基于变频率区间约束的结构材料优化设计

针对频率约束的结构材料优化问题,基于结构拓扑优化思想,提出变频率区间约束的结构材料优化方法。借鉴均匀化及ICM(独立、连续、映射)方法,以微观单元拓扑变量倒数为设计变量,导出宏观单元等效质量矩阵及导数,进而获得频率一阶近似展开式。结合变频率区间约束思想,获得以结构质量为目标函数、频率为约束条件的连续体微结构拓扑优化近似模型;采用对偶方法求解。通过算例验证该方法的有效性及可行性,表明考虑质量矩阵变化影响所得优化结果更合理。     

多工况下杆系结构的概率优化设计

研究了基于概率的杆系结构多工况优化设计问题。建立了以杆系截面积为设计变量,结构位移、单元应力可靠度和尺寸限为约束条件,使结构重量极小化设计的数学模型。通过将概率约束等价化处理使之转变为常规约束形式,以紧约束处理策略确定有效约束。最后利用齿行法求解。算例表明文中的模型和方法是合理的和有效的。     

应力约束全局化处理的连续体结构ICM拓扑优化方法

由于应力约束按单元计,加之多工况,使得连续体结构拓扑优化由于约束数目太多,导致应力敏度分析计算量太大而无法接受。基于第四强度理论提出了应力约束条件全局化处理的方法,化为全局替代约束——总应变能约束,用ICM方法对总应变能约束条件下的连续体结构拓扑优化进行建模及求解,其过程分为三步:第一步选择最大应变能对应的工况,在给定重量下求出最小结构总应变能;第二步提出一个数值经验公式,借助第一步的结果,计算出各工况下的许用总应变能;第三步以第二步计算出来的各工况的许用总应变能作为约束,以重量为目标建立模型并求解。顺便指出,第二步的处理方法可以处理载荷相差特别大的情况,即病态载荷情况。数值算例表明:全局性应力约束可以更好地得到传力路径,对于处理多工况问题具有优势。     

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.     

结构畸变比能处理的应力约束全局化的连续体结构拓扑优化

该文根据von Mises强度准则的畸变比能本质,计算单元畸变比能替代应力约束;依照应力全局化策略,定义结构畸变比能约束概念,求解应力约束下重量最小的连续体结构拓扑优化问题,急剧地减少了应力约束。构造许用应力和结构最大应力的比值含参数幂函数,对约束限进行动态修正。基于ICM(Independent Continuous and Mapping,独立、连续、映射)方法,采用指数型快滤函数建立了结构在畸变比能约束下的结构拓扑优化模型,并选取精确映射下的序列二次规划进行求解。数值算例表明:采用修正的结构畸变比能的应力全局化策略,对于结构拓扑优化问题的求解是有用和高效的。该文提出的方法对解决工况间存在病态载荷的问题也是有益的。     

基于应力约束全局化策略的板壳结构强度拓扑优化

应力约束下板壳结构的拓扑优化设计由于应力约束难以显式化和约束条件数量庞大等问题,存在建模和求解的困难。为了解决这些困难,首先,该文通过类比独立连续映射(Independent Continuous and Mapping, ICM)方法中的过滤函数,在实体各向同性材料惩罚(Solid Isotropic Material Penalization, SIMP)方法中引进了单元重量惩罚函数和材料许用应力惩罚函数,实现了对SIMP方法惩罚函数概念的拓广。然后,该文移植了ICM方法中行之有效的应力约束全局化策略,基于拓广后的SIMP方法,构造了代替应力约束后的结构畸变能约束的近似显式化函数,建立了多工况下板壳结构拓扑优化模型。最后,采用对偶序列二次规划算法求解该优化模型,并基于Python语言在ABAQUS软件平台进行了程序实现。数值算例都取得了很好的结果,这表明该文提出的方法是可行而有 效的。     

Optimum design of truss structures undergoing large deflections subject to a system stability constraint

A structural optimization algorithm is developed for shallow trusses undergoing large deflections subject to a system stability constraint. The method combines the non‐linear buckling analysis, through displacement control technique, with the optimality criteria approach. Four examples illustrate the procedure and allow the results obtained to be compared with those in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass for these types of structures so a design based on the linear buckling analysis can result in catastrophic failure. In one of the design examples the stresses in the elements, in the optimum design, exceed the allowable stresses, pointing out the need for a design that accounts for both non‐linear buckling and stress constraints. Copyright © 2000 John Wiley & Sons, Ltd.     

Heaviside projection–based aggregation in stress‐constrained topology optimization

The paper introduces an approach to stress‐constrained topology optimization through Heaviside projection–based constraint aggregation. The aggregation is calculated by integrating Heaviside projected local stresses over the design domain, and then, it is normalized over the total material volume. Effectively, the normalized integral measures the volume fraction of the material that has violated the stress constraint. Hence, with the Heaviside aggregated constraint, we can remove the stress failed material from the final design by constraining the integral to a threshold value near zero. An adaptive strategy is developed to select the threshold value for ensuring that the optimized design is conservative. By adding a stress penalty factor to the integrand, the Heaviside aggregated constraint can further penalize high stresses and becomes more stable and less sensitive to the selection of the threshold value. Our two‐dimensional and three‐dimensional numerical experiments demonstrate that the single Heaviside aggregated stress constraint can efficiently control the local stress level. Compared with the traditional approaches based on the Kreisselmeier‐Steinhauser and p‐norm aggregations, the Heaviside aggregation–based single constraint can substantially reduce computational cost on sensitivity analysis. These advantages make it possible to apply the proposed approach to large‐scale stress‐constrained problems.     

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.     

Stress constrained topology optimization simultaneously considering the uncertainty of load positions

This study presents a topological optimization method simultaneously considering the stress constraint and the uncertainty of the load positions for practical applications. The phase field design method is used to derive the topologically optimal structure shape. The stress penalization function, which makes intermediate design variable values disproportionately expensive, is employed to ensure numerical stability and avoid the singularity during the optimization process. The adaptive mesh refinement and a modified P-norm stress with correction factor are also employed to reduce the computational cost. As numerical examples, cantilever beam, L-shaped, and MBB-beam are presented to verify the proposed design method. The open-source code FreeFEM++ is used for the finite element analysis and the design process.     

A bidirectional evolutionary structural optimization algorithm for mass minimization with multiple structural constraints

A bidirectional evolutionary structural optimization algorithm is presented, which employs integer linear programming to compute optimal solutions to topology optimization problems with the objective of mass minimization. The objective and constraint functions are linearized using Taylor's first-order approximation, thereby allowing the method to handle all types of constraints without using Lagrange multipliers or sensitivity thresholds. A relaxation of the constraint targets is performed such that only small changes in topology are allowed during a single update, thus ensuring the existence of feasible solutions. A variety of problems are solved, demonstrating the ability of the method to easily handle a number of structural constraints, including compliance, stress, buckling, frequency, and displacement. This is followed by an example with multiple structural constraints and, finally, the method is demonstrated on a wing-box, showing that topology optimization for mass minimization of real-world structures can be considered using the proposed methodology.     

离散变量结构优化设计的最优综合效能法

针对结构优化问题的位移约束,引入关键约束的界约参数,提出了结构位移统一约束的缩减形式,从而简化了结构优化模型。根据离散变量结构优化问题的特点,提出了效能系数的概念,它衡量设计变量在离散邻域范围内变化对目标函数与约束函数值的影响,并研究了基于效能系数取值分类的四种主要调整方式。根据结构应力和位移约束的影响区域属性,以综合效能最大化为引导,提出了求解离散变量结构优化问题的最优综合效能法。算例结果显示该算法具有良好的优化效率,可求得问题的最优解或获得历史上的最优记录。     

Local and global searches of approximate optimal designs of regular frames

Methods of local and global searches of approximate optimal designs minimizing total structural volume under stress and displacement constraints are presented for regular frames subjected to static loads. Nonuniqueness of the optimal solution is extensively utilized for local search of approximate optimal solutions, where the search direction is computed from singular value decomposition of the stiffness matrix with respect to the cross‐sectional areas, or the sensitivity matrix of the constraints. The distance between the solutions is then defined, and the approximate optimal solutions are globally and consecutively found so as to maximize the distance from the already found solutions under upper bound constraint on the total structural volume. The effectiveness of the proposed method is demonstrated in application to a plane frame. Copyright © 2005 John Wiley & Sons, Ltd.     

A level set–based method for stress-constrained multimaterial topology optimization of minimizing a global measure of stress

In multimaterial topology optimization of minimizing a global measure of stress, the maximum stresses in different materials may not satisfy the strength design requirements simultaneously if stress constraints for different materials are not considered. In this paper, a level set–based method is presented to handle the stress-constrained multimaterial topology optimization of minimizing a global stress measure. Specifically, a multimaterial level set model is adopted to describe the structural topology, and a stress interpolation scheme is introduced for stress evaluation. Then, a stress penalty-based topology optimization model is presented. Meanwhile, an adaptive adjusting scheme of the stress penalty factor is employed to improve the control of the local stress level. To solve the stress-constrained multimaterial topology optimization problem minimizing the global measure of stress, the parametric level set method is employed, and the sensitivity analysis is carried out. Numerical examples are provided to demonstrate the effectiveness of the presented method. Results indicate that multimaterial structures with optimized global stress can be gained, and stress constraints for different materials can be satisfied simultaneously.     

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.     

Reliability‐based topology optimization of structures under stress constraints

In this paper, we propose an approach for reliability‐based design optimization where a structure of minimum weight subject to reliability constraints on the effective stresses is sought. The reliability‐based topology optimization problem is formulated by using the performance measure approach, and the sequential optimization and reliability assessment method is employed. This strategy allows for decoupling the reliability‐based topology optimization problem into 2 steps, namely, deterministic topology optimization and reliability analysis. In particular, the deterministic structural optimization problem subject to stress constraints is addressed with an efficient methodology based on the topological derivative concept together with a level‐set domain representation method. The resulting algorithm is applied to some benchmark problems, showing the effectiveness of the proposed approach.     

Reliability‐based design optimization with equality constraints

Equality constraints have been well studied and widely used in deterministic optimization, but they have rarely been addressed in reliability‐based design optimization (RBDO). The inclusion of an equality constraint in RBDO results in dependency among random variables. Theoretically, one random variable can be substituted in terms of remaining random variables given an equality constraint; and the equality constraint can then be eliminated. However, in practice, eliminating an equality constraint may be difficult or impossible because of complexities such as coupling, recursion, high dimensionality, non‐linearity, implicit formats, and high computational costs. The objective of this work is to develop a methodology to model equality constraints and a numerical procedure to solve a RBDO problem with equality constraints. Equality constraints are classified into demand‐based type and physics‐based type. A sequential optimization and reliability analysis strategy is used to solve RBDO with physics‐based equality constraints. The first‐order reliability method is employed for reliability analysis. The proposed method is illustrated by a mathematical example and a two‐member frame design problem. Copyright © 2007 John Wiley & Sons, Ltd.     

SAFE-FAIL LINE METHOD—AN ALGORITHM TO REDUCE THE CALCULATION OF COMPLEX CONSTRAINTS IN DESIGN OPTIMIZATION

A rather simple but quite effective algorithm is proposed for reducing the number of calculations of complex constraint functions in optimization problems. A type of monotonicity is used to set up a direct relationship between the design variables and the critical constraints. It can frequently be predicted whether or not the constraints are satisfied at a design point just by checking the variation in the design variables. It can be established in advance that a given constraint function varies with respect to a design variable in one of four ways, and rules can be defined, based on the previous history of function values, whether or not a constraint function need be evaluated. Trial calculations of a sample of problems suggest typical savings of 40-50%. The procedure is illustrated by stress constraints; but the same method can be used in other types.     

Trust region methods for structural optimization using exact second order sensitivity

The performance of multiplier algorithms for structural optimization has been significantly improved by using trust regions. The trust regions are constructed using analytical second order sensitivity, and within this region, the augmented Lagrangian ? is minimized subject to bounds. Evaluation of first and second derivatives of ? by the adjoint method does not require derivations of individual (implicit) constraint functions, which makes the method economical. Eight test problems are considered and a vast improvement over previously used multiplier algorithms has been noted. Also, the algorithm is robust with respect to scaling, input parameters and starting designs.