a level set method for shape and topology optimization of large displacement compliant mechanisms a parameterization level set method is presented for structural shape and topology optimization of compliant mechanisms involving large displacements a level set model is established mathematically as the hamilton jacobi equation to capture the motion of the free boundary of a continuum structure the structural design boundary is thus described implicitly as the zero level set of a level set scalar function of higher dimension the radial basis function with compact support is then applied to interpolate the level set function leading to a relaxation and separation of the temporal and spatial discretizations related to the original partial differential equation in doing so the more difficult shape and topology optimization problem is now fully parameterized into a relatively easier size optimization of generalized expansion coefficients as a result the optimization is changed into a numerical process of implementing a series of motions of the implicit level set function via an existing efficient convex programming method with the concept of the shape derivative the geometrical non linearity is included in the rigorous design sensitivity analysis to appropriately capture the large displacements of compliant mechanisms several numerical benchmark examples illustrate the effectiveness of the present level set method in particular its capability of generating new holes inside the material domain the proposed method not only retains the favorable features of the implicit free boundary representation but also overcomes several unfavorable numerical considerations relevant to the explicit scheme the reinitialization procedure and the velocity extension algorithm in the conventional level set method

digital computation of linear canonical transforms we deal with the problem of efficient and accurate digital computation of the samples of the linear canonical transform lct of a function from the samples of the original function two approaches are presented and compared the first is based on decomposition of the lct into chirp multiplication fourier transformation and scaling operations the second is based on decomposition of the lct into a fractional fourier transform followed by scaling and chirp multiplication both algorithms take n log n time where n is the time bandwidth product of the signals the only essential deviation from exactness arises from the approximation of a continuous fourier transform with the discrete fourier transform thus the algorithms compute lcts with a performance similar to that of the fast fourier transform algorithm in computing the fourier transform both in terms of speed and accuracy

numerical methods for the topology optimization of structures that exhibit snap through the inclusion of non linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response e g the geometric non linear response of compliant mechanisms in previous work the non linear response is computed by standard non linear elastic finite element analysis here we incorporate a load displacement constraint method to traverse non linear equilibrium paths with limit points to design structures that exhibit snap through behaviour to accomplish this we modify the basic arc length algorithm and embed this analysis into the topology optimization problem copyright 2002 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

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

the entire regularization path for the support vector machine the support vector machine svm is a widely used tool for classification many efficient implementations exist for fitting a two class svm model the user has to supply values for the tuning parameters the regularization cost parameter and the kernel parameters it seems a common practice is to use a default value for the cost parameter often leading to the least restrictive model in this paper we argue that the choice of the cost parameter can be critical we then derive an algorithm that can fit the entire path of svm solutions for every value of the cost parameter with essentially the same computational cost as fitting one svm model we illustrate our algorithm on some examples and use our representation to give further insight into the range of svm solutions

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

a newly developed qp relaxation method for element connectivity parameterization to achieve stress based topology optimization for geometrically nonlinear structures the aim of this work is to present a novel computational approach to employ the stress based topology optimization method stom to minimize the volume subject to the locally defined stress constraints of a geometrically nonlinear structure in the framework of the element connectivity parameterization ecp method considering the locally defined stress constraints in topology optimization to is a classic and challenging engineering problem and successful optimization procedures have recently been developed using the density based to method for linear elastic structures however no study has yet considered the static failure constraint when using to for a geometrically nonlinear structure therefore the present study develops a novel computational approach for the stom for a geometrically nonlinear structure to successfully optimize a geometrically nonlinear structure the unstable element issue must be properly addressed in addition to the stress singularity issue the existence of a large number of constraints and the highly nonlinear behavior of the local stress constraints to effectively resolve these issues this research adopts the ecp method to interpolate and optimize the connectivities among solid finite elements furthermore we find that a stress singularity issue linked to the local optima issue arises in the ecp method that is different from that of the density based to by investigating the singularity behavior in detail we develop a new qp relaxation method that is suitable for the ecp method to demonstrate the improved capability of the proposed ecp method with the modified qp relaxation several two dimensional to problems are solved

large strain phase field based multi material topology optimization summary a multi material topology optimization scheme is presented the formulation includes an arbitrary number of phases with different mechanical properties to ensure that the sum of the volume fractions is unity and in order to avoid negative phase fractions an obstacle potential function which introduces infinity penalty for negative densities is utilized the problem is formulated for nonlinear deformations and the objective of the optimization is the end displacement the boundary value problems associated with the optimization problem and the equilibrium equation are solved using the finite element method to illustrate the possibilities of the method it is applied to a simple boundary value problem where optimal designs using multiple phases are considered copyright 2015 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

topology optimization utilizing inverse motion based form finding topology optimization at finite strain setting using the concept of inverse motion based form finding is introduced this novel procedure allows boundary conditions and shape of the structure in the operating deformed state to be prescribed the outcome of the optimization algorithm will be the shape of the undeformed structure i e the state in which the structure should be manufactured the objective of the optimization considered is to find the stiffest structure for a given amount of material the problem is regularized using a helmholtz filter which is formulated in the deformed configuration both the elastic boundary value problem and the partial differential equation associated with the helmholtz filter are solved using the finite element method the optimization problem is solved using a sequence of convex separable approximations the paper is closed by 2d as well as 3d numerical examples that clearly illustrates that the method is able to find optimal solutions for inverse motion finite strain topology optimization problems

generalization of the fractional fourier transformation to an arbitrary linear lossless transformation an operator approach we develop the operator formalism to show how systematically the fractional fourier transformation of a wavefunction recently introduced by namias 1980 can be derived from the rotation of the corresponding wigner distribution function in phase space in this formalism the phase factor obtained by mcbride and kerr 1987 is seen to come from the caustics of the harmonic oscillator green function then the idea is generalized to the case of an arbitrary area preserving linear transformation in phase space and a concept of the special affine fourier transformation saft is introduced an explicit form of the integral representation of the saft is given and some simple examples are presented

topology optimization of compliant circular path mechanisms based on an aggregated linear system and singular value decomposition summary this paper proposes a topology optimization method for the design of compliant circular path mechanisms or compliant mechanisms having a set of output displacement vectors with a constant norm which is induced by a given set of input forces to perform the optimization a simple linear system composed of an input force vector an output displacement vector and a matrix connecting them is constructed in the context of a discretized linear elasticity problem using fem by adding two constraints 1 the dimensions of the input and the output vectors are equal 2 the euclidean norms of all local input force vectors are constant from the singular value decomposition of the matrix connecting the input force vector and the output displacement vector the optimization problem which specifies and equalizes the norms of all output vectors is formulated it is a minimization problem of the weighted summation of the condition number of the matrix and the least square error of the second singular value and the specified value this methodology is implemented as a topology optimization problem using the solid isotropic material with penalization method sensitivity analysis and method of moving asymptotes the numerical examples illustrate mechanically reasonable compliant circular path mechanisms and other mechanisms having multiple outputs with a constant norm copyright 2011 john wiley sons ltd

design sensitivity analysis and optimization for nonlinear buckling of finite dimensional elastic conservative structures the purpose of this review paper is to summarize the existing methods of design sensitivity analysis and optimization of elastic conservative finite dimensional systems with respect to nonlinear buckling behavior difficulties related to geometrical nonlinear singular behaviors are discussed in detail characteristics of optimized structures are demonstrated in reference to snapthrough behavior hill top branching and degenerate critical points a new optimization result of a flexible truss that fully utilizes the snapthrough behavior is also presented

nonsmooth optimization techniques for semisupervised classification we apply nonsmooth optimization techniques to classification problems with particular reference to the transductive support vector machine tsvm approach where the considered decision function is nonconvex and nondifferentiable hence difficult to minimize we present some numerical results obtained by running the proposed method on some standard test problems drawn from the binary classification literature

non probabilistic reliability based topology optimization of geometrically nonlinear structures using convex models this paper describes a non probabilistic reliability based topology optimization method for the design of continuum structures undergoing large deformations the variation of the structural system is treated with the multi ellipsoid convex model which is a realistic description of the parameters being inherently uncertain but bounded or lacking sufficient probabilistic data the formulation of the optimal design is established as a volume minimization problem with non probabilistic reliability constraints on the geometrically nonlinear structural behaviour in order to circumvent numerical difficulties in solving the nested double loop optimization problem a performance measure based approach is employed to transform the constraint on the reliability index into one on the concerned performance in conjunction with an efficient adjoint variable scheme for the sensitivity analysis of reliability constraints the optimization problem is solved by gradient based mathematical programming methods three numerical examples for the optimization design of planar structures are presented to illustrate the validity and applicability of the proposed method the obtained optimal solutions show the importance of incorporating various uncertainties in the design problem moreover it is also revealed that the geometrical nonlinearity needs to be accounted for to ensure satisfaction of the reliability constraints in the optimal design of structures with large deformation

the proximal trajectory algorithm in svm cross validation we propose a bilevel cross validation scheme for support vector machine svm model selection based on the construction of the entire regularization path since such path is a particular case of the more general proximal trajectory concept from nonsmooth optimization we propose for its construction an algorithm based on solving a finite number of structured linear programs our methodology differently from other approaches works directly on the primal form of svm numerical results are presented on binary data sets drawn from literature

minimizing nonconvex nonsmooth functions via cutting planes and proximity control we describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex not necessarily differentiable function of several variables the method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and is related to the use of the concept of proximal trajectory convergence to a stationary point is proved for weakly semismooth functions

topological design for vibrating structures the topological optimization technique using micro scale voids with the homogenization method for stress analysis of perforated structures has been applied to solve stiffness maximization problem with success here the homogenization design technique is extended to design problems concerning vibrating structures an extended optimization algorithm is also derived to maximize a set of eigenvalues as well as to identify the topology design for specified eigenvalues to characterize forced vibration of a structure

an algorithm for the topology optimization of geometrically nonlinear structures summary most papers on topology optimization consider that there is a linear relation between the strains and displacements of the structure implicitly assuming that the displacements of the structure are small however when the external loads applied to the structure are large the displacements also become large so it is necessary to suppose that there is a nonlinear relation between strains and displacements in this case we say that the structure is geometrically nonlinear in practice this means that the linear system that needs to be solved each time the objective function of the problem is evaluated is replaced by an ill conditioned nonlinear system of equations moreover the stiffness matrix and the derivatives of the problem also become harder to compute the objective of this work is to solve topology optimization problems under large displacements through a new optimization algorithm named sequential piecewise linear programming this method relies on the solution of convex piecewise linear programming subproblems that include second order information about the objective function to speed up the algorithm these subproblems are converted into linear programming ones the new algorithm is not only globally convergent to stationary points but our numerical experiments also show that it is efficient and robust copyright 2014 john wiley sons ltd

geometrical design of thermoelectric generators based on topology optimization summary this paper discusses an application of the topology optimization method for the design of thermoelectric generators the proposed methodology provides the optimized geometry in accordance with various arbitrary conditions such as the types of materials the volume of materials and the temperature and shape of the installation position by considering the coupled equations of state for the thermoelectric problem we introduce an analytical model subject to these equations which mimics the closed circuit composed of thermoelectric materials electrodes and a resistor the total electric power applied to the resistor and the conversion efficiency are formulated as objective functions to be optimized the proposed optimization method for thermoelectric generators is implemented as a geometrical optimization method using the solid isotropic material with penalization method used in topology optimizations simple relationships are formulated between the density function of the solid isotropic material with penalization method and the physical properties of the thermoelectric material a sensitivity analysis for the objective functions is formulated with respect to the density function and the adjoint equations required for calculating it depending on the sensitivity the density function is updated using the method of moving asymptotes finally numerical examples are provided to demonstrate the validity of the proposed method copyright 2012 john wiley sons ltd

a bundle modification strategy for convex minimization we present a new bundle algorithm for minimizing convex not necessarily smooth functions the novelty of our approach is based on a bundle modification strategy that we apply whenever the stability center is updated and which is aimed at substituting the points of the bundle by new points characterized by possibly better values of the objective function convergence of the algorithm is proved and numerical results are presented

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

solutions to shape and topology eigenvalue optimization problems using a homogenization method a solution strategy to find the shape and topology of structures that maximize a natural frequency is presented the methodology is based on a homogenization method and the representation of the shape of the structure as a material property the problem is formulated as a reinforcement problem in which a given structure is reinforced using a prescribed amount of material two dimensional plane elasticity problems are considered examples are presented for illustration

topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique abstract this paper presents a simple but effective additive hyperelasticity technique to circumvent numerical difficulties in solving the material density based topology optimization of elastic structures undergoing large displacements by adding a special hyperelastic material to the design domain excessive distortion and numerical instability occurred in the low density or intermediate density elements are thus effectively alleviated during the optimization process the properties of the additional hyperelastic material are established based on a new interpolation scheme which allows the nonlinear mechanical behaviour of the remodelled structure to achieve an acceptable approximation to the original structure in conjunction with the adjoint variable scheme for sensitivity analysis the topology optimization problem is solved by a gradient based mathematical programming algorithm numerical examples are given 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

fast bundle algorithm for multiple instance learning we present a bundle algorithm for multiple instance classification and ranking these frameworks yield improved models on many problems possessing special structure multiple instance loss functions are typically nonsmooth and nonconvex and current algorithms convert these to smooth nonconvex optimization problems that are solved iteratively inspired by the latest linear time subgradient based methods for support vector machines we optimize the objective directly using a nonconvex bundle method computational results show this method is linearly scalable while not sacrificing generalization accuracy permitting modeling on new and larger data sets in computational chemistry and other applications this new implementation facilitates modeling with kernels

generalized eigenvectors and fractionalization of offset dfts and dcts the offset discrete fourier transform dft is a discrete transform with kernel exp j2 spl pi m a n b n it is more generalized and flexible than the original dft and has very close relations with the discrete cosine transform dct of type 4 dct iv dct viii discrete sine transform dst iv dst viii and discrete hartley transform dht iv in this paper we derive the eigenvectors eigenvalues of the offset dft especially for the case where a b is an integer by convolution theorem we can derive the close form eigenvector sets of the offset dft when a b is an integer we also show the general form of the eigenvectors in this case then we use the eigenvectors eigenvalues of the offset dft to derive the eigenvectors eigenvalues of the dct iv dct viii dst iv dst viii and dht iv after the eigenvectors eigenvalues are derived we can use the eigenvectors decomposition method to derive the fractional operations of the offset dft dct iv dct viii dst iv dst viii and dht iv these fractional operations are more flexible than the original ones and can be used for filter design data compression encryption and watermarking etc