topology optimization with pressure load through a level set method abstract the topology optimization problem with pressure load is solved by using a level set method the free boundary and the pressure boundary of a structure are represented separately as two zero level sets of two level set functions and they are independently propagated during the optimization by solving two hamilton jacobi equations in order to prevent the two boundaries from touching or crossing each other the design velocities of the two boundaries that amount to the steepest descent directions are modified the optimization problem of minimum compliance with perimeter regularization is considered the shape derivatives of the two boundaries are derived by using the material derivative approach and the adjoint method the finite element analysis is done through an eulerian method by employing a fixed mesh and an artificial weak material that represents void numerical examples in two dimensions are investigated

minimum length scale control in structural topology optimization based on the moving morphable components mmc approach abstract a novel approach which can control the minimum length scale in topology optimization in a straightforward and explicit way is proposed in the present paper this approach is constructed under the so called moving morphable components mmc based solution framework where optimized structural topology can be found by changing the shapes and layout of a set of trapezoid shaped structural components on a fixed finite element mesh the extended finite element method xfem is employed for structural response analysis and shape sensitivity information is obtained by carrying out numerical integration along the structural boundary a precision definition of the minimum length scale of a structure is also suggested based on the proposed solution framework compared with existing approaches the proposed approach can achieve minimum length scale control by simply setting lower bounds on a set of geometric design variables only a small number of purely geometric constraints imposed on the sizes of intersection regions should be considered explicitly in order to get a complete minimum length scale control numerical examples demonstrate the effectiveness of the proposed method

color level sets a multi phase method for structural topology optimization with multiple materials in this paper we address the problem of structural shape and topology optimization in a multi material domain a level set method is employed as an alternative approach to the popular homogenization based methods of rule of mixtures for multi material modeling a multi phase level set model is adapted for material and topology representation this model eliminates the need for a material interpolation or phase mixing scheme it only requires m level set functions to represent a structure of n 2m different material phases in a principle similar to combining colors from the three primary colors therefore this multi phase model may be referred to as a color level set representation which has its unique benefits it is flexible to handle complex topologies it substantially reduces the number of model functions when n 3 it automatically avoids the problem of overlap between material phases of a conventional partitioning approach we describe numerical techniques for efficient and robust implementation of the method by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain this would separate the issues of accuracy in numerical calculations of the physical equation and in the level set model propagation a gradient projection method is described for incorporating multiple constraints in the problem finally the benefits and the advantages of the developed method are illustrated with several 2d examples of mean compliance minimization of multi material structures

topology optimization of incompressible media using mixed finite elements abstract we present an alternative formulation for the topology optimization of structures made of incompressible materials a topic that cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements that are well known to be affected by the locking phenomenon a way out of the problem has recently been proposed in sigmund and clausen 2007 o sigmund p m clausen topology optimization using a mixed formulation an alternative way to solve pressure load problems comput methods appl mech engrg 196 13 16 2007 1874 1889 based on a displacement pressure finite element discretization the approach presented here consists in a truly mixed variational formulation coupled to a mixed fem discretization that uses the composite element of johnson and mercier for the discretization of the stress field by so doing the continuous and discrete inf sup conditions of the problem are automatically met even in the presence of incompressible materials a few numerical studies are presented to validate the theoretical framework for which the well known method of moving asymptotes mma is adopted for the numerical optimization of the problem different topologies in plane stress and plane strain conditions are presented with particular attention to the convergence of the last ones to pure 0 1 designs some forthcoming investigations are eventually highlighted including the solution of stress constrained topology optimization problems that find in the truly mixed setting their natural environment and the extension of the present formulation to deal with pressure load structural problems

robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis abstract topology optimization often leads to structures consisting of slender elements which are particularly sensitive to geometric imperfections such imperfections might affect the structural stability and induce large displacement effects in these slender structures this paper therefore presents a robust approach to topology optimization which accounts for geometric imperfections and their potentially detrimental influence on the structural stability geometric nonlinear effects are incorporated in the optimization by means of a total lagrangian finite element formulation in the minimization of end compliance geometric imperfections are modeled as a vector valued random field in the design domain the resulting uncertain performance of the design is taken into account by minimizing a weighted sum of the mean and standard deviation of the compliance in the robust optimization problem these stochastic moments are typically estimated by means of sampling methods such as monte carlo simulation however these methods require multiple independent nonlinear finite element analyses in each design iteration of the optimization algorithm an efficient solution algorithm which uses adjoint differentiation in a second order perturbation method is therefore developed to estimate the stochastic moments during the optimization two applications with structures that exhibit different types of structural instabilities are examined in both cases it is demonstrated by means of an extensive monte carlo simulation that the deterministic design is very sensitive to imperfections while the design obtained by means of the proposed method is much more robust

image segmentation by variational methods mumford and shah functional and the discrete approximations in this paper we discuss the links between mumford and shah s variational problem for signal and image segmentation based on an energy functional of a continuous grey level function and the numerical algorithms proposed to solve it these numerical approaches are based on a discrete functional we recall that in one dimension this discrete functional is asymptotically equivalent to the continuous functional this can be summarized in a gamma convergence result we show that the same result holds in dimension two provided that the continuous energy is adapted to the anisotropy of the discrete approaches we display a few experimental results in dimensions one and two

modified conditions for minimum of compliance we consider the problem of finding a design with minimum compliance using a simp material in particular we present a method to obtain competitive designs that do not include composite like material distributions the idea of this method is to modify a necessary condition for the optimal design of an object we prove that the modified condition defines a design and present numerical solutions

control in the coefficients with variational crimes application to topology optimization of kirchhoff plates abstract we study convergence of discontinuous galerkin type discretizations of the problems of control in the coefficients of uniformly elliptic partial differential equations pdes as a model problem we use that of the optimal design of thin kirchhoff plates where the governing equations are of the fourth order methods which do not require approximation subspaces to conform to the smoothness requirements dictated by the pde are very attractive for such problems however variational formulations of such methods normally contain boundary integrals whose dependence on the small with respect to volumetric lebesgue norm changes of the coefficients is generally speaking not continuous we utilize the lifting formulation of the discontinuous galerkin method to deal with this issue our main result is that limit points of sequences of designs verifying discrete versions of stationarity can also be expected to satisfy stationarity for the limiting continuum mechanics problem we illustrate the practical behaviour of our discretization strategy on some benchmark type examples

topology optimization using regularized intermediate density control abstract we consider topology optimization of elastic continua the elasticity tensor is assumed to depend linearly on the design function density as in the variable thickness sheet problem in order to get black white design pictures the intermediate density values are controlled by an explicit constraint this constraint is regularized by including a compact and linear operator s to guarantee existence of solutions a proof of convergence of the finite element fe discretized optimization problem s solutions to exact ones is also given so the method is not prone to numerical anomalies such as mesh dependence or checkerboards the procedure is illustrated in some minimum compliance examples where s is chosen to be a classical convolution type operator the fe discretized optimization problems are solved by sequential convex approximations

cost approximation a unified framework of descent algorithms for nonlinear programs this paper describes and analyzes the cost approximation algorithm this class of iterative descent algorithms for nonlinear programs and variational inequalities places a large number of algorithms within a common framework and provides a means for analyzing relationships among seemingly unrelated methods a common property of the methods included in the framework is that their subproblems may be characterized by monotone mappings which replace an additive part of the original cost mapping in an iterative manner alternately a step is taken in the direction obtained in order to reduce the value of a merit function for the original problem the generality of the framework is illustrated through examples and the convergence characteristics of the algorithm are analyzed for applications to nondifferentiable optimization the convergence results are applied to some example methods demonstrating the strength of the analysis compared to existing results

explicit level set based topology optimization using an exact heaviside function and consistent sensitivity analysis summary this paper presents a level set based topology optimization method based on numerically consistent sensitivity analysis the proposed method uses a direct steepest descent update of the design variables in a level set method the level set nodal values an exact heaviside formulation is used to relate the level set function to element densities the level set function is not required to be a signed distance function and reinitialization is not necessary using this approach level set based topology optimization problems can be solved consistently and multiple constraints treated simultaneously the proposed method leads to more insight in the nature of level set based topology optimization problems the level set based design parametrization can describe gray areas and numerical hinges consistency causes results to contain these numerical artifacts we demonstrate that alternative parameterizations level set based or density based regularization can be used to avoid artifacts in the final results the effectiveness of the proposed method is demonstrated using several benchmark problems the capability to treat multiple constraints shows the potential of the method furthermore due to the consistency the optimizer can run into local minima a fundamental difficulty of level set based topology optimization more advanced optimization strategies and more efficient optimizers may increase the performance in the future copyright 2012 john wiley sons ltd

some convergence results in perimeter controlled topology optimization abstract the computation of optimal topologies of elastic continuum structures using a constraint on the perimeter is investigated predicting macroscopic black white topologies without the use of homogenization techniques this approach is presently one of the most attractive approaches in topology optimization mathematical justifications are given for both the relaxation of the discrete value constraint on the design variable and for the finite element discretizations it turns out that the way in which the perimeter has been calculated to date the numerical results will not approximate the intended original problem but one with a taxi cab perimeter which measures lengths of structural edges after projection onto the coordinate axes

higher order multi resolution topology optimization using the finite cell method this article presents a detailed study on the potential and limitations of performing higher order multi resolution topology optimization with the finite cell method to circumvent stiffness overestimation in high contrast topologies a length scale is applied on the solution using filter methods the relations between stiffness overestimation the analysis system and the applied length scale are examined while a high resolution topology is maintained the computational cost associated with nested topology optimization is reduced significantly compared with the use of first order finite elements this reduction is caused by exploiting the decoupling of density and analysis mesh and by condensing the higher order modes out of the stiffness matrix

convergence rates in forward backward splitting forward backward splitting methods provide a range of approaches to solving large scale optimization problems and variational inequalities in which structures conducive to decomposition can be utilized apart from special cases where the forward step is absent and a version of the proximal point algorithm comes out efforts at evaluating the convergence potential of such methods have so far relied on lipschitz properties and strong monotonicity or inverse strong monotonicity of the mapping involved in the forward step the perspective mainly being that of projection algorithms here convergence is analyzed by a technique that allows properties of the mapping in the backward step to be brought in as well for the first time in such a general setting global and local contraction rates are derived moreover they are derived in a form which makes it possible to determine the optimal step size relative to certain constants associated with the given problem insights are thereby gained into the effects of shifting strong monotonicity between the forward and backward mappings when a splitting is selected

computational simulation of trabecular adaptation progress in human proximal femur during growth abstract there are a large number of clinical and experimental studies that analyzed trabecular architecture as a result of bone adaptation however only a limited amount of quantitative data is currently available on the progress of trabecular adaptation during growth in this paper we proposed a two step numerical simulation method that predicts trabecular adaptation progress during growth using a recently developed topology optimization algorithm design space optimization dso under the hypothesis that the mechanisms of dso are functionally equivalent to those of bone adaptation we applied the proposed scheme to trabecular adaptation simulation in human proximal femur for the simulation the full trabecular architecture in human proximal femur was represented by a two dimensional fe model with 50 m resolution in step 1 we determined a reference value that regulates trabecular adaptation in human proximal femur in step 2 we simulated trabecular adaptation in human proximal femur during growth with the reference value derived in step 1 we analyzed the architectural and mechanical properties of trabecular patterns through iterations from the comparison with experimental data in the literature we showed that in the early growth stage trabecular adaptation was achieved mainly by increasing bone volume fraction or trabecular thickness while in the later stage of the development the trabecular architecture gained higher structural efficiency by increasing structural anisotropy with a relatively low level of bone volume fraction or trabecular thickness we demonstrated that the proposed numerical framework predicted the growing progress of trabecular bone that has a close correlation with experimental data

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

an enhanced genetic algorithm for structural topology optimization genetic algorithms gas have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures in this paper a versatile robust and enhanced ga is proposed for structural topology optimization by using problem specific knowledge the original discrete black and white 0 1 problem is directly solved by using a bit array representation method to address the related pronounced connectivity issue effectively the four neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns a simpler version of the perimeter control approach is developed to obtain a well posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge free design to handle the problem of representation degeneracy effectively a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner an efficient fem based function evaluation method is developed to reduce the computational cost a dynamic penalty method is presented for the ga to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy with all these enhancements and appropriate choice of the ga operators the present ga can achieve significant improvements in evolving into near optimum solutions and viable topologies with checkerboard free mesh independent and hinge free characteristics numerical results show that the present ga can be more efficient and robust than the conventional gas in solving the structural topology optimization problems of minimum compliance design minimum weight design and optimal compliant mechanisms design it is suggested that the present enhanced ga using problem specific knowledge can be a powerful global search tool for structural topology optimization copyright 2005 john wiley sons ltd

dynamic pricing of seasonal goods with spot and forward purchase demands this paper deals with the problem of jointly determining the order size and dynamic prices for a perishable inventory system over a finite time planning horizon when firms simultaneously provide customers with a prompt delivery option and a delivery schedule option customers are segmented into two types namely spot purchase customers and the forward purchase customers demands for both types of customers are assumed to be time and price dependent the decision maker of the inventory system is assumed to apply pricing policies to stimulate demand to improve revenues under the condition that customers with forward purchases may cancel their orders a mathematical model is developed to find the optimal number of price settings the optimal dynamic prices and the order quantity a solution procedure is found to determine the optimal decisions

on the continuity in bv ω of the l 2 projection into finite element spaces we show how to obtain continuity in the bv q seminorm of the l 2 projection of u e bv q into a large class of finite element spaces

a quadratic approximation for structural topology optimization in topology optimization it is customary to use reciprocal like approximations which result in monotonically decreasing approximate objective functions in this paper we demonstrate that efficient quadratic approximations for topology optimization can also be derived if the approximate hessian terms are chosen with care to demonstrate this we construct a dual sao algorithm for topology optimization based on a strictly convex diagonal quadratic approximation to the objective function although the approximation is purely quadratic it does contain essential elements of reciprocal like approximations for self adjoint problems our approximation is identical to the quadratic or second order taylor series approximation to the exponential approximation we present both a single point and a two point variant of the new quadratic approximation

stress energy minimization as a tool in the material layout design of shallow shells abstract the present research deals with the compliance minimization problem of an elastic thin shallow shell subjected to simultaneous in plane and bending loads in this context our goal is to lay out a given amount of material in the volume of a shell assuming that the distribution in the direction transversal to its middle surface s is homogeneous the discussion hence reduces to the question of finding the optimal material arrangement on s similar problems were solved in the framework of two dimensional elasticity or kirchhoff plate theory and the present research attempts to generalize these results following the pattern emerging from the above mentioned considerations our research starts from the minimum compliance problem of a structure made of two elastic materials whose volumetric fractions are fixed the existence of a solution to thus posed optimization task is guaranteed if the fine scale microstructural composites are admitted in the analysis their constitutive tensors can be obtained by certain averaging ensuing from the theory of homogenization for periodic media additionally by the castigliano theorem the compliance minimization problem is equivalent to the one for structural stress energy in turn the lower estimation of the energy is achieved in two steps i its modification by a certain energy like functional and ii utilizing the quasiconvexity property of thus obtained expression as a result formulae describing the effective stress energy of one material shallow shell and the material distribution function are explicitly derived

a mixed fem approach to stress constrained topology optimization we present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion the main idea is to adopt a mixed finite element discretization scheme wherein not only displacements as usual but also stresses are the variables entering the formulation by doing so any stress constraint may be handled within the optimization procedure without resorting to post processing operation typical of displacement based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed two dual variational principles of hellinger reissner type are presented in continuous and discrete form that which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes int j numer meth engng 1984 24 3 359 373 extensive numerical simulations are performed and ongoing extensions outlined including the optimization of elastoplastic and incompressible media copyright 2007 john wiley sons ltd

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

a finite element analysis of optimal variable thickness sheets a quasi mixed finite element fe method for maximum stiffness of variable thickness sheets is analyzed the displacement is approximated with nine node lagrange quadrilateral elements and the thickness is approximated as elementwise constant one is guaranteed that the fe displacement solutions will converge in hh but in an example it is shown that in general one cannot expect any subsequence of the fe thickness solutions to converge in any l p omega norm however under a regularity and biaxiality assumption on the optimal stress field uniqueness of the optimal thickness function as well as convergence in l p omega 1 leq p infty of fe thickness solutions are proven

constraints of distance from boundary to skeleton for the control of length scale in level set based structural topology optimization abstract a method is proposed for the control of minimum maximum length scale in the level set based structural topology optimization the minimum maximum length scale of structure is characterized by using the concept of smallest biggest maximal inscribable ball in order to prevent trivial zero minimum length scale the skeleton of structure is utilized and trimmed the control of length scale is realized by constraining the distance from boundary to skeleton and the distance is explicitly constructed by using the highly efficient fast marching method numerical examples in two dimensions are investigated

optimal selling price and lotsize with time varying deterioration and partial backlogging the article deals with an eoq economic order quantity model over an infinite time horizon for perishable items where demand is price dependent and partial backorder is permitted the rate of deterioration is taken to be time proportional and it is assumed that shortage occurs at starting of the inventory cycle based on the partial backlogging and lost sale cases the author develops the criterion for the optimal solution for the replenishment schedule and proves the optimal ordering policy is unique moreover the article suggests to new functions regarding price dependent demand and time varying deterioration rate finally numerical examples are illustrated to test the model in various issues

connected linear groups as differential galois groups theorem 1 1 let c be an algebraically closed field of characteristic zero g a connected linear algebraic group defined o er c and k a differential field containing c as its field of constants and of finite nonzero transcendence degree o er c then g can be realized as the galois group of a picard vessiot extension of k w x previous work by kovacic 17 18 reduced this problem to the case of powers of a simple connected linear algebraic group our contribution is to show that one is able to realize any connected semisimple group as a galois group and when k s c x x9 s 1 to control the number and types of singularities when one constructs a system y 9 s ay a g m c x realizing an arbitrary connected linear algebraic group as its n

phase field relaxation of topology optimization with local stress constraints we introduce a new relaxation scheme for structural topology optimization problems with local stress constraints based on a phase field method in the basic formulation we have a pde constrained optimization problem where the finite element and design analysis are solved simultaneously the starting point of the relaxation is a reformulation of the material problem involving linear and 0 1 constraints only the 0 1 constraints are then relaxed and approximated by a cahn hilliard type penalty in the objective functional which yields convergence of minimizers to 0 1 designs as the penalty parameter decreases to zero a major advantage of this kind of relaxation opposed to standard approaches is a uniform constraint qualification that is satisfied for any positive value of the penalization parameter the relaxation scheme yields a large scale optimization problem with a high number of linear inequality constraints we discretize the problem by finite elements and solve the arising finite dimensional programming problems by a primal dual interior point method numerical experiments for problems with local stress constraints based on different criteria indicate the success and robustness of the new approach

a new scheme for imposing a minimum length scale in topology optimization a new scheme for imposing a minimum length scale in topology optimization is presented it guarantees the existence of an optimal design for a large class of topology optimization problems of practical interest it is formulated as one constraint that is computationally cheap and for which sensitivities are also cheap to compute the constraint value is ideally zero but it can be relaxed to a positive value the effect of the method is illustrated in topology optimization for minimum compliance and design of compliant mechanisms notably the method produces compliant mechanisms with distributed flexibility something that has previously been difficult to obtain using topology optimization for the design of compliant mechanisms the term mole method is suggested for the method copyright 2003 john wiley sons ltd

real and p adic picard vessiot fields we consider differential modules over real and p adic differential fields k such that its field of constants k is real closed resp p adically closed using p deligne s work on tannakian categories and a result of j p serre on galois cohomology a purely algebraic proof of the existence and unicity of real resp p adic picard vessiot fields is obtained the inverse problem for real forms of a semi simple group is treated some examples illustrate the relations between differential modules picard vessiot fields and real forms of a linear algebraic group

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

topology optimization using a reaction diffusion equation abstract this paper presents a structural topology optimization method based on a reaction diffusion equation in our approach the design sensitivity for the topology optimization is directly employed as the reaction term of the reaction diffusion equation the distribution of material properties in the design domain is interpolated as the density field which is the solution of the reaction diffusion equation so free generation of new holes is allowed without the use of the topological gradient method our proposed method is intuitive and its implementation is simple compared with optimization methods using the level set method or phase field model the evolution of the density field is based on the implicit finite element method as numerical examples compliance minimization problems of cantilever beams and force maximization problems of magnetic actuators are presented to demonstrate the method s effectiveness and utility

a level set method for shape and topology optimization of both structure and support of continuum structures abstract we present a level set method for the shape and topology optimization of both structure and support two level set functions are used to implicitly represent a structure the traction free boundary and the dirichlet boundary are represented separately and are allowed to be continuously changed during the optimization the optimization problem of minimum compliance is considered the shape derivatives of both boundaries are derived by using a lagrangian function and the adjoint method the finite element analysis is done by modifying a fixed background mesh and we do not use the artificial weak material numerical examples in two dimensions are investigated

shape and topology optimization of compliant mechanisms using a parameterization level set method in this paper a parameterization level set method is presented to simultaneously perform shape and topology optimization of compliant mechanisms the structural shape boundary is implicitly embedded into a higher dimensional scalar function as its zero level set resultantly establishing the level set model by applying the compactly supported radial basis function with favorable smoothness and accuracy to interpolate the level set function the temporal and spatial hamilton jacobi equation from the conventional level set method is then discretized into a series of algebraic equations accordingly the original shape and topology optimization is now fully transformed into a parameterization problem namely size optimization with the expansion coefficients of interpolants as a limited number of design variables design of compliant mechanisms is mathematically formulated as a general optimization problem with a nonconvex objective function and two additionally specified constraints the structural shape boundary is then advanced as a process of renewing the level set function by iteratively finding the expansion coefficients of the size optimization with a sequential convex programming method it is highlighted that the present method can not only inherit the merits of the implicit boundary representation but also avoid some unfavorable features of the conventional discrete level set method such as the cfl condition restriction the re initialization procedure and the velocity extension algorithm finally an extensively investigated example is presented to demonstrate the benefits and advantages of the present method especially its capability of creating new holes inside the design domain

undercut and overhang angle control in topology optimization a density gradient based integral approach summary we present an approach for controlling the undercut and the minimal overhang angle in density based topology optimization which are useful for reducing support structures in additive manufacturing we cast both the undercut control and the minimal overhang angle control that are inherently constraints on the boundary shape into a domain integral of heaviside projected density gradient such a heaviside projection based integral of density gradient leads to a single constraint for controlling the undercut or controlling the overhang angle in the optimization it effectively corresponds to a constraint on the projected perimeter that has undercut or has slope smaller than the prescribed overhang angle in order to prevent trivial solutions of intermediate density to satisfy the density gradient constraints a constraint on density grayness is also incorporated into the formulations numerical results on messerschmitt bolkow blohm beams cantilever beams and 2d and 3d heat conduction problems demonstrate the proposed formulations are effective in controlling the undercut and the minimal overhang angle in the optimized designs copyright 2016 john wiley sons ltd

polygonal finite elements for topology optimization a unifying paradigm in topology optimization literature the parameterization of design is commonly carried out on uniform grids consisting of lagrangian type finite elements e g linear quads these formulations however suffer from numerical anomalies such as checkerboard patterns and one node connections which has prompted extensive research on these topics a problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members leading to mesh dependent sub optimal designs thus to address the geometric features of the spatial discretization we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions more specifically we consider polygonal meshes constructed from voronoi tessellations which in addition to possessing higher degree of geometric isotropy allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities copyright 2009 john wiley sons ltd

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

optimal design for polymer extrusion part i sensitivity analysis for nonlinear steady state systems we present a general framework to obtain analytical design sensitivities for steady state nonlinear systems where special emphasis is given to design variables that define prescribed boundary conditions the sensitivities are derived for a general response functional using both the direct and adjoint methods and are compared to show that the two sensitivity analysis methods yield identical expressions both adjoint and direct sensitivities for the generalized hele shaw system which models the flow of purely viscous non newtonian fluids through thin cavities are obtained using weak formulations and then discretized for the finite element analysis these sensitivities are used with numerical optimization to design polymer sheeting dies with minimum pressure drop and reduced velocity variation across the die exit

differential embedding problems over laurent series fields abstract we solve the inverse differential galois problem over the fraction field of k t x and use this to solve split differential embedding problems over k t x that are induced from k x the proofs use patching as well as prior results on inverse problems and embedding problems

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

parametric structural shape topology optimization with a variational distance regularized level set method abstract the signed distance function sdf gives the shortest distance from a given point to the boundary and the sign indicates whether this point is inside or outside the closed boundary or enclosed region the sdf property is highly preferred in classical level set methods to maintain the numerical stability during the topology optimization process and provide a metric for the distance based interpolation of different material properties in conventional level set methods a common way of achieving a level set function with the signed distance property is to periodically implement the so called reinitialization scheme by solving an additional hamilton jacobi partial differential equation however such reinitialization scheme is implemented outside the optimization loop with the optimization process suspended which may shift the optimization result and bring convergence issues in this paper a double well potential functional is employed for distance regularization inside the topology optimization loop which can enforce the signed distance property of the level set function in a narrow band along the design boundaries while keeping the level set function flat in the rest area of the computational domain the radial basis function rbf based parameterization technique is combined with mathematical programming to improve the performance of the proposed distance regularized topology optimization method in handling problems with non convex objective functions and multiple constraints the flatness of the level set function in the material region also enables easy creation of new holes to the design in the topology optimization process both 2d and 3d benchmark examples are employed to demonstrate the validity of the proposed method

interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems abstract the focus of this paper is on interpolation schemes for fictitious domain and topology optimization approaches with structures undergoing large displacements numerical instability in the finite element simulations can often be observed due to excessive distortion in low stiffness regions a new energy interpolation scheme is proposed in order to stabilize the numerical simulations the elastic energy density in the solid and void regions is interpolated using the elastic energy densities for large and small deformation theory respectively the performance of the proposed method is demonstrated for a challenging test geometry as well as for topology optimization of minimum compliance and compliant mechanisms the effect of combining the proposed interpolation scheme with different hyperelastic material models is investigated as well numerical results show that the proposed approach alleviates the problems in the low stiffness regions and for the simulated cases results in stable topology optimization of structures undergoing large displacements

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

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

stress based topology optimization of continuum structures under uncertainties abstract this work addresses the use of the topology optimization approach to the design of continuum structures with failure constraints under the hypothesis of uncertainties in the spatial distribution of young s modulus to this end the first order perturbation approach is used to model the response of the structure and the midpoint discretization technique is used to represent the random field the objective is the minimization of the amount of material used in the design subjected to local stress constraints under uncertainties the probability of failure is bounded by the one sided chebychev inequality since the exact probability distribution function of the stress constraints is not known in advance the effective probability of failure of the obtained optimal designs is validated with the use of the monte carlo simulation approach indicating that the probability of failures of the topologies obtained with the stochastic approach is within the bounds provided by the one sided chebychev inequality the optimization problem is solved by means of the augmented lagrangian method in order to address the large number of constraints associated to this kind of formulation it is shown that the correlation length and the number of standard deviations considered in the formulation play an important role in both the obtained topology and effective probability of failure

a hierarchical method for discrete structural topology design problems with local stress and displacement constraints in this paper we present a hierarchical optimization method for finding feasible true 0 1 solutions to finite element based topology design problems the topology design problems are initially modelled as non convex mixed 0 1 programs the hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design dependent stress constraints the method iteratively treats a sequence of problems of increasing size of the same type as the original problem the problems are defined on a design mesh which is initially coarse and then successively refined as needed at each level of design mesh refinement a neighbourhood optimization method is used to treat the problem considered the non convex topology design problems are equivalently reformulated as convex all quadratic mixed 0 1 programs this reformulation enables the use of methods from global optimization which have only recently become available for solving the problems in the sequence numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented copyright 2006 john wiley sons ltd

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

a gradient based parameter free approach to shape optimization in shape optimization the independent node movement approach wherein finite element node coordinates are used directly as design variables allows the most freedom for shape change and avoids the time consuming parameterization preprocess however this approach lacks a length scale control that is necessary to ensure a well posed shape optimization problem and avoid numerical instability motivated by the success of filtering techniques that impose minimum length scales in topology optimization we propose a scheme with consistent filtering to introduce a length scale and thereby ensure smoothness in shape optimization while preserving the advantages of the independent node movement approach

on the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem we study the classical topology optimization problem in which minimum compliance is sought subject to linear constraints using a dual statement we propose two separable and strictly convex subproblems for use in sequential approximate optimization sao algorithms respectively the subproblems use reciprocal and exponential intermediate variables in approximating the non linear compliance objective function any number of linear constraints or linearly approximated constraints are provided for the relationships between the primal variables and the dual variables are found in analytical form for the special case when only a single linear constraint on volume is present we note that application of the ever popular optimality criterion oc method to the topology optimization problem combined with arbitrary values for the heuristic numerical damping factor proposed by bendsoe results in an updating scheme for the design variables that is identical to the application of a rudimentary dual sao algorithm in which the subproblems are based on exponential intermediate variables what is more we show that the popular choice for the damping factor 0 5 is identical to the use of sao with reciprocal intervening variables finally computational experiments reveal that subproblems based on exponential intervening variables result in improved efficiency and accuracy when compared to sao subproblems based on reciprocal intermediate variables and hence the heuristic topology oc method hitherto used this is attributed to the fact that a different exponent is computed for each design variable in the two point exponential approximation we have used using gradient information at the previously visited point copyright 2007 john wiley sons ltd

tangent operators and design sensitivity formulations for transient non linear coupled problems with applications to elastoplasticity tangent operators and design sensitivities are derived for transient non linear coupled problems the solution process and the formation of tangent operators are presented in a systematic manner and sensitivities for a generalized response functional are formulated via both the direct differentiation and adjoint methods the derived formulations are suitable for finite element implementations analyses of systems with materials that exhibit history dependent response may be obtained directly by applying the analyses of transient non linear coupled systems rate independent elastoplasticity is investigated as a case study and a problem with an analytical solution is analysed for demonstration purposes

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

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

shape and topology optimization based on the phase field method and sensitivity analysis this paper discusses a structural optimization method that optimizes shape and topology based on the phase field method the proposed method has the same functional capabilities as a structural optimization method based on the level set method incorporating perimeter control functions the advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required structural shapes are represented by the phase field function defined in the design domain and optimization of this function is performed by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis the proposed method is applied to two dimensional linear elastic and vibration optimization problems such as the minimum compliance problem a compliant mechanism design problem and the eigenfrequency maximization problem the numerical examples provided illustrate the convergence of the various objective functions and the effect that perimeter control has on the optimal configurations

two metric projection methods for constrained optimization this paper is concerned with the problem min f x mid x in x where x is a convex subset of a linear space h and f is a smooth real valued function on h we propose the class of methods x k 1 p x k alpha k g k where p denotes projection on x with respect to a hilbert space norm cdot g k denotes the frechet derivative of f at x k with respect to another hilbert space norm cdot k on h and alpha k is a positive scalar stepsize we thus remove an important restriction in the original proposal of goldstein 1 and levitin and poljak 2 where the norms cdot and cdot k must be the same it is therefore possible to match the norm cdot with the structure of x so that the projection operation is simplified while at the same time reserving the option to choose cdot k on the basis of approximations to the hessian of f so as to attain a typically superlinear rate of convergence the resulting methods are particularly attrac

improving the efficiency of large scale topology optimization through on the fly reduced order model construction summarytopology optimization of large scale structures is computationally expensive notably because of the cost of solving the equilibrium equations at each iteration reduced order models by projection also known as reduced basis models have been proposed in the past for alleviating this cost we propose here a new method for coupling reduced basis models with topology optimization to improve the efficiency of topology optimization of large scale structures the novel approach is based on constructing the reduced basis on the fly using previously calculated solutions of the equilibrium equations the reduced basis is thus adaptively constructed and enriched based on the convergence behavior of the topology optimization a direct approach and an approach with adjusted sensitivities are described and their algorithms provided the approaches are tested and compared on various 2d and 3d minimum compliance topology optimization benchmark problems computational cost savings by up to a factor of 12 are demonstrated using the proposed methods copyright textcopyright 2014 john wiley sons ltd

optimization of structures with thin layer functional device on its surface through a level set based multiple type boundary method abstract functional structure i e native base structure equipped with thin layer functional devices on its surface is important in many engineering applications the performance of such a functional structure is seriously affected by the configuration of the base structure and the surface functional devices and the optimization of configuration is a simultaneous topology optimization of the boundaries that belong to different types in the present study load cell is taken as an example of the functional structure and strain gauge is taken as an example of the functional device the task of designing a load cell is accomplished by solving a multi objective topology optimization problem whose objective makes a compromise between stiffness and flexibility of the base structure the design variables include two types of boundaries functional boundary where a strain gauge is installed the free boundary where there does not exist any connection to the environment the optimization problem is solved by using a level set based multiple type boundary method two level set functions are used to represent a functional structure and the two types of boundaries evolution of the two boundaries is modeled by two independent hamilton jacobi equations re meshing is conducted in each iteration of optimization quadrilateral body fitted mesh is used for the finite element analysis and the artificial weak material is not used numerical examples in two dimensions are investigated

the stochastic eoq model with random sales price abstract the article deals with a stochastic economic order quantity eoq model over a finite time horizon where uniform demand over the replenishment period is price dependent the selling price is assumed to be a random variable that follows a probability density function as demand is probabilistic stock out situation may occur based on the partial backlogging and lost sale cases during stock out period the author develops the criterion for the optimal solution for the replenishment size such that the integrated expected profit is maximized moreover the article suggests a new function regarding price dependent demand finally numerical examples and its sensitivity analysis of key parameters are given to illustrate the proposed model

short note shape and topology optimization for electrothermomechanical microactuators using level set methods in this short note a shape and topology optimization method is presented for multiphysics actuators including geometrically nonlinear modeling based on an implicit free boundary parameterization method a level set model is established to describe structural design boundary by embedding it into the zero level set of a higher dimensional level set function the compactly supported radial basis functions csrbf are introduced to parameterize the implicit level set surface with a high level of accuracy and smoothness the original more difficult shape and topology optimization driven by the hamilton jacobi partial differential equation pde is transferred into a relatively easier parametric size optimization to which many well founded optimization algorithms can be applied thus the structural optimization is transformed to a numerical process that describes the design as a sequence of motions of the design boundaries by updating the expansion coefficients of the size optimization two widely studied examples are chosen to demonstrate the effectiveness of the proposed method

a closer look at consistent operator splitting and its extensions for topology optimization abstract in this work we explore the use of operator splitting algorithms for solving regularized structural topology optimization problems the context is a classical structural design problem e g compliance minimization and compliant mechanism design parametrized by means of density functions whose ill posedness is addressed by introducing a tikhonov regularization term the proposed forward backward splitting algorithm treats the constituent terms of the cost functional separately which allows for suitable approximations of the structural objective we will show that one such approximation inspired by the reciprocal expansions underlying the optimality criteria method improves the convergence characteristics and leads to an update scheme resembling the heuristic sensitivity filtering method we also discuss a two metric variant of the splitting algorithm that removes the computational overhead associated with bound constraints on the density field without compromising convergence and quality of optimal solutions we present several numerical results and investigate the influence of various algorithmic parameters

distributed parameter optimization and topology design for non linear thermoelasticity abstract topology optimization has been the focus of considerable attention in the shape optimization community in recent years since significant performance improvements can be obtained if the topology is allowed to vary in shape optimization problems attention has mainly focussed on the topology design of an elastic continuum for minimum compliance subject to a volume constraint the macroscopic version of this problem is not well posed if no restrictions are placed on the structure topology the most widely used method for making the problem has been the so called homogenization method which introduces microstructure to the design space though there are a number of advantages to this method it also suffers from some drawbacks optimal designs generated using an optimal microstructure are difficult to manufacture while using sub optimal microstructures reverts the problem back to the original ill posed problem in addition extension of the homogenization method to problems involving non linear material behavior is quite difficult to address some of these issues a new method for making the compliance optimization problem well posed has recently been proposed this method introduces an additional constraint on the perimeter of the solid regions in the design to make the problem well posed since microstructure is not introduced the designs are easily manufacturable preliminary results for topology design using the perimeter method for problems where the material behavior is linear elastic have already been reported in the literature we show that the perimeter method can be used even when the material behavior involves material and geometric non linearities we formulate the distributed parameter optimization and topology design problems using the perimeter method for non linear thermoelasticity finite element optimization procedures based on this formulation are developed and numerical solutions for the distributed parameter optimization problem are presented though we use the compliance as the objective function one could optimize any objective function by making minor modifications in the method outlined here

bilateral filtering for structural topology optimization filtering has been a major approach used in the homogenization based methods for structural topology optimization to suppress the checkerboard pattern and relieve the numerical instabilities in this paper a bilateral filtering technique originally developed in image processing is presented as an efficient approach to regularizing the topology optimization problem a non linear bilateral filtering process leads to a suitable problem regularization to eliminate the checkerboard instability pronounced edge preserving smoothing characteristics to favour the 0 1 convergence of the mass distribution and computational efficiency due to its single pass and non iterative nature thus we show that the application of the bilateral filtering brings more desirable effects of checkerboard free mesh independence crisp boundary computational efficiency and conceptual simplicity the proposed bilateral technique has a close relationship with the conventional domain filtering and range filtering the proposed method is implemented in the framework of a power law approach based on the optimality criteria and illustrated with 2d examples of minimum compliance design that has been extensively studied in the recent literature of topology optimization and its efficiency and accuracy are highlighted copyright 2005 john wiley sons ltd

modelling topology optimization problems as linear mixed 0 1 programs this paper deals with topology optimization of discretized continuum structures it is shown that a large class of non linear 0 1 topology optimization problems including stress and displacement constrained minimum weight problems can equivalently be modelled as linear mixed 0 1 programs the modelling approach is applied to some test problems which are solved to global optimality