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This article presents a simple yet efficient method for the topology optimization of continuum structures considering interval uncertainties in loading directions. Interval mathematics is employed to equivalently transform the uncertain topology optimization problem into a deterministic one with multiple load cases. An efficient soft-kill bi-directional evolutionary structural optimization (BESO) method is proposed to solve the problem, which only requires two finite element analyses per iteration for each external load with directional uncertainty regardless of the number of the multiple load cases. The presented algorithm leads to significant computational savings when compared with Monte Carlo-based optimization (MCBO) algorithms. A series of numerical examples including symmetric and non-symmetric loading variations demonstrate the considerable improvement of computational efficiency of the proposed approach as well as the significance of including uncertainties in topology optimization when to design a structure. Optimums obtained from the proposed algorithm are verified by MCBO method.

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... To date, many researchers have focused on the topology optimization of macroscale structure based on robust optimization theory, including the uncertainties of load magnitude and direction [27][28][29][30][31][32][33], load position [34,35], random field load [36,37], material properties [38][39][40], geometry [41][42][43][44], and others. In all the uncertain factors, load uncertainty has the most significant impact on structure topology optimization [30]. ...

... To date, many researchers have focused on the topology optimization of macroscale structure based on robust optimization theory, including the uncertainties of load magnitude and direction [27][28][29][30][31][32][33], load position [34,35], random field load [36,37], material properties [38][39][40], geometry [41][42][43][44], and others. In all the uncertain factors, load uncertainty has the most significant impact on structure topology optimization [30]. Magnitude, direction, and applied position are the three basic elements to determine a load and most of the existing works on robust topology optimization (RTO) with load uncertainty are carried out for these three aspects. ...

... Zhao et al. [28] solved RTO problem by using Monte Carlo method to quantify uncertainties when load magnitude and direction are uncertain. Csébfalvi [29] and Liu et al. [30] developed approaches for RTO when the only source of uncertainty are load directions. Jeong et al. [34] proposed the compliance minimization topology design method due to load position uncertainty using the phase field design method. ...

This study proposes a robust concurrent topology optimization method with considering dynamic load uncertainty for the design of structures composed of periodic microstructures based on the bi-directional evolutionary structural optimization (BESO) method. The objective function is formulated as the summation of the mean and standard deviation of the structural dynamic compliance modulus. The constraints are imposed on the macrostructure and material microstructure volumes, respectively. The hybrid dimension reduction method and Gauss integral (HDRG) method is proposed to quantify and propagate load uncertainty to estimate the objective function. By the HDRG method, robust topology optimization with uncertainty modeled by probabilistic methods can be handed uniformly. To reduce the computational burden, a decoupled sensitivity analysis method is proposed to calculate the sensitivities of objective function with respect to the microstructure design variables. Five numerical examples are used to validate the effectiveness of the proposed robust concurrent topology optimization method and demonstrate the influence of load uncertainty on the design results. Results illustrate that the proposed methods can obtain the clear topologies of macro and micro structures, and the dynamic load uncertainty has a significant impact on the design results.

... Optimal structures using deterministic parameter settings may be prone to premature failure in the presence of uncertain disturbances [37,38]. TO problems with consideration of uncertainties can be roughly divided into three categories: reliability-based topology optimization (RBTO), robust topology optimization (RTO), and equivalent topology optimization (ETO) [39][40][41][42][43][44]. RBTO often emphasizes high reliability in a topological design by satisfying probabilistic constraints at desired levels [39]. ...

... This method has become increasingly popular in recent years. ETO operates based on a criterion that transforms the original difficult TO problem into a problem that can be solved using a mature algorithm without distorting the attributes of the original problem [44]. ...

... However, if the lower and upper bounds of uncertain variable can be acquired easily than the accurate probability distribution in practical use, the structural performance under the worst condition can be used to define the objective function. The other problem dedicates to minimize the worst objective function associated with uncertain variables (worst-case method) [44], which is referred to as the interval optimization model. This paper focuses on the averaging problem because uncertain forces, including load magnitude and direction, are all assumed to follow Gaussian distributions, which are ubiquitous and very important probability distribution of continuous random variables. ...

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 semi‐analytical method combined with probability theory, which is important for RTO when uncertain variables follow probabilistic distributions. To obtain optimal topologies, the bi‐directional 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.

... In this section, we present an alternative scheme to figure out the conundrum based on our previous work (Liu et al. 2015;Liu et al. 2017). We employ interval mathematics to equivalently transform the topology optimization problem with the uncertainties of the direction of applied loads into a deterministic one with multiple load cases. ...

... The scheme is simply presented here. For more details, the readers are referred to references (Liu et al. 2015(Liu et al. , 2017. We define the uncertainty of load directions in 2D space in terms of their angles of the applied loads,θ = (θ 1 , θ 2 , ⋯ , θ i , ⋯ , θ s ), where θ i is regarded as an interval and s denotes the total number of external loading. ...

... Thus, the sensitivity number can be formulated by (Bendsøe and Sigmund 1999) optimization under multiple load cases directly (Liu et al. 2017). Fig. 7 and 8 show the optimal solutions for the design problems with symmetrical and asymmetrical loads, respectively, when the proposed scheme is implemented. ...

In some cases, topology optimization of continuum structures subjected to applied loads having a zero resultant force may result in an unpractical design without support. This phenomenon occurs because the original optimization problem neglects the possible change of the direction of applied load. This brief note sheds the light on avoiding such an unpractical design from the engineering viewpoint. In our work, this usually neglected phenomenon is systematically illustrated by employing a series of two-dimensional (2D) cantilever design problems using a simple and efficient Bi-directional Evolutionary Structural Optimization (BESO) method. An alternative scheme is further recommended to tackle the concerned conundrum. The proposed scheme not only can avoid unpractical designs without any support, but also takes into account the inherent uncertainty property in designing actual engineering structures.

... Improper evaluation of these bounds may result in either unsafe or over-conservative designs [9 , 10] . Most of the existed researches were devoted to construct convex models based on the presumed bounds of uncertain parameters by experience [11][12][13][14] , which is inappropriate and unreasonable for engineering applications in most cases. Currently, the rotation matrix method [15] and the correlation analysis technique [16] can construct convex models in a general setting. ...

... where α is the increment coefficient. Alternatively, according to Eqs. (10 )- (12) , the problem of finding suitable bounds of uncertain parameters can be transformed into an equivalent problem, as min α, ...

In this paper, a novel method for non-probabilistic convex modelling with the bounds to precisely encircle all the data of uncertain parameters extracted from practical engineering is developed. The method is based on the traditional statistical method and the correlation analysis technique. Mean values and correlation coefficients of uncertain parameters are first calculated by utilizing the information of all the given data. Then, a simple yet effective optimization procedure is first introduced in the mathematical modelling process for uncertain parameters to obtain their precise bounds. This procedure works by optimizing the area of the convex model, at the same time, covering all the given data. Thus, the effective mathematical expression of the convex models are finally formulated. To test the prediction capability and generalization ability of the proposed convex modelling method, evaluation criteria, i.e. volume ratio, standard volume ratio, and prediction accuracy are established. The performance of the proposed method is systematically studied and compared with other existing competitive methods through test standards. The results demonstrate the effectiveness and efficiency of the present method.

... Recently, there are three different ways to consider uncertainties in structural TO: reliability-based topology optimization (RBTO), robust topology optimization (RTO), and equivalent topology optimization (ETO) [11]. RBTO aims to achieve an optimal structure with its required functions subjected to a stated reliability index [12], whereas the RTO commits to obtain an optimized structure which can accomplish its intended function persistently even in the presence of parameter disturbance [13]. ...

... Besides, it is worth noting that significant works have been done to resolve the dynamic TO problems using ETO methods. More recently, Liu [11] employs ETO method to deal with the uncertain TO problems with the interval model, which is under an assumption that the distribution of uncertain load is unknown. RBTO approaches depend on the functions containing the uncertain parameters which represent one or more failure states. ...

Uncertainty is omnipresent in engineering design and manufacturing. This paper dedicates to present a robust topology optimization (RTO) methodology for structural compliance minimization problems considering load uncertainty, which includes magnitude and direction uncertainty subjected to Gaussian distribution. To this end, comprehensible semi-analytical formulations are derived to fleetly calculate the statistical data of structural compliance, which is critical to the probability-based RTO problem. In order to avoid the influence of numerical units on evaluating the robust results, this paper considers a generic coefficient of variation (GCV) as robust index which contains both the expected compliance and standard variance. In addition, the accuracy and efficiency of semi-analytical formulas are validated by the Monte Carlo (MC) simulation; comparison results provide higher calculation efficiency over the MC-based optimization algorithms. Four numerical examples are provided via density-based approach to demonstrate the effectiveness and robustness of the proposed method.

... The BESO approach has been recently developed to provide the optimal topologies for a wide range of engineering applications, incorporating multi-materials [41,42], additional displacement constraints [43], stiffness and frequency optimization [44,45], nonlinear materials and large deformation [46][47][48] and uncertainties in load directions [49,50]. In this study, we have developed the BESO algorithm for the determination of the optimal layout of two external plate stiffeners welded directly onto the SHS/RHS column faces at the top and bottom flange locations of an I-section beam. ...

In this paper, we propose a cost-effective optimal-topology retrofitting technique for hollow steel section columns to sufficiently support industrial running cranes. A so-called bi-directional evolutionary structural optimization (BESO) method was encoded within the MATLAB modeling framework, with a direct interface with an ANSYS commercial finite-element analysis program , to determine the optimal topology of double external steel plates connected to columns in a 3D space. For the initial ground structure, we have adopted standard uniform double U-shaped external stiffener plates located at the top and bottom flange layers of an I-beam to box-column connection (IBBC) area. The influences of inelastic materials and the incorporated nonlinear geometry can effectively describe the premature (local buckling) failures of the columns in an IBBC area. The applications of the proposed optimal-topology BESO-based stiffening method are illustrated through the retrofitting of three hollow-steel-section columns, characterized by non-slender and slender compression sections. Some concluding remarks are provided on the pre-and post-retrofitted responses of the columns, with the results showing both the accuracy and robustness of the proposed external stiffening schemes.

... A variety of uncertainty models are considered for topology optimization. For example, the random variable [129,231,97,96,225], random field [58,367,199,438,315,80,181,16], interval model [396,226,448,387], convex model [183,184,365], evidence theory [444,387], fuzzy-set [422] and the hybrid model [57,447,445,254,385]. ...

The objective of this thesis is to develop density based-topology optimization methods for several challenging dynamic structural problems. First, we propose a normalization strategy for elastodynamics to obtain optimized material distributions of the structures that reduces frequency response and improves the numerical stabilities of the bi-directional evolutionary structural optimization (BESO). Then, to take into account uncertainties in practical engineering problems, a hybrid interval uncertainty model is employed to efficiently model uncertainties in dynamic structural optimization. A perturbation method is developed to implement an uncertainty-insensitive robust dynamic topology optimization in a form that greatly reduces computational costs. In addition, we introduce a model of interval field uncertainty into dynamic topology optimization. The approach is applied to single material, composites and multi-scale structures topology optimization. Finally, we develop a topology optimization for dynamic brittle fracture structural resistance, by combining topology optimization with dynamic phase field fracture simulations. This framework is extended to design impact-resistant structures. In contrast to stress-based approaches, the whole crack propagation is taken into account into the optimization process.

... The BESO approach has been recently developed to provide the optimal topologies for a wide range of engineering applications, incorporating multi-materials [41,42], additional displacement constraints [43], stiffness and frequency optimization [44,45], nonlinear materials and large deformation [46][47][48] and uncertainties in load directions [49,50]. In this study, we have developed the BESO algorithm for the determination of the optimal layout of two external plate stiffeners welded directly onto the SHS/RHS column faces at the top and bottom flange locations of an I-section beam. ...

The local buckling phenomenon presents one of the main premature failures often prohibiting the steel hollow-section columns to attend the ultimate strength capacity. A strengthening method is then required to extend the service life of the member. This paper presents the optimal retrofitting design of standard steel hollow-section columns using external steel plates, such that the ultimate strength of the post-retrofitted column sufficiently resists the design load imposed by an industrial crane. The optimal design adopts a so-called bi-directional evolutionary structural optimization (BESO) algorithm that determines the cost-effective distribution of steel plate topology welded to the column. The proposed method realistically considers the influences of inelastic material properties and geometric nonlinearity, simultaneously. The BESO algorithm is encoded within the MATLAB modeling framework providing a direct application interface to ANSYS (a commercial-purposed finite element analysis software), which models the retrofitting joint between steel column and corbel using the comprehensive 3D finite elements. The robustness of the proposed scheme is illustrated through standard steel warehouse applications. The accuracy and integrity of the resulting design are validated by the full elastoplastic responses of the post-retrofitted column under applied forces.

... Du [8] and Du et al. [9] systematically studied the structural reliability analysis consisting of interval parameters. Various techniques have also proposed to solve this kind of reliability and design problem [10][11][12][13][14][15][16][17][18]. However, there could be the circumstance that the designing parameters of a structure have different kinds of uncertainties. ...

Nondeterministic parameters of certain distribution are employed to model structural uncertainties, which are usually assumed as stochastic factors. However, model parameters may not be precisely represented due to some factors in engineering practices, such as lack of sufficient data, data with fuzziness, and unknown-but-bounded conditions. To this end, interval and fuzzy parameters are implemented and an efficient approach to structural reliability analysis with random-interval-fuzzy hybrid parameters is proposed in this study. Fuzzy parameters are first converted to equivalent random ones based on the equal entropy principle. 3σ criterion is then employed to transform the equivalent random and the original random parameters to interval variables. In doing this, the hybrid reliability problem is transformed into the one only with interval variables, in other words, nonprobabilistic reliability analysis problem. Nevertheless, the problem of interval extension existed in interval arithmetic, especially for the nonlinear systems. Therefore, universal grey mathematics, which can tackle the issue of interval extension, is employed to solve the nonprobabilistic reliability analysis problem. The results show that the proposed method can obtain more conservative results of the hybrid structural reliability.

... Robust structural optimization has been subject of intense research over the last decade, as discussed in details by [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27], among others. However, robustness is an abstract concept, strongly linked to what is generally called 5 "physical intuition". ...

In probabilistic robust optimization problems we generally take as objective function the weighted sum of the expected value and the standard deviation of the performance. Unfortunately, it has been observed that if the weight given to the standard deviation is too high, then the designs obtained may have no physical meaning at all. If this occurs we say that the design obtained is not consistent from the physical point of view. In this work we present a probabilistic robust optimization approach aimed to preserve physical consistency of the problem being addressed. The objective function is written as the p-norm of a vector composed by the expected value and the weighted standard deviation. We then demonstrate that the standard approach is a particular case of this p-norm approach with p = 1. In the proposed approach we take p = 2 and prove that physical consistency is ensured provided the standard deviation is not given more weight than the expected value. A similar proof is not available for the standard approach. This makes statement of the robust optimization problem much easier with the proposed approach. The proposed approach can be adapted into existing computational routines, since the same statistical moments and sensitivities are required. The theoretical results are illustrated in the context of compliance-based topology optimization. In the numerical examples we show that the concept of physical consistency may have a significant impact on the designs obtained.

... This is because the U-shaped structure is subjected to applied loads having a zero resultant force, which is well studied by our previous studies. [36][37][38] The evolution history is also shown in Figure 15 to prove the convergence of our method. In Table 2, the iteration of presented method is decreased from 215 to 137. ...

... One more point should be emphasized that the studies on NRBTO are still in their primary stage, and many challenges (containing such uncertainty issue in loading direction) remain unsolved. As referred by [48][49][50], the current research works can be further expanded and investigated. ...

This study presents a novel non-probabilistic reliability-based topology optimization (NRBTO) framework to determine optimal material configurations for continuum structures under local stiffness and strength limits. Uncertainty quantification (UQ) analysis under unknown-but-bounded (UBB) inputs is conducted to determine the feasible bounds of structural responses by combining a material interpolation model with stress aggregation function and interval mathematics. For safety reasons, improved interval reliability indexes that correspond to displacement and stress constraints are applied in topological optimization issues. Meanwhile, an adjoint-vector based sensitivity analysis is further discussed from which the gradient features between reliability measures and design variables are mathematically deduced, and the computational difficulties in large-scale variable updating can be effectively overcome. Numerical examples are eventually given to demonstrate the validity of the developed NRBTO methodology.

... Zhao et al. constructed a stochastic response surface model for solving RBTO issues, in which two reliability methodologies were discussed and implemented, including the performance measure approach (PMA) and the sequential optimization and reliability assessment (SORA) (Zhao et al. 2015). As revealed by the investigations, it can be found that the RBTO model yields results that are more reliable than those produced by DTO (Liu et al. 2017). ...

This paper develops a non-probabilistic reliability-based topology optimization (NRBTO) framework for continuum structures under multi-dimensional convex uncertainties. Combined with the solid isotropic material with penalization (SIMP) model and the set-theoretical convex method, the uncertainty quantification (UQ) analysis is firstly conducted to obtain mathematical approximations and boundary laws of considered displacement responses. By normalization treatment of the limit-state function, a new quantified measure of the non-probabilistic reliability is then defined and further deduced by the principle of the hyper-volume ratio. For circumventing optimization difficulties arising from large-scale design variables, the adjoint vector scheme for sensitivity analysis of the reliability index with respect to design variables are discussed as well. Numerical applications eventually illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques.

... Alternatively, the non-probabilistic model [9][10][11][12] can be used to describe the soil characteristics while the upper and lower bounds of the soil parameters are easy to get. The seminal work of using non- probabilistic models to solve structural analysis and design problem was done by Ben-Haim and Elishakoff [9,13]. ...

The properties of the simulated lunar soil, which are of great importance to guide the investigation on the lunar due to the rarity and preciousness of the lunar soil, are extremely complex such that their precise probability distributions are hard to obtain but their bounds are easy to gain. In this study, an innovative non-random analysis method is developed to capture the ultimate soil shear stress based on the non-probabilistic convex model and Mohr–Coulomb failure criterion. Specifically, we extend the traditional Mohr–Coulomb failure criterion, which is usually in deterministic case, to an uncertain case by using a symmetrical positive definite matrix to characterize the correlation of the uncertain parameters related to the soil parameters. To obtain the upper and lower bounds of failure shear stress of the simulated lunar soil, the problem of calculating the stress bounds is converted into two optimization problems, which are solved by using Lagrangian method. Considering whether the time is involved or not in the uncertainties of the soil parameters, both time-invariant and time-variant interval uncertainties problems can be tackled by the proposed method. The effectiveness and accuracy of the proposed method is evidenced by numerical examples and data retrieved from the experimental results exactly between the upper and lower bounds. It indicates that the proposed method can provide important guidance for the design and optimization of lunar soil sampling mechanisms.

... The first two optimization techniques concern about the size and shape of a structure under the specified structural topology. The topology optimization technique can determine the best load transmission path and achieve the optimal material layout, which has great attention from both academia and industry [Eschenauer and Olhoff (2001); Gong et al. (2012); Sigmund and Maute (2013); Deaton and Grandhi (2014); Li et al. (2015); He et al. (2015); Luo and Tong (2016); Li et al. (2016); Liu et al. (2017); He et al. (2018)]. Until now, many methods have been developed to solve the topology optimization problems, such as the homogenization method [Bendsøe and Kikuchi (1988)], solid isotropic material with penalization (SIMP) method [Bendsøe (1989); Sigmund (2001)], level-set method (LSM) [Wang et al. (2003); Allaire et al. (2004)], evolutionary structural optimization (ESO) method [Xie and Steven (1993); Huang and Xie (2007)] and moving morphable components (MMC) method [Guo et al. (2014); Zhang et al. (2016)]. ...

In this study, a hybrid method for density-related topology optimization is proposed, which consists of two parts: the discrete level-set method (LSM) based on solid isotropic material with penalization (SIMP) and the structural boundary extraction method based on support vector machine (SVM). In the hybrid method, the SIMP method is implemented to create new holes which are inserted into the topological structure obtained by the discrete LSM. SVM is utilized for extracting the boundary of the structure obtained by the SIMP-based discrete LSM. Based on the clear boundary extracted by SVM, a smooth boundary can be further obtained after data filtering. Four numerical examples are used to test the advantages of the hybrid method.

... Stress is an important design criterion in practical engineering applications. As a challenging problem, stress-constrained topology optimization has been extensively studied in recent years [De Leon et al. (2015); Collet et al. (2016); Chu et al. (2017a); Liu et al. (2017)]. The common approaches to deal with stress-constrained topology optimization problems can be classified into three categories: the local method, the global method and the regional or block aggregation technique [Deaton and Grandhi (2014)]. ...

This paper focuses on two kinds of bi-objective topology optimization problems with uniform-stress constraints: compliance-volume minimization and local frequency response–volume minimization problems. An adaptive volume constraint (AVC) algorithm based on an improved bisection method is proposed. Using this algorithm, the bi-objective uniform-stress-constrained topology optimization problem is transformed into a single-objective topology optimization problem and a volume-decision problem. The parametric level set method based on the compactly supported radial basis functions is employed to solve the single-objective problem, in which a self-organized acceleration scheme based on shape derivative and topological sensitivity is proposed to adaptively adjust the derivative of the objective function and the step length during the optimization. To solve the volume-decision problem, an improved bisection method is proposed. Numerical examples are tested to illustrate the feasibility and effectiveness of the self-organized acceleration scheme and the AVC algorithm based on the improved bisection method. An extended application to the bi-objective stress-constrained topology optimization of a structure with stress concentration is also presented.

... Thore et al. [37] formulated a nonlinear SDP problem for topology optimization under load uncertainty with the objective and constraints being quadratic functions of the uncertain load-vector. Liu et al. [38] presented an effective algorithm by combining the interval mathematics and BESO method to solve the RTO problem with loading direction uncertainty. Wang et al. [39][40][41] further extended the convex models to practical reliability assessment and analysis of structures with various uncertainties. ...

The optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable, or even lead to catastrophic failures. A two-level optimization formulation is often used in the Robust Topology Optimization (RTO) under uncertain loads. Various approaches have been reported to identify the critical loads associated with the worst structure responses. Because Convex Model approaches apply convex approximations to the original non-convex model at the lower level, the optimal designs obtained by these methods are greatly dependent on the quality of the approximation. In this paper, a new formulation based on the Wolfe duality for the RTO problems with multiple independent unknown-but-bounded loads is proposed. Following the two-level formulation, the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the Wolfe duality. Both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non-convexity associated with the lower level formulation. Numerical examples are also presented to demonstrate the proposed approach.

... Meanwhile, incorporation of loading uncertainty into RTO framework has recently become a strong focus. Efficient RTO algorithms using the analytic sensitivity information in the presence of uncertainties in loading magnitude were proposed (Dunning et al. 2011; Dunning and Kim 2013).Liu et al. (2015Liu et al. ( , 2017a) developed an efficient algorithm to solve RTO design problems considering interval uncertainties in loading directions.Chen et al. (2009)proposed a robust topology optimization method with consideration of random field uncertainty in loading, in which the Karhunen–Loeve expansion was employed to characterize the high-dimensional random-field uncertainty with a reduced set of random variables. More recently, Papadimitriou andPapadimitriou (2016)solved RTO design problems considering loading uncertainties by the use of a continuous adjoint method.Thore et al. (2015)presented a deterministic robust formulation of topology optimization for maximum stiffness design which accounts for the variations around a set of nominal loads; interval variables were used to describe the uncertain loads.Csébfalvi (2017)investigated 2D/3D RTO design problems by employing convex models to manage the uncertain loading. ...

Few researches have paid attention to designing structures in consideration of the uncertainties in
the loading locations, which may significantly influence the structural performances. In this work,
the cloud models are employed to depict the uncertainties in the loading locations. A robust
algorithm is developed in 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 the low efficiency. An innovative strategy is then
implemented to enormously improve the computational efficiency. A modified soft-kill
Bi-directional Evolutionary Structural Optimization (BESO) method using the derived sensitivity
numbers is employed to output the robust novel configurations. Several numerical examples are
presented to demonstrate the effectiveness and the efficiency of the proposed algorithm.

We propose a time-variant reliability analysis framework to quantitatively predict the lifetime of the lattice structures fabricated by selective laser melting (SLM), including confirming hybrid uncertainties, establishing a hybrid model, and proposing an efficient time-variant reliability method. We first design and manufacture a representative and complex L-shaped body-centred cubic (BCC) lattice structure utilising the SLM method, followed by morphology and microstructure observations to indicate the necessity of accounting for material uncertainty. Further considering loading fluctuation, we develop an effective time-variant reliability analysis method utilising the mixed probability and convex set model. One benchmark numerical example has been employed to shed a light on the high computational efficiency and acceptable computational accuracy of the developed time-variant reliability method. Finally, the proposed framework is performed to a real L-shaped BCC lattice structure to predict its lifetime, finding that the failure probability after ten years can reach more than 40 times the initial design.

The effects of different design sensitivity schemes for an appropriate incorporation of the loading position uncertainty into a gradient-based topology optimization procedure are explored intensively, so that the robust optimal design of a continuum structure can be efficiently performed over the prescribed uncertain region. Three compliance sensitivity schemes are tested and the distinctive strategy offering the most reliable and competitive material layout of the load-bearing components is obtained from comprehensive comparisons. Three benchmark examples are used to demonstrate the diverse configurations of the material layouts. The numerical results show that the scheme with the largest absolute design sensitivity can most adequately integrate the load uncertainties into the topology optimization algorithm to make the structural performance insensitive to, or have less degradation in the presence of perturbations in the loading points.

This study presents a novel topology optimization method for the robust design of structures and material microstructures. Uncertainties are usually ubiquitous and of different sources, and especially hybrid uncertainties widely exist in structural designs including external loads and material properties. Firstly, an orthogonal decomposition and uniform sampling (ODUS) method will be proposed to avoid the time-consuming double loops, in terms of load uncertainties described by upper and lower bounds. Secondly, a non-intrusive polynomial chaos expansion (NIPCE) is implicitly implemented, in terms of base material uncertainties subjected to Gaussian distributions. In the optimization formulation, the robust objective function is defined according to both the expectation and standard variation of structural compliance, and the sensitivity information with respect to the two-scale design variables are given in detail. Finally, an effective evolutionary method is employed to iteratively find the optimal topologies of the design. In addition, this study also defines a dimensionless index to evaluate the robustness of deterministic and robust designs. Three numerical examples are provided to demonstrate the efficiency of the proposed method, and 3D design results are fabricated by using appropriate additive manufacturing techniques.

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 optimality criteria 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 material 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.

The available robust and reliable topology optimization methods provide quick and efficient design output in an uncertain environment. However, the whole domain of performance function remains hidden during this design process. In the interest of the designer, it is required to know the overall behavior of performance functions in deterministic as well as uncertain/realistic environment. The current work achieves this by proposing an integrated methodology, which combines the design of experiments approach and reliability-based topology optimization. The proposed method enables the designer to simulate performance functions in a desired design-factors space, including uncertainties, via reliability value. For this analysis, compliance, maximum deflection, mechanical advantage, and von Mises stress values are selected as performance functions. Volume fraction, applied force, and dimensions or aspect ratio are chosen as design/control factors. The uncertainties of these design factors are captured using reliability-based topology optimization. The uncertainties due to noncontrollable factors such as material property, load direction, and magnitude are incorporated using the design of experiments approach. Under these uncertainties, the performance of topologically optimized problem is simulated for different experimental combinations of the design factors. The experimental combinations for uncertainties and design factors are generated using Taguchi's orthogonal array. Simulated results are analyzed using techniques such as analysis of mean and variance, signal-to-noise ratio, and response surface method. These analyses help in identifying statistical significance of factors and uncertainties, performance variations, and equivalence relation of performance vs. factor. The proposed methodology is illustrated by selecting monolithic structures such as, on MBB, cantilever beam, and force inverter mechanism.

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 non‐probabilistic 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.

Purpose
The purpose of this paper is to propose a gauge for the convergence of the deterministic particle swarm optimization (PSO) algorithm to obtain an optimum upper bound for PSO algorithm and also developing a precise equation for predicting the rock fragmentation, as important aims in surface mines.
Design/methodology/approach
In this study, a database including 80 sets of data was collected from 80 blasting events in Shur river dam region, in Iran. The values of maximum charge per delay (W), burden (B), spacing (S), stemming (ST), powder factor (PF), rock mass rating (RMR) and D80, as a standard for evaluating the fragmentation, were measured. To check the performance of the proposed PSO models, artificial neural network was also developed. Accuracy of the developed models was evaluated using several statistical evaluation criteria, such as variance account for, R-square ( R ² ) and root mean square error.
Findings
Finding the upper bounds for the difference between the position and the best position of particles in PSO algorithm and also developing a precise equation for predicting the rock fragmentation, as important aims in surface mines.
Originality/value
For the first time, the convergence of the deterministic PSO is studied in this study without using the stagnation or the weak chaotic assumption. The authors also studied application of PSO inpredicting rock fragmentation.

When geometric uncertainties arising from manufacturing errors are comparable to the characteristic length or the product responses are sensitive to such uncertainties, the products of deterministic design cannot perform robustly. This paper presents a new level set-based framework for robust shape and topology optimization against geometric uncertainties. We first propose a stochastic level set perturbation model of uncertain topology/shape to characterize manufacturing errors in conjunction with Karhunen–Loève (K–L) expansion. We then utilize polynomial chaos expansion (PCE) to implement the stochastic response analysis. In this context, the mathematical formulation of the considered robust shape and topology optimization problem is developed, and the adjoint-variable shape sensitivity scheme is derived. An advantage of this method is that relatively large shape variations and even topological changes can be accounted for with desired accuracy and efficiency. Numerical examples are given to demonstrate the validity of the present formulation and numerical techniques. In particular, this method is justified by the observations in minimum compliance problems, where slender bars vanish when the manufacturing errors become comparable to the characteristic length of the structures. This article is protected by copyright. All rights reserved.

This contribution presents a computationally efficient method for reliability-based topology optimization for continuum domains under material properties uncertainty. Material Young’s modulus is assumed to be lognormally distributed and correlated within the domain. The computational efficiency is achieved through estimating the response statistics with stochastic perturbation of second order, using these statistics to fit an appropriate distribution that follows the empirical distribution of the response, and employing an efficient gradient-based optimizer. Two widely-studied topology optimization problems are examined and the changes in the optimized topology is discussed for various levels of target reliability and correlation strength. Accuracy of the proposed algorithm is verified using Monte Carlo simulation.

In this paper, we focus on incorporating a stochastic collocation method (SCM) into a topological shape optimization of a power semiconductor device, including material and geometrical uncertainties. This results in a stochastic direct problem and, in consequence, affects the formulation of an optimization problem. In particular, our aim is to minimize the current density overshoots, since the change of the shape and topology of a device layout is the proven technique for the reduction of a hotspot area. The gradient of a stochastic cost functional is evaluated using the topological asymptotic expansion and the continuous design sensitivity analysis with the SCM. Finally, we show the results of the robust optimization for the power transistor device, which is an example of a relevant problem in nanoelectronics, but which is also widely used in the automotive industry.

This paper reports an efficient approach for uncertain topology optimization in which the uncertain optimization problem is equivalent to that of solving a deterministic topology optimization problem with multiple load cases. Probabilistic and fuzzy property of the directional uncertainty of the applied loads is considered in the topology optimization; the cloud model is employed to describe that property which can also take the correlations of the probability and fuzziness into account. Convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) algorithms are utilized to obtain the final optimal solution. The proposed method is suitable for linear elastic problems with uncertain applied loads, subject to volume constraint. Several numerical examples are presented to demonstrate the capability and effectiveness of the proposed approach. In-depth discussions are also given on the effects of considering the probability and fuzziness of the directions of the applied loads on the final layout.

In this paper an approach to robust topology optimization for truss structures with material and loading uncertainties, and discrete design variables, is investigated. Uncertainties on the loading, and spatially correlated material stiffness, are included in the problem formulation, taking truss element length into account. A more realistic random field representation of the material uncertainties is achieved, compared to classical scalar random variable approaches. A multiobjective approach is used to generate Pareto optimal solutions showing how the mean and standard deviation of the compliance can be considered as separate objectives, avoiding the need for an arbitrary combination factor.

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.

This paper presents a 100-line Python code for general 3D topology optimization. The code adopts the Abaqus Scripting Interface that provides convenient access to advanced finite element analysis (FEA). It is developed for the compliance minimization with a volume constraint using the Bi-directional Evolutionary Structural Optimization (BESO) method. The source code is composed of a main program controlling the iterative procedure and five independent functions realising input model preparation, FEA, mesh-independent filter and BESO algorithm. The code reads the initial design from a model database (.cae file) that can be of arbitrary 3D geometries generated in Abaqus/CAE or converted from various widely used CAD modelling packages. This well-structured code can be conveniently extended to various other topology optimization problems. As examples of easy modifications to the code, extensions to multiple load cases and nonlinearities are presented. This code is intended for educational purposes and would be useful for researchers and students in the topology optimization field. With further extensions, the code could solve sophisticated 3D conceptual design problems in structural engineering, mechanical engineering and architecture practice. The complete code is given in the appendix section and can also be downloaded from the website: www.rmit.edu.au/research/cism/.

This paper examines the evolutionary structural optimisation (ESO) method and its shortcomings. By proposing a problem statement for ESO followed by an accurate sensitivity analysis a framework is presented in which ESO is mathematically justifiable. It is shown that when using a sufficiently accurate sensitivity analysis, ESO method is not prone to the problem proposed by Zhou and Rozvany (Struct Multidiscip Optim 21(1):80–83, 2001). A complementary discussion on previous communications about ESO and strategies to overcome the Zhou-Rozvany problem is also presented. Finally it is shown that even the proposed rigorous ESO approach can result in highly inefficient local optima. It is discussed that the reason behind this shortcoming is ESO’s inherent unidirectional approach. It is thus concluded that the ESO method should only be used on a very limited class of optimisation problems where the problem’s constraints demand a unidirectional approach to final solutions. It is also discussed that the Bidirectional ESO (BESO) method is not prone to this shortcoming and it is suggested that the two methods be considered as completely separate optimisation techniques.

In level set methods for structural topology and shape optimization, the level set function gradients at the design interface need to be controlled in order to ensure stability of the optimization process. One popular way to do this is to enforce the level set function to be a signed distance function by periodically using initialization schemes, which is commonly known as re-initialization. However, such re-initialization schemes are time-consuming, as additional partial differential equations need to be solved in every iteration step. Furthermore, the use of re-initialization brings some undesirable problems; for example, it may move the zero level set away from the expected position. This paper presents a level set method with distance-suppression scheme for structural topology and shape optimization. An energy functional is introduced into the level set equation to maintain the level set function to close to a signed distance function near the structural boundaries, meanwhile forcing the level set function to be a constant at locations far away from the structural boundaries. As a result, the present method not only can avoid the need for re-initialization but also can simplify the setting of the initial level set function. The validity of the proposed method is tested on the mean compliance minimization problem and the compliant mechanisms synthesis problem. Different aspects of the proposed method are demonstrated on a number of benchmarks from the literature of structural optimization.

IntroductionProblem Statement and Material Interpolation SchemeSensitivity Analysis and Sensitivity NumberExamplesConclusion
Appendix 4.1References

Robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies. This paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude, using exact formulations and analytically derived sensitivities. It shows that only a few additional load cases are required, which scales in polynomial time with the number of uncertain load cases. The approach is implemented using the level set topology optimization method. Shape sensitivities are derived using the adjoint method. Several examples are used to investigate the effect of including variance in robust compliance optimization. Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Describes development work to combine the basic ESO with the additive evolutionary structural optimisation (AESO) to produce bidirectional ESO whereby material can be added and can be removed. It will be shown that this provides the same results as the traditional ESO. This has two benefits, it validates the whole ESO concept and there is a significant time saving since the structure grows from a small initial one rather than contracting from a sometimes huge initial one where 90 per cent of the material gets removed over many hundreds of finite element analysis (FEA) evolutionary cycles. Presents a brief background to the current state of Structural Optimisation research. This is followed by a discussion of the strategies for the bidirectional ESO (BESO) algorithm and two examples are presented.

The objective of this work is to integrate reliability analysis into topology optimization problems. The new model, in which we introduce reliability constraints into a deterministic topology optimization formulation, is called Reliability-Based Topology Optimization (RBTO). Several applications show the importance of this integration. The application of the RBTO model gives a different topology relative to deterministic topology optimization. We also find that the RBTO model yields structures that are more reliable than those produced by deterministic topology optimization (for the same weight).

This research explores the usage of classification approaches in order to facilitate the accurate estimation of probabilistic
constraints in optimization problems under uncertainty. The efficiency of the proposed framework is achieved with the combination
of a conventional topology optimization method and a classification approach- namely, probabilistic neural networks (PNN).
Specifically, the implemented framework using PNN is useful in the case of highly nonlinear or disjoint failure domain problems.
The effectiveness of the proposed framework is demonstrated with three examples. The first example deals with the estimation
of the limit state function in the case of disjoint failure domains. The second example shows the efficacy of the proposed
method in the design of stiffest structure through the topology optimization process with the consideration of random field
inputs and disjoint failure phenomenon, such as buckling. The third example demonstrates the applicability of the proposed
method in a practical engineering problem.
KeywordsReliability-based topology optimization–Probabilistic neural network–Uncertainty–Classification approach

Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization
of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop
optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats
the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints.
For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability
constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint
variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The
structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present
formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions
can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present
problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic
reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches.
The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization.

A robust shape and topology optimization (RSTO) approach with consideration of random field uncertainty in loading and material
properties is developed in this work. The proposed approach integrates the state-of-the-art level set methods for shape and
topology optimization and the latest research development in design under uncertainty. To characterize the high-dimensional
random-field uncertainty with a reduced set of random variables, the Karhunen–Loeve expansion is employed. The univariate
dimension-reduction (UDR) method combined with Gauss-type quadrature sampling is then employed for calculating statistical
moments of the design response. The combination of the above techniques greatly reduces the computational cost in evaluating
the statistical moments and enables a semi-analytical approach that evaluates the shape sensitivity of the statistical moments
using shape sensitivity at each quadrature node. The applications of our approach to structure and compliant mechanism designs
show that the proposed RSTO method can lead to designs with completely different topologies and superior robustness.
KeywordsRobust design-Topology optimization-Shape optimization-Level set methods-Uncertainty-Random field-Dimension reduction

This paper presents an improved algorithm for the bi-directional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A mesh-independency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. To further enhance the convergence of the optimization process, the accuracy of elemental sensitivity numbers is improved by its historical information. The new approach is demonstrated by solving several compliance minimization problems and compared with the solid isotropic material with penalization (SIMP) method. Results show the effectiveness of the new BESO method in obtaining convergent and mesh-independent solutions.

The resulting optimal configuration, obtained by deterministic topology optimization, may represent a lower reliability level, and then lead to a higher failure rate. Therefore it is necessary to take into account reliability for topology optimization. By integrating reliability analysis into topology optimization problems, a simple reliability-based topology optimization methodology for continuum structures is investigated in this paper. The two-layer nesting involved in RBTO, which is time-consuming, is decoupled by the use of a particular optimization procedure. A topology description function approach (TOTDF) and a first order reliability method are employed for topology optimization and reliability calculation, respectively. The problem of the non-smoothness inherent in TOTDF is dealt with by using two different smoothed Heaviside functions and the corresponding topologies are compared. Numerical examples demonstrate the validity and efficiency of the proposed improved method. In-depth discussions are also given on the influence of different structural reliability index on the final layout.

This article presents an efficient approach for reliability-based topology optimization (RBTO) in which the computational effort involved in solving the RBTO problem is equivalent to that of solving a deterministic topology optimization (DTO) problem. The methodology presented is built upon the bidirectional evolutionary structural optimization (BESO) method used for solving the deterministic optimization problem. The proposed method is suitable for linear elastic problems with independent and normally distributed loads, subjected to deflection and reliability constraints. The linear relationship between the deflection and stiffness matrices along with the principle of superposition are exploited to handle reliability constraints to develop an efficient algorithm for solving RBTO problems. Four example problems with various random variables and single or multiple applied loads are presented to demonstrate the applicability of the proposed approach in solving RBTO problems. The major contribution of this article comes from the improved efficiency of the proposed algorithm when measured in terms of the computational effort involved in the finite element analysis runs required to compute the optimum solution. For the examples presented with a single applied load, it is shown that the CPU time required in computing the optimum solution for the RBTO problem is 15-30% less than the time required to solve the DTO problems. The improved computational efficiency allows for incorporation of reliability considerations in topology optimization without an increase in the computational time needed to solve the DTO problem.

Many structures in the real world show nonlinear responses. The nonlinearity may be due to some reasons, such as nonlinear material (material nonlinearity), large deformation of the structures (geometric nonlinearity), or contact between the parts (contact nonlinearity). Conventional optimization algorithms considering the nonlinearities are fairly difficult and expensive because many nonlinear analyses are required. It is quite difficult to perform topology optimization considering nonlinear static behavior because of the many design variables. In the current element density based topology optimization considering nonlinear behavior, low-density finite elements cause serious numerical problems due to excessive mesh distortion. Updating the material of the finite elements based on the density is considerably complicated because of the relationship between the element density and structural material. The equivalent static loads method for nonlinear static response structural optimization (ESLSO) has been proposed for size and shape optimization. The equivalent static loads (ESLs) are defined as the linear static load sets which generate the same displacement field from nonlinear static analysis. In this research, a new algorithm is proposed for topology optimization considering all kinds of nonlinearities by modifying the existing ESLSO. The new ESLSO can overcome the difficulties which may occur in topology optimization with nonlinear static behavior. A nonlinear static response optimization problem is converted to cyclic use of linear static response optimization with ESLs. Therefore, the new ESLSO can generate results of nonlinear static response topology optimization by using well established nonlinear static analysis and linear static response topology optimization methods. Four structural examples are demonstrated using the finite element method. Different kinds of nonlinearities are involved in each example.

Topology optimization often leads to structures consisting of slender elements which are particularly sensitive to geometric imperfections. Such imperfections might affect the structural stability and induce large displacement effects in these slender structures. This paper therefore presents a robust approach to topology optimization which accounts for geometric imperfections and their potentially detrimental influence on the structural stability. Geometric nonlinear effects are incorporated in the optimization by means of a Total Lagrangian finite element formulation in the minimization of end-compliance. Geometric imperfections are modeled as a vector-valued random field in the design domain. The resulting uncertain performance of the design is taken into account by minimizing a weighted sum of the mean and standard deviation of the compliance in the robust optimization problem. These stochastic moments are typically estimated by means of sampling methods such as Monte Carlo simulation. However, these methods require multiple independent nonlinear finite element analyses in each design iteration of the optimization algorithm. An efficient solution algorithm which uses adjoint differentiation in a second-order perturbation method is therefore developed to estimate the stochastic moments during the optimization. Two applications with structures that exhibit different types of structural instabilities are examined. In both cases, it is demonstrated by means of an extensive Monte Carlo simulation that the deterministic design is very sensitive to imperfections, while the design obtained by means of the proposed method is much more robust.

Most topology optimization techniques find the optimal layout of a structure under static loads. Some studies are focused on dynamic response topology optimization because dynamic forces act in the real world. Dynamic response topology optimization is solved in the time or frequency domain. A method for dynamic response topology optimization in the time domain is proposed using equivalent static loads. Equivalent static loads are static loads that generate the same displacement field as dynamic loads at each time step. The equivalent static loads are made by multiplying the linear stiffness matrix and the displacement field from dynamic analysis and used as multiple loading conditions for linear static topology optimization. The results of topology optimization are again used in dynamic analysis and a cyclic process is used until the convergence criterion is satisfied. The paradigm of the method was originally developed for size and shape optimizations. A new objective function is defined to minimize the peaks of the compliance in the time domain and a convergence criterion is newly defined considering that there are many design variables in topology optimization. The developed method is verified by solving some examples and the results are discussed.

A computational method for topology optimization in the presence of uncertainty is proposed. The method combines the spectral stochastic approach for the representation and propagation of uncertainties with an existing deterministic topology optimization technique. The idea in spectral stochastic formulations is to add an extra dimension, random dimension, to the problem where the stochastic variability of the input parameters (and outputs of interest) can be modeled. The resulting compact representations for the response quantities allow for efficient and accurate calculation of sensitivities of response statistics with respect to the design variables which are then fed into a gradient-based optimizer that searches for the optimum design. From a variety of frameworks for uncertainty-informed optimization, robust topology optimization is chosen to demonstrate the applicability of the method. Examples from continuum topology optimization under uncertainty in material properties are presented. It is also shown that results obtained from the proposed method are in excellent agreement with those obtained from a Monte Carlo-based optimization algorithm.

This paper describes a non-probabilistic reliability-based topology optimization method for the design of continuum structures undergoing large deformations. The variation of the structural system is treated with the multi-ellipsoid convex model, which is a realistic description of the parameters being inherently uncertain-but-bounded or lacking sufficient probabilistic data. The formulation of the optimal design is established as a volume minimization problem with non-probabilistic reliability constraints on the geometrically nonlinear structural behaviour. In order to circumvent numerical difficulties in solving the nested double-loop optimization problem, a performance measure-based approach is employed to transform the constraint on the reliability index into one on the concerned performance. In conjunction with an efficient adjoint variable scheme for the sensitivity analysis of reliability constraints, the optimization problem is solved by gradient-based mathematical programming methods. Three numerical examples for the optimization design of planar structures are presented to illustrate the validity and applicability of the proposed method. The obtained optimal solutions show the importance of incorporating various uncertainties in the design problem. Moreover, it is also revealed that the geometrical nonlinearity needs to be accounted for to ensure satisfaction of the reliability constraints in the optimal design of structures with large deformation.

Recently, an algorithm for dynamic response optimization transforming dynamic loads into equivalent static loads has been proposed. In later research, it was proved that the solution obtained by the algorithm satisfies the Karush-Kuhn-Tiucker necessary conditions. In the present research, the proposed algorithm is applied to the optimization of flexible multibody dynamic systems. The equivalent static load is derived from the equations of motion for a flexible multibody dynamic system. The equivalent load is utilized in sequential static response optimization of the flexible mutibody dynamic system. In the end, the converged solution of the sequential static response optimization is the solution of the original dynamic response optimization. Some standard examples are solved to show the feasibility and efficiency of the proposed method. The control arm of an automobile suspension system is optimized as a practical problem. The results are discussed regarding the application of the proposed algorithm to flexible multibody dynamic systems.

This paper is concerned with the comparison of two non-probabilistic set-theoretical models for dynamic response measures of an infinitely long beam. The beam is on an uncertain foundation and subjected to a moving force with constant speed. The steady state vibration is analyzed with finite element method. The dynamic responses of the beam are approximated to the first-order respect of the uncertainty variables. As a rule, in convex models and interval analysis, the uncertainties are considered to be unknown, but they give out their allowable vector space. Comparing the convex models with interval analysis in mathematical proofs and numerical calculations, it’s shows that under the condition of transform an interval vector to an outer enclosed ellipsoid, the dynamic response of the infinitely long beam predicted by interval analysis is smaller than that by convex models; under the condition of transform a hyperellipsoid to an outer enclosed interval vector, the dynamic response of the infinitely long beam calculated by convex models is smaller than that by interval analysis method.

A simple evolutionary procedure is proposed for shape and layout optimization of structures. During the evolution process low stressed material is progressively eliminated from the structure. Various examples are presented to illustrate the optimum structural shapes and layouts achieved by such a procedure.

This paper presents a robust approach for the design of macro-, micro-, or nano-structures by means of topology optimization, accounting for spatially varying manufacturing errors. The focus is on structures produced by milling or etching; in this case over- or under-etching may cause parts of the structure to become thinner or thicker than intended. This type of error is modeled by means of a projection technique: a density filter is applied, followed by a Heaviside projection, using a low projection threshold to simulate under-etching and a high projection threshold to simulate over-etching. In order to simulate the spatial variation of the manufacturing error, the projection threshold is represented by a (non-Gaussian) random field. The random field is obtained as a memoryless transformation of an underlying Gaussian field, which is discretized by means of an EOLE expansion. The robust optimization problem is formulated in a probabilistic way: the objective function is defined as a weighted sum of the mean value and the standard deviation of the structural performance. The optimization problem is solved by means of a Monte Carlo method: in each iteration of the optimization scheme, a Monte Carlo simulation is performed, considering 100 random realizations of the manufacturing error. A more thorough Monte Carlo simulation with 10000 realizations is performed to verify the results obtained for the final design. The proposed methodology is successfully applied to two test problems: the design of a compliant mechanism and a heat conduction problem.

The aim of this paper was to present a topology optimization methodology for obtaining robust designs insensitive to small uncertainties in the geometry. The variations are modeled using a stochastic field. The model can represent spatially varying geometry imperfections in devices produced by etching techniques. Because of under-etching or over-etching parts of the structure may become thinner or thicker than a reference design supplied to the manufacturer. The uncertainties are assumed to be small and their influence on the system response is evaluated using perturbation techniques. Under the above assumptions, the proposed algorithm provides a computationally cheap alternative to previously introduced stochastic optimization methods based on Monte Carlo sampling. The method is demonstrated on the design of a minimum compliance cantilever beam and a compliant mechanism.

The structural optimization presented in this paper is based on an
evolutionary procedure, developed recently, in which the low stressed part of
a structure is removed from the structure step-by-step until an
optimal design is obtained. Various tests have shown the effectiveness of this
evolutionary procedure. The purpose of this paper is to present applications
of such an evolutionary procedure to the optimal design of structures with
multiple load cases or with a traffic (moving) load.

A computational strategy is proposed for robust structural topology optimization in the presence of uncertainties with known second order statistics. The strategy combines deterministic topology optimization techniques with a perturbation method for the quantification of uncertainties associated with structural stiffness, such as uncertain material properties and/or structure geometry. The use of perturbation transforms the problem of topology optimization under uncertainty to an augmented deterministic topology optimization problem. This in turn leads to significant computational savings when compared with Monte Carlo-based optimization algorithms which involve multiple formations and inversions of the global stiffness matrix. Examples from truss structures are presented to show the importance of including the effect of controlling the variability in the final design. It is also shown that results obtained from the proposed method are in excellent agreement with those obtained from a Monte Carlo-based optimization algorithm.

A reliability-based design optimization method is developed to apply to topology design problems. Using the total Lagrangian formulation, the spatial domain is discretized using Mindlin plate elements with the von Kármán strain–displacement relation. The topology optimization problem is reformulated as a volume minimization problem having probabilistic displacement constraints using the performance measure approach. For the efficient computation of the sensitivity with respect to the design and random variables, an adjoint variable method for geometrically nonlinear structures is employed. Since the converged tangent stiffness is available from the response analysis, the computing cost for the sensitivity analysis is trivial. The uncertainties such as material property and external loads are considered. Numerical results show that the developed sensitivity analysis method is very efficient and the topology optimization method effectively yields reliable designs.

This paper presents algorithms for solving structural topology optimization problems with uncertainty in the magnitude and location of the applied loads and with small uncertainty in the location of the structural nodes. The second type of uncertainty would typically arise from fabrication errors where the tolerances for the node locations are small in relation to the length scale of the structural elements. We first review the discrete form of the uncertain loads problem, which has been previously solved using a weighted average of multiple load patterns. With minor modifications, we extend this solution to include loads described by continuous joint probability density functions. We then proceed to the main contribution of this paper: structural optimization under uncertainty in the nodal locations. This optimization problem is computationally difficult because it involves variations of the inverse of the structural stiffness matrix. It is shown, however, that for small uncertainties the problem can be recast into a simpler but equivalent structural optimization problem with equivalent uncertain loads. By expressing these equivalent loads in terms of continuous random variables, we are able to make use of the extended form of the uncertain loads problem presented in the first part of this paper. The optimization algorithms are developed in the context of minimum compliance (maximum stiffness) design. Simple examples are presented. The results demonstrate that load and nodal uncertainties can have dramatic impact on optimal design. For structures containing thin substructures under axial loads, it is shown that these uncertainties (a) are of first-order significance, influencing the linear elastic response quantities, and (b) can affect designs by avoiding unrealistically optimistic and potentially unstable structures. The additional computational cost associated with the uncertainties scales linearly with the number of uncertainties and is insignificant compared to the cost associated with solving the deterministic structural optimization problem.

This paper presents a methodology for the design of micro-electro-mechanical systems (MEMS) by topology optimization accounting for stochastic loading and boundary conditions as well as material properties. This methodology combines recent advances in material-based topology optimization for compliant mechanisms undergoing large displacements and design optimization under uncertainties using first order reliability analysis methods. The performance measure approach is applied to the formulation of the optimization problem. The structural response is predicted by a co-rotational finite element formulation and the design and imperfection sensitivities are evaluated by an adjoint method. The methodology is illustrated by the topology optimization of a compliant mechanism. The results show the importance of accounting for the stochastic nature of the micro-system in the topology optimization process.

This paper presents a new approach to structural topology optimization. We represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension. Such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure. Furthermore, a well-founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints. The result is a 3D topology optimization technique that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation. We have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods. The benefit and the advantages of the proposed method are illustrated with several 2D examples that are widely used in the recent literature of topology optimization, especially in the homogenization based methods.

This paper discusses the homogenization method to determine the effective average elastic constants of linear elasticity of general composite materials by considering their microstructure. After giving a brief theory of the homogenization method, a finite element approximation is introduced with convergence study and corresponding error estimate. Applying these, computer programs PREMAT and POSTMAT are developed for preprocessing and postprocessing of material characterization of composite materials. Using these programs, the homogenized elastic constants for macroscopic stress analysis are obtained for typical composite materials to show their capability. Finally, the adaptive finite element method is introduced to improve the accuracy of the finite element approximation.

Shape and topology optimization of a linearly elastic structure is discussed using a modification of the homogenization method introduced by Bendsoe and Kikuchi together with various examples which may justify validity and strength of the present approach for plane structures.

Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, isotropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

The ESO method revisited Structural

- K Ghabraie
- A C C E P T E D Multidisciplinary
- Optimization

Ghabraie, K. (2015). The ESO method revisited. Structural and Multidisciplinary
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Optimization, 51(6), 1211-1222.

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- J Int
- Comput

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