aerodynamic shape optimization of aircraft wings using panel methods panel methods are frequently applied to aerodynamic shape optimization problems due to their fast turnaround time and ability to model arbitrary geometries despite being advantageous for design op

topology optimization for effective energy propagation in rate independent elastoplastic material systems abstract a topology optimization framework for effective energy management in dynamically loaded structures with rate independent elastoplastic material behavior is presented the elastoplastic constitutive equations are integrated using an implicit numerical scheme based on a multi level newton procedure this primal analysis procedure requires the computation of the incremental algorithmic consistent tangent operator derived from the mappings that enforce the evolving constitutive relations for our gradient based optimization we develop discrete adjoint sensitivity expressions finite element implementation is discussed and the optimization framework is exemplified via the design of three dimensional structures subjected to dynamic loads

coercive boundary value problems for regular degenerate differential operator equations abstract this study focuses on nonlocal boundary value problems bvp for degenerate elliptic differential operator equations doe that are defined in banach valued function spaces where boundary conditions contain a degenerate function and a principal part of the equation possess varying coefficients several conditions obtained that guarantee the maximal l p regularity and fredholmness these results are also applied to nonlocal bvp for regular degenerate partial differential equations on cylindrical domain to obtain the algebraic conditions that ensure the same properties

articulated mechanism design with a degree of freedom constraint this paper deals with design of articulated mechanism using a truss based ground structure representation the proposed method can accommodate extremely large displacement by considering geometric non linearity in addition it can also control the mechanical degrees of freedom dof of the resultant mechanism by using a dof equation based on maxwell s rule the optimization is based on a relaxed formulation of an original integer problem and also involves developments directed at handling the redundancy inherent in the ground structure representation one planar test example is selected as the basis for the developments so as to compare the proposed method with other alternative approaches including a graph theoretical enumeration approach which guarantees the identification of the globally optimal solution also an inverter problem is treated where a continuation method is required in order to direct the optimization algorithm towards an integer solution copyright 2004 john wiley sons ltd

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

path generation of articulated mechanisms by shape and topology variations in non linear truss representation this paper presents studies on an optimization based method for path generation of articulated mechanisms an extended truss ground structure approach is taken in which both the shape and topology of the truss are designed using cross sectional areas and nodal positions as design variables this leads to a technique for simultaneous type and dimensional synthesis of articulated mechanisms for the analysis part it is essential to control the mechanism configuration so that the mechanism remains within a given configuration space thus stabilizing the optimization process and resulting in realistic solutions this can be achieved by using the levenberg marquardt method the design method is illustrated by a number of design cases for both closed and open input and output paths

a meshfree level set method for topological shape optimization of compliant multiphysics actuators abstract this paper proposes a topology optimization method for compliant multiphysics actuators of geometrically nonlinear structures using meshfree galerkin weak forms and level set methods the design boundary is implicitly represented as the zero level set of a higher dimensional level set function leading to a level set model capable of handling complex shape and topological changes with flexibilities a family of compactly supported radial basis functions csrbfs is firstly used to interpolate the level set function of lipschitz continuity and then augmented to construct the shape function for meshless approximation by satisfying basic requirements in particular the predetermined consistency and the kronecker delta function property a meshless galerkin method mgm with global weak forms is established to implement the discretization of the state equations the design of actuators is transformed into an easier size optimization from a more difficult shape and topology optimization the design boundary evolution is just a question of advancing the discrete level set function in time by updating the design variables of the size optimization compared to most conventional level set methods the proposed meshless level set method is able to implement the free moving boundary discontinuities without remeshing and unify two different numerical procedures in propagating the discrete level set function e g eulerian grid and approximating the state equation e g lagrangian mesh respectively this method can also avoid numerical difficulties in solving a series of complicate hamilton jacobi partial differential equations pdes with explicit time schemes two typical numerical examples are used to demonstrate the effectiveness of the proposed method

structural topology optimization based on non local shepard interpolation of density field abstract this paper presents a non local density interpolation strategy for topology optimization based on nodal design variables in this method design variable points can be positioned at any locations in the design domain and may not necessarily coincide with elemental nodes by using the shepard family of interpolants the density value of any given computational point is interpolated by design variable values within a certain circular influence domain of the point the employed interpolation scheme has an explicit form and satisfies non negative and range restricted properties required by a physically significant density interpolation since the discretizations of the density field and the displacement field are implemented on two independent sets of points the method is well suited for a topology optimization problem with a design domain containing higher order elements or non quadrilateral elements moreover it has the ability to yield mesh independent solutions if the radius of the influence domain is reasonably specified numerical examples demonstrate the validity of the proposed formulation and numerical techniques it is also confirmed that the method can successfully avoid checkerboard patterns as well as islanding phenomenon

improved constraint aggregation methods abstract many constraints in design optimization problems take the form of bounds on a physical quantity over a domain these infinite dimensional constraints are often addressed through aggregation methods that approximate the bound in a differentiable manner however the most common constraint aggregation methods exhibit mesh dependence as the mesh spacing decreases the aggregation function diverges in addition there are no established methods to assess the impact of constraint aggregation on the optimized design to address these deficiencies we propose a new class of constraint aggregation method that we call induced aggregates we examine the properties of these aggregation methods describe their numerical implementation and test their numerical accuracy on a pressure loaded circular cylinder test case furthermore we propose a post optimality estimation technique to assess the impact of the constraint aggregation method on the design we use this method to examine the optimized mass obtained from two stress constrained mass minimization problems a variable thickness sheet problem and a wingbox design problem

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

topology design optimization of geometrically non linear structures using meshfree method a topology optimization method using meshfree methods is developed for geometrically non linearly structures a density approach is used to optimally distribute the bulk material densities in a continuum design domain the reproducing kernel rk shape function is employed to discretize both displacement and bulk density fields thus the points can be located at any part of the design domain and be easily included in or excluded from the response analysis the well known convergence difficulty which normally occurs in the sub domain of low bulk density can be avoided by excluding the sub domain the numerical instability of bulk density solution due to the rank deficiency of system matrix is effectively prevented by properly selecting the number of independent design variables through numerous numerical examples we confirm that the developed method works very well even in the large deformation problems the sub domain exclusion scheme is found to be greatly beneficial in terms of computational costs compared with the results without the scheme

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

optimizing the layout of discrete objects in structures and materials a projection based topology optimization approach abstract this paper uses topology optimization to optimize the layout of discrete objects such as fixed shape void stiffening or functional inclusions in structures and periodic materials the methodology follows a material distribution approach and the heaviside projection method hpm is manipulated to ensure that the designed objects are discrete of fixed size length scale and shape and satisfy minimum spacing requirements these restrictions are achieved implicitly through projection without additional constraints or heuristics the methodology is built in the spirit of free form topology optimization and thus discrete objects are shown to appear vanish or translate across the design domain during the design evolution the technique is demonstrated on problems governed by mechanical stiffness including the design of structures and material microstructures whose elastic stiffness is to be enhanced by optimizing the location of stiff circular inclusions

voigt reuss topology optimization for structures with nonlinear material behaviors this work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization in the proposed framework alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain since two or more materials are permitted to simultaneously occupy local regions of the design domain small strain integration algorithms for general two material mixtures of solids are developed for the voigt isostrain and reuss isostress assumptions and hybrid combinations thereof structural topology optimization problems involving non linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis dsa of energy type functionals are presented the consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials elastoplastic materials and viscoelastic materials the good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures it is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance 1997 john wiley sons ltd

the stability of difference schemes of second order of accuracy for hyperbolic parabolic equations a nonlocal boundary value problem for hyperbolic parabolic equations in a hilbert space h is considered difference schemes of second order of accuracy difference schemes for approximate solution of this problem are presented stability estimates for the solution of these difference schemes are established

a simple model for a symmetrical theory of generalized functions i definition of singular generalized functions abstract a new theory of generalized functions which is more suitable for physical applications than distribution theory is developed the necessary definitions are given and the scalar product is defined

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

a geometry projection method for continuum based topology optimization with discrete elements abstract this article describes a method for the continuum based topology optimization of structures made of discrete elements in particular we examine the optimization of linearly elastic planar structures made of bars of fixed width and semicircular ends the design space for the optimization consists of the endpoint locations of the bar s medial axes and their out of plane thicknesses to circumvent re meshing upon design changes we project the design onto a fixed analysis grid using a differentiable geometry projection that results in a density field indicating the fraction of solid material anywhere in the design space as in density based topology optimization methods the out of plane thickness is penalized so that the optimizer is capable of removing bars during the optimization the differentiability of the projection allows for the computation via the chain rule of design sensitivities of responses of interest and therefore it allows for the use of robust and efficient gradient based optimization methods notably this approach makes it easier to fabricate optimal designs by using off the shelf stock material furthermore the method considers the case where bars overlap at their joints i e their thicknesses are added at the joint and when they do not finally our proposed method naturally accommodates the imposition of several fixed length scales we demonstrate the proposed approach on classical problems of compliance based topology optimization and identify its benefits as well as research directions to be addressed in the future

material cloud method for topology optimization material cloud method mcm a new approach for topology optimization is presented in mcm an optimal structure can be obtained by manipulating the sizes and positions of material clouds which are material patches with finite sizes and constant material densities the optimal distributions of material clouds can be obtained by mcm using fixed background finite element meshes in the numerical analysis procedure only active elements where more than one material cloud is contained are treated optimal material distribution can be element wise extracted from the distribution of material clouds with mcm an expansion reduction procedure of design domain can be naturally realized through movements of material clouds so that a true optimal solution can be found without any significant increase of computational costs it is also shown that a clear material distribution with narrow region of intermediate density can be obtained with relatively fast convergence several numerical examples are shown some of the results are compared with those of the traditional density distribution method ddm copyright 2005 john wiley sons ltd

topology optimization of viscoelastic structures using a time dependent adjoint method abstract we present a mathematical framework for producing optimal designs of structures that exhibit viscoelastic creep deformation using a linear viscoelastic finite element model we implement a corresponding time dependent adjoint sensitivity formulation the resulting formula is then incorporated into a computational topology optimization framework in order to achieve optimal topologies based on the expected lifespan or operating cycle of the structure designs are optimized for minimum mass subject to a constraint on the maximum local deflection we validate the method using several two dimensional numerical examples that include design dependent gravitational loading self weight and time dependent applied loads creep plots are used to quantify the impact of the viscoelastic optimization method the results show that the design of the optimal structure is highly dependent on the load duration and the complete load history this result supports the design premise that it is necessary to account for the full viscoelastic response of the structure when designing for optimal long term performance

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

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

topology optimization under uncertainty via non intrusive polynomial chaos expansion abstract this paper presents a systematic approach for topology optimization under uncertainty that integrates non intrusive polynomial chaos expansion with design sensitivity analysis for reliability based and robust topology optimization uncertainty is introduced in loading and in geometry to address the manufacturing variability the manufacturing variability is modeled via a thresholding technique in which the threshold field is represented by a reduced dimensional random field response metrics such as compliance and volume are characterized as polynomial chaos expansions of the underlying uncertain parameters thus allowing accurate and efficient estimation of statistical moments failure probabilities and their sensitivities the number of simulations is reduced for linear structures under loading uncertainty by means of superposition efficiency of the non intrusive polynomial chaos approach is highlighted by comparison with the monte carlo method in terms of the number of simulations to demonstrate the effect of uncertainty optimized designs that consider uncertainty are compared to those that do not comparisons of polynomial chaos expansion to existing analytical methods on a benchmark numerical example are also provided for reliability based and worst case designs

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

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

an operator difference scheme for abstract cauchy problems an abstract cauchy problem for second order hyperbolic differential equations containing the unbounded self adjoint positive linear operator a t with domain in an arbitrary hilbert space is considered a new second order difference scheme generated by integer powers of a t is developed the stability estimates for the solution of this difference scheme and for the first and second order difference derivatives are established in hilbert norms with respect to space variable to support the theoretical statements for the solution of this difference scheme the numerical results for the solution of one dimensional wave equation with variable coefficients are presented

cad inspired hierarchical partition of unity constructions for nurbs based meshless design analysis and optimization in this paper we develop a unified representational paradigm for design and analysis that is inspired by the set theoretic boolean operations of the constructive solid geometry csg procedure of computer aided design cad we develop the notion of a primitive design state corresponding to a csg primitive region which is characterized by a description of the geometrical shape of the primitive and by a description of the material distribution within the primitive we analogously define a primitive behaviour state that is associated with a primitive design state and that is determined through a global analysis problem we define a global multi level design problem to determine the primitive design states we define boolean operations on the fields belonging to primitive design and behaviour spaces and show that the compositions of these fields amount to a hierarchical partition of unity construction we propose to use non uniform rational b splines nurbs to discretize the geometry material and behavioural fields local approximations defined on the primitive regions we show that this leads to recursive partition of unity constructions on the knot spaces sub local approximations due to the partition of unity property of nurbs basis functions since the primitive shapes are parametrically defined using nurbs the numerical quadrature is carried out on the parametric sub regions of the nurbs entities this eliminates the need for either a background mesh or a pseudo mesh for quadrature the developed methodology is implemented in a symbolic meshless computational framework written using the java language jnurbs and was demonstrated on two classes of problems the first class is the simultaneous optimal design of shape and multi material distributions the procedure is argued as being a generalization of classical topology optimization to designing objects made of functionally graded materials with pre defined geometrical constraints such as holes or fixed boundary shape the second class of problems demonstrated in the paper is optimal design to mitigate the effects of cracks here we identify the optimal locations of holes introduced to reduce the stress intensity factor of an edge crack copyright 2007 john wiley sons ltd

isogeometric topology optimization using trimmed spline surfaces in the present work a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed in the proposed approach the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by cad systems is employed for structural response analysis and sensitivity calculation in the topology optimization the outer and inner boundaries of design models are represented by a spline surface and trimming curves design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves new sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified the creation of new inner fronts during optimization is allowed for the topological flexibility an inner front merging algorithm is also presented the proposed spline based topology optimization is used to solve some benchmarking problems design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one it is also shown that post processing effort for converting to cad model is eliminated by using the same spline information in numerical analysis and design optimization

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

stability of difference schemes for hyperbolic parabolic equations the stable difference schemes approximately solving the nonlocal boundary value problem for hyperbolic parabolic equationd 2u t dt 2 au t f t 0 t 1 u 1 au m bu l fdu t dt au t g t 1 t 0 a b 1 0 m l 1in a hilbert space h with self adjoint positive definite operator a are presented the stability estimates for the solutions of the difference schemes of the mixed type boundary value problems for hyperbolic parabolic equations are obtained the theoretical statements for the solution of these difference schemes for hyperbolic parabolic equation are supported by the results of numerical experiments

improving multiresolution topology optimization via multiple discretizations summary unlike the traditional topology optimization approach that uses the same discretization for finite element analysis and design optimization this paper proposes a framework for improving multiresolution topology optimization imtop via multiple distinct discretizations for 1 finite elements 2 design variables and 3 density this approach leads to high fidelity resolution with a relatively low computational cost in addition an adaptive multiresolution topology optimization amtop procedure is introduced which consists of selective adjustment and refinement of design variable and density fields various two dimensional and three dimensional numerical examples demonstrate that the proposed schemes can significantly reduce computational cost in comparison to the existing element based approach copyright 2012 john wiley sons ltd

minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation adjoint equations of differential equations have seen widespread applications in optimization inverse problems and uncertainty quantification a major challenge in solving adjoint equations for time dependent systems has been the need to use the solution of the original system in the adjoint calculation and the associated memory requirement in applications where storing the entire solution history is impractical checkpointing methods have frequently been used however traditional checkpointing algorithms such as revolve require a priori knowledge of the number of time steps making these methods incompatible with adaptive time stepping we propose a dynamic checkpointing algorithm applicable when the number of time steps is a priori unknown our algorithm maintains a specified number of checkpoints on the fly as time integration proceeds for an arbitrary number of time steps the resulting checkpoints at any snapshot during the time integration have the optimal repetition number the efficiency of our algorithm is demonstrated both analytically and experimentally in solving adjoint equations this algorithm also has significant advantage in automatic differentiation when the length of execution is variable

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

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

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

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

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

megapixel topology optimization on a graphics processing unit we show how the computational power and programmability of modern graphics processing units gpus can be used to efficiently solve large scale pixel based material distribution problems using a gradient based optimality criterion method to illustrate the principle a so called topology optimization problem that results in a constrained nonlinear programming problem with over 4 million decision variables is solved on a commodity gpu

failure mitigation in optimal topology design using a coupled nonlinear continuum damage model abstract this paper presents a novel framework for performing topology optimization of structures while accounting for material damage a quasi static non local brittle damage model is coupled to a linear finite element analysis code for modeling the structural response and the initiation and propagation of damage the optimization problem is parameterized using simp variables and failure mitigation is achieved through the enforcement of constraints on the maximum local damage intensity we derive a computationally efficient adjoint sensitivity formulation that accounts for the path dependence of the coupled analysis problem the algorithm is validated through a series of numerical examples

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

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

a simple model for a symmetrical theory of generalised functions vi analytical limits and convergence abstract the symmetrical theory of generalised functions is extended to include an appropriate limit concept for generalised functions this concept allows the derivation of the fundamental theorem of the calculus for generalised functions which can be used as a convenient tool for the evaluation of integrals the physically important limit properties associated with the limit to an infinite system are derived

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

a simple model for a symmetrical theory of generalized functions iv some extensions and conclusions abstract periodic generalized functions are introduced the theory of generalized functions is compared with distribution theory and functional analysis

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