Target-matching test problem for multiobjective topology optimization using genetic algorithms

This paper describes the multiobjective topology optimization of continuum structures solved as a discrete optimization problem using a multiobjective genetic algorithm (GA) with proficient constraint handling. Crucial to the effectiveness of the methodology is the use of a morphological geometry representation that defines valid structural geometries that are inherently free from checkerboard patterns, disconnected segments, or poor connectivity. A graph- theoretic chromosome encoding, together with compatible reproduction operators, helps facilitate the transmission of topological/shape characteristics across generations in the evolutionary process. A multicriterion target-matching problem developed here is a novel test problem, where a predefined target geometry is the known optimum solution, and the good results obtained in solving this problem provide a convincing demonstration and a quantitative measure of how close to the true optimum the solutions achieved by GA methods can be. The methodology is then used to successfully design a path-generating compliant mechanism by solving a multicriterion structural topology optimization problem.     

A two phase approach based on skeleton convergence and geometric variables for topology optimization using genetic algorithm

This paper introduces a set of skeleton operators for characterizing topologies evolving in a bit-array represented structural topology optimization problem. It is shown that the design generally converges to a stable skeleton fairly early in the optimization process. It is observed that further optimization is more about finding optimal gross shape for the various branches of the converged skeleton and the bit-array representation is not appropriate. A two-phase approach to topology optimization is proposed in which the first phase, where bit-array is used to represent the topology, ends with the detection of stabilization of skeleton, and the second phase proceeds further with the geometry based representation that directly addresses gross variation in shape of the branches of the converged skeleton. Genetic Algorithm has been used for optimization in both the phases. The efficiency and effectiveness of the use of skeleton operators and geometric variables for identification of convergence in the first phase and optimization in the second phase respectively is demonstrated.     

Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows

Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. Several approaches based on the displacement of a set of control points have been developed in the last decades, such as the so-called free-form deformations. In this paper we present a new theoretical result which allows to recast free-form deformations into the general class of perturbation of identity maps, and to guarantee the compactness of the set of admissible shapes. Moreover, we address both a general optimization framework based on the continuous shape gradient and a numerical procedure for solving efficiently three-dimensional optimal design problems. This framework is applied to the optimal design of immersed bodies in Stokes flows, for which we consider the numerical solution of a benchmark case study from literature.     

Conceptual design of aeroelastic structures by topology optimization

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.     

A hybrid genetic algorithm for constrained multi-objective optimization under uncertainty and target matching problems

This work presents a new approach for interval-based uncertainty analysis. The proposed approach integrates a local search strategy as the worst-case-scenario technique of anti-optimization with a constrained multi-objective genetic algorithm. Anti-optimization is a term for an approach to safety factors in engineering structures which is described as pessimistic and searching for least favorable responses, in combination with optimization techniques but in contrast to probabilistic approaches. The algorithm is applied and evaluated to be efficient and effective in producing good results via target matching problems: a simulated topology and shape optimization problem where a ‘target’ geometry set is predefined as the Pareto optimal solution and a constrained multiobjective optimization problem formulated such that the design solutions will evolve and converge towards the target geometry set.     

Topology optimization with displacement constraints: a bilevel programming approach

We consider the minimum-compliance formulation of the truss topology problem with additional linear constraints on the displacements: the so-called displacement constraints. We propose a new bilevel programming approach to this problem. Our primal goal (upper-level) is to satisfy the displacement constraint as well as possible — we minimize the gap between the actual and prescribed displacement. Our second goal (lower level) is to minimize the compliance — we still want to find the stiffest structure satisfying the displacement constraints. On the lower level we solve a standard truss topology problem and hence we can solve it in the formulation suitable for the fast interior point alogrithms. The overall bilevel problem is solved by means of the so-called implicit programming approach. This approach leads to a nonsmooth optimization problem which is finally solved by a nonsmooth solver.     

On optimal topology of grillage structures

Two problems of optimum topological design of grillages are discussed: (1) the Equilibrium Linear Programming (ELP), where the analysis model is based only on equilibrium conditions and (2) the Nonlinear Program (NLP), where the ELP formulation is extended to include compatibility conditions. The structural topology is optimized by allowing elimination of elements. Three different force method formulations are presented for each of the problems. It is shown that the optimal topology for the NLP problem might correspond to a singular point in the design space. The optimal topology for the ELP problem is obtained by solving a linear program (LP).

Conditions for selecting a geometry of Multiple Optimal Topologies (MOT) are derived. The objective function for the MOT geometry is shown to be independent of the redundant forces, and some of the optimal topologies are usually statically determinate structures. In such cases the lower bound on the optimal value obtained by the ELP solution is equal to the final global optimum. Examples are given to illustrate how the optimal topology and its corresponding load path change with the geometric parameters. Design procedures that combine automated optimization and CAD techniques are most suitable for solving the presented problems.

     

Visual processing and classification of items on a moving conveyor: a selective perception approach

Many industrial applications require some sort of automated visual processing and classification of items placed on a moving conveyor. In this paper, we present a selective perception based approach to visual processing. The novelty of this approach is that instead of processing the whole image, only areas that are deemed ‘‘interesting’’ and hence calling for attention are analyzed. The attentional sequences thus constructed can then be used for a variety of tasks including shape determination. Since only a small portion of the whole image is processed, visual processing can be real-time and flexible without requiring special hardware. Two different applications based on this approach are described. In a defective item detection task, we explain in detail how attentional sequences can be used. As a second application, the approach has been implemented in an automated remote controller sorter in a TV manufacturing plant—thus confirming its practical applicability.     

A linear-complexity reparameterisation strategy for the hierarchical bootstrapping of capabilities within perception–action architectures

Perception–action (PA) architectures are capable of solving a number of problems associated with artificial cognition, in particular, difficulties concerned with framing and symbol grounding. Existing PA algorithms tend to be ‘horizontal’ in the sense that learners maintain their prior percept–motor competences unchanged throughout learning. We here present a methodology for simultaneous ‘horizontal’ and ‘vertical’ perception–action learning in which there additionally exists the capability for incremental accumulation of novel percept–motor competences in a hierarchical fashion.The proposed learning mechanism commences with a set of primitive ‘innate’ capabilities and progressively modifies itself via recursive generalising of parametric spaces within the linked perceptual and motor domains so as to represent environmental affordances in maximally-compact manner. Efficient reparameterising of the percept domain is here accomplished by the exploratory elimination of dimensional redundancy and environmental context.Experimental results demonstrate that this approach exhibits an approximately linear increase in computational requirements when learning in a typical unconstrained environment, as compared with at least polynomially-increasing requirements for a classical perception–action system.     

Hybrid multi-objective shape design optimization using Taguchi’s method and genetic algorithm

This research is based on a new hybrid approach, which deals with the improvement of shape optimization process. The objective is to contribute to the development of more efficient shape optimization approaches in an integrated optimal topology and shape optimization area with the help of genetic algorithms and robustness issues. An improved genetic algorithm is introduced to solve multi-objective shape design optimization problems. The specific issue of this research is to overcome the limitations caused by larger population of solutions in the pure multi-objective genetic algorithm. The combination of genetic algorithm with robust parameter design through a smaller population of individuals results in a solution that leads to better parameter values for design optimization problems. The effectiveness of the proposed hybrid approach is illustrated and evaluated with test problems taken from literature. It is also shown that the proposed approach can be used as first stage in other multi-objective genetic algorithms to enhance the performance of genetic algorithms. Finally, the shape optimization of a vehicle component is presented to illustrate how the present approach can be applied for solving multi-objective shape design optimization problems.     

A new level-set based approach to shape and topology optimization under geometric uncertainty

Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems and partial differential equations that define mappings between different manifolds. There are several contributions of this work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity field characterized with a reduced set of random variables using the Karhunen–Loeve expansion. Multivariate Gauss quadrature is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts.     

A performance index for topology and shape optimization of plate bending problems with displacement constraints

This paper presents a performance index for topology and shape optimization of plate bending problems with displacement constraints. The performance index is developed based on the scaling design approach. This performance index is used in the Performance-Based Optimization (PBO) method for plates in bending to keep track of the performance history when inefficient material is gradually removed from the design and to identify optimal topologies and shapes from the optimization process. Several examples are provided to illustrate the validity and effectiveness of the proposed performance index for topology and shape optimization of bending plates with single and multiple displacement constraints under various loading conditions. The topology optimization and shape optimization are undertaken for the same plate in bending, and the results are evaluated by using the performance index. The proposed performance index is also employed to compare the efficiency of topologies and shapes produced by different optimization methods. It is demonstrated that the performance index developed is an effective indicator of material efficiency for bending plates. From the manufacturing and efficient point of view, the shape optimization technique is recommended for the optimization of plates in bending. Received November 27, 1998?Revised version received June 6, 1999     

连续体结构的参数化水平集统一优化

针对传统分步式结构优化设计的不足,提出一种同时进行结构拓扑、形状和尺寸统一优化的设计方法.首先采用水平集函数描述统一的结构优化模型和几何尺寸边界,通过引入紧支径向插值基函数将结构拓扑优化变量、形状优化变量和尺寸优化变量变换为基函数的扩展系数;然后取该扩展系数为设计变量,借助一种参数的变化表达3种优化要素对结构性能的影响,将复杂的多变量优化问题变换为相对简单的参数优化问题,有利于与相对成熟的优化算法相结合提高求解效率;进一步用R函数将其融合为一个整体,构造出统一优化模型,并用最优化准则法进行求解.最后通过数值案例证明了该方法的有效性和精确性.     

Force recovery procedures in nonlinear analysis

Conventionally, incremental-iterative schemes have been used in solving nonlinear problems. There are two phases involved in such analyses. The first or ‘predictor’ phase relates to solution of the structural displacement increments from the total incremental equations of equilibrium, while the second or ‘corrector’ phase is concerned with recovery of the element forces from the element displacement increments obtained in the first phase. In this paper, it will be demonstrated that accuracy of the numerical solutions depends primarily on the corrector or procedure for recovering the element forces. The expression used in the predictor can only affect the number of iterations required at each incremental step, but not the final shape of the load-deflection curves. A key factor in selecting the procedure for force recovery is that higher order nonlinear effects must be included, based on rigorous continuum mechanics formulations.     

Block aggregation of stress constraints in topology optimization of structures

Structural topology optimization problems have been traditionally stated and solved by means of maximum stiffness formulations. On the other hand, some effort has been devoted to stating and solving this kind of problems by means of minimum weight formulations with stress (and/or displacement) constraints. It seems clear that the latter approach is closer to the engineering point of view, but it also leads to more complicated optimization problems, since a large number of highly non-linear (local) constraints must be taken into account to limit the maximum stress (and/or displacement) at the element level. In this paper, we explore the feasibility of defining a so-called global constraint, which basic aim is to limit the maximum stress (and/or displacement) simultaneously within all the structure by means of one single inequality. Should this global constraint perform adequately, the complexity of the underlying mathematical programming problem would be drastically reduced. However, a certain weakening of the feasibility conditions is expected to occur when a large number of local constraints are lumped into one single inequality. With the aim of mitigating this undesirable collateral effect, we group the elements into blocks. Then, the local constraints corresponding to all the elements within each block can be combined to produce a single aggregated constraint per block. Finally, we compare the performance of these three approaches (local, global and block aggregated constraints) by solving several topology optimization problems.     

Topological sensitivity analysis

The so-called topological derivative concept has been seen as a powerful framework to obtain the optimal topology for several engineering problems. This derivative characterizes the sensitivity of the problem when a small hole is created at each point of the domain. However, the greatest limitation of this methodology is that when a hole is created it is impossible to build a homeomorphic map between the domains in study (because they have not the same topology). Therefore, some specific mathematical framework should be developed in order to obtain the derivatives. This work proposes an alternative way to compute the topological derivative based on the shape sensitivity analysis concepts. The main feature of this methodology is that all the mathematical procedure already developed in the context of shape sensitivity analysis may be used in the calculus of the topological derivative. This idea leads to a more simple and constructive formulation than the ones found in the literature. Further, to point out the straightforward use of the proposed methodology, it is applied for solving some design problems in steady-state heat conduction.     

Evaluation of key shapes on a touchscreen simulation of a specialised keypad

The effect of key shape on a touchscreen simulation of a flat touch-sensitive keypad was investigated by rapid prototyping. A software prototype was developed of a hardware keypad that controlled chemical analysis equipment. The prototype was used to answer some basic ergonomics questions concerning the design of the device keys. In a target selection task, keys shaped as equilateral triangles were most precise, and least precise was a compound shape comprising of a rectangular lower part and a triangular ‘hat’ as the upper part. No significant differences in the times taken to complete the task were found. It is suggested that the use of a touchscreen prototype is suitable when designing flat-keypad layouts. Due to the visual nature of flat keypad use, where perception of the target must counter reduced tactile feedback, the design of shapes to aid the selection of keys is of paramount importance. It is concluded that rapid prototyping of hardware with graphical designs and touchscreens is a powerful tool for the ergonomic design of interfaces.     

Pareto frontier exploration in multiobjective topology optimization using adaptive weighting and point selection schemes

Topology optimization has been used in many industries and applied to a variety of design problems. In real-world engineering design problems, topology optimization problems often include a number of conflicting objective functions, such to achieve maximum stiffness and minimum mass of a design target. The existence of conflicting objective functions causes the results of the topology optimization problem to appear as a set of non-dominated solutions, called a Pareto-optimal solution set. Within such a solution set, a design engineer can easily choose the particular solution that best meets the needs of the design problem at hand. Pareto-optimal solution sets can provide useful insights that enable the structural features corresponding to a certain objective function to be isolated and explored. This paper proposes a new Pareto frontier exploration methodology for multiobjective topology optimization problems. In our methodology, a level set-based topology optimization method for a single-objective function is extended for use in multiobjective problems, using a population-based approach in which multiple points in the objective space are updated and moved to the Pareto frontier. The following two schemes are introduced so that Pareto-optimal solution sets can be efficiently obtained. First, weighting coefficients are adaptively determined considering the relative position of each point. Second, points in sparsely populated areas are selected and their neighborhoods are explored. Several numerical examples are provided to illustrate the effectiveness of the proposed method.     

Optimization of perforated domains through homogenization

We begin by explaining briefly why some shape/topology optimization problems need to be relaxed through homogenization. In Section 2 we present, from a mechanical viewpoint, the formula for the homogenized coefficients for a periodic infinitesimal perforation, and then briefly discuss the locally periodic ones (Section 3). Sections 4–6 describe a program which minimizes a certain functional over the set of model holes, and then its integration into a larger program, intended to treat topology and shape optimization problems. Numerical results are presented.     

Topology optimization of beam cross-section considering warping deformation

The beam cross-section optimization problems have been very important as beams are widely used as efficient load-carrying structural components. Most of the earlier investigations focus on the dimension and shape optimization or on the topology optimization along the axial direction. An important problem in beam section design is to find the location and direction of stiffeners, for the introduction of a stiffener in a closed beam section may result in a topologically different configuration from the original; the existing section shape optimization theory cannot be used. The purpose of this paper is to formulate a section topology optimization technique based on an anisotropic beam theory considering warping of sections and coupling among deformations. The formulation and corresponding solving method for the topology optimization of beam cross-sections are proposed. In formulating the topology optimization problem, the minimum averaged compliance of the beam is taken as objective, and the material density of every element is used as design variable. The schemes to determine the rigidity matrix of the cross-sections and the sensitivity analysis are presented. Several kinds of topologies of the cross-section under different load conditions are given, and the effect of load condition on the optimum topology is analyzed.