基于映射的拓扑优化最大尺寸控制方法

拓扑优化方法经过几十年的发展,已成功应用于机械工程、航空航天、电磁等领域的构型设计中。然而,由于制造工艺的限制,拓扑优化结果通常无法直接应用,需根据工艺要求进行修改,因此在拓扑优化模型中考虑制造约束成为重要的研究方向。其中,尺寸控制广泛存在于大部分制造工艺中,主要包括最小尺寸控制与最大尺寸控制。该文提出了一种基于映射的拓扑优化最大尺寸控制方法,构造了一种新的映射模型,对结构中不满足最大尺寸约束的中心单元密度进行惩罚,在不引入任何约束条件的情况下实现了对结构最大尺寸的控制。此外,该文将该方法中的惩罚转变为一个全局约束条件后与具有最小尺寸控制功能的拓扑优化鲁棒列式相结合,实现了对构件的最大最小尺寸协同控制。数值算例表明了该方法的有效性。     

模糊参数平面连续体结构的拓扑优化设计

研究了具有模糊参数的连续体结构在模糊载荷作用下的拓扑优化设计问题。利用信息熵将模糊变量转换为随机变量,构建了随机载荷作用下的随机参数的连续体结构的拓扑优化设计数学模型,以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,满足单元应力可靠性为约束条件,利用分布函数法对应力可靠性约束进行了等价显式化处理。基于随机因子法,利用代数综合法导出了应力响应的数字特征的计算表达式。采用双方向渐进结构优化(BESO)方法求解。通过两个算例验证了该文模型及求解方法的合理性和有效性。     

Optimization of truss topology using tabu search

A design procedure for integrating topological considerations in the framework of structural optimization is presented. The proposed approach is capable of considering multiple load conditions, stress, displacement and local/global buckling constraints, and multiple objective functions in the problem formulation. Further, since the proposed method permits members to be added to or deleted from an existing topology and the topology is not defined by member areas, the difficulty of not being able to reach singular optima is also avoided. These objectives are accomplished using a discrete optimization procedure which uses 0–1 topological variables to optimize alternate designs. Since the topological variables are discrete in nature and the member cross-sections are assumed to be continuous, the topological optimization problem has mixed discrete-continuous variables. This non-linear programming problem is solved using a memory-based combinatorial optimization technique known as tabu search. Numerical results obtained using tabu search for single and multiobjective topological optimization of truss structures are presented. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. The results indicate that the optimum topologies obtained using tabu search compare favourably, and in some instances, outperform the results obtained using the ground–structure approach. However, this improvement occurs at the expense of a significant increase in computational burden owing to the fact that the proposed approach necessitates that the geometry of each trial topology be optimized.     

Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm

Transient heat conduction analysis involves extensive computational cost. It becomes more serious for multi-material topology optimization, in which many design variables are involved and hundreds of iterations are usually required for convergence. This article aims to provide an efficient quadratic approximation for multi-material topology optimization of transient heat conduction problems. Reciprocal-type variables, instead of relative densities, are introduced as design variables. The sequential quadratic programming approach with explicit Hessians can be utilized as the optimizer for the computationally demanding optimization problem, by setting up a sequence of quadratic programs, in which the thermal compliance and weight can be explicitly approximated by the first and second order Taylor series expansion in terms of design variables. Numerical examples show clearly that the present approach can achieve better performance in terms of computational efficiency and iteration number than the solid isotropic material with penalization method solved by the commonly used method of moving asymptotes. In addition, a more lightweight design can be achieved by using multi-phase materials for the transient heat conductive problem, which demonstrates the necessity for multi-material topology optimization.     

Robust topology optimization under multiple independent uncertainties of loading positions

The topology optimization problem of a continuum structure is further investigated under the independent position uncertainties of multiple external loads, which are now described with an interval vector of uncertain-but-bounded variables. In this study, the structural compliance is formulated with the quadratic Taylor series expansion of multiple loading positions. As a result, the objective gradient information to the topological variables can be evaluated efficiently upon an explicit quadratic expression as the loads deviate from their ideal application points. Based on the minimum (largest absolute) value of design sensitivities, which corresponds to the most sensitive compliance to the load position variations, a two-level optimization algorithm within the non-probabilistic approach is developed upon a gradient-based optimization method. The proposed framework is then performed to achieve the robust optimal configurations of four benchmark examples, and the final designs are compared comprehensively with the traditional topology optimizations under the loading point fixation. It will be observed that the present methodology can provide a remarkably different structural layout with the auxiliary components in the design domain to counteract the load position uncertainties. The numerical results also show that the present robust topology optimization can effectively prevent the structural performance from a noticeable deterioration than the deterministic optimization in the presence of load position disturbances.     

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。     

Optimal design of dynamically loaded continuous structures

A computational algorithm is developed and applied for optimization of beam and plate structures, subject to constraints on transient dynamic response. A continuous design formulation is retained, with dynamic response governed by partial differential operator equations. Adjoint equations are employed for sensitivity analysis and a function space gradient projection optimization approach is presented. Finite element analysis methods are applied for solution of the system dynamic and adjoint differential equations. Displacement constrained beam and plate minimum weight examples are solved, with a variety of boundary conditions.     

Topology optimization of structures under variable loading using a damage superposition approach

We present an original algorithm and accompanying mathematical formulation for topology optimization of structures that can sustain material damage and are subject to multiple load cases with varying configurations. Damage accumulation is simulated using a coupled, non‐linear brittle damage model. The structures are optimized for minimum mass subject to stiffness constraints defined as the compliance evaluated at the end of each loading sequence. To achieve robustness of the optimized structures, the respective damage fields caused by each individual load case are computed and combined using superposition to simulate a worst‐case damage field. All load cases are then run a second time using the worst‐case damage distribution as a starting point. In this way, one effectively accounts for the spectrum of possible load sequences to which the structure may be subjected. Results from this method are compared with an exhaustive, brute‐force approach in which all non‐repeating load sequences are analyzed individually. For each method, the corresponding sensitivities are derived and implemented analytically using a path‐dependent adjoint method. The two approaches are implemented on a series of numerical examples, which demonstrate that the superposition method produces structures that are as robust as those obtained using the exhaustive method but require significantly less computational effort. Copyright © 2014 John Wiley & Sons, Ltd.     

基于水平集描述的结构拓扑与构件布局联合优化新方法

该文提出了一种结构拓扑与内嵌构件布局联合优化的新颖方法。这种方法突出的特点是利用水平集函数隐式地描述不规则的构件形状,因此可以非常方便地处理构件之间的互不覆盖约束条件。数值算例表明:较之文献中已有的方法,该文算法能够以更小的计算量有效地实现结构拓扑与内嵌构件布局的联合优化。     

Improving multiresolution topology optimization via multiple discretizations

Unlike the traditional topology optimization approach that uses the same discretization for finite element analysis and design optimization, this paper proposes a framework for improving multiresolution topology optimization (iMTOP) via multiple distinct discretizations for: (1) finite elements; (2) design variables; and (3) density. This approach leads to high fidelity resolution with a relatively low computational cost. In addition, an adaptive multiresolution topology optimization (AMTOP) procedure is introduced, which consists of selective adjustment and refinement of design variable and density fields. Various two‐dimensional and three‐dimensional numerical examples demonstrate that the proposed schemes can significantly reduce computational cost in comparison to the existing element‐based approach. Copyright © 2012 John Wiley & Sons, Ltd.     

Temperature-constrained topology optimization of thermo-mechanical coupled problems

Temperature-constrained topology optimization for thermo-mechanical coupled problems under a design-dependent temperature field considering the thermal expansion effect remains an open problem. A temperature-constrained topology optimization method is proposed for thermo-mechanical coupled problems. In this article, the temperature values at the heat sources are constrained. The numerical results reveal that the temperature constraints play an important role in topology optimization of thermo-mechanical coupled problems. The optimized structure obtained by the presented method not only has certain strength but also decreases the temperature significantly compared with structures obtained by other methods without considering temperature constraints. The proposed method is applied to the design of the cooling system of a battery package. Numerical examples verify the efficiency of the presented method.     

OPTIMAL STRUCTURAL TOPOLOGY DESIGN USING THE HOMOGENIZATION METHOD WITH MULTIPLE CONSTRAINTS

In applications of the homogenization method for optimal structural topology design the solution is obtained by solving the optimahty conditions directly. This reduces the computational burden by taking advantage of closed-form solutions but it restricts the optimization model to having only one constraint. The article develops a generalized class of convex approximation methods for mathematical programming that can be used for the optimal topology homogenization problem with multiple constraints in-eluded in the model, without substantial reduction in computational efficiency. A richer class of design models can be then addressed using the hotnogenization method. Design examples illustrate the performance of the proposed solution strategy.     

A stress-influence-function with adaptive strength feature approach for stress constrained continuum topology optimization via small-loop sequential strategy

Recently, the stress-influence-function (SIF) approach is presented for stress constrained continuum topology optimization. The SIF approach provides an alternative for continuum topology optimization with stress constraints. However, the SIF approach is not good at controlling the maximum stress of the elements compared to the conventional approach. In the study, the stress-influence-function with adaptive strength feature (SIF-ASF) approach via small-loop sequential strategy is proposed to achieve better control on the maximum elemental stress. First, the stress constrained continuum topology optimization formulation is given and the SIF approach is briefly introduced. Then the SIF-ASF approach is proposed for stress constrained continuum topology optimization, in which the strength feature in the stress influence function is adjusted in each iterative step of the optimization process. The adjoint-vector based sensitivity analysis to the design variables is further discussed. Three numerical examples are given to illustrate the applicability and validity of the proposed SIF-ASF approach. It is shown that the proposed SIF-ASF approach can achieve better control on the maximum elemental stress than the SIF approach. Moreover, the proposed SIF-ASF approach may obtain a lighter structure than the conventional approach.     

Robust topology optimization under load position uncertainty

The topology optimization problem of a continuum structure on the compliance minimization objective is investigated under consideration of the external load uncertainty in its application position with a nonprobabilistic approach. The load position is defined as the uncertain-but-bounded parameter and is represented by an interval variable with a nominal application point. The structural compliance due to the load position deviation is formulated with the quadratic Taylor series expansion. As a result, the objective gradient information to the topological variables can be evaluated efficiently in a quadratic expression. Based on the maximum design sensitivity value, which corresponds to the most sensitive compliance to the uncertain loading position, a single-level optimization approach is suggested by using a popular gradient-based optimality criteria method. The proposed optimization scheme is performed to gain the robust topology optimizations of three benchmark examples, and the final configuration designs are compared comprehensively with the conventional topology optimizations under the loading point fixation. It can be observed that the present method can provide remarkably different material layouts with auxiliary components to accommodate the load position disturbances. The numerical results of the representative examples also show that the structural performances of the robust topology optimizations appear less sensitive to the load position perturbations than the traditional designs.     

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

This study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two- and three-dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.     

稳态热传导结构非概率可靠性拓扑优化设计

研究具有区间参数的稳态热传导结构在散热弱度非概率可靠性约束下的拓扑优化设计问题。建立了以单元相对导热系数为设计变量,导热材料体积极小化为目标函数,满足散热弱度非概率可靠性为约束条件的稳态热传导结构的拓扑优化设计数学模型。基于区间因子法,推导出散热弱度的均值及离差的计算表达式。采用渐进结构优化法的求解策略与方法,并利用过滤技术消除优化过程中的数值不稳定性现象。通过算例验证文中模型及求解策略、方法的合理性和有效性。     

An algorithm for optimization of non-linear shell structures

An algorithm for optimal design of non-linear shell structures is presented. The algorithm uses numerical optimization techniques and nonlinear finite element analysis to find a minimum weight structure subject to equilibrium conditions, stability constraints and displacement constraints. A barrier transformation is used to treat an apparent non-smoothness arising from posing the stability constraints in terms of the eigenvalues of the Hessian of the potential energy of the structure. A sequential quadratic programming strategy is used to solve the resulting non-linear optimization problem. Matrix sparsity in the constraint Jacobian is exploited because of the large number of variables. The usefulness of the proposed algorithm is demonstrated by minimizing the weight of a number of stiffened thin shell structures.     

Explicit level‐set‐based topology optimization using an exact Heaviside function and consistent sensitivity analysis

This paper presents a level‐set‐based topology optimization method based on numerically consistent sensitivity analysis. The proposed method uses a direct steepest‐descent update of the design variables in a level‐set method; the level‐set nodal values. An exact Heaviside formulation is used to relate the level‐set function to element densities. The level‐set function is not required to be a signed‐distance function, and reinitialization is not necessary. Using this approach, level‐set‐based topology optimization problems can be solved consistently and multiple constraints treated simultaneously. The proposed method leads to more insight in the nature of level‐set‐based topology optimization problems. The level‐set‐based design parametrization can describe gray areas and numerical hinges. Consistency causes results to contain these numerical artifacts. We demonstrate that alternative parameterizations, level‐set‐based or density‐based regularization can be used to avoid artifacts in the final results. The effectiveness of the proposed method is demonstrated using several benchmark problems. The capability to treat multiple constraints shows the potential of the method. Furthermore, due to the consistency, the optimizer can run into local minima; a fundamental difficulty of level‐set‐based topology optimization. More advanced optimization strategies and more efficient optimizers may increase the performance in the future. Copyright © 2012 John Wiley & Sons, Ltd.     

Topology optimization for unsteady flow

A computational methodology for optimizing the conceptual layout of unsteady flow problems at low Reynolds numbers is presented. The geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity. The flow is predicted by a stabilized finite element formulation of the incompressible Navier–Stokes equations. A Brinkman penalization is used to enforce zero‐velocities in solid material. The resulting parameter optimization problem is solved by a non‐linear programming method. The paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems. The derivation of the governing equations and the adjoint sensitivity analysis are presented. A design‐dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material. The emergence of non‐physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the Brinkman penalization. Two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady‐state counterparts. Copyright © 2011 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.