Topology design of structures using a dual algorithm and a constraint on the perimeter

Dual optimization algorithms are well suited for the topology design of continuum structures in discrete variables, since in these problems the number of constraints is small in comparison to the number of design variables. The ‘raw’ dual algorithm, which was originally proposed for the minimum compliance design problem, worked well when a perimeter constraint was added in addition to the volume constraint. However, if the perimeter constraint was gradually relaxed by increasing the upper bound on the allowable perimeter, the algorithm tended to behave erratically. Recently, a simple strategy has been suggested which modifies the raw dual algorithm to make it more robust in the absence of the perimeter constraint; in particular the problem of checkerboarding which is frequently observed with the use of lower‐order finite elements is eliminated. In this work, we show how the perimeter constraint can be incorporated in this improved algorithm, so that it not only provides a designer with a control over the topology, but also generates good topologies irrespective of the value of the upper bound on the perimeter. Copyright © 2002 John Wiley & Sons, Ltd.     

A dual algorithm for the topology optimization of non‐linear elastic structures

Dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of Lagrange multipliers associated with the constraints. To date, dual algorithms have been applied only for linear structures. Here we extend this methodology to the case of non‐linear structures. The perimeter constraint is used to make the topology problem well‐posed. We show that the proposed algorithm yields a value of perimeter that is close to that specified by the user. We also address the issue of manufacturability of these designs, by proposing a variant of the standard dual algorithm, which generates designs that are two‐dimensional although the loading and the geometry are three‐dimensional. Copyright © 2008 John Wiley & Sons, Ltd.     

On the equivalence of optimality criterion and sequential approximate optimization methods in the classical topology layout problem

We study the ‘classical’ topology optimization problem, in which minimum compliance is sought, subject to linear constraints. Using a dual statement, we propose two separable and strictly convex subproblems for use in sequential approximate optimization (SAO) algorithms. Respectively, the subproblems use reciprocal and exponential intermediate variables in approximating the non‐linear compliance objective function. Any number of linear constraints (or linearly approximated constraints) are provided for. The relationships between the primal variables and the dual variables are found in analytical form. For the special case when only a single linear constraint on volume is present, we note that application of the ever‐popular optimality criterion (OC) method to the topology optimization problem, combined with arbitrary values for the heuristic numerical damping factor η proposed by Bendsøe, results in an updating scheme for the design variables that is identical to the application of a rudimentary dual SAO algorithm, in which the subproblems are based on exponential intermediate variables. What is more, we show that the popular choice for the damping factor η=0.5 is identical to the use of SAO with reciprocal intervening variables. Finally, computational experiments reveal that subproblems based on exponential intervening variables result in improved efficiency and accuracy, when compared to SAO subproblems based on reciprocal intermediate variables (and hence, the heuristic topology OC method hitherto used). This is attributed to the fact that a different exponent is computed for each design variable in the two‐point exponential approximation we have used, using gradient information at the previously visited point. Copyright © 2007 John Wiley & Sons, Ltd.     

An enhanced genetic algorithm for structural topology optimization

Genetic algorithms (GAs) have become a popular optimization tool for many areas of research and topology optimization an effective design tool for obtaining efficient and lighter structures. In this paper, a versatile, robust and enhanced GA is proposed for structural topology optimization by using problem‐specific knowledge. The original discrete black‐and‐white (0–1) problem is directly solved by using a bit‐array representation method. To address the related pronounced connectivity issue effectively, the four‐neighbourhood connectivity is used to suppress the occurrence of checkerboard patterns. A simpler version of the perimeter control approach is developed to obtain a well‐posed problem and the total number of hinges of each individual is explicitly penalized to achieve a hinge‐free design. To handle the problem of representation degeneracy effectively, a recessive gene technique is applied to viable topologies while unusable topologies are penalized in a hierarchical manner. An efficient FEM‐based function evaluation method is developed to reduce the computational cost. A dynamic penalty method is presented for the GA to convert the constrained optimization problem into an unconstrained problem without the possible degeneracy. With all these enhancements and appropriate choice of the GA operators, the present GA can achieve significant improvements in evolving into near‐optimum solutions and viable topologies with checkerboard free, mesh independent and hinge‐free characteristics. Numerical results show that the present GA can be more efficient and robust than the conventional GAs in solving the structural topology optimization problems of minimum compliance design, minimum weight design and optimal compliant mechanisms design. It is suggested that the present enhanced GA using problem‐specific knowledge can be a powerful global search tool for structural topology optimization. Copyright © 2005 John Wiley & Sons, Ltd.     

Simultaneous optimal design of structural topology, actuator locations and control parameters for a plate structure

 Simultaneous optimization with respect to the structural topology, actuator locations and control parameters of an actively controlled plate structure is investigated in this paper. The system consists of a clamped-free plate, a H ₂ controller and four surface-bonded piezoelectric actuators utilized for suppressing the bending and torsional vibrations induced by external disturbances. The plate is represented by a rectangular design domain which is discretized by a regular finite element mesh and for each element the parameter indicating the presence or absence of material is used as a topology design variable. Due to the unavailability of large-scale 0–1 optimization algorithms, the binary variables of the original topology design problem are relaxed so that they can take all values between 0 and 1. The popular techniques in the topology optimization area including penalization, filtering and perimeter restriction are also used to suppress numerical problems such as intermediate thickness, checkerboards, and mesh dependence. Moreover, since it is not efficient to treat the structural and control design variables equally within the same framework, a nested solving approach is adopted in which the controller syntheses are considered as sub processes included in the main optimization process dealing with the structural topology and actuator locations. The structural and actuator variables are solved in the main optimization by the method of moving asymptotes, while the control parameters are designed in the sub optimization processes by solving the Ricatti equations. Numerical examples show that the approach used in this paper can produce systems with clear structural topology and high control performance. Received 16 November 2001 / Accepted 26 February 2002     

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.     

周期性多材料结构稳态热传导拓扑优化设计

提出一种考虑周期性约束的多材料结构稳态热传导拓扑优化设计方法。针对多材料结构,提出基于有序有理近似材料属性模型（ordered rational approximation of material properties,Ordered-RAMP）的多材料插值模型。以结构散热弱度最小化为目标函数,体积为约束条件,将设计区域划分为有限个相同的子多材料区域。通过重新分配单元散热弱度基值,实现周期性几何约束,借助优化准则法推导设计变量的迭代格式。通过典型2D与3D数值算例,分析不同子区域个数对宏观结构与微观子区域多材料拓扑构型的影响。结果表明:所提方法可实现面向多材料结构的周期性微观构型设计,且各材料分布合理边界清晰,具有良好的稳健性;当子区域个数不同时,均可得到具有周期性的拓扑构型,且所获拓扑形式具有差异性。     

A mixed FEM approach to stress‐constrained topology optimization

We present an alternative topology optimization formulation capable of handling the presence of stress constraints in a straightforward fashion. The main idea is to adopt a mixed finite‐element discretization scheme wherein not only displacements (as usual) but also stresses are the variables entering the formulation. By doing so, any stress constraint may be handled within the optimization procedure without resorting to post‐processing operation typical of displacement‐based techniques that may also cause a loss in accuracy in stress computation if no smoothing of the stress is performed. Two dual variational principles of Hellinger–Reissner type are presented in continuous and discrete form that, which included in a rather general topology optimization problem in the presence of stress constraints that is solved by the method of moving asymptotes (Int. J. Numer. Meth. Engng. 1984; 24 (3):359–373). Extensive numerical simulations are performed and ongoing extensions outlined, including the optimization of elastoplastic and incompressible media. Copyright © 2007 John Wiley & Sons, Ltd.     

A novel algorithm using an orthotropic material model for topology optimization

This article presents a novel algorithm for topology optimization using an orthotropic material model. Based on the virtual work principle, mathematical formulations for effective orthotropic material properties of an element containing two materials are derived. An algorithm is developed for structural topology optimization using four orthotropic material properties, instead of one density or area ratio, in each element as design variables. As an illustrative example, minimum compliance problems for linear and nonlinear structures are solved using the present algorithm in conjunction with the moving iso-surface threshold method. The present numerical results reveal that: (1) chequerboards and single-node connections are not present even without filtering; (2) final topologies do not contain large grey areas even using a unity penalty factor; and (3) the well-known numerical issues caused by low-density material when considering geometric nonlinearity are resolved by eliminating low-density elements in finite element analyses.     

基于密度分布曲线法的复合材料变刚度铺层优化

基于梯度的优化方法对复合材料层合板进行了变刚度铺层优化设计。在优化过程中需确定铺层中各单元的密度以及角度。为了使优化结果具有可制造性,优化结果需满足制造工艺约束并且铺层角度需从预定角度中选取。为了避免在优化问题中引入过多的约束并减少设计变量的数目,提出密度分布曲线法(DDCM)对层合板中各单元的密度进行参数化。根据各单元的密度以及角度设计变量并基于Bi-value Coding Parameterization(BCP)方法中的插值公式确定各单元的弹性矩阵。优化过程中以结构柔顺度作为优化目标,结构体积作为约束,优化算法采用凸规划对偶算法。对碳纤维复合材料的算例结果表明:采用DDCM可得到较理想的优化结果,并且收敛速率较快。     

基于遗传算法和拓扑优化的结构多孔洞损伤识别

鉴于拓扑优化和遗传算法在结构损伤识别中各自的优点,本文将遗传算法、有限元和拓扑优化三种方法相结合,提出了一种用于二维结构多损伤识别的新方法。这种方法将拓扑优化的设计变量和遗传算法的参数统一化,将拓扑优化中的目标函数和约束方程与遗传算法的适应度函数联系起来,并以拓扑优化的约束方程作为控制条件参与整个遗传运算的控制。采用二进制编码遗传算法代替连续变量拓扑优化的方式对发生孔洞损伤形式的二维结构进行损伤识别,避免了利用连续变量拓扑优化进行损伤识别时参数阈值的确定可能给识别结果带来的不良影响。通过对两个二维结构模型的多损伤识别仿真计算,结果显示本方法能够很好地识别二维结构中多个位置的损伤,对于仅用拓扑优化法很难识别的轻微孔洞损伤情况,该方法也能得出与实际情况吻合良好的结果。     

On sequential approximate simultaneous analysis and design in classical topology optimization

We study the simultaneous analysis and design (SAND) formulation of the ‘classical’ topology optimization problem subject to linear constraints on material density variables. Based on a dual method in theory, and a primal‐dual method in practice, we propose a separable and strictly convex quadratic Lagrange–Newton subproblem for use in sequential approximate optimization of the SAND‐formulated classical topology design problem. The SAND problem is characterized by a large number of nonlinear equality constraints (the equations of equilibrium) that are linearized in the approximate convex subproblems. The availability of cheap second‐order information is exploited in a Lagrange–Newton sequential quadratic programming‐like framework. In the spirit of efficient structural optimization methods, the quadratic terms are restricted to the diagonal of the Hessian matrix; the subproblems have minimal storage requirements, are easy to solve, and positive definiteness of the diagonal Hessian matrix is trivially enforced. Theoretical considerations reveal that the dual statement of the proposed subproblem for SAND minimum compliance design agrees with the ever‐popular optimality criterion method – which is a nested analysis and design formulation. This relates, in turn, to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal (and exponential) intervening variables and the optimality criterion method. Copyright © 2016 John Wiley & Sons, Ltd.     

Techniques to suppress intermediate density in topology optimization of compliant mechanisms

Discrete topological problems are often relaxed with continuous design variables so that they can be solved using continuous mathematical programming. Such practice prevails because large-scale discrete 0–1 mathematical programming is not generally available. Although the relaxed problems become tractable, they may cause the appearance of intermediate density in the optimum topologies, especially those of structures and compliant mechanisms. Various penalty schemes have been proposed to suppress the intermediate density. Most of the past works assumed that the same penalty schemes could be effectively applied to both problems of stiffest structure design and compliant mechanism design. Differences in nature between the problems are generally neglected. This work distinguished the two problems, and observed that complaint mechanism (CM) problem does not suffer intermediate density as seriously as minimum compliance (MC) problem does. Besides allocating more material, explicit and implicit penalties were pursued to suppress intermediate density. To ensure mesh-independence and not to complicate the nonconvex objective function in CM problem, a new technique using a constraint of explicit penalty with variable bound is proposed to suppress intermediate density in topology optimization of compliant mechanisms. Together with a perimeter constraint, the new technique is also applied to MC problem. Received 13 March 2000     

A nodal variable method of structural topology optimization based on Shepard interpolant

A method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated. The original discrete‐valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain. With the use of the Shepard family of interpolants, this density field is mapped onto the design space defined by a finite number of nodal density variables. The employed interpolation scheme has an explicit form and satisfies range‐restricted properties that makes it applicable for physically meaningful density interpolation. Its ability to resolve more complex spatial distribution of the material density within an individual element, as compared with the conventional elementwise design variable approach, actually provides certain regularization to the topology optimization problem. Numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques. Copyright © 2011 John Wiley & Sons, Ltd.     

Dual methods for discrete structural optimization problems

The purpose of this paper is to present a mathematical programming method developed to solve structural optimization problems involving discrete variables. We work in the following context: the structural responses are computed by the finite elements method and convex and separable approximation schemes are used to generate a sequence of explicit approximate subproblems.Each of them is solved in the dual space with a subgradient‐based algorithm (or with a variant of it) specially developed to maximize the not everywhere differentiable dual function. To show that the application field is large, the presented applications are issued from different domains of structural design, such as sizing of thin‐walled structures, geometrical configuration of trusses, topology optimization of membrane or 3‐D structures and welding points numbering in car bodies. The main drawback of using the dual approach is that the obtained solution is generally not the global optimum. This is linked to the presence of a duality gap, due to the non‐convexity of the primal discrete subproblems. Fortunately, this gap can be quantified: a maximum bound on its value can be computed. Moreover, it turns out that the duality gap is decreasing for higher number of variables; the maximum bound on the duality gap is generally negligible in the treated applications. The developed algorithms are very efficient for 2‐D and 3‐D topology optimization, where applications involving thousands of binary design variables are solved in a very short time. Copyright © 2000 John Wiley & Sons, Ltd.     

Topology optimization of periodic cellular solids based on a superelement method

It is well known that the structural performance of lightweight cellular solids depends greatly on the design of the representative volume element (RVE). In this article, an integrated topology optimization procedure is developed for the global stiffness maximization of 2D periodic and cyclic-symmetry cellular solids. A design variable linking technique and a superelement method are applied to model the structural periodicity and to reduce the computing time. In order to prevent the numerical instabilities associated with checkerboards in the design process, the quadratic perimeter constraint is used. Finally, the topology optimization problem is solved by the dual optimization algorithm. Several numerical examples are used to test the efficiency of the optimization procedure. Results show that the optimal topology of the RVE is not unique. It greatly depends on the size of the RVE. The computing efficiency can be greatly improved by means of the superelement technique. Also, for the optimal solution, the equivalent torsional rigidity has been compared with what is in the literature, to check the structural efficiency of the obtained topology. It has been observed that the current topology solution has the strongest rigidity when the same volume fraction of solid-phase materials is used.     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

A level set approach for topology optimization with local stress constraints

The purpose of this work is to present a level set‐based approach for the structural topology optimization problem of mass minimization submitted to local stress constraints. The main contributions are threefold. First, the inclusion of local stress constraints by means of an augmented Lagrangian approach within the level set context. Second, the proposition of a constraint procedure that accounts for a continuous activation/deactivation of a finite number of local stress constraints during the optimization sequence. Finally, the proposition of a logarithmic scaling of the level set normal velocity as an additional regularization technique in order to improve the minimization sequence. A set of benchmark tests in two dimensions achieving successful numerical results assesses the good behavior of the proposed method. In these examples, it is verified that the algorithm is able to identify stress concentrations and drive the design to a feasible local minimum. Copyright © 2014 John Wiley & Sons, Ltd.     

An engineering method for complex structural optimization involving both size and topology design variables

This work presents an engineering method for optimizing structures made of bars, beams, plates, or a combination of those components. Corresponding problems involve both continuous (size) and discrete (topology) variables. Using a branched multipoint approximate function, which involves such mixed variables, a series of sequential approximate problems are constructed to make the primal problem explicit. To solve the approximate problems, genetic algorithm (GA) is utilized to optimize discrete variables, and when calculating individual fitness values in GA, a second-level approximate problem only involving retained continuous variables is built to optimize continuous variables. The solution to the second-level approximate problem can be easily obtained with dual methods. Structural analyses are only needed before improving the branched approximate functions in the iteration cycles. The method aims at optimal design of discrete structures consisting of bars, beams, plates, or other components. Numerical examples are given to illustrate its effectiveness, including frame topology optimization, layout optimization of stiffeners modeled with beams or shells, concurrent layout optimization of beam and shell components, and an application in a microsatellite structure. Optimization results show that the number of structural analyses is dramatically decreased when compared with pure GA while even comparable to pure sizing optimization.     

Structural shape and topology optimization of cast parts using level set method

This paper presents a level set‐based shape and topology optimization method for conceptual design of cast parts. In order to be successfully manufactured by the casting process, the geometry of cast parts should satisfy certain moldability conditions, which poses additional constraints in the shape and topology optimization of cast parts. Instead of using the originally point‐wise constraint statement, we propose a casting constraint in the form of domain integration over a narrowband near the material boundaries. This constraint is expressed in terms of the gradient of the level set function defining the structural shape and topology. Its explicit and analytical form facilitates the sensitivity analysis and numerical implementation. As compared with the standard implementation of the level set method based on the steepest descent algorithm, the proposed method uses velocity field design variables and combines the level set method with the gradient‐based mathematical programming algorithm on the basis of the derived sensitivity scheme of the objective function and the constraints. This approach is able to simultaneously account for the casting constraint and the conventional material volume constraint in a convenient way. In this method, the optimization process can be started from an arbitrary initial design, without the need for an initial design satisfying the cast constraint. Numerical examples in both 2D and 3D design domain are given to demonstrate the validity and effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.