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.     

Robust topology optimization of continuum structures with loading uncertainty using a perturbation method

An approach for the robust topology optimization (RTO) of continuum structures with loading uncertainty is investigated. The loading uncertainties are quantified using the second order Taylor series expansion of uncertain loading magnitudes and directions, and then the response statistic mean and standard deviation of compliance are calculated using the uncertain perturbation propagation method. A robust design Lagrange function considering the compliance objective and finite element constraints is developed, and a sensitivity analysis is performed to calculate the Lagrange coefficients. The Lagrange objective function is optimized using the modified solid isotropic material with penalization (SIMP) algorithm; thus, the optimum material distribution under loading uncertainty is acquired. The proposed methodology is used for the RTO of two examples, revealing its efficiency under both concentrated and distributed uncertain loadings. The accuracy of the results is verified by comparison with similar cases found in the literature where a different modelling approach was used.     

Robust topology optimization for periodic structures by combining sensitivity averaging with a semianalytical method

Uncertainty factors play an important role in the design of periodic structures because structures with small periodic design spaces are extremely sensitive to loading uncertainty. Therefore, for the first time, this paper proposes a framework for robust topology optimization (RTO) of periodic structures assuming that load uncertainties follow a Gaussian distribution. In this framework, the expected value and variance of structural compliance can be easily computed using a semianalytical method combined with probability theory, which is important for RTO when uncertain variables follow probabilistic distributions. To obtain optimal topologies, the bidirectional evolutionary structural optimization method is used. Structural periodicity is calculated using a strategy of sensitivity averaging and consistency constraints. To eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, a generic coefficient of variation is defined as the robust index, which contains both the expected value and variance. The proposed framework is verified through the optimization of both 2D and 3D structures with periodicity. Computational results demonstrate the feasibility and effectiveness of the proposed framework for designing robust periodic structures under loading uncertainties.     

Robust concurrent topology optimization of multiscale structure under single or multiple uncertain load cases

Concurrent topology optimization of macrostructure and material microstructure has attracted significant interest in recent years. However, most of the existing works assumed deterministic load conditions, thus the obtained design might have poor performance in practice when uncertainties exist. Therefore, it is necessary to take uncertainty into account in structural design. This article proposes an efficient method for robust concurrent topology optimization of multiscale structure under single or multiple load cases. The weighted sum of the mean and standard deviation of the structural compliance is minimized and constraints are imposed to both the volume fractions of macrostructure and microstructure. The effective properties of the microstructure are calculated via the homogenization method. An efficient sensitivity analysis method is proposed based on the superposition principle and orthogonal similarity transformation of real symmetric matrices. To further reduce the computational cost, an efficient decoupled sensitivity analysis method for microscale design variables is proposed. The bidirectional evolutionary structural optimization method is employed to obtain black and white designs for both macrostructure and microstructure. Several two-dimensional and three-dimensional numerical examples are presented to demonstrate the effectiveness of the proposed approach and the effects of load uncertainty on the optimal design of both macrostructure and microstructure.     

A novel minimum weight formulation of topology optimization implemented with reanalysis approach

In this paper, we develop an efficient diagonal quadratic optimization formulation for minimum weight design problem subject to multiple constraints. A high-efficiency computational approach of topology optimization is implemented within the framework of approximate reanalysis. The key point of the formulation is the introduction of the reciprocal-type variables. The topology optimization seeking for minimum weight can be transformed as a sequence of quadratic program with separable and strictly positive definite Hessian matrix, thus can be solved by a sequential quadratic programming approach. A modified sensitivity filtering scheme is suggested to remove undesirable checkerboard patterns and mesh dependence. Several typical examples are provided to validate the presented approach. It is observed that the optimized structure can achieve lighter weight than those from the established method by the demonstrative numerical test. Considerable computational savings can be achieved without loss of accuracy of the final design for 3D structure. Moreover, the effects of multiple constraints and upper bound of the allowable compliance upon the optimized designs are investigated by numerical examples.     

拓扑相关荷载作用下结构拓扑优化的水平集方法

发展了一种利用水平集演化技术求解拓扑相关荷载作用下结构拓扑优化问题的数值方法。通过引入水平集函数,我们以隐含的方式对结构的拓扑和形状作了描述,从而把拓扑优化问题转化为了寻求最优水平集函数的数学规划问题。利用基于连续体概念的灵敏度分析技术,构造了用于驱动水平集演化的速度场。由于结构的边界可以用零水平集加以描述,因此利用适当的数学变换,我们可以方便地处理施加在结构上的拓扑相关荷载,这样就避免了以往算法中繁复的边界提取工作以及为了处理拓扑相关荷载所采取的特殊技巧。文末的数值算例表明了提出的优化方法在处理此类问题时所具有的独到的优越性。     

Robust compliance topology optimization based on the topological derivative concept

In this paper, we present an approach for robust compliance topology optimization under volume constraint. The compliance is evaluated considering a point‐wise worst‐case scenario. Analogously to sequential optimization and reliability assessment, the resulting robust optimization problem can be decoupled into a deterministic topology optimization step and a reliability analysis step. This procedure allows us to use topology optimization algorithms already developed with only small modifications. Here, the deterministic topology optimization problem is addressed with an efficient algorithm based on the topological derivative concept and a level‐set domain representation method. The reliability analysis step is handled as in the performance measure approach. Several numerical examples are presented showing the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Global optimization of robust truss topology via mixed integer semidefinite programming

This paper discusses a global optimization method of robust truss topology under the load uncertainties and slenderness constraints of the member cross-sectional areas. We consider a non-stochastic uncertainty of the external load, and attempt to minimize the maximum compliance corresponding to the most critical load. A design-dependent uncertainty model in the external load is proposed in order to consider the variation of truss topology rigorously. It is shown that this optimization problem can be formulated as a 0–1 mixed integer semidefinite programming (0–1MISDP) problem. We propose a branch-and-bound method for computing the global optimal solution of the 0–1MISDP. Numerical examples illustrate that the topology of robust optimal truss depends on the magnitude of uncertainty. The presented method can provide global optimal solutions for benchmark examples, which can be used for evaluating the performance of any other local optimization method for robust structural optimization.     

Continuum topology optimization considering uncertainties in load locations based on the cloud model

Few researchers have paid attention to designing structures in consideration of uncertainties in the loading locations, which may significantly influence the structural performance. In this work, cloud models are employed to depict the uncertainties in the loading locations. A robust algorithm is developed in the context of minimizing the expectation of the structural compliance, while conforming to a material volume constraint. To guarantee optimal solutions, sufficient cloud drops are used, which in turn leads to low efficiency. An innovative strategy is then implemented to enormously improve the computational efficiency. A modified soft-kill bi-directional evolutionary structural optimization method using derived sensitivity numbers is used to output the robust novel configurations. Several numerical examples are presented to demonstrate the effectiveness and efficiency of the proposed algorithm.     

Robust topology optimization for cellular composites with hybrid uncertainties

This paper will develop a new robust topology optimization method for the concurrent design of cellular composites with an array of identical microstructures subject to random‐interval hybrid uncertainties. A concurrent topology optimization framework is formulated to optimize both the composite macrostructure and the material microstructure. The robust objective function is defined based on the interval mean and interval variance of the corresponding objective function. A new uncertain propagation approach, termed as a hybrid univariate dimension reduction method, is proposed to estimate the interval mean and variance. The sensitivity information of the robust objective function can be obtained after the uncertainty analysis. Several numerical examples are used to validate the effectiveness of the proposed robust topology optimization method.     

An approach to robust optimization of impact problems using random samples and meta-modelling

Conventionally optimized structures may show a tendency to be sensitive to variations, for instance in geometry and loading conditions. To avoid this, research has been carried out in the field of robust optimization where variations are taken into account in the optimization process. The overall objective is to create solutions that are optimal both in the sense of mean performance and minimum variability. This work presents an alternative approach to robust optimization, where the robustness of each design is assessed through multiple sampling of the stochastic variables at each design point. Meta-models for the robust optimization are created for both the mean value and the standard deviation of the response. Furthermore, the method is demonstrated on an analytical example and an example of an aluminium extrusion with quadratic cross-section subjected to axial crushing. It works well for the chosen examples and it is concluded that the method is especially well suited for problems with a large number of random variables, since the computational cost is essentially independent of the number of random variables. In addition, the presented approach makes it possible to take into consideration variations that cannot be described with a variable. This is demonstrated in this work by random geometrical perturbations described with the use of Gaussian random fields.     

复合材料层合板屈曲稳定性优化设计及其灵敏度计算方法

笔者在有限元分析基础上研究了以屈曲稳定性作为约束条件或优化目标的复合材料层合板结构优化设计及其灵敏度分析方法,重点讨论了屈曲临界荷载灵敏度对内力场和载荷的依赖关系及其在铺层优化、尺寸优化和形状优化问题中的不同计算方法,并在JIFEX软件中实现了复杂结构复合材料层合板优化设计方法。数值算例验证了本文算法和程序的有效性。     

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.     

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.     

高层建筑抗风优化设计和风振控制相关问题研究

高层建筑由于自振周期长、阻尼小,其高柔的特征使其对风荷载特别敏感,风荷载是沿海地区超高层建筑的主要水平控制荷载,因此在强/台风作用下,其抗风设计须在满足规范安全要求的前提下,同时又要经济实用和结构性能高效,为此,开展高层建筑抗风优化和风振控制方面的研究具有十分重要的现实意义。该文在对高层建筑抗风优化设计和风振控制研究现状做简要介绍的基础上,首先根据风荷载的特点,着重研究了考虑风速风向联合概率分布和基于可靠度及性能化的高层建筑抗风设计方法,采用最优准则法,以结构的总重或总造价为目标函数,以顶部位移、层间侧移以及顶部风致加速度为约束条件,对高层建筑结构杆件截面抗风优化设计的相关问题进行了研究。同时为提高基因遗传智能优化算法的收敛速度和获得最可能优化解,该文提出了传统基因遗传的改进算法(如基于改进罚函数及分级遗传算法)用于结构抗风优化设计。在结构拓扑抗风优化方面,则主要引入分层优化的概念,对变密度法和改进动态进化率的双向渐进拓扑优化方法,应用于抗风结构的拓扑构型优化算法进行了相关研究。通过实例分析验证了上述结构抗风优化算法的高效和正确性。在风振控制方面,该文结合摩擦摆系统和调谐质量阻尼器各自的优点,提出了摩擦摆调谐质量阻尼器(FPS-TMD)被动控制系统,对其力学和动力特性,以及高层建筑顶部带FPS-TMD系统的风振控制理论,进行了相关研究。以结构控制第三代Benchmark模型为实例,研究顶部带FPS-TMD系统的高层建筑风振控制效果,同时结合该文开发的基于小型电振动台的实时混合实验测试平台,采用风振控制实时混合实验结果与理论模拟计算结果的对比,验证了该文提出的FPS-TMD被动控制系统,应用于高层建筑风振控制的有效性。     

Material nonlinear topology optimization considering the von Mises criterion through an asymptotic approach: Max strain energy and max load factor formulations

This paper addresses material nonlinear topology optimization considering the von Mises criterion by means of an asymptotic analysis using a fictitious nonlinear elastic model. In this context, we consider the topology optimization problem subjected to prescribed energy, which leads to robust convergence in nonlinear problems. Two nested formulations are considered. In the first, the objective is to maximize the strain energy of the system in equilibrium, and in the second, the objective is to maximize the load factor. In both cases, a volume constraint is imposed. The sensitivity analysis is quite effective and efficient in the sense that there is no extra adjoint equation. In addition, the nonlinear structural equilibrium problem is solved using direct minimization of the structural strain energy using Newton's method with an inexact line-search strategy. Four numerical examples demonstrate the features of the proposed material nonlinear topology optimization framework for approximating standard von Mises plasticity.     

Truss layout design under nonprobabilistic reliability-based topology optimization framework with interval uncertainties

A nonprobabilistic reliability-based topology optimization (NRBTO) method for truss structures with interval uncertainties (or unknown-but-bounded uncertainties) is proposed in this paper. The cross-sectional areas of levers are defined as design variables, while the material properties and external loads are regard as interval parameters. A modified perturbation method is applied to calculate structural response bounds, which are the prerequisite to obtain structural reliability. A deviation distance between the current limit state plane and the objective limit state plane, of which the expression is explicit, is defined as the nonprobabilistic reliability index, which serves as a constraint function in the optimization model. Compared with the deterministic topology optimization problem, the proposed NRBTO formulation is still a single-loop optimization problem, as the reliability index is explicit. The sensitivity results are obtained from an analytical approach as well as a direct difference method. Eventually, the NRBTO problem is solved by a sequential quadratic programming method. Two numerical examples are used to testify the validity and effectiveness of the proposed method. The results show significant effects of uncertainties to the topology configuration of truss structures.     

Bi-directional evolutionary structural optimization for design-dependent fluid pressure loading problems

This article presents an evolutionary topology optimization method for compliance minimization of structures under design-dependent pressure loads. In traditional density based topology optimization methods, intermediate values of densities for the solid elements arise along the iterations. Extra boundary parametrization schemes are demanded when these methods are applied to pressure loading problems. An alternative methodology is suggested in this article for handling this type of load. With an extended bi-directional evolutionary structural optimization method associated with a partially coupled fluid–structure formulation, pressure loads are modelled with hydrostatic fluid finite elements. Due to the discrete nature of the method, the problem is solved without any need of pressure load surfaces parametrization. Furthermore, the introduction of a separate fluid domain allows the algorithm to model non-constant pressure fields with Laplace's equation. Three benchmark examples are explored in order to show the achievements of the proposed method.     

Topology optimization using bi-directional evolutionary structural optimization based on the element-free Galerkin method

In this article, the bi-directional evolutionary structural optimization (BESO) method based on the element-free Galerkin (EFG) method is presented for topology optimization of continuum structures. The mathematical formulation of the topology optimization is developed considering the nodal strain energy as the design variable and the minimization of compliance as the objective function. The EFG method is used to derive the shape functions using the moving least squares approximation. The essential boundary conditions are enforced by the method of Lagrange multipliers. Several topology optimization problems are presented to show the effectiveness of the proposed method. Many issues related to topology optimization of continuum structures, such as chequerboard patterns and mesh dependency, are studied in the examples.     

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.