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.     

Hyperstatic and redundancy thresholds in truss topology optimization considering progressive collapse due to aleatory and epistemic uncertainties

Optimization leads to specialized structures which are not robust to disturbance events like unanticipated abnormal loading or human errors. Typical reliability-based and robust optimization mainly address objective aleatory uncertainties. To date, the impact of subjective epistemic uncertainties in optimal design has not been comprehensively investigated. In this paper, we use an independent parameter to investigate the effects of epistemic uncertainties in optimal design: the latent failure probability. Reliability-based and risk-based truss topology optimization are addressed. It is shown that optimal risk-based designs can be divided in three groups: (A) when epistemic uncertainty is small (in comparison to aleatory uncertainty), the optimal design is indifferent to it and yields isostatic structures; (B) when aleatory and epistemic uncertainties are relevant, optimal design is controlled by epistemic uncertainty and yields hyperstatic but nonredundant structures, for which expected costs of direct collapse are controlled; (C) when epistemic uncertainty becomes too large, the optimal design becomes redundant, as a way to control increasing expected costs of collapse. The three regions above are divided by hyperstatic and redundancy thresholds. The redundancy threshold is the point where the structure needs to become redundant so that its reliability becomes larger than the latent reliability of the simplest isostatic system. Simple truss topology optimization is considered herein, but the conclusions have immediate relevance to the optimal design of realistic structures subject to aleatory and epistemic uncertainties.     

Microstructural topology optimization for minimizing the sound pressure level of structural–acoustic coupled systems with multi-scale random parameters

The main aim of this article is to present a robust microstructural topology optimization methodology for structural–acoustic coupled systems with multi-scale random parameters. During the microstructural topology optimization, both the uncertainty at the macro-scale, which comes from the physical parameters of the acoustic medium or the external load, and the uncertainty existing in the constituent material properties of the microstructure at the micro-scale are considered as random parameters. A homogenization-based probabilistic finite element method (HPFEM) is first developed for quantifying the structural–acoustic system with multi-scale random parameters. The use of the HPFEM transforms the problem of microstructural topology optimization with multi-scale random parameters to an augmented deterministic microstructural topology optimization problem. This provides a computationally cheap alternative to Monte Carlo-based optimization algorithms. A numerical example of a hexahedral box is given to demonstrate the efficiency of the proposed method.     

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.     

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.     

Global optimization of minimum weight truss topology problems with stress,displacement, and local buckling constraints using branch‐and‐bound

We present a convergent continuous branch‐and‐bound algorithm for global optimization of minimum weight truss topology problems with displacement, stress, and local buckling constraints. Valid inequalities which strengthen the problem formulation are derived. The inequalities are generated by solving well‐defined convex optimization problems. Computational results are reported on a large collection of problems taken from the literature. Most of these problems are, for the first time, solved with a proof of global optimality. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

A GA‐based technique for layout optimization of truss with stress and displacement constraints

Various developments of increasing complexity involved in layout optimization are discussed. The use of conventional GA in layout optimization is briefly mentioned with emphasis on its limitations and conditions imposed in finding the optimal design. The proposed new technique is applied to the benchmark example of Michell's truss for verification. The approach has also been applied to new examples of bridge truss and crane truss problems in order to demonstrate the generality and robustness for topology optimization. The approach is extended to include dual stress‐displacements constraints since many practical problems involve these two constraints simultaneously. Two‐bar and 10‐bar trusses are solved as examples for layout optimization with both stress and displacement constraints with satisfactory results. The effect of mutation on the final topology is also discussed. The major drawbacks of the ground structure approach are overcome in this proposed new method. The optimal designs obtained demonstrate the ability, robustness and generality of using the proposed new technique in layout optimization problems. Copyright © 2001 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.     

Multifidelity approaches for optimization under uncertainty

It is important to design robust and reliable systems by accounting for uncertainty and variability in the design process. However, performing optimization in this setting can be computationally expensive, requiring many evaluations of the numerical model to compute statistics of the system performance at every optimization iteration. This paper proposes a multifidelity approach to optimization under uncertainty that makes use of inexpensive, low‐fidelity models to provide approximate information about the expensive, high‐fidelity model. The multifidelity estimator is developed based on the control variate method to reduce the computational cost of achieving a specified mean square error in the statistic estimate. The method optimally allocates the computational load between the two models based on their relative evaluation cost and the strength of the correlation between them. This paper also develops an information reuse estimator that exploits the autocorrelation structure of the high‐fidelity model in the design space to reduce the cost of repeatedly estimating statistics during the course of optimization. Finally, a combined estimator incorporates the features of both the multifidelity estimator and the information reuse estimator. The methods demonstrate 90% computational savings in an acoustic horn robust optimization example and practical design turnaround time in a robust wing optimization problem. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

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.     

Comparison of robust,reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints

It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: (1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; (2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; (3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Results are computed for two reference problems: similarities and differences between optimized topologies obtained with the three formulations are exploited and discussed.     

Level set topology optimization of fluids in Stokes flow

We propose the level set method of topology optimization as a viable, robust and efficient alternative to density‐based approaches in the setting of fluid flow. The proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process, leading to significant advantages over density‐based topology optimization algorithms. Specifically, the no‐slip boundary condition is implemented directly—this is accurate, removes the need for interpolation schemes and continuation methods, and gives significant computational savings by only requiring flow to be modeled in fluid regions. Topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two‐dimensional power dissipation minimization problems are solved successfully. Computational efficiency of the algorithm is also clearly demonstrated on large‐scale three‐dimensional problems. Copyright © 2009 John Wiley & Sons, Ltd.     

A unified stochastic framework for robust topology optimization of continuum and truss-like structures

In this article, a unified framework is introduced for robust structural topology optimization for 2D and 3D continuum and truss problems. The uncertain material parameters are modelled using a spatially correlated random field which is discretized using the Karhunen–Loève expansion. The spectral stochastic finite element method is used, with a polynomial chaos expansion to propagate uncertainties in the material characteristics to the response quantities. In continuum structures, either 2D or 3D random fields are modelled across the structural domain, while representation of the material uncertainties in linear truss elements is achieved by expanding 1D random fields along the length of the elements. Several examples demonstrate the method on both 2D and 3D continuum and truss structures, showing that this common framework provides an interesting insight into robustness versus optimality for the test problems considered.     

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.     

Path‐generation of articulated mechanisms by shape and topology variations in non‐linear truss representation

This paper presents studies on an optimization‐based method for path‐generation of articulated mechanisms. An extended truss ground‐structure approach is taken in which both the shape and topology of the truss are designed using cross‐sectional areas and nodal positions as design variables. This leads to a technique for simultaneous type and dimensional synthesis of articulated mechanisms. For the analysis part it is essential to control the mechanism configuration so that the mechanism remains within a given configuration space, thus stabilizing the optimization process and resulting in realistic solutions. This can be achieved by using the Levenberg–Marquardt method. The design method is illustrated by a number of design cases for both closed and open input and output paths. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Topology optimization of resonating structures using SIMP method

The present paper is concerned with the layout optimization of resonating actuators using topology optimization techniques. The goal of the optimization is a maximization of the magnitude of steady‐state vibrations for a given excitation frequency. The problem formulation includes an external viscous damper at the output port which models a working load on the structure. Copyright © 2002 John Wiley & Sons, Ltd.     

A new method to generate the ground structure in truss topology optimization

The aim of this article is to provide an effective method to generate the ground structure in truss topology optimization. The core of this method is to place nodal points for the ground structure at the intersection of the first and third principal stress trajectories, which are obtained by solving the equivalent static problem in the design domain with a homogeneous isotropic material property. It is applicable to generate the ground structure for arbitrary regular and irregular geometric design domains. The proposed method is tested on some benchmark examples in truss topology optimization. The optimization model is a standard linear programming problem based on plastic design and solved by the interior point algorithm. Compared with other methods, the proposed method may use a well-defined ground structure with fewer nodes and bars, resulting in faster solution convergence, which shows it to be efficient.