Element exchange method for topology optimization

This paper presents a stochastic direct search method for topology optimization of continuum structures. In a systematic approach requiring repeated evaluations of the objective function, the element exchange method (EEM) eliminates the less influential solid elements by switching them into void elements and converts the more influential void elements into solid resulting in an optimal 0–1 topology as the solution converges. For compliance minimization problems, the element strain energy is used as the principal criterion for element exchange operation. A wider exploration of the design space is assured with the use of random shuffle while a checkerboard control scheme is used for detection and elimination of checkerboard regions. Through the solution of multiple two- and three-dimensional topology optimization problems, the general characteristics of EEM are presented. Moreover, the solution accuracy and efficiency of EEM are compared with those based on existing topology optimization methods.     

Some aspects of truss topology optimization

The present paper studies some aspects of formulations of truss topology optimization problems. The ground structure approach-based formulations of three types of truss topology optimization problems, namely the problems of minimum weight design for a given compliance, of minimum weight design with stress constraints and of minimum weight design with stress constraints and local buckling constraints are examined. The common difficulties with the formulations of the three problems are discussed. Since the continuity of the constraint or/and objective function is an important factor for the determination of the mathematical structure of optimization problems, the issue of the continuity of stress, displacement and compliance functions in terms of the cross-sectional areas at zero area is studied. It is shown that the bar stress function has discontinuity at zero crosssectional area, and the structural displacement and compliance are continuous functions of the cross-sectional area. Based on the discontinuity of the stress function we point out the features of the feasible domain and global optimum for optimization problems with stress and/or local buckling constraints, and conclude that they are mathematical programming with discontinuous constraint functions and that they are essentially discrete optimization problems. The difference between topology optimization with global constraints such as structural compliance and that with local constraints on stress or/and local buckling is notable and has important consequences for the solution approach.     

A new PSO-based algorithm for multi-objective optimization with continuous and discrete design variables

This paper presents a new multi-objective optimization algorithm called FC-MOPSO for optimal design of engineering problems with a small number of function evaluations. The proposed algorithm expands the main idea of the single-objective particle swarm optimization (PSO) algorithm to deal with constrained and unconstrained multi-objective problems (MOPs). FC-MOPSO employs an effective procedure in selection of the leader for each particle to ensure both diversity and fast convergence. Fifteen benchmark problems with continuous design variables are used to validate the performance of the proposed algorithm. Finally, a modified version of FC-MOPSO is introduced for handling discrete optimization problems. Its performance is demonstrated by optimizing five space truss structures. It is shown that the FC-MOPSO can effectively find acceptable approximations of Pareto fronts for structural MOPs within very limited number of function evaluations.     

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.     

SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems

This paper introduces a surrogate model based algorithm for computationally expensive mixed-integer black-box global optimization problems with both binary and non-binary integer variables that may have computationally expensive constraints. The goal is to find accurate solutions with relatively few function evaluations. A radial basis function surrogate model (response surface) is used to select candidates for integer and continuous decision variable points at which the computationally expensive objective and constraint functions are to be evaluated. In every iteration multiple new points are selected based on different methods, and the function evaluations are done in parallel. The algorithm converges to the global optimum almost surely. The performance of this new algorithm, SO-MI, is compared to a branch and bound algorithm for nonlinear problems, a genetic algorithm, and the NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search) algorithm for mixed-integer problems on 16 test problems from the literature (constrained, unconstrained, unimodal and multimodal problems), as well as on two application problems arising from structural optimization, and three application problems from optimal reliability design. The numerical experiments show that SO-MI reaches significantly better results than the other algorithms when the number of function evaluations is very restricted (200–300 evaluations).     

Topology optimization of members of flexible multibody systems under dominant inertia loading

The topology optimization of members of flexible multibody systems is considered for energy-efficient lightweight design, where the gradient calculation has an essential role. In topology optimization of flexible multibody systems, where the function evaluations are very time consuming, the gradient information is necessary to accelerate the optimization process. Different approaches have been introduced and tested for the gradient calculation in the fully coupled topology optimization of flexible multibody systems. However, the computation and implementation costs of these methods are high, which limits the optimization size and the possible number of design variables. In this work, we present a modified gradient calculation based on the equivalent static load (ESL) method, which combines the time efficiency of gradient calculation of the ESL method with the higher accuracy of gradient calculation in the fully coupled methods. This modified approach, which takes into account the linear dependencies of inertial loads on acceleration, is tested on the application example of a flexible slider-crank mechanism, and the results are compared with the weakly coupled ESL method and a fully coupled optimization where gradients are calculated using the adjoint variable method.     

A level-set-based topology and shape optimization method for continuum structure under geometric constraints

Recent advances in level-set-based shape and topology optimization rely on free-form implicit representations to support boundary deformations and topological changes. In practice, a continuum structure is usually designed to meet parametric shape optimization, which is formulated directly in terms of meaningful geometric design variables, but usually does not support free-form boundary and topological changes. In order to solve the disadvantage of traditional step-type structural optimization, a unified optimization method which can fulfill the structural topology, shape, and sizing optimization at the same time is presented. The unified structural optimization model is described by a parameterized level set function that applies compactly supported radial basis functions (CS-RBFs) with favorable smoothness and accuracy for interpolation. The expansion coefficients of the interpolation function are treated as the design variables, which reflect the structural performance impacts of the topology, shape, and geometric constraints. Accordingly, the original topological shape optimization problem under geometric constraint is fully transformed into a simple parameter optimization problem; in other words, the optimization contains the expansion coefficients of the interpolation function in terms of limited design variables. This parameterization transforms the difficult shape and topology optimization problems with geometric constraints into a relatively straightforward parameterized problem to which many gradient-based optimization techniques can be applied. More specifically, the extended finite element method (XFEM) is adopted to improve the accuracy of boundary resolution. At last, combined with the optimality criteria method, several numerical examples are presented to demonstrate the applicability and potential of the presented method.     

Explicit level set and density methods for topology optimization with equivalent minimum length scale constraints

The goal of this paper is to introduce local length scale control in an explicit level set method for topology optimization. The level set function is parametrized explicitly by filtering a set of nodal optimization variables. The extended finite element method (XFEM) is used to represent the non-conforming material interface on a fixed mesh of the design domain. In this framework, a minimum length scale is imposed by adopting geometric constraints that have been recently proposed for density-based topology optimization with projections filters. Besides providing local length scale control, the advantages of the modified constraints are twofold. First, the constraints provide a computationally inexpensive solution for the instabilities which often appear in level set XFEM topology optimization. Second, utilizing the same geometric constraints in both the density-based topology optimization and the level set optimization enables to perform a more unbiased comparison between both methods. These different features are illustrated in a number of well-known benchmark problems for topology optimization.

     

An optimality criteria based method for discrete design optimization taking into account buildability constraints

This paper presents a heuristic design optimization method specifically developed for practicing structural engineers. Practical design optimization problems are often governed by buildability constraints. The majority of optimization methods that have recently been proposed for design optimization under buildability constraints are based on evolutionary computing. While these methods are generally easy to implement, they require a large number of function evaluations (finite element analyses), and they involve algorithmic parameters that require careful tuning. As a consequence, both the computation time and the engineering time are high. The discrete design optimization algorithm presented in this paper is based on the optimality criteria method for continuous optimization. It is faster than an evolutionary algorithm and it is free of tuning parameters. The algorithm is successfully applied to two classical benchmark problems (the design of a ten-bar truss and an eight-story frame) and to a practical truss design optimization problem.     

Evolutionary topology optimization of continuum structures with smooth boundary representation

This paper develops an extended bi-directional evolutionary structural optimization (BESO) method for topology optimization of continuum structures with smoothed boundary representation. In contrast to conventional zigzag BESO designs and removal/addition of elements, the newly proposed evolutionary topology optimization (ETO) method, determines implicitly the smooth structural topology by a level-set function (LSF) constructed by nodal sensitivity numbers. The projection relationship between the design model and the finite element analysis (FEA) model is established. The analysis of the design model is replaced by the FEA model with various elemental volume fractions, which are determined by the auxiliary LSF. The introduction of sensitivity LSF results in intermediate volume elements along the solid-void interface of the FEA model, thus contributing to the better convergence of the optimized topology for the design model. The effectiveness and robustness of the proposed method are verified by a series of 2D and 3D topology optimization design problems including compliance minimization and natural frequency maximization. It has been shown that the developed ETO method is capable of generating a clear and smooth boundary representation; meanwhile the resultant designs are less dependent on the initial guess design and the finite element mesh resolution.     

Structural topology design optimization using Genetic Algorithms with a bit-array representation

In this paper, a bit-array representation method for structural topology optimization using the Genetic Algorithm (GA) is implemented. The importance of structural connectivity in a design is further emphasized by considering the total number of connected objects of each individual explicitly in an equality constraint function. To evaluate the constrained objective function, Deb’s constraint handling approach is further developed to ensure that feasible individuals are always better than infeasible ones in the population to improve the efficiency of the GA. A violation penalty method is proposed to drive the GA search towards the topologies with higher structural performance, less unusable material and fewer separate objects in the design domain. An identical initialization method is also proposed to improve the GA performance in dealing with problems with long narrow design domains. Numerical results of structural topology optimization problems of minimum weight and minimum compliance designs show the success of this bit-array representation method and suggest that the GA performance can be significantly improved by handling the design connectivity properly.     

Stress-based topology optimization

Previous research on topology optimization focussed primarily on global structural behaviour such as stiffness and frequencies. However, to obtain a true optimum design of a vehicle structure, stresses must be considered. The major difficulties in stress based topology optimization problems are two-fold. First, a large number of constraints must be considered, since unlike stiffness, stress is a local quantity. This problem increases the computational complexity of both the optimization and sensitivity analysis associated with the conventional topology optimization problem. The other difficulty is that since stress is highly nonlinear with respect to design variables, the move limit is essential for convergence in the optimization process. In this research, global stress functions are used to approximate local stresses. The density method is employed for solving the topology optimization problems. Three numerical examples are used for this investigation. The results show that a minimum stress design can be achieved and that a maximum stiffness design is not necessarily equivalent to a minimum stress design.     

A new multi-objective programming scheme for topology optimization of compliant mechanisms

This paper presents an alternative method in implementing multi-objective optimization of compliant mechanisms in the field of continuum-type topology optimization. The method is designated as “SIMP-PP” and it achieves multi-objective topology optimization by merging what is already a mature topology optimization method—solid isotropic material with penalization (SIMP) with a variation of the robust multi-objective optimization method—physical programming (PP). By taking advantages of both sides, the combination causes minimal variation in computation algorithm and numerical scheme, yet yields improvements in the multi-objective handling capability of topology optimization. The SIMP-PP multi-objective scheme is introduced into the systematic design of compliant mechanisms. The final optimization problem is formulated mathematically using the aggregate objective function which is derived from the original individual design objectives with PP, subjected to the specified constraints. A sequential convex programming method, the method of moving asymptotes (MMA) is then utilized to process the optimization evolvement based on the design sensitivity analysis. The main findings in this study include distinct advantages of the SIMP-PP method in various aspects such as computation efficiency, adaptability in convex and non-convex multi-criteria environment, and flexibility in problem formulation. Observations are made regarding its performance and the effect of multi-objective optimization on the final topologies. In general, the proposed SIMP-PP method is an appealing multi-objective topology optimization scheme suitable for “real world” problems, and it bridges the gap between standard topological design and multi-criteria optimization. The feasibility of the proposed topology optimization method is exhibited by benchmark examples.     

Structural topology synthesis with dynamics and nonlinearities using equivalent linear systems

Topology optimization for nonlinear and dynamic problems is expensive because of the necessity to solve the equations of motion at every optimization iteration in order to evaluate the objective function and constraints. In this work, an iterative methodology is developed using the concept of an equivalent linear system for the topology synthesis of structures undergoing nonlinear and dynamic response, using minimal nonlinear response evaluations. The approach uses equivalent loads obtained from nonlinear dynamic analysis to perform optimization iterations, during the course of which the nonlinear and dynamic system is continuously approximated. In this process, the optimization is decoupled from the nonlinear dynamic analysis. Results are presented for various kinds of nonlinear and dynamic problems showing the effectiveness of the developed approach.     

多种群并行协作的粒子群算法

针对高维复杂优化问题在求解时容易产生维数灾难导致算法极易陷入局部最优的问题,提出一种能够综合考虑高维复杂优化问题的特性,动态调整进化策略的多种群并行协作的粒子群算法。该算法在分析高维复杂问题求解过程中的粒子特点的基础上,建立融合环形拓扑、全连接形拓扑和冯诺依曼拓扑结构的粒子群算法的多种群并行协作的网络模型。该模型结合3种拓扑结构的粒子群算法在解决高维复杂优化问题时的优点,设计一种基于多群落粒子广播-反馈的动态进化策略及其进化算法,实现高维复杂优化环境中拓扑的动态适应,使算法在求解高维单峰函数和多峰函数时均具有较强的搜索能力。仿真结果表明,该算法在求解高维复杂优化问题的寻优精度和收敛速度方面均有良好的性能。     

An efficient second-order SQP method for structural topology optimization

This article presents a Sequential Quadratic Programming (SQP) solver for structural topology optimization problems named TopSQP. The implementation is based on the general SQP method proposed in Morales et al. J Numer Anal 32(2):553–579 (2010) called SQP+. The topology optimization problem is modelled using a density approach and thus, is classified as a nonconvex problem. More specifically, the SQP method is designed for the classical minimum compliance problem with a constraint on the volume of the structure. The sub-problems are defined using second-order information. They are reformulated using the specific mathematical properties of the problem to significantly improve the efficiency of the solver. The performance of the TopSQP solver is compared to the special-purpose structural optimization method, the Globally Convergent Method of Moving Asymptotes (GCMMA) and the two general nonlinear solvers IPOPT and SNOPT. Numerical experiments on a large set of benchmark problems show good performance of TopSQP in terms of number of function evaluations. In addition, the use of second-order information helps to decrease the objective function value.     

Stress-based design of thermal structures via topology optimization

The design of thermal structures in the aerospace industry, including exhaust structures on embedded engine aircraft and hypersonic thermal protection systems, poses a number of complex design challenges. These challenges are particularly well addressed by the material layout capabilities of structural topology optimization; however, no topology optimization methods are readily available with the necessary thermoelastic considerations for these problems. This is due in large part to the emphasis on cases of maximum stiffness design for structures subjected to externally applied mechanical loads in the majority of topology optimization applications. In addition, while limited work in the literature has investigated thermoelastic topology optimization, a direct treatment of thermal stresses is not well documented. Such a treatment is critical in the design of thermal structures where excessive thermal stresses are a primary failure mode. In this paper, we present a method for the topology optimization of structures with combined mechanical and thermoelastic (temperature) loads that are subject to stress constraints. We present the necessary steps needed to address both the design-dependent thermal loads and accommodate the challenges of stress-based design criteria. A relaxation technique is utilized to remove the singularity phenomenon in stresses and the large number of stress constraints is handled using a scaled aggregation technique that has been shown previously to satisfy prescribed stress limits in mechanical problems. Finally, the stress-based thermoelastic formulation is applied to two numerical example problems to demonstrate its effectiveness.     

Topology optimization in OpenMDAO

Recently, topology optimization has drawn interest from both industry and academia as the ideal design method for additive manufacturing. Topology optimization, however, has a high entry barrier as it requires substantial expertise and development effort. The typical numerical methods for topology optimization are tightly coupled with the corresponding computational mechanics method such as a finite element method and the algorithms are intrusive, requiring an extensive understanding. This paper presents a modular paradigm for topology optimization using OpenMDAO, an open-source computational framework for multidisciplinary design optimization. This provides more accessible topology optimization algorithms that can be non-intrusively modified and easily understood, making them suitable as educational and research tools. This also opens up further opportunities to explore topology optimization for multidisciplinary design problems. Two widely used topology optimization methods—the density-based and level-set methods—are formulated in this modular paradigm. It is demonstrated that the modular paradigm enhances the flexibility of the architecture, which is essential for extensibility.

     

An efficient decoupled sensitivity analysis method for multiscale concurrent topology optimization problems

The conventional coupled sensitivity analysis method for concurrent topology optimization problems is computationally expensive for microscale design variables. This study thus proposes an efficient decoupled sensitivity analysis method for concurrent topology optimization based on the chain differentiation rule. Two numerical studies are performed to demonstrate the effectiveness of the decoupled sensitivity analysis method for concurrent topology optimization problems with single or multiple porous materials. It can be concluded from the results that the decoupled method is computationally much more efficient than the coupled method, while they are mathematically equivalent. The outstanding merits of the decoupled method are two-fold: (1) computational efficiency of sensitivity analysis with respect to the microscale design variables; and (2) applicability to concurrent topology optimization problems with single or multiple porous materials as well as with composite microstructure and multi-phase materials.     

Knowledge discovery in databases for determining formulation in topology optimization

Whereas topology optimization has achieved immense success, it involves an intrinsic difficulty. That is, optimized structures obtained by topology optimization strongly depend on the settings of the objective and constraint functions, i.e., the formulation. Nevertheless, the appropriate formulation is not usually obvious when considering structural design problems. Although trial-and-error to determine appropriate formulations are implicitly performed in several studies on topology optimization, it is important to explicitly support the process of trial-and-error. Therefore, in this study, we propose a new framework for topology optimization to determine appropriate formulations. The basic idea of this framework is incorporating knowledge discovery in databases (KDD) and topology optimization. Thus, we construct a database by collecting various and numerous material distributions that are obtained by solving various structural design problems with topology optimization, and find useful knowledge with respect to appropriate formulations from the database on the basis of KDD. An issue must be resolved when realizing the above idea, namely the material distribution in the design domain of a data record must be converted to conform to the design domain of the target design problem wherein an appropriate formulation should be determined. For this purpose, we also propose a material distribution-converting method termed as design domain mapping (DDM). Several numerical examples are used to demonstrate that the proposed framework including DDM successfully and explicitly supports the process of trial-and-error to determine the appropriate formulation.