Topology Optimization Procedure Based on Structure Stress History

The structure design points stress analysis for succeeding optimization steps is the base of the topology optimization algorithm. Because the material density of the material is proportional to the strain energy we can predict how to change the density for each design point for the next optimization steps. Proposed procedure make the optimization process faster and final topology is finally more optimal than topology obtained using standard optimization procedures. This stress analyzing procedure, can be treated as a hardware of the sensors being the part of the smart structure for the real time structure reconstruction. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Isogeometric structural shape optimization using a fictitious energy regularization

In isogeometric analysis, NURBS basis functions are used as shape functions in an isoparametric finite-element-type discretization. Among other advantageous features, this approach is able to provide exact and smooth representations of a broad class of computational domains with curved boundaries. Therefore, this discretization method seems to be especially convenient for computational shape optimization, where a smooth and CAD-like parametrization of the optimal geometry is desired. Choosing boundary control point coordinates of an isogeometric discretization as design variables, an additional design model can be avoided. However, for a higher number of design variables, typical drawbacks like oscillating boundaries as known from early node-based shape optimization methods appear. To overcome this problem, we propose to use a fictitious energy regularization: the strain energy of a fictitious deformation, which maps the initial to the optimized domain, is employed as a regularizing term in the optimization problem. Moreover, this deformation is used for efficiently moving the dependent nodes within the domain in each step of the optimization process. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topology and material orientation optimization based on evolution equations

Many modern high-performance materials have inherent anisotropic elastic properties and its local material orientation can be considered to be an additional design variable for the topology optimization [1–3]. We extend our previous model for topology optimization with variational controlled growth [4–6] for linear elastic anisotropic materials, for which the material orientation is introduced as an additional design variable. We solve the optimization problem purely with the principles of thermodynamics by minimizing the Gibbs energy. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bi-criteria optimization problems for decision rules

We consider bi-criteria optimization problems for decision rules and rule systems relative to length and coverage. We study decision tables with many-valued decisions in which each row is associated with a set of decisions as well as single-valued decisions where each row has a single decision. Short rules are more understandable; rules covering more rows are more general. Both of these problems—minimization of length and maximization of coverage of rules are NP-hard. We create dynamic programming algorithms which can find the minimum length and the maximum coverage of rules, and can construct the set of Pareto optimal points for the corresponding bi-criteria optimization problem. This approach is applicable for medium-sized decision tables. However, the considered approach allows us to evaluate the quality of various heuristics for decision rule construction which are applicable for relatively big datasets. We can evaluate these heuristics from the point of view of (i) single-criterion—we can compare the length or coverage of rules constructed by heuristics; and (ii) bi-criteria—we can measure the distance of a point (length, coverage) corresponding to a heuristic from the set of Pareto optimal points. The presented results show that the best heuristics from the point of view of bi-criteria optimization are not always the best ones from the point of view of single-criterion optimization.     

Displacement field analysis of the topology optimized structure

In this paper for the topology optimization process the minimum compliance approach is used with FEM as very useful method for numerical realization of the problem. During the optimization process homogenized domain changes into discrete structure which means the final structure consists of the many optimal placed bars. The analysis of the deformed structure and the deformed finite elements is done from the displacement field point of view. It can be noticed, some of the finite elements reduce their size, some of them increase their size. It depends on the element status (void-empty, stressed or not stressed). The question arises: is the topology optimization process cause of the negative Poisson ratio for some parts of the structure? (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An evolution equation based approach to topology optimization

The objective of topology optimization is to find a mechanical structure with maximum stiffness and minimal amount of used material for given boundary conditions [2]. There are different approaches. Either the structure mass is held constant and the structure stiffness is increased or the amount of used material is constantly reduced while specific conditions are fulfilled. In contrast, we focus on the growth of a optimal structure from a void model space and solve this problem by introducing a variational problem considering the spatial distribution of structure mass (or density field) as variable [3]. By minimizing the Gibbs free energy according to Hamilton's principle in dynamics for dissipative processes, we are able to find an evolution equation for the internal variable describing the density field. Hence, our approach belongs to the growth strategies used for topology optimization. We introduce a Lagrange multiplier to control the total mass within the model space [1]. Thus, the numerical solution can be provided in a single finite element environment as known from material modeling. A regularization with a discontinuous Galerkin approach for the density field enables us to suppress the well-known checkerboarding phenomena while evaluating the evolution equation within each finite element separately [4]. Therefore, the density field is no additional field unknown but a Gauß-point quantity and the calculation effort is strongly reduced. Finally, we present solutions of optimized structures for different boundary problems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Una formulación de mínimo peso con restricciones en tensión en optimización topológica de estructuras

Optimum design of structures has been traditionally focused on the analysis of shape and dimensions optimization problems. However, more recently a new discipline has emerged: the topology optimization of the structures. This discipline states innovative models that allow to obtain optimal solutions without a previous definition of the type of structure being considered. These formulations obtain the optimal topology and the optimal shape and size of the resulting elements. The most usual formulations of the topology optimization problem try to obtain the structure of maximum stiffness. These approaches maximize the stiffness for a given amount of material to be used. These formulations have been widely analyzed and applied in engineering but they present considerable drawbacks from a numerical and from a practical point of view. In this paper the author propose a different formulation, as an alternative to maximum stiffness approaches, that minimizes the weight and includes stress constraints. The advantages of this kind of formulations are crucial since the cost of the structure is minimized, which is the most frequent objective in engineering, and they guarantee the structural feasibility since stresses are constrained. In addition, this approach allows to avoid some of the drawbacks and numerical instabilities related to maximum stiffness approaches. Finally, some practical examples have been solved in order to verify the validity of the results obtained and the advantages of the proposed formulation.     

Distributed multi-agent optimization with state-dependent communication

We study distributed algorithms for solving global optimization problems in which the objective function is the sum of local objective functions of agents and the constraint set is given by the intersection of local constraint sets of agents. We assume that each agent knows only his own local objective function and constraint set, and exchanges information with the other agents over a randomly varying network topology to update his information state. We assume a state-dependent communication model over this topology: communication is Markovian with respect to the states of the agents and the probability with which the links are available depends on the states of the agents. We study a projected multi-agent subgradient algorithm under state-dependent communication. The state-dependence of the communication introduces significant challenges and couples the study of information exchange with the analysis of subgradient steps and projection errors. We first show that the multi-agent subgradient algorithm when used with a constant stepsize may result in the agent estimates to diverge with probability one. Under some assumptions on the stepsize sequence, we provide convergence rate bounds on a “disagreement metric” between the agent estimates. Our bounds are time-nonhomogeneous in the sense that they depend on the initial starting time. Despite this, we show that agent estimates reach an almost sure consensus and converge to the same optimal solution of the global optimization problem with probability one under different assumptions on the local constraint sets and the stepsize sequence.     

Multi-objective combinatorial optimization for selecting best available techniques (BAT) in the industrial sector: the COMBAT tool

The introduction of best available techniques (BAT) with the European Commission's directive 96/61, created a new framework for ‘cleaner’ production in the industrial sector. BATs practically constitute recommended techniques for each of the steps in the manufacturing process. Thus, the industries must decide on which BATs are most appropriate for their processes. In the current study, an integrated approach is applied in order to find the mixture of BATs for the entire industrial sector that satisfies as much as possible the economic and the environmental criteria. The former represent the industry owner's point of view expressed by the Net Present Value of the projects and the latter represent the society's point of view quantified by the emission reduction in some major pollutants. The developed multi-objective optimization model is addressed using two methods: (1) goal programming and (2) generation of the Pareto optimal solutions using an augmented version of the ε-constraint method followed by an interactive filtering process in order to select the most preferred Pareto optimal solution. The generation of the Pareto optimal solutions is performed using an improved version of the widely used ε-constraint method that overcomes some of its known drawbacks. The COMBAT tool (combinatorial optimization with multiple criteria for BAT selection) that is developed for implementing these methods is also described and the results from its application in the industrial sector of the greater Athens area are presented.     

Beyond polyconvexity: an existence result for a class of quasiconvex hyperelastic materials

This article contains an existence result for a class of quasiconvex stored energy functions satisfying the material non‐interpenetrability condition, which primarily obstructs applying classical techniques from the vectorial calculus of variations to nonlinear elasticity. The fundamental concept of reversibility serves as the starting point for a theory of nonlinear elasticity featuring the basic duality inherent to the Eulerian and Lagrangian points of view. Motivated by this concept, split‐quasiconvex stored energy functions are shown to exhibit properties, which are very alluding from a mathematical point of view. For instance, any function with finite energy is automatically a Sobolev homeomorphism; existence of minimizers can be readily established and first variation formulae hold. Copyright © 2016 John Wiley & Sons, Ltd.     

Estimating allocations for Value-at-Risk portfolio optimization

Value-at-Risk, despite being adopted as the standard risk measure in finance, suffers severe objections from a practical point of view, due to a lack of convexity, and since it does not reward diversification (which is an essential feature in portfolio optimization). Furthermore, it is also known as having poor behavior in risk estimation (which has been justified to impose the use of parametric models, but which induces then model errors). The aim of this paper is to chose in favor or against the use of VaR but to add some more information to this discussion, especially from the estimation point of view. Here we propose a simple method not only to estimate the optimal allocation based on a Value-at-Risk minimization constraint, but also to derive—empirical—confidence intervals based on the fact that the underlying distribution is unknown, and can be estimated based on past observations.     

Shape and topology optimization for elliptic boundary value problems using a piecewise constant level set method

The aim of this paper is to propose a variational piecewise constant level set method for solving elliptic shape and topology optimization problems. The original model is approximated by a two-phase optimal shape design problem by the ersatz material approach. Under the piecewise constant level set framework, we first reformulate the two-phase design problem to be a new constrained optimization problem with respect to the piecewise constant level set function. Then we solve it by the projection Lagrangian method. A gradient-type iterative algorithm is presented. Comparisons between our numerical results and those obtained by level set approaches show the effectiveness, accuracy and efficiency of our algorithm.     

求解多项式规划的一个全局最优化算法

提出一个求解带箱子约束的一般多项式规划问题的全局最优化算法, 该算法包含两个阶段, 在第一个阶段, 利用局部最优化算法找到一个局部最优解. 在第二阶段, 利用一个在单位球上致密的向量序列, 将多元多项式转化为一元多项式, 通过求解一元多项式的根, 找到一个比当前局部最优解更好的点作为初始点, 回到第一个 阶段, 从而得到一个更好的局部最优解, 通过两个阶段的循环最终找到问题的全局最优解, 并给出了算法收敛性分析. 最后, 数值结果表明了算法是有效的.     

A dynamic programming approach to adjustable robust optimization

In this paper we consider the adjustable robust approach to multistage optimization, for which we derive dynamic programming equations. We also discuss this from the point of view of risk averse stochastic programming. We consider as an example a robust formulation of the classical inventory model and show that, like for the risk neutral case, a basestock policy is optimal.     

Topology optimization of periodic Mindlin plates

Periodic structures exhibit unique dynamic characteristics that make them act as tunable mechanical filters for wave propagation. As a result, waves can propagate along the periodic structures only within specific frequency bands called the ‘pass bands’ and wave propagation is completely blocked within other frequency bands called the ‘stop bands’ or ‘band gaps’. The spectral width of these bands can be optimized using topology optimization. In this paper, topology optimization is used to maximize the fundamental natural frequency of Mindlin plates while enforcing periodicity. A finite element model for Mindlin plates is presented and used along with an optimization algorithm that accounts for the periodicity constraint in order to determine the optimal topologies of plates with various periodic configurations. The obtained results demonstrate the effectiveness of the proposed design optimization approach in generating periodic plates with optimal natural frequency and wide stop bands. The presented approach can be invaluable design tool for many structures in order to control the wave propagation in an attempt to stop/confine the propagation of undesirable disturbances.     

Diagonalized multiplier methods and quasi-Newton methods for constrained optimization

Two approaches to quasi-Newton methods for constrained optimization problems inR ⁿ are presented. These approaches are based on a class of Lagrange multiplier approximation formulas used by the author in his previous work on Newton's method for constrained problems. The first approach is set in the framework of a diagonalized multiplier method. From this point of view, a new update rule for the Lagrange multipliers which depends on the particular quasi-Newton method employed is given. This update rule, in contrast to most other update rules, does not require exact minimization of the intermediate unconstrained problem. In fact, the optimal convergence rate is attained in the extreme case when only one step of a quasi-Newton method is taken on this intermediate problem. The second approach transforms the constrained optimization problem into an unconstrained problem of the same dimension.The author would like to thank J. Moré and M. J. D. Powell for comments related to the material in Section 13. He also thanks J. Nocedal for the computer results in Tables 1–3 and M. Wright for the results in Table 4, which were obtained via one of her general programs. Discussions with M. R. Hestenes and A. Miele regarding their contributions to this area were very helpful. Many individuals, including J. E. Dennis, made useful general comments at various stages of this paper. Finally, the author is particularly thankful to R. Byrd, M. Heath, and R. McCord for reading the paper in detail and suggesting many improvements.This work was supported by the Energy Research and Development Administration, Contract No. E-(40-1)-5046, and was performed in part while the author was visiting the Department of Operations Research, Stanford University, Stanford, California.     

Efficient topology optimization of thermo-elasticity problems using coupled field adjoint sensitivity analysis method

We develop a unified and efficient adjoint design sensitivity analysis (DSA) method for weakly coupled thermo-elasticity problems. Design sensitivity expressions with respect to thermal conductivity and Young's modulus are derived. Besides the temperature and displacement adjoint equations, a coupled field adjoint equation is defined regarding the obtained adjoint displacement field as the adjoint load in the temperature field. Thus, the computing cost is significantly reduced compared to other sensitivity analysis methods. The developed DSA method is further extended to a topology design optimization method. For the topology design optimization, the design variables are parameterized using a bulk material density function. Numerical examples show that the DSA method developed is extremely efficient and the optimal topology varies significantly depending on the ratio of mechanical and thermal loadings.     

Multi-train trajectory optimization for energy-efficient timetabling

This paper proposes a novel approach for energy-efficient timetabling by adjusting the running time allocation of given timetables using train trajectory optimization. The approach first converts the arrival and departure times to time window constraints in order to relax the given timetable. Then a train trajectory optimization method is developed to find optimal arrival/departure times and optimal energy-efficient speed profiles within the relaxed time windows. The proposed train trajectory optimization method includes two types, a single-train trajectory optimization (STTO), which focuses on optimizing individual train movements within the relaxed arrival and departure time windows, and a multi-train trajectory optimization (MTTO), which computes multi-train trajectories simultaneously with a shared objective of minimizing multi-train energy consumption and an additional target of eliminating conflicts between trains. The STTO and MTTO are re-formulated as a multiple-phase optimal control problem, which has the advantage of accurately incorporating varying gradients, curves and speed limits and different train routes. The multiple-phase optimal control problem is then solved by a pseudospectral method. The proposed approach is applied in case studies to fine-tune two timetables, for a single-track railway corridor and a double-track corridor of the Dutch railway. The results suggest that the proposed approach is able to improve the energy efficiency of a timetable.     

Topology optimization considering the fatigue constraint of variable amplitude load based on the equivalent static load approach

In this research, a new layout optimization method is developed to consider high cycle fatigue constraints which occur due to variable amplitude mechanical loading. Although fatigue is a very important property in terms of safety when designing mechanical components, it has rarely been considered in topology optimization with the lack of concept and the difficulty of sensitivity analysis for fatigue constraints calculated from multiaxial cycle counting. For the topology optimization for fatigue constraint, we use transient stress analysis to extract effective stress cycles and Miner's cumulative damage rule to calculate total damage at every spatial element. Because the calculation of the exact sensitivities of a transient system is complex and time consuming for the topology optimization application, this research proposes to use the pseudo-sensitivities of fatigue constraints calculated by applying equivalent static load approach. In addition, as an aggregated fatigue constraint is very sensitive to the changes in stress value which causes some unstable convergences in optimization process, a new scaling approach of the aggregated fatigue damage constraint is developed. To validate the usefulness of the developed approaches, we solved some benchmark topology optimization problems and found that the present method provides physically appropriate layouts with stable optimization convergence.     

Shape optimization by the homogenization method

Summary. In the context of shape optimization, we seek minimizers of the sum of the elastic compliance and of the weight of a solid structure under specified loading. This problem is known not to be well-posed, and a relaxed formulation is introduced. Its effect is to allow for microperforated composites as admissible designs. In a two-dimensional setting the relaxed formulation was obtained in [6] with the help of the theory of homogenization and optimal bounds for composite materials. We generalize the result to the three dimensional case. Our contribution is twofold; first, we prove a relaxation theorem, valid in any dimensions; secondly, we introduce a new numerical algorithm for computing optimal designs, complemented with a penalization technique which permits to remove composite designs in the final shape. Since it places no assumption on the number of holes cut within the domain, it can be seen as a topology optimization algorithm. Numerical results are presented for various two and three dimensional problems. Received July 4, 1995