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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 . ...

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]. ...

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. ...

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. ...

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. ...

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. ...

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. ...

In this paper, a new non-probabilistic reliability-based topology optimization (NRBTO) method is proposed to account for interval uncertainties considering parametric correlations. Firstly, a reliability index is defined based on a newly developed multidimensional parallelepiped (MP) convex model, and the reliability-based topology optimization problem is formulated to optimize the topology of the structure, to minimize material volume under displacement constraints. Secondly, an efficient decoupling scheme is applied to transform the double-loop NRBTO into a sequential optimization process, using the sequential optimization & reliability assessment (SORA) method associated with the performance measurement approach (PMA). Thirdly, the adjoint variable method is used to obtain the sensitivity information for both uncertain and design variables, and a gradient-based algorithm is employed to solve the optimization problem. Finally, typical numerical examples are used to demonstrate the effectiveness of the proposed topology optimization method.

... 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. ...

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.

... 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.

... 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. ...

Compliant mechanisms with multiple inputs and multiple outputs have a wide range of applications in precision mechanics, e.g., cell manipulations, electronic microscopes, and etc. The movement uncoupling and maximum desired output displacements among these devices all become critical because many inputs and outputs are involved. The topology optimization design of compliant mechanisms, which can solve output coupling and input coupling problems, hinge and gray region problems, and the multiple objective optimal problem, is an important topic of researches for achieving fully decoupled motion. It is also a challenge area of research due to serious conflicts of between the four of output and input uncoupling constraints, volume constraint, the hinge‐free requirement, and the good black/white solution requirement. In order to comprehensively solve these problems, a simple optimization model overcoming these serious conflicts is posed, which includes small change rate constraints of structural compliances corresponding to the driving input loads and output point virtual loads. Then, the multiple output displacement functions of the model are equivalently converted into non‐negative functions. The multiple objective model is converted into a single‐objective optimization model by using a bound variable. The MMA (Method of Moving Asymptotes) algorithm is adopted to solve it. Several examples are presented to demonstrate the validity of the proposed method.

Additive manufacturing provides more freedom for the design of structures but also exhibits prominent local uncertainties of material properties, which bring potential challenges. The performance of structures should depend not only on uncertain variations of material properties but also on the spatial occurrence frequency of the extreme material properties. This paper proposes a non-probabilistic reliability-based topology optimization algorithm by considering local material uncertainties in additive manufacturing. The whole design domain is divided into several uncertainty regions (URs), whose size is proportional to the spatial occurrence frequency of extreme material properties. Within each UR, these uncertain-but-bounded variations of materials are correlated by the multi-dimensional ellipsoid model. Then, the multi-ellipsoid model for all URs is established by considering the overall material uncertainties of the structure. Thereafter, a non-probabilistic reliability-based topology optimization (NRBTO) is proposed for minimizing structural volume against displacement constraints by considering material uncertainties during additive manufacturing. Several 2D and 3D examples are presented to illustrate the effectiveness of the proposed method. Compared with solutions resulting from the deterministic topology optimization (DTO), NRBTO provides conservative designs with a larger volume fraction due to material uncertainties. When smaller URs are assigned to indicate the high occurrence frequency of extreme material properties, the NRBTO design becomes even conservative. The extreme case is equivalent to the deterministic topology optimization using the lower bound material.

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.

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

Static and dynamic multiobjective topology optimization of trusses with interval parameters is investigated. The uncertain parameters of the trusses are described by an interval model. The multiobjective topology optimization model of trusses with interval parameters is constructed. On the basis of Taylor expansion and natural interval extension, the stress and displacement response intervals under static loads and inherent natural frequency interval of truss are deduced. The non-deterministic optimization problem is transformed into a deterministic programming problem by minimizing maximum standards and the concept of interval possibility degree. The Pareto CMOPGA (genetic algorithm for constrained multiobjective optimization problem) embedding structural stability examination on the basis of ranking is adopted to solve the constrained multiobjective optimization problem. Two numerical examples show that the proposed method is effective and reasonable. © The Author(s) 2012 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav.

In this paper, non-probabilistic reliability indices for frequency and static displacement constraints are analyzed based on the ellipse convex model of elastic modulus and mass density. The dynamic non-probabilistic reliability-based topology optimization model of a truss is built, where the cross-sectional areas and nodal topology variables are taken as design variables. The objective is to minimize the structural total mass. Constraints are imposed on static stresses and non-probabilistic reliability indices of static displacement and natural frequency. A genetic algorithm is used as the optimization method to find optimal solutions in the outer loop and an analysis method is adopted to seek non-probabilistic reliability index according to implicit forms of the limit state function in the inner loop. Results of numerical examples show that the optimal mass of a non-probabilistic reliability-based topology optimization is larger than that of the deterministic topology optimization and the optimal mass increases with the increase of the non-probabilistic reliability requirement in order to ensure structural safety.

The present paper studies the integrated size and topology optimization of skeletal structures under natural frequency constraints. It is found that, unlike the con-ventional compliance-oriented topology optimization prob-lems, the considered problem may be strongly singular in the sense that the corresponding feasible domain may be disconnected and the global optimal solutions are often located at the tips of some separated low dimensional sub-domains when the cross-sectional areas of the structural components are used as design variables. As in the case of stress-constrained topology optimization, this unpleasant behavior may prevent the gradient-based numerical opti-mization algorithms from finding the true optimal topolo-gies. To overcome the difficulties posed by the strongly singular optima, some particular forms of area/moment of inertia-density interpolation schemes, which can restore the connectedness of the feasible domain, are proposed. Based on the proposed optimization model, the probability of find-ing the strongly singular optimum with gradient-based algo-rithms can be increased. Numerical examples demonstrate the effectiveness of the proposed approach.

Topology optimization problem requires repeated evaluations of the objective function and design sensitivity in the design domain with various density distributions. A repeated computations in the optimization process, requires a large amount of computing time and resources. These issues have inspired the development of optimization techniques combined with a system reduction. In order to reduce the system, this study employs a system dynamic condensation method based on selected primary degrees of freedom. Based on a system reduction, this study performs a topology optimization to maximize the eigenvalue and linear summation of each eigenvalue. In the optimization procedure, mode tracking method, called MAC, is used to pursue target modes, and the design sensitivity is calculated by a method of the rigid body mode separation assuring the reliability of sensitivity regardless of the design variable perturbation size. Each result of the numerical examples based on the reduction system is compared to that of the full system. Through a few numerical examples, it is demonstrated that the proposed method can provide efficient and reliable results in topology optimization.

Crashworthiness of automotive structures is most often engineered after an optimal topology has been arrived at using other design considerations. This study is an attempt to incorporate crashworthiness requirements upfront in the topology synthesis process using a mathematically consistent framework. It proposes the use of equivalent linear systems from the nonlinear dynamic simulation in conjunction with a discrete-material topology optimizer. Velocity and acceleration constraints are consistently incorporated in the optimization set-up. Issues specific to crash problems due to the explicit solution methodology employed, nature of the boundary conditions imposed on the structure, etc. are discussed and possible resolutions are proposed. A demonstration of the methodology on two-dimensional problems that address some of the structural requirements and the types of loading typical of frontal and side impact is provided in order to show that this methodology has the potential for topology synthesis incorporating crashworthiness requirements.

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.

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.

In this paper we show how the dynamic response of a multi-story structure can be improved by finding an appropriate layout for the bracing system. This is performed by the use of a topology optimization scheme while a suitable norm is defined and used for reduction of the structural response during a time interval. A consistent sensitivity analysis is presented in order to employ a mathematical programming approach. The well-known SIMP method is used for the optimization procedure. The use of SIMP in a realistic time dependent problem requires an effective removal of intermediate densities in the final topology. We show that clear layouts may be found through an approach recently proposed by the authors based on the use of a nodal based interpolation of the density and a sequence of mesh design refining. Several structural examples are given to demonstrate the performance of the approach presented.

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.

Using the extended evolutionary structural optimization method, an investigation into the effect of an initial nondesign domain on the optimal topology of a two-dimensional structure during the structural natural frequency optimization has been carried out under plane stress conditions. From the related results, it is concluded that (1) the optimal topology of a target structure produced by the problem domain of a small initial nondesign domain is obviously different from that produced by the same problem domain of a large initial nondesign domain, although the material used for the target structure is exactly the same in both cases; (2) in most cases, for the same amount of material, the material efficiency of a target structure in relation to the small nondesign domain is higher than that in relation to the large initial nondesign domain; (3) the size of the initial nondesign domain affects the minimum weight of a target structure during the structural natural frequency optimization. Thus, it is suggested that the initial nondesign domain be selected carefully to support the nonstructural lumped masses during the structural natural frequency optimization.

A practical methodology based on a topology group concept is presented for finding optimal topologies of trusses. The trusses are subjected to natural frequency, stress, displacement and Euler buckling constraints. Multiple loading conditions are considered, and a constant nodal mass is assumed for each existing node. The nodal cost as well as the member cost is incorporated in the cost function. Starting with a ground structure, a sequence of substructures with different node distribution, called topology group, is generated by using the binary number combinatorial algorithm. Before optimizing a certain topology, its meaningfulness should be examined. If a topology is meaningless, it is then excluded; otherwise, it is optimized as a sectional area optimization problem. In order to avoid a singular solution, the dimension of the structure for a given topology is kept unchanged in the optimization process by giving the member to be removed a tiny sectional area. A parabolic interpolation method is used to solve a non-linear constrained problem, which forms the part of the algorithm. The efficiency of the proposed method is demonstrated by two typical examples of truss.

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.

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.