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... The proposed work will focus on the microscale RBTO under heterogeneous materials with multi-phases. The related research works that consider multi-phase materials were reported in [16,17]. However, only the deterministic case or the randomness in external loadings were investigated. ...
... Fig. 13 The volume fraction and constraint histories are compared in Figs. [14][15][16][17][18]. For these two materials, the proposed RBTO-AIS achieves better designs. ...
With advanced manufacturing processes, many modern structural designs need to include microscale uncertainties of material properties and microstructures that can affect overall performances, such as microelectromechanical systems, micro–opto–electro–mechanical systems, and micro-optical electronics systems. Topology design optimization at this scale must consider uncertainties from material microstructures as the scale of the structure and material microstructure is comparable. Very few studies consider microstructure uncertainties in topology design due to their scales are much different in classical mechanical/civil engineering. A novel framework of reliability-based topology optimization is proposed to specifically address this gap for the design. The microscale uncertainties of heterogeneous materials are quantified by the explicit mixture random field model. Then, the material distribution is optimized by a heuristic updating scheme, and sensitivities of reliability constraints are calculated using the adjoint design-point-based importance sampling method. The proposed methodology simultaneously considers microscale hierarchical uncertainties in microstructures and material property variations for each phase. The feasibility of the framework is demonstrated by several numerical examples with different multi-phase materials. Compared with deterministic topology optimization and classical topology optimization with first-order reliability method approximations, the optimal design obtained from the proposed method can achieve accurate target reliability with minimized limit state function evaluations.
github code link https://github.com/ymlasu/Reliability-Based-Topology-Optimization/tree/main
... Chun et al. (2016) optimized the structure under stochastic excitations while ensuring that the instantaneous failure probability reached the target failure probability. Xu et al. (2016) and Hu et al. (2018) approximated the first-passage probability in terms of the out-crossing rates of the responses of interest, which assumes a Poisson distribution for the out-crossing events. Chun et al. (2019) computed the first-passage probability with the sequential compounding method and developed the sensitivity analysis approach to use gradient-based optimization algorithms. ...
In current engineering practice, the vibration serviceability performance is usually checked after the safety design is finished. Such process does not guarantee the optimized design, especially for those large-span truss structures, where the vibration serviceability performance is the controlling factor that dominates the design process. The randomness of the human-induced loads and structures significantly affects the design results. To address such problem, a dynamic-reliability-based topology optimization (DRBTO) method for the vibration serviceability design of large-span truss structures under random human-induced loads is proposed, which employs the probability density evolution method (PDEM) for dynamic reliability analysis and the density approach for solving topology optimization. To reduce the calculation of multiple order force vectors, the conjugation of the force vectors is utilized to reduce half of the calculation in finite element analysis and sensitivity analysis. Unlike other types of dynamic loads, the load positions of human-induced loads also have strong randomness, so the load positions are simulated as uniformly distributed random variables. To better implement the optimization, sensitivity analysis function is adopted for efficient calculation of the structural response sensitivity. Gray-scale operator is adopted to avoid the occurrence of intermediate density. Numerical results with different volume fractions and acceleration thresholds show that the proposed method could give better design results from the perspectives of structural responses and dynamic probability, thus demonstrating the effectiveness and stability of this optimization method.
... To better meet the demands of structural applications, the applications of multiple materials are developing day by day. Xu et al. [23] proposed a dynamic response RBTO method based on the BESO approach and validated the effectiveness of his method. Zhao et al. [24] considered the influences of factors such as incomplete measurements, inaccurate information, and limited knowledge, and developed a reliability-constrained multi-material topology optimization method. ...
This paper proposes a multi-material topology optimization method based on the hybrid reliability of the probability-ellipsoid model with stress constraint for the stochastic uncertainty and epistemic uncertainty of mechanical loads in optimization design. The probabilistic model is combined with the ellipsoidal model to describe the uncertainty of mechanical loads. The topology optimization formula is combined with the ordered solid isotropic material with penalization (ordered-SIMP) multi-material interpolation model. The stresses of all elements are integrated into a global stress measurement that approximates the maximum stress using the normalized p-norm function. Furthermore, the sequential optimization and reliability assessment (SORA) is applied to transform the original uncertainty optimization problem into an equivalent deterministic topology optimization (DTO) problem. Stochastic response surface and sparse grid technique are combined with SORA to get accurate information on the most probable failure point (MPP). In each cycle, the equivalent topology optimization formula is updated according to the MPP information obtained in the previous cycle. The adjoint variable method is used for deriving the sensitivity of the stress constraint and the moving asymptote method (MMA) is used to update design variables. Finally, the validity and feasibility of the method are verified by the numerical example of L-shape beam design, T-shape structure design, steering knuckle, and 3D T-shaped beam.
... Moreover, traditional structures usually adopt a single-material design, which is challenging to satisfy the lightweight requirement and high load-bearing capacity simultaneously. In the last decade, multi-material design has been a hot topic (Kim et al. 2021;Yu et al. 2012;Bandyopadhyay and Heer 2018;Vogiatzis et al. 2017;Han and Lee 2020;Wang et al. 2019;Ma et al. 2020aMa et al. , 2020bChen et al. 2021;Xu et al. 2016), especially with the development of additive manufacturing, making it achievable to apply multi-material design in engineering structures. In recent aerospace engineering, space missions such as deep space exploration have put forward higher requirements for the lightweight and dynamic design of space equipment (Omidi et al. 2015;Wang et al. 2022). ...
Topology optimization methods for structures subjected to random excitations are difficult to widely apply in aeronautic and aerospace engineering, primarily due to the high computational cost of frequency response analysis for large-scale systems. Conventional methods are either unsuitable or inefficient for large-scale engineering structures, especially for structures consisting of multi-materials with non-proportional damping systems. To address this challenge, an accurate and highly efficient reduced-order method (ROM) based on the second-order Krylov subspace and the multigrid method is proposed in this paper, which is applicable to non-proportional damping systems. Moreover, a novel multigrid reduced-order topology optimization scheme for structures subjected to stationary random excitations is proposed based on the pseudo-excitation method (PEM). Two 3D numerical examples demonstrate the accuracy and efficiency of the proposed scheme for multi-material topology optimization. For a cantilever beam with about degrees of freedom (DOFs), compared against the original reduced-order method, the efficiency of pseudo-harmonic analysis of the multigrid reduced-order method is improved by about 91% with sufficient accuracy, and the efficiency of the whole optimization process of the multigrid reduced-order method is improved by more than 71%. For a pedestal structure with about DOFs, compared against the original reduced-order method, the efficiency of pseudo-harmonic analysis of the multigrid reduced-order method is improved by about 61%.
... Compared to RBTO under static loads, relatively few investigations have been reported in DRBTO. In Xu et al. [24] and Hu et al. [25], first-passage probability measures were approximated in terms of the out-crossing rates of the responses of interest, which assumes a Poisson distribution for the out-crossing events [26]. Similarly, Chun et al. [27] employed FORM with the sequential compounding method to approximate the first-passage probability. ...
The present paper explores the feasibility of topology optimization of stochastic dynamical systems in the framework of the probability density evolution method (PDEM). A new method is proposed for solving dynamic-reliability-based topology optimization (DRBTO) problems by combining the PDEM, the ground structure approach and the solid isotropic material with penalization (SIMP) model. In the investigated optimization problems, the first-passage probability is considered as an objective or constraint function. To obtain a clear layout of the optimized structure, the topology of the structure is described by the ground structure approach together with the SIMP model. The PDEM is employed as an efficient approach to assess the first-passage probability. For improved numerical efficiency, an approximate formulation of the first-passage probability based on the important representative points (IRPs) is implemented. On the basis of the approximate formulation of the first-passage probability, a relationship between the sensitivity of the first-passage probability and the transient response is obtained. The adjoint sensitivity analysis of the transient response is introduced to avoid extra numerical efforts. Then, by incorporating the first-passage probability and its sensitivity into the method of moving asymptotes (MMA), the investigated DRBTO problems are solved in an effective manner. The DRBTO of a braced frame structure is presented to demonstrate the availability and effectiveness of the proposed method.
... With the density-based method, recursive multiphase (RM) [29][30][31] and discrete material optimization (DMO) [32] are proposed to extend the solid isotropic material with a penalization (SIMP) scheme to accommodate the arbitrary number of material phases. The multi-material bi-directional evolutionary structural optimization (BESO) method has also been derived and used to solve the optimization problems in the microstructure design [33] and dynamics field [34]. From the perspective of stability and efficiency of the algorithm, the multi-material optimizations with a single mass constraint [35] or design variable [36] are presented. ...
Topology optimization for multi-material structures has become an important and hot research topic due to their great application potential in modern industries. Unfortunately, most existing multi-material topology optimization methods assume that the interface is perfectly bonded, and those that do not, ignore the strength difference between different interfaces, limiting their application in the design of multi-material structures with more than two solid phases. In this study, a multi-material topology optimization method considering a tension/compression-asymmetric piecewise interface stress constraint is proposed to improve the overall interface strengths of the multi-material structure under the SIMP method framework. Firstly, a new interpolation scheme is developed to identify interfaces between two arbitrary materials and classify the interfaces into different pieces based on the two neighbor materials. Then, a novel piecewise interface stress constraint is developed to describe different targeted strengths at multiple material interfaces. In this constraint, the equivalent interfacial stress considering the tension/compression-asymmetric characteristics is defined by tensile stresses and shear stresses at material interfaces. The maximum stresses at material interfaces are measured by the global p-norm aggregation function which makes imposing the constraint easy. After that, the proposed piecewise tension/compression-asymmetric interface stress constraint is introduced into multi-material topology optimization for compliance minimization. The sensitivity analyses are derived by the adjoint method. Several benchmark 2D examples are provided to discuss the influence of some parameters on optimization problems, such as the interface width, strength scaling factor and the filter radii. Other numerical examples include complex 3D problems and problems involving more than three materials. In all examples, the results show that the proposed algorithm with the tension/compression-asymmetric piecewise interface stress constraint can effectively control interface stress levels and improve structural safety.
... Yunzhen He et al. [17] integrated BESO with genetic algorithms to create diverse and efficient structural designs. ESO/BESO is recognized to be a powerful and promising technique and the application has been extended to the design of energy-absorbing structure [18,19], multiple constraints' optimization problem [20], dynamic reliabilitybased topology optimization [21], concurrent topology optimization on both macro-and microscale [22], and numerous others. However, it has been criticized that the results from ESO are likely to be local optimums [23] and even the BESO cannot help. ...
Genetic evolutionary structural optimization (GESO) method is an integration of the genetic algorithm (GA) and evolutionary structural optimization (ESO). It has proven to be more powerful in searching for global optimal response and requires less computational efforts than ESO or GA. However, GESO breaks down in the Zhou-Rozvany problem. Furthermore, GESO occasionally misses the optimum layout of a structure in the evolution for its characteristic of probabilistic deletion. This paper proposes an improved strategy that has been realized by MATLAB programming. A penalty gene is introduced into the GESO strategy and the performance index (PI) is monitored during the optimization process. Once the PI is less than the preset value which means that the calculation error of some element’s sensitivity is too big or some important elements are mistakenly removed, the penalty gene becomes active to recover those elements and reduce their selection probability in the next iterations. It should be noted that this improvement strategy is different from “freezing,” and the recovered elements could still be removed, if necessary. The improved GESO performs well in the Zhou-Rozvany problem. In other numerical examples, the results indicate that the improved GESO has inherited the computational efficiency of GESO and more importantly increased the optimizing capacity and stability.
... Advances in Engineering Software 148 (2020) 102834where r 1 and r k + 1 are varying constraint limits in the optimization processaccording to Eq. (41) at each outer loop iteration. As a consequence, the optimization model(35) can be reformulated as: ...
... Rong et al. investigated continuum structural topological optimization with dynamic displacement response constraints under random excitations (Rong et al. 2013). Xu et al. proposed a methodology for maximizing the dynamic response reliability of continuum structures involving multi-phase materials Xu et al. 2016). Zhang and Guo et al. presented a novel approach for damage identification of continuum structures based on their dynamic performances (Zhang et al. 2017). ...
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.
... Chun et al. (2016) employed instantaneous failure probability and first-passage probability as constraints to study optimal topologies of continuum structures under stationary stochastic excitations. Based on bi-directional evolutionary structural optimization method, Xu et al. (2016) proposed a topology optimization method for maximizing reliability indexes at specified time instants for structures subjected to stationary stochastic excitations. However, to the best of the authors' knowledge, there has been no report on the topology optimization of structures subjected to non-stationary stochastic excitations yet, despite the fact that most dynamic excitations in natural environment, such as earthquake, wind and wave, are non-stationary in their nature. ...
This paper studies the optimum structural design considering non-stationary stochastic excitations. The topology optimization of the lateral bracing system of frame structures is conducted and the first-passage probability of a displacement response is minimized under the material volume constraint. The concept of Solid Isotropic Material with Penalization (SIMP) model is employed for describing the material distribution. An efficient optimization algorithm based on explicit time-domain method is developed. Numerical examples of frame structures subjected to non-stationary seismic excitations are investigated to demonstrate the effectiveness of the proposed approach.
... Wang et al. ) gave a time-dependent RBDO method considering both the time-dependent kinematic reliability and the time-dependent structural reliability as constrains. Furthermore, several researches about the timedependent RBDO (Singh et al. 2010;Spence and Gioffrè 2011;Wang et al. 2011a) as well as the instantaneous design (Chun et al. 2015;Xu et al. 2016;Kayedpour et al. 2016;Babykina et al. 2016) were also conducted. ...
Uncertainty with characteristics of time-dependency, multi-sources and small-samples extensively exists in the whole process of structural design. Associated with frequent occurrences of material aging, load varying, damage accumulating, traditional reliability-based design optimization (RBDO) approaches by combination of the static assumption and the probability theory will be no longer applicable when dealing with the design problems for lifecycle structural models. In view of this, a new non-probabilistic time-dependent RBDO method under the mixture of time-invariant and time-variant uncertainties is investigated in this paper. Enlightened by the first-passage concept, the hybrid reliability index is firstly defined, and its solution implementation relies on the technologies of regulation and the interval mathematics. In order to guarantee the stability and efficiency of the optimization procedure, the improved ant colony algorithm (ACA) is then introduced. Moreover, by comparisons of the models of the safety factor-based design as well as the instantaneous RBDO design, the physical means of the proposed optimization policy are further discussed. Two numerical examples are eventually presented to demonstrate the validity and reasonability of the developed methodology.
... They tackled structural compliance minimization problems with a probability of failure as a constraint and regarded the geometry and the applied loads as random parameters. Later on, more and more researchers have explored this field, and a mass of success works have been accomplished (Liu et al., 2016a;Patel and Choi, 2012;Jung and Cho, 2004;Kharmanda et al., 2004;Yoo et al., 2011;Chen et al., 2010;Maute and Frangopol, 2003;Kang and Luo, 2009;Xu et al., 2016). ...
Purpose
To tackle the challenge topic of continuum structural layout in presence of random loads, and to develop an efficient robust method.
Design/methodology/approach
An innovative robust topology optimization approach for continuum structures with random applied loads is reported. Simultaneous minimization of the expectation and the variance of the structural compliance is performed. Uncertain load vectors are dealt with by using additional uncertain pseudo random load vectors. The sensitivity information of the robust objective function is obtained approximately via the Taylor expansion technique. The design problem is solved by Bi-directional Evolutionary Structural Optimization (BESO) method utilizing the derived sensitivity numbers.
Findings
The numerical examples show the significant topological changes of the robust solutions compared with the equivalent deterministic solutions.
Originality/value
A simple yet efficient robust topology optimization approach for continuum structures with random applied loads is developed. The computational time scales linearly with the number of applied loads with uncertainty, which is very efficient when compared with Monte Carlo-based optimization method.
... 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. ...
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.
... Remarkable works have been done to solve RBTO problems in the presence of uncertainties in geometry dimensions or/and material properties or/and loadings [Kharmanda et al. (2004); Maute and Frangopol (2003); Jung and Cho (2004); Kang and Luo (2009) ;Luo et al. (2009) ;Patel and Choi (2012); Kanakasabai and Dhingra (2016); Xu et al. (2016); ]. And there are few such that the loading uncertainties are purely considered. ...
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.
... Remarkable works have been done to solve RBTO problems in the presence of uncertainties in geometry dimensions or/and material properties or/and loadings [Kharmanda et al. (2004); Maute and Frangopol (2003); Jung and Cho (2004); Kang and Luo (2009);Luo et al. (2009);Patel and Choi (2012); Kanakasabai and Dhingra (2016); Xu et al. (2016); ]. And there are few such that the loading uncertainties are purely considered. ...
Uncertainty in applied loads is a vital factor that needs to be considered in the design of the engineering structures. This paper concerns the minimization of mean compliance for continuum structures subjected to directional uncertain applied loads. Due to the uncertain behaviour of this type of the optimization problem, the existed deterministic topology optimization methods are not able to solve such problems. The loading directional uncertainty is described by directional interval variables which are divided into many small intervals, and then the uncertain small interval variables are approximated by their deterministic midpoints. In doing so, the uncertain topology optimization problem is transformed into deterministic multiple load case one. The optimization problem is then formulated as minimizing the mean compliance under multiple load cases, subject to material volume constraint. A soft-kill bi-directional evolutionary structural optimization (BESO) method is developed to solve the problem, which requires very few changes to the BESO computer code. The results reported in this work show that the proposed methodology suits engineering design and represents an improvement over existing topology optimization methods.
This contribution focuses on addressing the challenging problem of dynamic-reliability-based topology optimization (DRBTO) of engineering structures involving uncertainties by synthesizing the probability density evolution method (PDEM) and the bi-directional evolutionary structural optimization (BESO) approach. The considered optimization problem aims at minimizing the first-passage probability under the constraint of material volume. Generally, the double-loop essence of DRBTO involving dynamic reliability evaluation and topology searching makes the computational efforts prohibitively large. To this end, the PDEM is adopted to efficiently assess the first-passage probability of structures under earthquake actions. In particular, by reformulating the first-passage probability under the framework of the PDEM, the sensitivity of the first-passage probability is derived. To further improve the efficiency, a strategy taking advantage of important representative points (IRPs) is employed to achieve a robust estimate of sensitivity of the first-passage probability. The adjoint variable method (AVM) for the sensitivity analysis of transient response considering given modal damping ratios is incorporated to considerably improve the computational efficiency when the reliability sensitivity analysis in terms of multiple design variables is needed. To drive the topology towards the optimum, the above highly efficient reliability assessment and sensitivity analysis are embedded into BESO. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed method, illustrating significant improvement in computational efficiency compared to direct implementation. Additionally, the necessity of introducing seismic reliability in topology optimization is also discussed based on the numerical results.
Hydrogels with intricate 3D networks and high hydrophilicity have qualities resembling those of biological tissues, making them ideal candidates for use as smart biomedical materials. Reactive oxygen species (ROS) responsive hydrogels are an innovative class of smart hydrogels, and are cross-linked by ROS-responsive modules through covalent interactions, coordination interactions, or supramolecular interactions. Due to the introduction of ROS response modules, this class of hydrogels exhibits a sensitive response to the oxidative stress microenvironment existing in organisms. Simultaneously, due to the modularity of the ROS-responsive structure, ROS-responsive hydrogels can be manufactured on a large scale through additive manufacturing. This review will delve into the design, fabrication, and applications of ROS-responsive hydrogels. The main goal is to clarify the chemical principles that govern the response mechanism of these hydrogels, further providing new perspectives and methods for designing responsive hydrogel materials.
This paper, a multi-material topology optimization method (MMTO) is presented for two-dimensional structures using modified ordered solid isotropic material with penalization (SIMP) under multiple constraints such as mass, stiffness, frequency, and stress. The element density is obtained by a density filtering and the Heaviside approximation, which are combined to avoid gray elements and numerical issues of stress. Effective improvement of multiphase material topology optimization was performed including the criteria for stiffness, stress, and free vibration. 𝑃-norm stress aggregation is employed to calculate the maximum von Mises stress and sensitivity of von Mises stress is formulated by the adjoint method. The natural frequency and stress constraints are solved simultaneously to enhance the stability and reduce the stress concentration of the optimized structures, which are not much covered in the literature. The multi-mode phenomenon in the eigenfrequency problem is treated by the simply proposed technique. Results indicate that the multi-material designs with multi-constraints can be illustrated and optimized effectively. The optimized topologies, for stress minimization, indicate that the maximum stress can be significantly reduced compared with compliance minimization. The maximum von Mises stress of the topological results considering the maximum stiffness with natural frequency and mass constraints is higher than that maximum stiffness with only mass constraints, which indicates that the optimized structure created considering the eigenvalues is safer. The proposed approach can obtain a reasonable result that effectively controls the stress level and reduces the stress concentration at the high stress regions of multi-phase material structures with multi-constraints. Benchmark numerical examples are presented to confirm the effectiveness of the presented method.
Unsymmetrical complex plate and shell structure is one of the common engineering structures. In practice, more redundant materials exist because of the irrationality of this kind of structure with heavy load and multiple working conditions, and the study of its topology optimization has become an engaging topic. Using the SIMP model, topological results show that one side of the main web is a hollow structure, and the other side of the auxiliary web is a truss structure. According to the topological results and considering manufacturable processing, a new structure is redesigned, the size and shape of the redesigned structure is secondary optimized, and the final structure is obtained. The method in this paper not only meets the performance requirements of the unsymmetrical complex plate and shell structures, but also realizes the topology and lightweight. The effectiveness scientific research value of the proposed method is verified by engineering examples.
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.
This work is focused on the topology optimization related to harmonic responses for large-scale problems. A comparative study is made among mode displacement method (MDM), mode acceleration method (MAM) and full method (FM) to highlight their effectiveness. It is found that the MDM results in the unsatisfactory convergence due to the low accuracy of harmonic responses, while MAM and FM have a good accuracy and evidently favor the optimization convergence. Especially, the FM is of superiority in both accuracy and efficiency under the excitation at one specific frequency; MAM is preferable due to its balance between the computing efficiency and accuracy when multiple excitation frequencies are taken into account.
A technique is proposed for determining the material distribution of a structure to obtain desired eigenmode shapes for problems of maximizing the fundamental eigenfrequency. The design objective is achieved using the solid isotropic method with penalization (SIMP) for topology optimization. Weighted constraints added in bound formulation are proposed to maximize the fundamental natural frequency, which provides an easy and straightforward way to prevent mode switching in the optimization process. Aside from maximizing the fundamental frequency, a method to modify existing eigenmodes to continuously evolve and assume the same shapes as the desired modes within the optimization process is proposed. The topology layout of a structure with desired eigenmodes is obtained by adding the modal assurance criterion (MAC) as additional constraints in the bound formulation optimization. Examples are presented to illustrate the proposed method, and a potential application of the proposed technique in decoupling a mechanical system is demonstrated.
A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external
excitation frequencies. This can be achieved by, e.g., maximizing the fundamental eigenfrequency, an eigenfrequency of higher
order, or the gap between two consecutive eigenfrequencies of given order. This problem is often complicated by the fact that
the eigenfrequencies in question may be multiple, and this is particularly the case in topology optimization. In the present
paper, different approaches are considered and discussed for topology optimization involving simple and multiple eigenfrequencies
of linearly elastic structures without damping. The mathematical formulations of these topology optimization problems and
several illustrative results are presented.
A topology and shape optimization technique using the homogenization method was developed for stiffness of a linearly elastic structure by Bendsøe and Kikuchi (1988), Suzuki and Kikuchi (1990, 1991), and others. This method has also been extended to deal with an optimal reinforcement problem for a free vibration structure by Diaz and Kikuchi (1992). In this paper, we consider a frequency response optimization problem for both the optimal layout and the reinforcement of an elastic structure. First, the structural optimization problem is transformed to an Optimal Material Distribution problem (OMD) introducing microscale voids, and then the homogenization method is employed to determine and equivalent “averaged” structural analysis model. A new optimization algorithm, which is derived from a Sequential Approximate Optimization approach (SAO) with the dual method, is presented to solve the present optimization problem. This optimization algorithm is different from the CONLIN (Fleury 1986) and MMA (Svanderg 1987), and it is based on a simpler idea that employs a shifted Lagrangian function to make a convex approximation. The new algorithm is called “Modified Optimality Criteria method (MOC)” because it can be reduced to the traditional OC method by using a zero value for the shift parameter. Two sensitivity analysis methods, the Direct Frequency Response method (DFR) and the Modal Frequency Response method (MFR), are employed to calculate the sensitivities of the object functions. Finally, three examples are given to show the feasibility of the present approach. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/47814/1/466_2004_Article_BF00370133.pdf
Contents: Preface.- Introduction.- Basic Evolutionary Structural Optimization.- ESO for Multiple Load Cases and Multiple Support Environments.- Structures with Stiffness or Desplacement Contraints.- Frequency Optimization.- Optimization Against Buckling.- ESO for Pin- and Rigid-Jointed Frames.- ESO for Shape Optimization and the Reduction of Stress Concentrations.- ESO Computer Program Evolve97.- Author Index.- Subject Index.
Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsøe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.
This paper presents a solution method, which can be regarded as the further extension of the generalized evolutionary method (Zhao et al. 1998a), for the simultaneous optimization of several different natural frequencies of a structure in general and a two dimensional structure in particular. The main function of the present method is to optimize the topology of a structure so as to simultaneously make several different natural frequencies of interest to be of the corresponding different desired values for the target structure. In order to develop the present method, the new contribution factor of an element is proposed to consider the contribution of an element to the gaps between the currently calculated values for the different natural frequencies of interest and their corresponding desired values in a weighted manner. Using this new contribution factor of an element, the most inefficiently used material can be detected and removed gradually from the design domain of a structure. Through applying the present method to optimize two and three different natural frequencies of a two dimensional structure, it has been demonstrated that it is possible and applicable to use the generalized evolutionary method for tackling the simultaneous optimization of several different natural frequencies of a structure in the structural design.
More and more stringent structural performance requirements are imposed in advanced engineering application, only a limited number of works have been devoted to the topology optimization of the structures with random vibration response requirements. In this study, the topology optimization problem with the objective function being the structural weight and the constraint functions being structural random vibration responses is investigated. An approximate topological optimization model for suppressing ‘localized modes’ of vibrating Cauchy solids is established in this paper. Based on moving asymptotes approximate functions, approximated-approximations expressions of the dynamic responses are constructed. In order to control the change quantity of topologic design variables, new dynamic response constraint limits are formed and introduced into the optimization model at the beginning of each sub-loop iteration. Then, an optimization sequential quadratic programming is introduced, and a set of iteration formulas for Lagrange multipliers is developed. Two examples are provided to demonstrate that the proposed method is feasible and effective for obtaining optimal topology.
Topological design with multiple constraints is of great importance in practical engineering design problems. The present work extends the bi-directional evolutionary structural optimization (BESO) method to multiple constraints of displacement and frequency in addition to the amount of material usage. Besides the binary design variables, the Lagrange multipliers for constraints are considered as additional continuous variables and determined by a search scheme. The enhanced approach can include a number of constraints besides the simple volume constraint. To demonstrate the effectiveness of the proposed BESO approach, several examples are presented for the maximization of structural overall stiffness subject to the material volume, displacement and frequency constraints.
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 method to maximize the natural frequencies of vibration of truss-like continua with the constraint of material volume is presented. Truss-like is a kind of particular anisotropic continuum, in which there are finite numbers of members with infinitesimal spaces. Structures are analyzed by finite element method. The densities and orientations of members at nodes are taken as design variables. The densities and orientations of members in elements are interpolated by these values at nodes; therefore they vary continuously in design domain. For no intermediate densities being suppressed, there is no numerical instability, such as checkerboard patterns and one-node connected hinges. The natural frequency and its sensitivities of truss-like continuum are derived. Optimization is achieved by the techniques of moving asymptotes and steepest descent. Several numerical examples are provided to demonstrate this optimization method.
From the energy conservation principle, the contribution factor of an element to the natural frequency of the finite element discretized system of a membrane vibration problem is presented in this paper. Since the contribution factor of an element is derived from the general finite element formulation of structural eigenvalue problems, it can be used for solving the natural frequency optimization problem of any structure. Therefore, the method associated with it is called as a generalized evolutionary method. Using the contribution factor of an element, the relevant evolutionary criterion which bridges between the structural optimization and the conventional finite, element analysis can be easily established for the evolutionary natural frequency optimization of a structure. Physically, the contribution factor of an element implies the direct contribution of an element to the natural frequency of the structure. A square membrane with four sides fixed has been chosen as an example to show the usefulness and robustness of the generalized evolutionary method for natural frequency optimization of membrane vibration problems in the finite element analysis.
In recent years, the Evolutionary Structural Optimization (ESO) method has been developed into an effective tool for engineering design. However, no attempts have been made to incorporate random dynamic response constraints. The optimum design of structures with dynamic response constraints is of great importance, particularly in the aeronautical and automotive industries. This paper considers the extension and modification of the ESO method to control the structural random dynamic responses. The random dynamic theory is applied to build an expression of random dynamic response constraints considering engineering requirements. Based on the modal truncation method of eigenderivatives and some approximate process, a set of formulations for sensitivity numbers of mean square random dynamic responses is derived. The algorithm is implemented in optimization software. Several examples are provided to demonstrate the validity and effectiveness of the proposed method.
Frequency optimization is of great importance in the design of machines and structures subjected to dynamic loading. When the natural frequencies of considered structures are maximized using the solid isotropic material with penalization (SIMP) model, artificial localized modes may occur in areas where elements are assigned with lower density values. In this paper, a modified SIMP model is developed to effectively avoid the artificial modes. Based on this model, a new bi-directional evolutionary structural optimization (BESO) method combining with rigorous optimality criteria is developed for topology frequency optimization problems. Numerical results show that the proposed BESO method is efficient, and convergent solid-void or bi-material optimal solutions can be achieved for a variety of frequency optimization problems of continuum structures.
This paper pertains to the use of topology optimization based on the internal element connectivity parameterization (I-ECP) method for nonlinear dynamic problems. When standard density-based topology optimization methods are used for nonlinear dynamic problems, they typically suffer from two main numerical difficulties, element instability and localized vibration modes. As an alterative approach, the I-ECP method is employed to avoid element instability and a new patch mass model in the I-ECP formulation is developed to control the problem of localized vibration modes. After the I-ECP based formulation is developed, the advantages of the proposed method are checked with several numerical examples.
This paper presents an evolutionary method for structural topology optimization subject to frequency constraints. The evolutionary structural optimization (ESO) method is based on the idea that by gradually removing inefficient material, the residual shape of structure evolves toward an optimum. The method is further developed by allowing the material to be added as well as removed, and this new approach is called the bidirectional ESO method (BESO). BESO has been successfully used for problems of stress and stiffness/displacement constraints. Its application to frequency optimization is addressed in this paper. Three kinds of optimization objectives, namely, maximizing a single frequency, maximizing multiple frequencies, and designing structures with prescribed frequencies are considered. Four examples are tested by BESO and ESO. The objective functions yielded by the two methods are close, and BESO is computationally more efficient in most cases.
Although a lot of attention in the topology optimization literature has focused on the optimization of eigenfrequencies in free vibration problems, relatively little work has been done on the optimization of structures subjected to periodic loading. In this paper, we propose two measures, one global and the other local, for the minimization of vibrations of structures subjected to periodic loading. The global measure which we term as the “dynamic compliance” reduces the vibrations in an overall sense, and thus has important implications from the viewpoint of reducing the noise radiated from a structure, while the local measure reduces the vibrations at a user-defined point. Both measures bring about a reduction in the vibration level by moving the natural frequencies which contribute most significantly to the measures, away from the driving frequencies, although, as expected, in different ways. Quite surprisingly, the structure of the dynamic compliance optimization problem turns out to be very similar to the structure of the static compliance optimization problem. The availability of analytical sensitivities results in an efficient algorithm for both measures. We show the effectiveness of the measures by presenting some numerical examples.
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.
An optimum design method to minimize the weight of a linear elastic structural system subjected to random excitations is
presented. It is focused on the constraints of the first passage failure and displacement response mean square (RMS) at certain
degree of freedoms. Constraints on natural frequencies and bounds of design variables are also considered in the optimization.
Both correlated and un-correlated generalized random excitations are considered in the present formulation. The sensitivities
of the expected number of crossings as well as the displacement RMS with respect to the design variables are also derived.
The present method is applicable to stationary Guassian random excitations. Computational examples show the feasibility and
efficiency of the proposed method.
This paper presents a simple method for structural optimization with frequency constraints. The structure is modelled by a fine mesh of finite elements. At the end of each eigenvalue analysis, part of the material is removed from the structure so that the frequencies of the resulting structure will be shifted towards a desired direction. A sensitivity number indicating the optimum locations for such material elimination is derived. This sensitivity number can be easily calculated for each element using the information of the eigenvalue solution. The significance of such an evolutionary structural optimization (ESO) method lies in its simplicity in achieving shape and topology optimization for both static and dynamic problems. In this paper, the ESO method is applied to a wide range of frequency optimization problems, which include maximizing or minimizing a chosen frequency of a structure, keeping a chosen frequency constant, maximizing the gap of arbitrarily given two frequencies, as well as considerations of multiple frequency constraints. The proposed ESO method is verified through several examples whose solutions may be obtained by other methods.
This paper presents a simple solution strategy to find the shape and topology of a general structure that maximize or minimize the natural frequency. The structure is modelled with a fine mesh of finite elements. During an evolutionary process, a small part of the material is removed from the structure at the end of each finite element analysis. A criterion is established as to which elements should be eliminated so that the frequency of the resulting structure can be increased or reduced. It is found that the proposed simple method is effective in solving frequency optimization problems which usually require sophisticated mathematical programming techniques to solve.
This study uses model reduction (MR) schemes such as the mode superposition (MS), Ritz vector (RV), and quasi-static Ritz vector (QSRV) methods, which reduce the size of the dynamic stiffness matrix of dynamic structures, to calculate dynamic responses and sensitivity values with adequate efficiency and accuracy for topology optimization in the frequency domain. The calculation of structural responses to dynamic excitation using the framework of the finite element (FE) procedure usually requires a significant amount of computation time; that is mainly attributable to repeated inversions of dynamic stiffness matrices depending on time or frequency intervals, which hastens the dissemination of the MR schemes in the analysis. However, using well-established MR schemes in topology optimization has not been prevalent. Therefore, this study conducted a comprehensive investigation to highlight the drawbacks and advantages of these MR schemes for topology optimization. In the results, the MS method, which generates reduction bases by considering some of the lowest eigenmodes, can lose the accuracy in both approximated structural responses and sensitivity values due to locally vibrating eigenmodes and higher mode truncation in the solid isotropic material with penalization (SIMP) approach. In addition, the RV and QSRV methods, which generate reduction bases by considering the external force, mass, and stiffness matrices of a structure, can be used as alterative model reduction schemes for stable optimization. Through several analysis and design examples, the efficiency and reliability of the model reduction schemes for topology optimization are compared and validated.
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.