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Topology optimization of continuum structures with uncertain-but-bounded parameters for maximum non-probabilistic reliability of frequency requirement

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Abstract

A method for the non-probabilistic reliability optimization on frequency of continuum structures with uncertain-but-bounded parameters is proposed. The objective function is to maximize the non-probabilistic reliability index of frequency requirement.The corresponding bi-level optimization model is built, where the constraints are applied on the material volume in the outer loop and the limit state equation in the inner loop. The non-probabilistic reliability index of frequency requirement is derived by the analytical method for the continuum structure with the uncertain elastic module and mass density. Further, the sensitivity of the non-probabilistic reliability index with respect to the design variables is analyzed. The topology optimization in the outer loop is performed by a bi-directional evolutionary structural optimization (BESO) method, where the numerical techniques and the optimization procedure of BESO method are presented. Numerical results show that the proposed BESO method is efficient, and convergent optimal solutions can be achieved for a variety of optimization problems on frequency non-probabilistic reliability of continuum structures.

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... aircraft, helicopter, missile, and many others [1,2]. In general, engineers attempt to maximize the fundamental frequency or the gap between the first two natural frequencies of one structure [3][4][5][6][7][8][9][10][11][12], with the aid of continuum topology optimization methods [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], to avoid the resonance which is normally critically harmful to the real-life structure . ...
... aircraft, helicopter, missile, and many others [1,2]. In general, engineers attempt to maximize the fundamental frequency or the gap between the first two natural frequencies of one structure [3][4][5][6][7][8][9][10][11][12], with the aid of continuum topology optimization methods [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], to avoid the resonance which is normally critically harmful to the real-life structure . ...
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Structures and/or materials with engineered functionality, capable of achieving targeted mechanical responses reacting to changes in external excitation, have various potential engineering applications, e.g. aerospace, oceanographic engineering, soft robot, and several others. Yet tunable mechanical performance is normally realized through carefully designing the architecture of structures, which is usually porous, leading to the complexity of the fabrication of the structures even using the recently emerged 3D printing technique. In this study we show that origami technique can provide an alternative solution to achieving the aim by carefully stacking the classical Miura sheets into the Miura-ori tube metamaterial and tuning the geometric parameters of the origami metamaterial. By combining numerical and experimental methods, we have demonstrated that an extremely broad range of natural frequency and dynamic response of the metamaterial can be achieved. The proposed structure can be easily fabricated from a single thin sheet made of one material and simultaneously owns better mechanical properties than the Miura sheet.
... Topology optimization has become one of the important research topics [1][2][3][4][5][6][7] in the structural optimization community in recent years. A large effort has been done and numerous approaches have been presented [1][2][3][4][5][6][7], such as the homogenization approach [1,8], the solid isotropic microstructure with penalization (SIMP) approach [9,10], the level set approach [11,12], evolutionary approach [13][14][15][16], the moving morphable component (MMC) approach [17][18][19], and others. With these technical advancements, topology optimization has become an effective tool for obtaining a high-performance and innovative structure, and been successfully applied in civil, mechanical, aircraft and aerospace structures' design [7,[19][20][21][22][23]. ...
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Minimum length scale control on real and void material phases in topology optimization is an important topic of research with direct implications on numerical stability and solution manufacturability. And it also is a challenge area of research due to serious conflicts of both the solid and the void phase element densities in phase mixing domains of the topologies obtained by existing methods. Moreover, there is few work dealing with controlling distinct minimum feature length scales of real and void phase materials used in topology designs. A new method for solving the minimum length scale controlling problem of real and void material phases, is proposed. Firstly, we introduce two sets of coordinating design variable filters for these two material phases, and two distinct smooth Heaviside projection functions to destroy the serious conflicts in the existing methods (e.g. Guest Comput Methods Appl Mech Eng 199(14):123–135, 2009). Then, by introducing an adaptive weighted 2-norm aggregation constraint function, we construct a coordinating topology optimization model to ensure distinct minimum length scale controls of real and void phase materials for the minimum compliance problem. By adopting a varied volume constraint limit scheme, this coordinating topology optimization model is transferred into a series of coordinating topology optimization sub-models so that the structural topology configuration can stably and smoothly changes during an optimization process. The structural topology optimization sub-models are solved by the method of moving asymptotes (MMA). Then, the proposed method is extended to the compliant mechanism design problem. Numerical examples are given to demonstrate that the proposed method is effective and can obtain a good 0/1 distribution final topology.
... Kang and Luo [28,29] firstly introduced the convex model into NRBTO, and recently attracted much attention in multidimensional convex model and non-probabilistic analysis under material property and external load uncertainties [30,31]. Moreover, Xu et al. [32] developed non-probabilistic reliability optimization on the frequency of continuum structures with uncertain-but-bounded parameters considering elastic modulus and mass density; and Wang et al. [33] presented a novel measuring index for NRBTO considering the unknown but bounded interval uncertainties existing in material and external loads simultaneously. Currently, Yin et al. [34] addressed an NRBTO approach for the uncertain structural design with fuzzy uncertainties in material properties and loading conditions. ...
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It is essential to consider the effects of incomplete measurement, inaccurate information and inadequate cognition on structural topology optimization. For the multi-material structural topology optimization with non-probability uncertainty, the multi-material interpolation model is represented by the ordered rational approximation of material properties (ordered RAMP). Combined with structural compliance minimization, the multi-material topology optimization with reliability constraints is established. The corresponding non-probability uncertainties are described by the evidence theory, and the uniformity processing method is introduced to convert the evidence variables into random variables. The first-order reliability method is employed to search the most probable point under the reliability index constraint, and then the random variables are equivalent to the deterministic variables according to the geometric meaning of the reliability index and sensitivity information. Therefore, the non-probabilistic reliability-based multi-material topology optimization is transformed into the conventional deterministic optimization format, followed by the ordered RAMP method to solve the optimization problem. Finally, through numerical examples of 2D and 3D structures, the feasibility and effectiveness of the proposed method are verified to consider the geometrical dimensions and external loading uncertainties.
... Later, Liu et al. proposed a methodology for designing porous fibrous material with optimal sound absorption under set frequency bands (Liu et al. 2014). Xu et al. presented a bi-level optimization methodology for the non-probabilistic reliability optimization on frequency of continuum structures with uncertain-but-bounded parameters (Xu et al. 2017). Moreover, another classic criterion of dynamic topology optimization problems, such as minimization of the dynamic compliance or dynamic displacements, has been considered by many researchers. ...
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This paper proposes a methodology for maximizing dynamic stress response reliability of continuum structures involving multi-phase materials by using a bi-directional evolutionary structural optimization (BESO) method. The topology optimization model is built based on a material interpolation scheme with multiple materials. The objective function is to maximize the dynamic stress response reliability index subject to volume constraints on multi-phase materials. To solve the defined topology optimization problems, the sensitivity of the dynamic stress response reliability index with respect to the design variables is derived for iteratively updating the structural topology. Subsequently, an optimization procedure based on the BESO method is developed. Finally, a series of numerical examples of both 2D and 3D structures are presented to demonstrate the effectiveness of the proposed approach.
... 35 As an alternative uncertain analysis method, the interval method has since been introduced in topology optimization for years. For example, on the basis of ellip- soid convex models, interval uncertainties were considered in topology optimization for continuum, 36 dynamic, 37,38 and multiscale 39 structural design. In the work of Wu et al, 40 a nonprobabilistic RTO approach was developed for structures under interval uncertainty by using the Chebyshev interval method. ...
Article
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This paper will develop a new robust topology optimization method based on level sets for structures subject to hybrid uncertainties, with a more efficient Karhunen‐Loeve hyperbolic Polynomial Chaos‐Chebyshev Interval (KL‐hPCCI) method to conduct the hybrid uncertain analysis. The loadings and the material properties are considered as hybrid uncertainties in structures. The parameters with sufficient information are regarded as random fields, while the parameters without sufficient information are treated as intervals. The KL expansion is applied to discretise random fields into a finite number of random variables, and then the original hybrid uncertainty analysis is transformed to a new process with random and interval parameters, to which the hPCCI is employed for the uncertainty analysis. The robust topology optimization is formulated to minimize a weighted sum of the mean and standard variance of the structural objective function under the worst‐case scenario. Several numerical examples are employed to demonstrate the effectiveness of the proposed robust topology optimization, and the Monte Carlo Simulation (MCS) is used to validate the numerical accuracy of our proposed method.
... [51][52][53][54] Interval uncertainties have also been considered in structural topology optimization problems, such as the ellipsoid-based models. [55][56][57] For example, Guo et al 58 investigated the robust concurrent topology optimization with load uncertainties by the ellipsoid model. Recently, there have also been a few works for topology optimization considering hybrid uncertainties. ...
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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 interval mean and interval variance of the corresponding objective function. A new uncertain propagation approach, termed as a hybrid univariate dimension reduction (HUDR) 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.
... In some circumstances, it may be difficult to acquire the precise probability distributions when samples are insufficient, however, in general we can easily get the variation intervals [9,10] for uncertain parameters based on limited information and engineering experience. Therefore, Non-probabilistic RBTO (NRBTO) can be regarded as an attractive alternative and can provide worthy information for designers [11,12]. ...
... For instance, Luo et al. (2009b) proposed a non-probabilistic reliability-based topology optimization method for design of structures, based on the definition of a non-probabilistic reliability index. Xu et al. (2015) developed a method for the nonprobabilistic reliability topology optimization of dynamic structures with uncertain-but-bounded parameters using a bi-directional evolutionary structural optimization (BESO) method. ...
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... Recently, Jiang et al. [31] presented a multidimensional parallelepiped model to combine the commonly-used interval and ellipsoidal convex models in a unified form. Xu et al. [32] established a bi-level non-probabilistic model to optimize the required frequency of continuum structures with uncertain-but-bounded parameters. Kang et al. [33][34][35] developed a multi-ellipsoid convex model and introduced the affine invariance for reliability analysis. ...
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... Bendsøe and Kikuchi presented a method which makes the optimal shape design as the material distribution problem based on the theory of homogenization (Bendsøe and Kikuchi, 1988;Bendsøe, M. P. 1989). In addition, there is another research branch such as incorporating uncertainties into structural topology optimization (Guest et al., 2008;Asadpoure et al., 2011;Chen et al., 2011;Jung et al., 2004;Schevenels et al., 2011;Xu et al., 2016;Xu et al., 2015). The method that we proposed in this article is to resist the structural local failure that may be caused by those uncertainties or possible structural fatigue. ...
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In the article, a new approach considering structural local failure for topology optimization of continuum structure is proposed. It aims at not only lowering the risk of local failure in the concerned structural regions, but also ensuring a good stiffness of the structure. The local failure may be caused by the structural uncertainties or possible structural fatigue. To this end, a criterion to evaluate the effect of one local failure on the structure is introduced. This criterion is minimized to reduce the probability of structural damage based on a initialized structure whose compliance is optimized. Solid Isotropic with Material Penalization (SIMP) method and Optimality Criteria (OC) method are combined to solve the design problem. The effectiveness of the proposed algorithm is verified by a series of numerical examples. Furthermore, experiments merging with additive manufacturing technique are taken to prove the practical ability of the method in actual engineering.
... Yoon [13] used the topology optimization based on the internal element connectivity parameterization method for nonlinear dynamic problems, where element instability is avoided and localized vibration modes is controlled. Xu et al. [14] presented a bi-level optimization methodology for the non-probabilistic reliability optimization on frequency of continuum structures with uncertain-but-bounded parameters. ...
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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.
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An efficient procedure for topology optimization of dynamics problems is proposed. The method is based on frequency responses represented by Padé approximants and analytical sensitivity analysis derived using the adjoint method. This gives an accurate approximation of the frequency response over wide frequency ranges and a formulation that allows for design sensitivities to be computed at low computational cost also for a large number of design variables. Two examples that deal with optimization of forced vibrations are included. Copyright © 2007 John Wiley & Sons, Ltd.
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In vibration optimization problems, eigenfrequencies are usually maximized in the optimization since resonance phenomena in a mechanical structure must be avoided, and maximizing eigenfrequencies can provide a high probability of dynamic stability. However, vibrating mechanical structures can provide additional useful dynamic functions or performance if desired eigenfrequencies and eigenmode shapes in the structures can be implemented. In this research, we propose a new topology optimization method for designing vibrating structures that targets desired eigenfrequencies and eigenmode shapes. Several numerical examples are presented to confirm that the method presented here can provide optimized vibrating structures applicable to the design of mechanical resonators and actuators. Copyright © 2006 John Wiley & Sons, Ltd.