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Topology optimization of finite similar periodic continuum structures based on a density exponent interpolation model
Abstract
Abstract: Similar periodic structures have been widely used in engineering. In order to obtaining the optimal similar periodic structures, a topology optimization method of similar periodic structures with multiple displacement constraints is proposed in this paper. Firstly, in the proposed method, the design domain is divided into sub-domains. Secondly, a penalty term considering discrete conditions of density variables is introduced into the objective function, and the reciprocal density exponents of structural elements are taken as design variables. A topological optimization model of a similar periodic continuum structure with the objective function being the structural mass and the constraint functions being structural displacements is constructed in the proposed method. The optimization dual method is introduced and a set of iteration formula for Lagrange multipliers is built. Then, virtual sub-domain design variables are introduced to establish the relation of corresponding variables between all the sub-domains of the similar periodic continuum structure in order to enforce structurally similar periodic requirement. Three examples are provided to demonstrate that the proposed method is feasible and effective for obtaining optimal similar periodic structures.
Keywords: similar periodic structure, topological optimization, displacement constraint. quadratic programming, Lagrange multiplier
... [113], the stress-constrained problem for continuum structure subject to harmonic excitation [114], forced vibration structure containing multiple materials [115], transient heat transfer problem [116], concurrent design considering load carrying capabilities and thermal insulation [117], large-scale computing problem resort to reanalysis technique [118], fail-safe design combined with the load uncertainty [119], etc. [120]. Rong et al. introduced a design space expansion strategy to stabilize the ICM optimization process [121][122][123]. ...
This paper offers an extensive overview of the utilization of sequential approximate optimization approaches in the context of numerically simulated large-scale continuum structures. These structures, commonly encountered in engineering applications, often involve complex objective and constraint functions that cannot be readily expressed as explicit functions of the design variables. As a result, sequential approximation techniques have emerged as the preferred strategy for addressing a wide array of topology optimization challenges. Over the past several decades, topology optimization methods have been advanced remarkably and successfully applied to solve engineering problems incorporating diverse physical backgrounds. In comparison to the large-scale equation solution, sensitivity analysis, graphics post-processing, etc., the progress of the sequential approximation functions and their corresponding optimizers make sluggish progress. Researchers, particularly novices, pay special attention to their difficulties with a particular problem. Thus, this paper provides an overview of sequential approximation functions, related literature on topology optimization methods, and their applications. Starting from optimality criteria and sequential linear programming, the other sequential approximate optimizations are introduced by employing Taylor expansion and intervening variables. In addition, recent advancements have led to the emergence of approaches such as Augmented Lagrange, sequential approximate integer, and non-gradient approximation are also introduced. By highlighting real-world applications and case studies, the paper not only demonstrates the practical relevance of these methods but also underscores the need for continued exploration in this area. Furthermore, to provide a comprehensive overview, this paper offers several novel developments that aim to illuminate potential directions for future research.
... If the cell size is not an order of magnitude smaller than the characteristic lengthscale of the entire structure, a macroscopic approach will be more appropriate as it resolves all structural details of the periodic pattern and also captures influences of inhomogeneous load distributions. One or twodimensional translational periodicity has been studied in the context of cellular core design of sandwich structures (Zhang and Sun 2006;Huang and Xie 2008), bridges (Thomas et al. 2020;Huang and Xie 2008), roof frames (Rong et al. 2013) and thermo-elastic problems (Chen et al. 2010). Application of cyclic periodicity has been investigated for the design of car wheels (Zuo et al. 2011;Jiang et al. 2020), flywheels (Jiang and Wu 2017) and spur gears (Schmitt et al. 2019). ...
Building structures from identical components organized in a periodic pattern is a common design strategy to reduce design effort, structural complexity and cost. However, any periodic pattern will impose certain design restrictions often leading to lower structural efficiency and heavier weight. Much research is available for periodic structures with connected components. This paper addresses minimal compliance design for periodic arrangements of unconnected components. The design problem discussed here is relevant for many applications where a tightly nested, space-saving arrangement of identical components is required. We formulate an optimal design problem for a component being part of a periodic arrangement. The orientation and position of the component relatively to its neighbours are prescribed. The component design is computed by topology optimization on a design domain possibly shared by several neighbouring components. Additional constraints prevent components from overlapping. Constraint aggregation is employed to reduce the computational cost of many local constraints. The effectiveness of the method is demonstrated by a series of 2D and 3D examples with an ever-smaller distance between the components. Moreover, problem-specific ranges with only little to no increase in compliance are reported.
... Due to the characteristics of light mass, high tensile rigidity, and compact structure space, helix twisting structure (HTS) has been widely used in engineering fields such as mine hoists field, port loading, and unloading field [1][2][3]. The parametric model plays an important role in the design and analysis system of HTS, and a quick modeling approach for helix twisting structure is a challenging issue and has received increasing interest among researchers in recent years [4][5][6]. ...
In order to improve the modeling efficiency and realize the deformation simulation of helix twisting structure, a computer-aided design system based on mass-spring model is developed. The geometry structure of helix twisting structure is presented and mass-spring model is applied in the deformation simulation. Moreover, the key technologies such as coordinate mapping, system render and system architecture are elaborated. Finally, a prototype system is developed with Visual C++ and OpenGL, and the proposed system is proved efficient through a comparison experiment.
Abstract Data with unbalanced multivariate nominal attributes collected from a large number of users provide a wealth of knowledge for our society. However, it also poses an unprecedented privacy threat to participants. Local differential privacy, a variant of differential privacy, is proposed to eliminate the privacy concern by aggregating only randomized values from each user, with the provision of plausible deniability. However, traditional local differential privacy algorithms usually assign the same privacy budget to attributes with different dimensions, leading to large data utility loss and high communication costs. To obtain highly accurate results while satisfying local differential privacy, the aggregator needs a reasonable privacy budget allocation scheme. In this paper, the Lagrange multiplier (LM) algorithm was used to transform the privacy budget allocation problem into a problem of calculating the minimum value from unconditionally constrained convex functions. The solution to the nonlinear equation obtained by the Cardano formula (CF) and Newton-Raphson (NS) methods was used as the optimal privacy budget allocation scheme. Then, we improved two popular local differential privacy mechanisms by taking advantage of the proposed privacy budget allocation techniques. Extension simulations on two different data sets with multivariate nominal attributes demonstrated that the scheme proposed in this paper can significantly reduce the estimation error under the premise of satisfying local differential privacy.
Uncertainty factors play an important role in the design of periodic structures because structures with small periodic design spaces are extremely sensitive to loading uncertainty. Therefore, for the first time, this paper proposes a framework for robust topology optimization (RTO) of periodic structures assuming that load uncertainties follow a Gaussian distribution. In this framework, the expected value and variance of structural compliance can be easily computed using a semianalytical method combined with probability theory, which is important for RTO when uncertain variables follow probabilistic distributions. To obtain optimal topologies, the bidirectional evolutionary structural optimization method is used. Structural periodicity is calculated using a strategy of sensitivity averaging and consistency constraints. To eliminate the influence of numerical units when comparing the optimal results to deterministic and RTO solutions, a generic coefficient of variation is defined as the robust index, which contains both the expected value and variance. The proposed framework is verified through the optimization of both 2D and 3D structures with periodicity. Computational results demonstrate the feasibility and effectiveness of the proposed framework for designing robust periodic structures under loading uncertainties.
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