation problem via a dual technique requiring nested optimization

a algorithm for convex minimization for convex minimization we introduce an algorithm based on space decomposition the method uses a bundle subroutine to generate a sequence of approximate proximal points when a primal dual track leading to a solution and zero subgradient pair exists these points approximate the primal track points and give the algorithm s or corrector steps the subroutine also approximates dual track points that are gradients needed for the method s newton predictor steps with the inclusion of a simple line search the resulting algorithm is proved to be globally convergent the convergence is superlinear if the primal dual track points and the objective s hessian are approximated well enough

topology optimization for turbulent flow with spalart allmaras model abstract in this research a new finite element fe based topology optimization to for turbulent flow was developed using the spalart allmaras s a equation which is one of the reynolds averaged navier stokes rans equations despite many innovative works on the subject of fluidic to it is rare to consider the influence of turbulent flow in to to consider the effect of complex turbulent fluid motion this research considered the s a model developed mainly for aerodynamic flows for a successful to one primary issue is modification of the s a turbulent model to account for topological evolution during the optimization process to address this issue we proposed the addition of penalization terms to the original governing equations to show the validity of the present approach and the effect of turbulent flow on optimal layouts some two dimensional benchmark designs studied for laminar flow were reconsidered by considering the effect of turbulent flow the eddy viscosity values were increased at some local regions due to the boussinesq hypothesis and naturally optimal layouts affected by the spatially varying viscosity were obtained in turbulent flow it was also found that the balance between the inertia force and the viscosity force plays an important role in topological designs

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

relaxation method for large scale linear programming using decomposition we propose a new decomposition method for large scale linear programming this method dualizes an arbitrary subset of the constraints and then maximizes the resulting dual functional by dual ascent the ascent directions are chosen from a finite set and are generated by a truncated version of the painted index algorithm of rockafellar central to this method is the novel notion of e complementary slackness e 0 0 1 which allows each lagrangian subproblem to be solved only approximately with o e accuracy and provides a lower bound of e 1 on the amount of improvement per dual ascent by dynamically adjusting e the subproblems can be solved with increasing accuracy we show that i the method terminates finitely provided that e and are bounded away from 0 and 1 respectively ii the final solution produced by the method is feasible and is within o e in cost of the optimal cost and iii the final solution produced by the method is optimal for all e sufficiently small

augmented lagrangian methods under the constant positive linear dependence constraint qualification two augmented lagrangian algorithms for solving kkt systems are introduced the algorithms differ in the way in which penalty parameters are updated possibly infeasible accumulation points are characterized it is proved that feasible limit points that satisfy the constant positive linear dependence constraint qualification are kkt solutions boundedness of the penalty parameters is proved under suitable assumptions numerical experiments are presented

a rapidly convergent five point algorithm for univariate minimization this paper presents an algorithm for minimizing a function of one variable which uses function but not derivative values at five points to generate each iterate it employs quadratic and polyhedral approximations together with a safeguard the basic method without the safeguard exhibits a type of better than linear convergence for certain piecewise twice continuously differentiable functions the safeguard guarantees convergence to a stationary point for very general functions and preserves the better than linear convergence of the basic method

an active set method for second order conic constrained quadratic programming we consider the minimization of a convex quadratic objective subject to second order cone constraints this problem generalizes the well studied bound constrained quadratic programming qp problem we propose a new two phase method in the first phase a projected gradient method is used to quickly identify the active set of cones and in the second phase newton s method is applied to rapidly converge given the subsystem of active cones computational experiments confirm that the conically constrained qp is solved more efficiently by our method than by a specialized conic optimization solver and more robustly than by general nonlinear programming solvers

a globally convergent stabilized sqp method sequential quadratic programming sqp methods are a popular class of methods for nonlinearly constrained optimization they are particularly effective for solving a sequence of related problems such as those arising in mixed integer nonlinear programming and the optimization of functions subject to differential equation constraints recently there has been considerable interest in the formulation of stabilized sqp methods which are specifically designed to handle degenerate optimization problems existing stabilized sqp methods are essentially local in the sense that both the formulation and analysis focus on the properties of the methods in a neighborhood of a solution a new sqp method is proposed that has favorable global convergence properties yet under suitable assumptions is equivalent to a variant of the conventional stabilized sqp method in the neighborhood of a solution the method combines a primal dual generalized augmented lagrangian function with a flexible line search to obtain a seque

a survey on the continuous nonlinear resource allocation problem our problem of interest consists of minimizing a separable convex and differentiable function over a convex set defined by bounds on the variables and an explicit constraint described by a separable convex function applications are abundant and vary from equilibrium problems in the engineering and economic sciences through resource allocation and balancing problems in manufacturing statistics military operations research and production and financial economics to subproblems in algorithms for a variety of more complex optimization models this paper surveys the history and applications of the problem as well as algorithmic approaches to its solution the most common techniques are based on finding the optimal value of the lagrange multiplier for the explicit constraint most often through the use of a type of line search procedure we analyze the most relevant references especially regarding their originality and numerical findings summarizing with remarks on possible extensions and future research 2006 elsevier b v all rights reserved

structural topology and shape optimization using a level set method with distance suppression scheme abstract in level set methods for structural topology and shape optimization the level set function gradients at the design interface need to be controlled in order to ensure stability of the optimization process one popular way to do this is to enforce the level set function to be a signed distance function by periodically using initialization schemes which is commonly known as re initialization however such re initialization schemes are time consuming as additional partial differential equations need to be solved in every iteration step furthermore the use of re initialization brings some undesirable problems for example it may move the zero level set away from the expected position this paper presents a level set method with distance suppression scheme for structural topology and shape optimization an energy functional is introduced into the level set equation to maintain the level set function to close to a signed distance function near the structural boundaries meanwhile forcing the level set function to be a constant at locations far away from the structural boundaries as a result the present method not only can avoid the need for re initialization but also can simplify the setting of the initial level set function the validity of the proposed method is tested on the mean compliance minimization problem and the compliant mechanisms synthesis problem different aspects of the proposed method are demonstrated on a number of benchmarks from the literature of structural optimization

filters in topology optimization based on helmholtz type differential equations the aim of this paper is to apply a helmholtz type partial differential equation as an alternative to standard density filtering in topology optimization problems previously this approach has been successfully applied as a sensitivity filter the usual filtering techniques in topology optimization require information about the neighbor cells which is difficult to obtain for fine meshes or complex domains and geometries the complexity of the problem increases further in parallel computing when the design domain is decomposed into multiple non overlapping partitions obtaining information from the neighbor subdomains is an expensive operation the proposed filter technique requires only mesh information necessary for the finite element discretization of the problem the main idea is to define the filtered variable implicitly as a solution of a helmholtz type differential equation with homogeneous neumann boundary conditions the properties of the filter are demonstrated for various 2d and 3d topology optimization problems in linear elasticity solved on serial and parallel computers copyright 2010 john wiley sons ltd

topology optimization based on a two dimensional swirl flow model of tesla type pump devices abstract tesla pumps are composed of rotating disks without blades and are based on the boundary layer effect i e viscous friction forces acting on the fluid and coand effect the tesla pump has various applications however the efficiency of its operation is quite low which makes room for the optimization of its design this way novel configurations of devices based on this working principle can be developed thus in this work a topology optimization formulation is proposed to optimize the rotor of a tesla type pump device by using a 2d swirl flow model which is a specific model that presents an axisymmetric flow with or without flow rotation around the axisymmetric axis the 2d swirl laminar fluid flow modeling is solved by using the finite element method a traditional material model is adopted by considering nodal design variables and is extended to take into account the fact that the optimization is performed in a rotating reference frame and the relative tangential velocity behaves differently than the other velocity components since the 2d swirl flow model is axisymmetric a multi objective function is defined in order to minimize energy dissipation and vorticity an extra term is proposed to the vorticity function to reduce the generation of grayscale results an interior point optimization algorithm ipopt is applied to solve the optimization problem numerical results taking into account different aspects of the design of the rotor of a tesla type device are presented

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

a derivative based bracketing scheme for univariate minimization and the conjugate gradient method abstract a derivative based scheme for univariate minimization is developed this scheme has a quadratic convergence rate and requires an evaluation of the function and its derivative in each iteration an application involving the conjugate gradient method is presented

on superlinear convergence in univariate nonsmooth minimization this note demonstrates a new result on superlinear convergence in nonsmooth univariate minimization it also gives other concepts of rapid convergence for minimization of functions that may have discontinuous derivatives

a constraint linearization method for nondifferentiable convex minimization this paper presents a readily implementable algorithm for solving constrained minimization problems involving possibly nonsmooth convex functions the constraints are handled as in the successive quadratic approximations methods for smooth problems an exact penalty function is employed for stepsize selection a scheme for automatic limitation of penalty growth is given global convergence of the algorithm is established as well as finite termination for piecewise linear problems numerical experience is reported

optimization based simulation of nonsmooth rigid multibody dynamics we present a time stepping method to simulate rigid multibody dynamics with inelastic collision contact and friction the method progresses with fixed time step without backtracking for collision and solves at every step a strictly convex quadratic program we prove that a solution sequence of the method converges to the solution of a measure differential inclusion we present numerical results for a few examples and we illustrate the difference between the results from our scheme and previous linear complementarity based time stepping schemes

an operator splitting algorithm for tikhonov regularized topology optimization in this work we investigate a tikhonov type regularization scheme to address the ill posedness of the classical compliance minimization problem we observe that a semi implicit discretization of the gradient descent flow for minimization of the regularized objective function leads to a convolution of the original gradient descent step with the green s function associated with the modified helmholtz equation the appearance of filtering in this update scheme is different from the current density and sensitivity filtering techniques in the literature the next iterate is defined as the projection of this provisional density onto the space of admissible density functions for a particular choice of projection mapping we show that the algorithm is identical to the well known forward backward splitting algorithm an insight that can be further explored for topology optimization also of interest is that with an appropriate choice of the projection parameter nearly all intermediate densities are eliminated in the optimal solution using the common density material models we show examples of near binary solutions even for large values of the regularization parameter

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

on the global convergence of an slp filter algorithm that takes eqp steps a global convergence proof is presented for a class of trust region filter type methods for nonlinear programming such methods are characterized by their use of the dominance concept of multiobjective optimization instead of a penalty parameter whose adjustment can be problematic the methods are based on successively solving linear programming subproblems for which effective software is readily available the methods also permit the use of steps calculated on the basis of an equality constrained quadratic programming model which enables rapid convergence to take place for problems in which second order information is important the proof technique is presented in a fairly general context allowing a range of specific algorithm choices associated with choosing the quadratic model updating the trust region radius and with feasibility restoration

global and superlinear convergence of an algorithm for one dimensional minimization of convex functions this paper studies an algorithm for minimizing a convex function based upon a combination of polyhedral and quadratic approximation the method was given earlier but without a good specification for updating the algorithm s curvature matrix here for the case of onedimensional minimization we provide a specification that insures convergence even in cases where the curvature scalar tends to zero or infinity under mild additional assumptions we show that the convergence is superlinear

on the convergence of the affine scaling algorithm the affine scaling algorithm first proposed by dikin is presently enjoying great popularity as a potentially effective means of solving linear programs an outstanding question about this algorithm concerns its convergence in the presence of degeneracy in this paper we give new convergence results for this algorithm that do not require any non degeneracy assumption on the problem in particular we show that if the stepsize choice of either dikin or barnes or vanderbei et al is used then the algorithm generates iterates that converge at least linearly with a convergence ratio of 1 beta sqrt n wheren is the number of variables and 0 1 is a certain stepsize ratio for one particular stepsize choice which is an extension of that of barnes we show that the cost of the limit point is within o 1 of the optimal cost and for sufficiently small roughly proportional to how close the cost of the nonoptimal vertices are to the optimal cost is exactly optimal we prove the latter result by using an unusual proof technique that of analyzing the ergodic convergence of the corresponding dual vectors for the special case of network flow problems with integer data we show that it suffices to take 1 6mc wherem is the number of constraints andc is the sum of the cost coefficients to attain exact optimality

a consistent relaxation of optimal design problems for coupling shape and topological derivatives in this article we introduce and analyze a general procedure for approximating a black and white shape and topology optimization problem with a density optimization problem allowing for the presence of grayscale regions our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem and on an interpolation scheme which endows the intermediate density regions with fictitious material properties under mild hypotheses on the smoothing operator and on the interpolation scheme we prove that the features of the approximate density optimization problem material properties objective function etc converge to their exact counterparts as the smoothing parameter vanishes in particular the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective these results shed new light on the connections between these two different notions of sensitivities for functions of the domain and they give rise to different numerical algorithms which are illustrated by several experiments

on the maximization of a concave quadratic function with box constraints a new method for maximizing a concave quadratic function with bounds on the variables is introduced the new algorithm combines conjugate gradients with gradient projection techniques as the algorithm of more and toraldo siam j optimization 1 1991 pp 93 113 and other well known methods do a new strategy for the decision of leaving the current face is introduced that makes it possible to obtain finite convergence even for a singular hessian and in the presence of dual degeneracy numerical experiments are presented

nonlinear programming and nonsmooth optimization by successive linear programming methods are considered for solving nonlinear programming problems using an exactl1 penalty function lp like subproblems incorporating a trust region constraint are solved successively both to estimate the active set and to provide a foundation for proving global convergence in one particular method second order information is represented by approximating the reduced hessian matrix and coleman conn steps are taken a criterion for accepting these steps is given which enables the superlinear convergence properties of the coleman conn method to be retained whilst preserving global convergence and avoiding the maratos effect the methods generalize to solve a wide range of composite nonsmooth optimization problems and the theory is presented in this general setting a range of numerical experiments on small test problems is described

on r linear convergence of semi monotonic inexact augmented lagrangians for bound and equality constrained quadratic programming problems with application new convergence results for a variant of the inexact augmented lagrangian algorithm smalbe z dostal an optimal algorithm for bound and equality constrained quadratic programming problems with bounded spectrum computing 78 2006 311 328 for the solution of strictly convex bound and equality constrained quadratic programming problems are presented the algorithm smalbe m presented here uses a fixed regularization parameter and controls the precision of the solution of auxiliary bound constrained problems by a multiple of the norm of violation of the equality constraints and a constant which is updated in order to enforce the increase of lagrangian function a nice feature of smalbe m is its capability to find an approximate solution of important classes of problems in a number of iterations that is independent of the conditioning of the equality constraints here we prove the r linear rate of convergence of the outer loop of smalbe m for any positive regularization parameter after the strong active constraints of the solution are identified the theoretical results are illustrated by solving two benchmarks including the contact problem of elasticity discretized by two million of nodal variables the numerical experiments indicate that the inexact solution of auxiliary problems in the inner loop results in a very small increase of the number of outer iterations as compared with the exact algorithm the results do not assume independent equality constraints and remain valid when the solution is dual degenerate

a boundary piecewise constant level set method for boundary control of eigenvalue optimization problems this paper investigates optimization of the least eigenvalue of d with the constraint of one dimension hausdorff measure of dirichlet boundary we propose the boundary piecewise constant level set bpcls method based on the regularity technique to combine two types of boundary conditions into a single robin boundary condition we derive the first variation of the least eigenvalue w r t the bpcls function and propose a penalty bpcls algorithm and an augmented lagrangian bpcls algorithm numerical results are reported for experiments on ellipse and l shape domains

stability of finite element models for distributed parameter optimization and topology design abstract we address a problem of numerical instability that is often encountered in finite element solutions of distributed parameter optimization and variable topology shape design problems we show that the cause of this problem is numerical rather than physical in nature we consider a two field distributed parameter optimization problem involving a design field and a response field and show that the optimization problem corresponds to a mixed variational problem an improper selection of the discrete function spaces for these two fields leads to grid scale anomalies in the numerical solutions to optimization problems similar to those that are sometimes encountered in mixed formulations of the stokes problem we present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models the general theoretical framework is specialized to analyze the stability of specific optimization problems and stability results for various mixed finite element models are presented we propose patch tests that are useful in identifying unstable elements

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

a quadratic approximation method for minimizing a class of quasidifferentiable functions an unconstrained nonlinear programming problem with nondifferentiabilities is considered the nondifferentiabilities arise from terms of the form max f 1 x f n x which may enter nonlinearly in the objective function local convex polyhedral upper approximations to the objective function are introduced these approximations are used in an iterative method for solving the problem the algorithm proceeds by solving quadratic programming subproblems to generate search directions approximate line searches ensure global convergence of the method to stationary points the algorithm is conceptually simple and easy to implement it generalizes efficient variable metric methods for minimax calculations

a quasi second order proximal bundle algorithm this paper introduces an algorithm for convex minimization which includes quasi newton updates within a proximal point algorithm that depends on a preconditioned bundle subalgorithm the method uses the hessian of a certain outer function which depends on the jacobian of a proximal point mapping which in turn depends on the preconditioner matrix and on a lagrangian hessian relative to a certain tangent space convergence is proved under boundedness assumptions on the preconditioner sequence

an e out of kilter method for monotropic programming the out of kilter method was first proposed by fulkerson 1961 for linear cost network flow and independently by minty 1961 for piecewise linear cost network flow and was then extended by rockafellar 1984 section 11k to piecewise linear cost monotropic programming we propose an extension of this method based on rockafellar s work and on ideas from e relaxation methods for convex cost network flow bertsekas et al 1997a de leone et al 1999 to monotropic programming in general the proposed method is amenable to a complexity analysis and its convergence is based on essentially a monotone decrease in vertical distance to the characteristic curve for each index

relaxation methods for network flow problems with convex arc costs we consider the standard single commodity network flow problem with both linear and strictly convex possibly nondifferentiable arc costs for the case where all arc costs are strictly convex we study the convergence of a dual gauss seide type relaxation method that is well suited for parallel computation we then extend this method to the case where some of the arc costs are linear as a special case we recover a relaxation method for the linear minimum cost network flow problem proposed in bertsekas 1 and bertsekas and tseng 2

topology optimization of regions of darcy and stokes flow this paper treats the topology optimization problem of obtaining an optimal layout of regions of darcy and stokes flow where the objective is the total potential power functional representing aver

relaxation methods for minimum cost ordinary and generalized network flow problems we propose a new class of algorithms for linear cost network flow problems with and without gains these algorithms are based on iterative improvement of a dual cost and operate in a manner that is reminiscent of coordinate ascent and gauss seidel relaxation methods we compare our coded implementations of these methods with mature state of the art primal simplex and primal dual codes and find them to be several times faster on standard benchmark problems and faster by an order of magnitude on large randomly generated problems our experiments indicate that the speedup factor increases with problem dimension

a multidimensional filter algorithm for nonlinear equations and nonlinear least squares we introduce a new algorithm for the solution of systems of nonlinear equations and nonlinear least squares problems that attempts to combine the efficiency of filter techniques and the robustness of trust region methods the algorithm is shown under reasonable assumptions to globally converge to zeros of the system or to first order stationary points of the euclidean norm of its residual preliminary numerical experience is presented that shows substantial gains in efficiency over the traditional monotone trust region approach

a variational level set method for the topology optimization of steady state navier stokes flow the smoothness of topological interfaces often largely affects the fluid optimization and sometimes makes the density based approaches though well established in structural designs inadequate this paper presents a level set method for topology optimization of steady state navier stokes flow subject to a specific fluid volume constraint the solid fluid interface is implicitly characterized by a zero level contour of a higher order scalar level set function and can be naturally transformed to other configurations as its host moves a variational form of the cost function is constructed based upon the adjoint variable and lagrangian multiplier techniques to satisfy the volume constraint effectively the lagrangian multiplier derived from the first order approximation of the cost function is amended by the bisection algorithm the procedure allows evolving initial design to an optimal shape and or topology by solving the hamilton jacobi equation two classes of benchmarking examples are presented in this paper 1 periodic microstructural material design for the maximum permeability and 2 topology optimization of flow channels for minimizing energy dissipation a number of 2d and 3d examples well demonstrated the feasibility and advantage of the level set method in solving fluid solid shape and topology optimization problems

descent methods for composite nondifferentiable optimization problems we present a framework for the development of globally defined descent algorithms for the minimization of non differentiable objective functionsf h o f withh convex within our structure the global convergence properties of the cauchy modified newton gauss newton and variable metric methods are easily established along with that of several new approaches examples illustrating the calculational techniques are provided

an immersed boundary element method for level set based topology optimization summary in this paper we propose a new bem for level set based topology optimization in the proposed bem the nodal coordinates of the boundary element are replaced with the nodal level set function and the nodal coordinates of the eulerian mesh that maintains the level set function because this replacement causes the nodal coordinates of the boundary element to disappear the boundary element mesh appears to be immersed in the eulerian mesh therefore we call the proposed bem an immersed bem the relationship between the nodal coordinates of the boundary element and the nodal level set function of the eulerian mesh is clearly represented and therefore the sensitivities with respect to the nodal level set function are strictly derived in the immersed bem furthermore the immersed bem completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization by using the immersed bem we construct a concrete topology optimization method for solving the minimum compliance problem we provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results copyright 2012 john wiley sons ltd

a density gradient approach to topology optimization under design dependent boundary loading abstract the paper proposes a density gradient based approach to topology optimization under design dependent boundary loading since there is no explicit boundary representation in density based topology optimization the design dependent boundary loads are implicitly imposed through a domain integration of spatial gradient of the density field the density gradient based loading is formally derived and justified based on the smoothed boundary method for numerical solution of pde the heaviside projection density filtering and local refinement are combined to efficiently control the thickness of the loading interface and the number of elements within it for problems with both loading and non loading boundary an auxiliary density field is introduced to keep track of the loading boundary both 2d and 3d heat conduction problems linear elasticity problems and coupled thermoelastic problems under design dependent boundary loading are presented to illustrate the effectiveness and efficiency of the method

a descent algorithm for nonsmooth convex optimization this paper presents a new descent algorithm for minimizing a convex function which is not necessarily differentiable the algorithm can be implemented and may be considered a modification of the e subgradient algorithm and lemarechal s descent algorithm also our algorithm is seen to be closely related to the proximal point algorithm applied to convex minimization problems a convergence theorem for the algorithm is established under the assumption that the objective function is bounded from below limited computational experience with the algorithm is also reported

topology optimization in wavelet space we investigate the use of wavelet methods for the design description in topology optimization this is a new setting for topology optimization that does not take the material densities in the elements of the finite element model as the design space this new approach has advantages and limitations which are discussed it is shown how the control of wavelet coefficients can be used to prevent the formation of checkerboard patterns and to obtain some geometry control problems related to overlapping basis functions are analysed and solution methods are proposed examples of optimized designs using the described methods are given copyright 2001 john wiley sons ltd

an active set method for solving linearly constrained nonsmooth optimization problems an algorithm for solving linearly constrained optimization problems is proposed the search direction is computed by a bundle principle and the constraints are treated through an active set strategy difficulties that arise when the objective function is nonsmooth require a clever choice of a constraint to relax a certain nondegeneracy assumption is necessary to obtain convergence

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

topology optimization for three dimensional electromagnetic waves using an edge element based finite element method this paper develops a topology optimization procedure for three dimensional electromagnetic waves with an edge element based finite element method in contrast to the two dimensional case three dimensional electromagnetic waves must include an additional divergence free condition for the field variables the edge element based finite element method is used to both discretize the wave equations and enforce the divergence free condition for wave propagation described in terms of the magnetic field in the widely used class of non magnetic materials the divergence free condition is imposed on the magnetic field this naturally leads to a nodal topology optimization method when wave propagation is described using the electric field the divergence free condition must be imposed on the electric displacement in this case the material in the design domain is assumed to be piecewise homogeneous to impose the divergence free condition on the electric field this results in an element wise topology optimization algorithm the topology optimization problems are regularized using a helmholtz filter and a threshold projection method and are analysed using a continuous adjoint method in order to ensure the applicability of the filter in the element wise topology optimization version a regularization method is presented to project the nodal into an element wise physical density variable

an aggregate subgradient method for nonsmooth and nonconvex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

multimaterial topology optimization by volume constrained allen cahn system and regularized projected steepest descent method abstract a new computational algorithm is introduced in the present study to solve multimaterial topology optimization problems it is based on the penalization of the objective functional by the multiphase volume constrained ginzburg landau energy functional the update procedure is based on the gradient flow of the objective functional by a fractional step projected steepest descent method in the first step the new design is found based on the projected steepest descent method to ensure the reduction in the objective functional simultaneously satisfying the control constraints in the second step regularization step an h 1 regularity of the solution is ensured while keeping the feasibility of solution with respect to the set of control constraints the presented algorithm could be accounted as a constrained h 1 optimization algorithm which according to our knowledge has not been reported to solve such kind of problems yet the success and efficiency of the presented method are shown through several test problems numerical results show that the presented algorithm ends with a near 0 1 topology and its computational cost scales sub linearly by the number of phases for the sake of reader convenience and the ease of further extension the matlab implementation of the presented algorithm is included in the appendix

free steering relaxation methods for problems with strictly convex costs and linear constraints we consider dual coordinate ascent methods for minimizing a strictly convex possibly nondifferentiable function subject to linear constraints such methods are useful in large scale applications e g entropy maximization quadratic programming network flow because they are simply can exploit sparsity and in certain cases are highly parallelizable we establish their global convergence under weak conditions and a free steering order of relaxation previous comparable results were restricted to special problems with separable costs and equality constraints our convergence framework unifies to a certain extent the approaches of bregman censor and lent de pierro and iusem and luo and tseng and complements that of bertsekas and tseng

an e relaxation method for separable convex cost generalized network flow problems cover title the extended abstract of this article appeared in the proceedings of the 5th international ipco conference vancouver june 1996 p 1

multi resolution multi scale topology optimization a new paradigm abstract the purpose of this work is to present a new concept multi resolution multi scale topology optimization the key idea of the present strategy is that design optimization should be performed progressively from low to high resolution not at a single resolution level to achieve the multi resolution strategy design optimization is formulated in a wavelet based variable space not in a direct density variable space the major advantages of the multi resolution design optimization include 1 topologically simple and close to the global optimum structures may be obtained without any explicit constraint and 2 the convergence is not sensitive to mathematical programming methods for the efficient numerical implementation of the multi resolution approach the side constraints imposed on the direct density variables are removed by mapping the density variables into intermediate variables these intermediate variables are then wavelet transformed to new design variables it is addressed that the present multi resolution topology optimization can resolve major numerical instability problems such as mesh dependencies and local minima the usefulness of the multi scale nature of the wavelets in the present multi resolution multi scale optimization formulation is also discussed

global and finite termination of a two phase augmented lagrangian filter method for general quadratic programs we present a two phase algorithm for solving large scale quadratic programs qps in the first phase gradient projection iterations approximately minimize a bound constrained augmented lagrangian function and provide an estimate of the optimal active set in the second phase an equality constrained qp defined by the current active set is approximately minimized in order to generate a second order search direction a filter determines the required accuracy of the subproblem solutions and provides an acceptance criterion for the search directions the resulting algorithm is globally and finitely convergent the algorithm is suitable for large scale problems with many degrees of freedom and provides an alternative to interior point methods when iterative methods must be used to solve the underlying linear systems numerical experiments on a subset of the cuter qp test problems demonstrate the effectiveness of the approach