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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.

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... Representative works were mostly limited to problems with a small number of degrees of freedom (DOFs). For example, Rong et al. [7,8] optimized the structural topology using the ESO method with stationary random responses constrained in design. Zhang et al. [9] dealt with topology optimization of multi-component structures under both static loads and stationary random excitations using density method. ...

... The beam structure has a size of 0. Here, the first l = 30 modes are employed in both examples. Notice that Structure 1 represents the common problem studied previously with a discretization of about thousands of DOFs [7][8][9]15]. Structure 2 is adopted to illustrate large-scale problems of huge numbers of DOFs that are rarely studied in dynamic topology optimization. ...

... Other problems with the similar order of DOFs can also be found in [7][8][9]15]. Therefore, both the conventional PEM and the improved PEM are effective when the number of DOFs is small. ...

... Rong et. al.[91] proposed a new topology optimization model which suppress the localized modes of vibrating structures which originates from low stiffness regions during optimizatation of material distribution.The 2D material distribution was obtained under random vibraiton load and response constraints for limited material volume fraction. ...

... al.[90] shows the effectiveness of topology optimization compared to modal strain energy method.Rong et. al.[91] finds 2D material distribution under dynamic loads. Chen and Liu ...

With the advance of technology the radar antenna structures are being smaller and their design alternatives are quite numerous that they can be produced in different shapes and can be conformed to original structures such as body panel of an aircraft or car which are composed of light weight thin shell structures. Radar antennas as an integral part of the air or ground vehicles are subjected to various dynamic loadings which effects its overall radiation pattern which results overall degradation of antenna performance, especially at high amplitude resonance conditions due to low stiffness of host structures. The passive vibration control, namely surface damping treatment methodology is one of the measurement technique that can be taken account at the initial design phase of such integration process which is based on increasing the damping capacity of host structure by adding viscoelastic materials between contacting surfaces. However adding high density materials results increase of overall weight. Therefore an extensive research activity has been carried out in order to design of surface damping treatment with spacer layer with minimum weight and maximum damping constraints. In this study, for simplicity and to verify the design methodology, a four layer cantilever beam that represents the host structure, was designed, analyzed and tested for optimum dynamic behavior. Mainly topology and parametric optimization methods are used in order to find best material layout of uniform spacer and best slotted configuration of spacer layer that maximize the damping performance of the design with minimum material condition. Experimental study is also conducted for layered cantilever beam with developed concept design of slotted configuration under vibration load to verify the methodology used.

... Sequential Quadratic Programming (SQP) method is one of the most powerful techniques for the numerical solution of constrained nonlinear optimization problems and engineering problems [45,46]. In other words, SQP is an iterative method for finding the solution of nonlinear optimization problems. ...

... Generally, for most application it can be chosen as a k 2 ½0; 1. If L is introduced as the average scale of the problem of interest, c can be chosen as 1= ffiffi ffi L p [46][47][48][49]. It should be considered that the order of computation cost of the FA is OðN 2 G N I Þ. ...

Preserving cultural heritage against earthquake and ambient vibrations can be an attractive topic in the field of vibration control. This paper proposes a passive vibration isolator methodology based on inerters for improving the performance of the isolation system of the famous statue of Michelangelo Buonarroti Pietà Rondanini. More specifically, a five-degree-of-freedom (5DOF) model of the statue and the anti-seismic and anti-vibration base is presented and experimentally validated. The parameters of this model are tuned according to the experimental tests performed on the assembly of the isolator and the structure. Then, the developed model is used to investigate the impact of actuation devices such as tuned mass-damper (TMD) and tuned mass-damper-inerter (TMDI) in vibration reduction of the structure. The effect of implementation of TMDI on the 5DOF model is shown based on physical limitations of the system parameters. Simulation results are provided to illustrate effectiveness of the passive element of TMDI in reduction of the vibration transmitted to the statue in vertical direction. Moreover, the optimal design parameters of the passive system such as frequency and damping coefficient will be calculated using two different performance indexes. The obtained optimal parameters have been evaluated by using two different optimization algorithms: the sequential quadratic programming method and the Firefly algorithm. The results prove significant reduction in the transmitted vibration to the structure in the presence of the proposed tuned TMDI, without imposing a large amount of mass or modification to the structure of the isolator.

... Being referred to the varied constraint limit approach in Refs. [26][27][28], the original optimization model (6) is transferred into the approximate model (12), which replaces the model (10). ...

... If the conventional dual method in Refs. [26][27][28] is adopted to solve the Lagrange multipliers, the second order matrix of the dual programming may become an ill matrix, so that it is very difficult to obtain the Lagrange multipliers. Here, a smooth dual solving method is proposed to solve the approximate sub-problem. ...

... Being referred to the varied constraint limit approach in Refs. [26][27][28], the original optimization model (6) is transferred into the approximate model (12), which replaces the model (10). ...

... If the conventional dual method in Refs. [26][27][28] is adopted to solve the Lagrange multipliers, the second order matrix of the dual programming may become an ill matrix, so that it is very difficult to obtain the Lagrange multipliers. Here, a smooth dual solving method is proposed to solve the approximate sub-problem. ...

Stress-related problems have not been given the same attention as the minimum compliance topological optimization
problem in the literature. Continuum structural topological optimization with stress constraints is
of wide engineering application prospect, in which there still are many problems to solve, such as the stress
concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization
method of continuum structures with the stress constraints and the objective function being the
structural volume has been presented in this paper. To solve the stress concentration issue, an approximate
stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient
constraint is constructed for the optimized structure under the r�th load case. To obtain stable convergent
series solutions and enhance the control on the stress level, two p-norm global stress constraint functions
with different indexes are adopted, and some weighting p-norm global stress constraint functions are introduced
for any load case. And an equivalent topological optimization model with reduced stress constraints is
constructed,being incorporated with the rational approximation for material properties, an active constraint
technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve
the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed,
based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level
optimization problem with artificial variables and many possible non-active design variables is proposed by
adopting an inequality constrained nonlinear programming method with simple trust regions, based on the
primal-dual theory, in which the non-smooth expressions of the design variable solutions are reformulated
as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a twolevel
optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed
to deal with the aggregation constraints that always are of loose constraint (non active constraint) features
in the conventional structural optimization method. A novel structural topological optimization method with
stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed
method is feasible and very effective.

... Being referred to the varied constraint limit approach in Refs. [26][27][28], the original optimization model (6) is transferred into the approximate model (12), which replaces the model (10). 8 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ : min W V s:t: Q f r where ...

... However, because there are design variable small trust regions x k;L v 6 x v 6 x k;U v , there may be many non-active design variables in the model (34). If the conventional dual method in Refs.[26][27][28]is adopted to solve the Lagrange multipliers, the second order matrix of the dual programming may become an ill matrix, so that it is very difficult to obtain the Lagrange multipliers. Here, a smooth dual solving method is proposed to solve the approximate sub-problem. ...

Stress related problems have not been given the same attention as the minimum compliance problem in the literature. A topological optimization method of continuum structures with the stress constraints and any initial configuration requirements has been presented in this paper. In order to greatly reduce stress sensitivity cost, all element stress constraints of the structure being optimized under a load case are replaced by the most potential active stress constraints of several design sub-domains and a generalized average stress constraint. And the whole optimization process is divided into two optimization adjustment phases and a phase transferring step operation. Firstly, an optimization model dealing with varying stress limits and design space adjustments, is built. Secondly, a procedure is proposed to solve the optimization problem of the first optimization adjustment phase. This design space adjustment capability will not affect the property of the proposed method convergence. Then, a heuristic algorithm, that is based on structural gradient information, is given to make the design structural topology be of solid/empty property during the second optimization adjustment phase. The two examples show that different initial design structure can be selected, and this method is robust and practicable.

... Compared with other nonlinear programming problems, QP problems are more attractive. Etman et al. [15] transformed the sequential convex programming subproblem into a Newton Lagrange QP subproblem, and Rong et al. [16] directly used the Hessian matrix in the MMA subproblem to construct the quadratic programming subproblem. In practice, the Hessian matrix of most optimization problems is difficult to calculate. ...

When applying the sequential quadratic programming (SQP) algorithm to topology optimization, using the quasi-Newton methods or calculating the Hessian matrix directly will result in a considerable amount of calculation, making it computationally infeasible when the number of optimization variables is large. To solve the above problems, this paper creatively proposes a method for calculating the approximate Hessian matrix for structural topology optimization with minimum compliance problems. Then, the second-order Taylor expansion transforms the original problem into a series of separable and easy-to-solve convex quadratic programming (QP) subproblems. Finally, the quadratic programming optimality criteria (QPOC) method and the QP solver of MATLAB are used to solve the subproblems. Compared with other sequential quadratic programming methods, the advantage of the proposed method is that the Hessian matrix is diagonally positive definite and its calculation is simple. Numerical experiments on an MBB beam and cantilever beam verify the feasibility and efficiency of the proposed method.

... Gomez and Spencer (2019) also developed a topology optimization framework for linear structures subjected to stationary filter white noise excitation by integration of MMA, SIMP and the Lyapunov moment equation method. More literatures on the topology optimization of linear structures under stationary random excitation can be found in (Rong et al. 2013;Zhang et al. 2015;Chun et al. 2016;He et al. 2019;Zhao et al. 2021). However, to the best knowledge of the authors, there has been no report on the topology optimization of nonlinear structures exposed to random excitation, especially in the presence of non-stationary random excitation. ...

Non-linear fluid viscous dampers have found widespread applications in engineering practice for seismic mitigation of civil structures. Topology optimization has emerged as an appealing means to achieve the optimal design of non-linear viscous dampers in terms of both layouts and parameters. However, the conventional methodologies are mainly restricted to deterministic dynamic excitations. This research is devoted to the topology optimization of non-linear viscous dampers for energy-dissipating structures with consideration of non-stationary random seismic excitation. On the basis of the equivalent linearization—explicit time-domain method (EL-ETDM), which has been recently proposed for non-stationary stochastic response analysis of non-linear systems, an adjoint variable method-based (AVM-based) EL-ETDM is further proposed for non-stationary stochastic sensitivity analysis of energy-dissipating structures with non-linear viscous dampers. The stochastic response and sensitivity results obtained by EL-ETDM with high efficiency are utilized for topology optimization of non-linear viscous dampers with the gradient-based method of moving asymptotes. The optimization problem is formulated as the minimization of the maximum standard deviation of a critical response subjected to a specified maximum number of viscous dampers, and the p-norm function is employed for approximation of the non-smooth objective function. The existence information of each potential viscous damper as well as the damper parameters are characterized by continuous design variables, and the solid isotropic material with penalization technique is utilized to achieve clear existences of viscous dampers. Two numerical examples are presented to illustrate the feasibility of the proposed topology optimization framework.

... Zhang et al. (2012) performed topology optimization for multi-component structures subjected to stationary stochastic excitations based on the density-based approach. Rong et al. (2013) further studied topology optimization under stationary stochastic response constraints using the Sequential Quadratic Programming (SQP) method. However, all these works employed the conventional Complete Quadratic Combination (CQC) method (Wilson, Der Kiureghian, and Bayo 1981) to evaluate the variance of the stationary random responses in the topology optimization process. ...

Topology optimization subjected to stochastic excitations is usually computationally expensive. This work addresses this challenge by first transforming stochastic response and sensitivity analysis problems at each iteration into a series of frequency response analysis problems, and then an adaptive hybrid expansion method is employed to compute the displacement and adjoint vectors. The proposed method has two features: (1) the derivatives of the eigenpairs are not involved when computing the sensitivities of the responses, thus no special treatment is needed when repeated eigenfrequencies are present; (2) the numbers of lower-order eigenvectors and basis vectors that need to be computed can be determined adaptively according to the given accuracy. The accuracy and efficiency of the proposed method as well as the effect of the excitation frequency on the optimum designs are demonstrated by three numerical examples.

... Recently, topology optimisation subject to random excitations has attracted more and more attention due to the practical engineering requirements for complex working conditions, such as seismic hazards and wind damages. For example, the topology optimisation for continuum structures (Zhang et al., 2015;Yang et al., 2017;Bobby et al., 2017;Rong et al., 2013) and outriggers applied in structures under stationary random excitations are studied. To meet the actual working conditions, the layout optimisation of multi-component structures considered both static loads and white noise stochastic loads is investigated (Zhang et al., 2012). ...

... Recently, topology optimisation subject to random excitations has attracted more and more attention due to the practical engineering requirements for complex working conditions, such as seismic hazards and wind damages. For example, the topology optimisation for continuum structures (Zhang et al., 2015;Yang et al., 2017;Bobby et al., 2017;Rong et al., 2013) and outriggers applied in structures under stationary random excitations are studied. To meet the actual working conditions, the layout optimisation of multi-component structures considered both static loads and white noise stochastic loads is investigated (Zhang et al., 2012). ...

... In order to ensure the safety of engineering structure, it is necessary to consider the dynamic effect in topology optimization. Up to now, most dynamic topology optimizations dealt with the dynamic compliance minimization or dynamic displacement response minimization of continuum structure [13][14][15][16]. Recently, Zhao et al. proposed a methodology for maximizing dynamic stress response reliability of continuum structures involving multi-phase materials on the macro scale [17]. ...

This paper proposes a concurrent topology optimization method of macrostructural material distribution and periodic microstructure considering dynamic stress response under random excitations. The optimization problem is the minimization of the dynamic stress response of the macrostructure subject to volume constraints in both macrostructure and microstructure. To ensure the safety of the macrostructure, a new relaxation method is put forward to establish a relationship between the dynamic stress limit and the mechanical properties of microstructure. The sensitivities of the dynamic stress response with respect to the design variables in two scales, i.e., macro and micro scales, are derived. Then, the aforementioned optimization problem is solved by the bi-directional evolutionary structural optimization (BESO) method. Finally, several numerical examples are presented to demonstrate the feasibility and effectiveness of the proposed method.

... Zhang and Kang et al. investigated the optimal distribution of damping material in vibrating structures [31,32]. Rong et al. investigated continuum structural topological optimization with dynamic displacement response constraints under random excitations [33]. Zhang et al. researched structural topology optimization related to dynamic responses under stationary random force excitation by integrating the pseudo excitation method (PEM) with mode acceleration method (MAM) [34]. ...

... Although the compliance minimization problem with a density filter is a convex problem for the penalization power exponent p = 1 [31,37], a lot of example simulations made by using a MMA approximated-approximation [56] for the compliance have illustrated that the approximate twoorder derivatives of structural compliance with respect to design variables at some design points with partial small relative density variable values for the penalization power exponent p > 1 [48,49,[57][58][59][60][61], may be negative. And the smooth Heaviside projections inherently possess a nonlinear property. ...

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.

... In the element-based topology optimization method, the optimization procedure is to find the topology of a structure by determining for every point in the design domain whether there should be material (solid element) or not (void element). This type of method including homogenization method ( [3]), solid isotropic material with penalization (SIMP) method ( [4,8]), evolutionary structural optimization (ESO) method ( [6]), and its improved algorithm C bidirectional evolutionary structural optimization (BESO) method ( [9][10][11][12]) has tackled various problems, including uncertain design ( [13][14][15][16][17]), dynamic problems ( [18][19][20]), and designing metamaterials ( [21]). The aforementioned elementbased topology optimization methods have applications successfully in a number of fields. ...

Optimal geometries extracted from traditional element-based topology optimization outcomes usually have zigzag boundaries, leading to being difficult to fabricate. In this study, a fairly accurate and efficient topology description function method (TDFM) for topology optimization of linear elastic structures is developed. By employing the modified sigmoid function, a simple yet efficient strategy is presented to tackle the computational difficulties because of the nonsmoothness of Heaviside function in topology optimization problem. The optimization problem is to minimize the structural compliance, with highest stiffness, while satisfying the volume constraint. The design problem is solved by a Sequential Linear Programming method. Convergent, crisp, and smooth final layouts are obtained, which can be fabricated without postprocessing, demonstrated by a series of numerical examples. Further, the proposed method has a rather higher accuracy and efficiency compared with traditional TDFM, when the classical topology optimization methods, such as bidirectional evolutionary structural optimization (BESO) and solid isotropic material with penalization (SIMP) method, are taken as benchmark.

... The optimization presented in Equation (3) can be solved by utilizing many methods, e.g., the method of moving asymptotes (MMA) [43], optimality criterion (OC) method [44], linear or sequential quadratic programming methods [45,46], the ESO method [47,48], etc. In this study, Solid Isotropic Material with Penalty (SIMP)and OC methods were used to optimize the AS, in which OC is the solver. ...

In this study, a lightweight sandwich aircraft spoiler (AS) with a high stiffness-to-weight ratio was designed. Excellent mechanical properties were achieved by the synthetic use of topology optimization (TO), lattice structure techniques, and high-performance materials, i.e., titanium alloy and aluminum alloy. TO was first utilized to optimize the traditional aircraft spoiler to search for the stiffest structure with a limited material volume, where titanium alloy and aluminum alloy were used for key joints and other parts of the AS, respectively. We then empirically replaced the fine features inside the optimized AS with 3D kagome lattices to support the shell, resulting in a lightweight sandwich AS. Numerical simulations were conducted to show that the designed sandwich AS exhibited good mechanical properties, e.g., high bending rigidity, with a reduction in weight by approximately 80% when compared with that of the initial design model. Finally, we fabricated the designed model with photosensitive resin using a 3D printing technique.

... On one hand, considerable effort has been made for the frequency optimization with a single excitation frequency by minimizing the dynamic compliance [42][43][44][45][46][47]. On the other hand, it is also important to improve dynamic performance for a structure that is subject to an excitation frequency range [48][49][50][51][52]. Particularly, when the excitation force is within a wide frequency range, the model reduction (MR) technique [48,49,52] is often required to reduce the degree of freedoms of the finite element (FE) system, so as to alleviate the prohibitive computation cost caused by the repetitive FE simulations during each iteration. ...

This paper proposes a new multiscale topology optimization method for the concurrent design of multiphase composite structures under a certain range of excitation frequencies. Distinguished from the existed studies, a general concurrent design formulation for the dynamic composite structures with more than two material phases is developed. The macro structure and its microstructures with multiple material phases are optimized simultaneously. The integral of the dynamic compliances over an interval of frequencies is formulated as the optimization objective, so as to minimize the frequency response within the concerned excitation range. The effective properties of the multiphase microstructures are evaluated by using the numerical homogenization method, which actually serves as a link to bridge the macro and micro finite element analyses. Furthermore, to describe the boundaries of multiple material phases for the microstructure, a parametric color level set method (PCLSM) is developed by using an efficient interpolation scheme. In this way, L level set functions can represent at most 2L material phases without any overlaps. Moreover, these “color” level sets are updated by directly using the well-established gradient-based algorithm, which can greatly facilitate the proposed method to solve the multi-material optimizations with multiple design constraints. Several 2D and 3D numerical examples are used to demonstrate the effectiveness of the proposed method in the concurrent design of the dynamic composite structures under the excitation frequency ranges.

... This method has been widely applied in the structural dynamics 56,57 and topology optimization. [58][59][60][61] We will investigate its performance in the topology optimization for acoustic-structure interaction problems. Some basic formulations are presented here. ...

The cover image, by Wenchang Zhao et al., is based on the Research Article Topology optimization of exterior acoustic ‐ structure interaction systems using the coupled FEM ‐ BEM method. https://doi.org/10.1002/nme.6055.

... Meanwhile, the existing results have mainly focused on the topology optimization under random force excitations. For example, Rong 17 and Rong 18 et al. presented the topology optimization of continuous structures using ESO method where stationary random responses were regarded as design constraints. Yang et al. 19 studied the topology optimization under static loads and narrow-band random excitations, where the static and dynamic response analyses were processed independently without any superposition. ...

Structural topology optimization subjected to stationary random base acceleration excitations is investigated in this paper. In the random response analysis, the Large Mass Method (LMM) which attributes artificial large mass values at each driven nodal Degree Of Freedom (DOF) to transforming the base acceleration excitations into force excitations is proposed. The Complete Quadratic Combination (CQC) which is commonly used to calculate the random responses in previous optimization has been proven to be computationally expensive especially for large-scale problems. In order to conquer this difficulty, the Pseudo Excitation Method (PEM) and the combined method of PEM and Mode Acceleration Method (MAM) are adopted into the dynamic topology optimization, and random responses are calculated using these two methods to ascertain a high efficiency over the CQC. A density-based topology optimization method minimizing dynamic responses is then formulated based on the integration of LMM and PEM or the combined method of PEM and MAM. Numerical examples are presented to verify the accuracy of the proposed schemes in dynamic response analysis and the quality of the optimized design in improving dynamic performance.

... The optimization presented in Eq. (3) can be solved by utilizing many methods, e.g., the method of moving asymptotes (MMA) [40], optimality criterion (OC) method [41], linear or sequential quadratic programming method [42][43][44], and ESO method [45], et al. ...

By combing continuum topology optimization (TO) method and lattice structure technique, a sandwich aircraft spoiler with a high stiffness-to-weight is designed. TO method is served to produce the shell of the aircraft spoiler and the lattice structure, used as cores, is employed to support the shell. TO problem is established as maximizing the stiffness of the structure with limited material volume. Density-based method is utilized to achieve a 0/1 solution. We then empirically replace the core of the aircraft spoiler by using 3D kagome lattice structure. Two different materials, i.e., aluminum alloy and titanium alloy, are synthetically applied to further reduce the weight and simultaneously improve the strength of the aircraft spoiler. Numerical simulations are conducted to show that the designed aircraft spoiler can meet the service environment with a reduction of its weight by approximately 80% when compared with that of the initial design model. Finally, we have fabricated the designed model with photosensitive resin by using 3D printing technique.

... For multi-frequencies problems, a more efficient solution technique is to map the finite element model into a reduced space, i.e., the mode superposition method. This method has been widely applied in the structural dynamics [56,57] and topology optimization [58,59,60,61]. We will investigate its performance in the topology optimization for acoustic-structure interaction problems. ...

This paper presents an efficient topology optimization procedure for exterior acoustic–structure interaction problems, in which the coupled systems are formulated by the boundary element method (BEM) and the finite element method (FEM). So far, the topology optimization based on the coupled FEM‐BEM still faces several issues needed to be addressed, especially the efficient design sensitivity analysis for the coupled systems. In this work, we contribute to these issues in two main aspects. Firstly, the adjoint variable method (AVM) formulations are derived for sensitivity analysis of arbitrary objective function, and the feedback coupling between the structural and acoustic domains are taken into consideration in the sensitivity analysis. Secondly, in addition to the application of fast multipole method (FMM) in the acoustic BEM response analysis, the FMM is now updated to adapt to the arising different multiplications in the AVM equations. These accelerations save considerable computing time and memory. Numerical tests show that the developed approach permits its application to large‐scale problems. Finally, some basic observations for the optimized designs are drawn from the numerical investigations.

... Yoon [54] compared the performances of different MR schemes in dynamic structural analysis. More recently, Shu et al. [55], Rong et al. [56] and Liu et al. [57] also investigated the frequency response optimization problems subject to excitation frequency ranges. Nevertheless, most studies are based on the elemental density distribution approaches, which may have difficulties in capturing the structural boundary in geometry. ...

In conventional parametric level set methods, the compactly supported radial basis functions (CSRBF) are used to approximate the level set function due to their unique properties, such as the sparsity of the interpolation matrix. The CSRBFs only consider the contributions of knots within a narrow sub-region, which sacrifices accuracy for efficiency in the interpolation. However, the accuracy loss in the CSRBF-based method may prolong
the iteration and gradually lead the topology optimization towards a worse local optimum or even an unfeasible design, especially when the allowable material usage in the design domain is relatively low. This will significantly affect the performance of the optimization method. This paper proposes an improved parametric level set method (iPLSM), which is more efficient and effective in topology optimization designs. In this method, the Gaussian radial basis function with global support is used to parameterize the level set surface, to ensure a high numerical accuracy due to the consideration of all interpolation knots in the global domain. Then, a discrete wavelet transform scheme is incorporated into the parametric form to compress the full interpolation matrix and save the computational cost. The proposed method is applied to both the global and local frequency response optimization problems under wide excitation frequency ranges, to validate its efficiency and effectiveness.

... Kang et al. investigated the optimal distribution of damping material in vibrating structures subject to harmonic excitations by using topology optimization method (Kang et al. 2012). 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). ...

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.

... Only a limited number of works have been devoted to the topology optimization of the structures with random excitation. Rong et al. [15] used the ESO method and the sequential quadratic programming (SQP) method to optimize continuum structures under random excitations. Zhang et al. [16] investigated the optimal placements of the components and the configuration of the structure to improve the structural static and random dynamic responses simultaneously. ...

This paper deals with an optimal layout design of the constrained layer damping (CLD) treatment of vibrating structures subjected to stationary random excitation. The root mean square (RMS) of random response is defined as the objective function as it can be used to represent the vibration level in practice. To circumvent the computationally expensive sensitivity analysis, an efficient optimization procedure integrating the pseudoexcitation method (PEM) and the double complex modal superposition method is introduced into the dynamic topology optimization. The optimal layout of CLD treatment is obtained by using the method of moving asymptote (MMA). Numerical examples are given to demonstrate the validity of the proposed optimization procedure. The results show that the optimized CLD layouts can effectively reduce the vibration response of the structures subjected to stationary random excitation.

... In fact, a number of investigations have been made by integrating random vibration theories with topology optimization. Rong et al. (2000;2013) used the evolutionary structural optimization method and sequential quadratic programming to get optimal topology designs of continuum structures subjected to stationary stochastic excitations. Zhang et al. (2012) studied the topology optimization problem of multicomponent structures subjected to static and stationary stochastic excitations. ...

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.

... Although the sparsity pattern of A and sparse direct solver are considered, the solution of Ax = b is still computationally expensive. A more efficient solution technique is to map the system equations of the finite element model into a reduced space, which has been widely applied in topology optimization (e.g., Zhang and Kang 2013;Rong et al. 2013;Liu et al. 2015;Zhu et al. 2017). ...

A topology optimization approach is proposed for the optimal design of bi-material distribution on underwater shell structures. The coupled finite element method (FEM) / boundary element method (BEM) scheme is used for the system response analysis, where the strong interaction between the structural and the acoustic domain is considered. The Burton-Miller formulation is used to overcome the fictitious eigen-frequency problem when using a single Helmholtz boundary integral equation for exterior acoustic problems. The design variables are the artificial densities of design material elements in a bi-material model constructed by the solid isotropic material with penalization (SIMP) method, and the minimization of sound power level (SWL) is chosen to be the design objective. In this study, the adjoint operator method is employed to calculate the sensitivity of the objective function with respect to the design variables. Based on the sensitivity information, the gradient-based optimization solver is finally applied for updating the design variables during the optimization process. Numerical tests are provided to illustrate the correctness of the sensitivity analysis approach and the validity of the proposed optimization procedure. Results show that the heavy fluid feedback has a big impact on the final design, and thus it is necessary to conduct a strong coupling scheme between the fluid and structures. In addition, the optimal design is strongly frequency dependent, and performing an optimization in a frequency band is generally needed.

... Qiao et al. [43] considered both static loads and white noise stochastic loads for the layout optimization of multi-component structures. Rong et al. [44] developed Sequential Quadratic Programming method for the topology optimization of structures under random white noise excitation. ...

This work addresses the challenge of topology optimization for continuum structures subjected to non-white noise stochastic excitations. The objective is to manipulate the variance of the stochastic dynamic response of a linear continuum structure through design. The excitation is modeled as filtered white noise and the solution and sensitivities are obtained using the augmented state space formulation in the frequency domain. To reduce the computational cost and maintain high accuracy, closed-form solutions are utilized for evaluating the response variance and modal truncation is adopted to reduce dimension (the effect of which is investigated). The Solid Isotropic Material with Penalization Method is used together with the Heaviside Projection Method to achieve a clear distinction between solid and void regions in the structure, and the gradient-based optimizer Method of Moving Asymptotes informed by sensitivities of the real and complex eigenvectors is used to evolve the design. The algorithm is demonstrated on design problems of minimizing the variance of stochastic response under a volume constraint and several numerical examples are provided to show the effectiveness of the method. Solutions are compared to conventional maximum stiffness solutions and, as to be expected, show markedly improved response under the considered stochastic dynamic excitation.

... Qiao et al.[44]considered both static loads and white noise stochastic loads for the layout optimization of multi-component structures. Rong et al.[45]also developed a sequential quadratic programming (SQP) method for topology optimization of structures under white noise excitation. Those that are not confined to white noise excitations, such as the works by Taflanidis and Scruggs[46]and Gidaris and Taflanidis[47]do not explicitly consider the structural topology itself in the optimization. ...

... Therefore, the dynamic response based topology optimization has been considered [21]. Rong et al. [22,23] used the ESO method and SQP method to obtain the optimal topology of the continuum structures under random excitations. Zhang et al. [24] proposed an efficient optimization procedure integrating pseudo excitation method and mode acceleration method for the topology optimization of large-scale structures subjected to stationary random excitation. ...

... And then the topology optimization considering dynamic response has gradually attracted attention. Rong et al. [24,25] investigated the topology optimization of continuum structure under random excitations using BESO method and a sequential quadratic programming (SQP) method. Jog [26] pro-posed the global and the local measures for the minimization of vibrations of structures subjected to periodic loading by topology optimization. ...

... 1988 年， Bendsøe 和 Kikuchi [1] 将均匀化理论引 入连续体结构的拓扑优化设计中，提出了开创性的 结构拓扑优化方法。随后又发展出了材料密度惩罚 法(SIMP) [2][3] 、渐进优化方法(ESO) [4] 、独立连续映 射方法(ICM) [5][6][7] 以及水平集方法(LSM) [8] 等。 结构拓扑优化理论的发展，也为工程中的振动 控制问题提供了更为有效的解决途径。荣见华等 [9] 研究了随机激励下结构拓扑优化的 SQP 方法， 杨锐 振和杜建镔等 [10] 对微结构的相关声学性能进行优 化，朱继宏等 [11] 对频率优化过程中结构的局部模态 问题进行了研究分析，郑玲等 [12] 研究了结构的模态 阻尼比及相关的优化问题。振动结构优化研究还处 理了最小化结构重量 [13][14] ， 优化结构刚度 [15] 以及基 频和高阶频率 [16] 等目标。在结构动力学性能拓扑优 化研究中，大多数工作以结构稳态响应为优化目 标 [17][18] Fig.1 A thin-plate structure with surface damping material 假定阻尼材料层和基体板为理想连接，这一假 定在阻尼材料层的拓扑优化问题中经常被使用 [22] ...

This paper investigates the optimal distribution of damping material in thin-plate structures considering transient response by using structural topology optimization method. Therein, an artificial damping penalty model similar to the SIMP model is adopted. In the topology optimization model, the relative densities of the damping material are taken as design variables and the volume constraint of damping material is considered. The design objective is to minimize the time integration of the structural transient response at specified positions. Since the structure exhibits a non-proportional damping effect, the vibration equations of the structure are solved by the time integration method. The design sensitivities of the vibrating structure under applied loads are calculated using the adjoint variable method. Then the topology optimization problem is solved with the method of moving asymptote algorithm, which is a gradient-based method. Numerical examples are presented for demonstrating the validity and effectiveness of the proposed optimization model and numerical techniques.

... Then the size of topological design was further optimized under the constraints of dynamic responses, which avoid the difficulties of the direct dynamic topology optimization. The design sensitivities of the dynamic responses were derived within the methodologies of evolutionary structure optimization (ESO) techniques [98,99]. Clear structural topologies can be obtained by removing inefficient material from the design domain iteratively. ...

Topology optimization has become an effective tool for least-weight and performance design, especially in aeronautics and aerospace engineering. The purpose of this paper is to survey recent advances of topology optimization techniques applied in aircraft and aerospace structures design. This paper firstly reviews several existing applications: (1) standard material layout design for airframe structures, (2) layout design of stiffener ribs for aircraft panels, (3) multi-component layout design for aerospace structural systems, (4) multi-fasteners design for assembled aircraft structures. Secondly, potential applications of topology optimization in dynamic responses design, shape preserving design, smart structures design, structural features design and additive manufacturing are introduced to provide a forward-looking perspective.

... Among early works of topology optimization considering dynamic response are those by Diaz and Kikuchi [8], Ma et al. [9,10] and Xie and Steven [11,12]. Since then, many other studies have been carried out to develop topology optimization techniques for dynamic problems [13][14][15][16][17][18][19][20][21][22]. ...

... The development of topology optimization techniques considering dynamic loading is very limited. Structural optimization with dynamic loading is important due to the fact that dynamic response is likely to happen in reality [31][32][33]. External dynamic forces acting on structures, such as flying aircraft, moving high-velocity train, submarines etc., are generally wide band random. ...

... Topology optimization enables designers to find a suitable structural layout for the required performance. It has attracted considerable attention over the past decades and many different techniques such as the homogenization method [7], Solid Isotropic Material with Penalization (SIMP) method [8][9][10], level-set method [11,12] and evolutionary structural optimization (ESO) method [13,14] and others [15][16][17] have been developed. The ESO method is based on a simple concept that inefficient material is gradually removed from the design domain so that the resulting topology evolves toward an optimum. ...

This paper develops a bi-directional evolutionary structural optimization (BESO) method for topological design of compliant mechanisms. The design problem is reformulated as maximizing the flexibility of the compliant mechanism subject to the mean compliance and volume constraints. Based on the finite element analysis, a new BESO algorithm is established for solving such an optimization problem by gradually updating design variables until a convergent solution is obtained. Several 2D and 3D examples are presented to demonstrate the effectiveness of the proposed BESO method. A series of optimized mechanism designs with or without hinge regions are obtained. Numerical results also indicate that the flexibility and hinge-related property of the optimized compliant mechanisms can be controlled by the desired structural stiffness.

Although sensitivity analysis provides valuable information for structural optimization, it is often difficult to use the Hessian in large models since many methods still suffer from inaccuracy, inefficiency, or limitation issues. In this context, we report the theoretical description of a general sensitivity procedure that calculates the diagonal terms of the Hessian matrix by using a new variant of hyper‐dual numbers as derivative tool. We develop a diagonal variant of hyper‐dual numbers and their arithmetic to obtain the exact derivatives of tensor‐valued functions of a vector argument, which comprise the main contributions of this work. As this differentiation scheme represents a general black‐box tool, we supply the computer implementation of the hyper‐dual formulation in Fortran. By focusing on the diagonal terms, the proposed sensitivity scheme is significantly lighter in terms of computational costs, facilitating the application in engineering problems. As an additional strategy to improve efficiency, we highlight that we perform the derivative calculation at the element‐level. This work can contribute to many studies since the sensitivity scheme can adapt itself to numerous finite element formulations or problem settings. The proposed method promotes the usage of second‐order optimization algorithms, which may allow better convergence rates to solve intricate problems in engineering applications.

This paper focuses on the topological optimization of structures subjected to stationary random excitations. A new topology optimization scheme based on the pseudo excitation method (PEM) for calculating structural random responses in a frequency domain is proposed. In this method, the Sturm sequence is applied to adaptively determine the number of lower-order modes used for mode superposition analysis. The contribution of unknown higher-order modes is approximated by the partial sum of a constructed convergent series. Since the method can offer an approximate expression of structural response solutions, not only it can enhance the flexibility of implementation and also improve the computational effort and accuracy. In addition, derivatives of the objective function are derived by means of the adjoint method. They can be achieved by solving an adjoint problem that is similar to the original governing equation of the system. Two illustrative examples are presented to affirm the proposed scheme in terms of computational accuracy and efficiency.

This work develops some foundations of topology optimization for the robust design of structural systems subjected to general stationary stochastic dynamic loads. Three methods are explored to evaluate the dynamic response – the time domain, frequency domain, and state space methods – and the associated design variable sensitivities are derived analytically. The resulting stochastic dynamic topology optimization problem is solved using the gradient-based optimizer Method of Moving Asymptotes (MMA). Sensitivities are computed using the adjoint method and the popular Solid Isotropic Material with Penalization (SIMP) is used to achieve clear existence of structural members. The approach is used to design the lateral load systems of structures that minimize the variance of the system response to stationary stochastic ground motion excitation. Numerical results are presented to illustrate the differences between topologies optimized for stochastic ground motion and topologies optimized for equivalent static loading.

This paper presents a study on maximizing single and multiple modal damping ratios (MDR) of a structure by finding the optimal layouts of damping and/or base materials using an extended moving iso-surface threshold (MIST) topology optimization. Firstly, by using the Lagrange’s equation, a general formulation of loss factor is derived and then used to extract MDR for classical and non-classical damping. Secondly, in the extended MIST, new general formulations for the physical response functions of individual mode are derived for the classical non-local damping and the non-classical damping, in which for non-classical damping modal strain energy (MSE) is used to estimate MDR and derive physical response function. Thirdly, to maximize multiple MDRs simultaneously or overcome the mode switching issue, a multi-objective optimization strategy is developed; and a concurrent design optimization of both base and damping layer is proposed to maximize MDR. Finally several numerical examples are presented to validate and illustrate the efficiency of the present extended MIST approach.

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.

Damping treatments have been extensively used as a powerful means to damp out structural resonant vibrations. Usually, damping materials are fully covered on the surface of plates. The drawbacks of this conventional treatment are also obvious due to an added mass and excess material consumption. Therefore, it is not always economical and effective from an optimization design view. In this paper, a topology optimization approach is presented to maximize the modal damping ratio of the plate with constrained layer damping treatment. The governing equation of motion of the plate is derived on the basis of energy approach. A finite element model to describe dynamic performances of the plate is developed and used along with an optimization algorithm in order to determine the optimal topologies of constrained layer damping layout on the plate. The damping of visco-elastic layer is modeled by the complex modulus formula. Considering the vibration and energy dissipation mode of the plate with constrained layer damping treatment, damping material density and volume factor are considered as design variable and constraint respectively. Meantime, the modal damping ratio of the plate is assigned as the objective function in the topology optimization approach. The sensitivity of modal damping ratio to design variable is further derived and Method of Moving Asymptote (MMA) is adopted to search the optimized topologies of constrained layer damping layout on the plate. Numerical examples are used to demonstrate the effectiveness of the proposed topology optimization approach. The results show that vibration energy dissipation of the plates can be enhanced by the optimal constrained layer damping layout. This optimal technology can be further extended to vibration attenuation of sandwich cylindrical shells which constitute the major building block of many critical structures such as cabins of aircrafts, hulls of submarines and bodies of rockets and missiles as an invaluable design tool.

This paper investigates the optimal distribution of damping material in vibrating structures subject to harmonic excitations by using topology optimization method. Therein, the design objective is to minimize the structural vibration level at specified positions by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is suggested and the relative densities of the damping material are taken as design variables. The vibration equation of the structure has a non-proportional damping matrix. A system reduction procedure is first performed by using the eigenmodes of the undamped system. The complex mode superposition method in the state space, which can deal with the non-proportional damping, is then employed to calculate the steady-state response of the vibrating structure. In this context, an adjoint variable scheme for the response sensitivity analysis is developed. Numerical examples are presented for illustrating validity and efficiency of this approach. Impacts of the excitation frequency as well as the damping coefficients on topology optimization results are also discussed.

This paper demonstrates a novel two-phase approach to the preliminary structural design of grid shell structures, with the objective of material minimization and improved structural performance. The two-phase approach consists of: (i) a form-finding technique that uses dynamic relaxation with kinetic damping to determine the global grid shell form, (ii) a genetic algorithm optimization procedure acting on the grid topology and nodal positions (together called the ‘grid configuration’ in this paper). The methodology is demonstrated on a case study minimizing the mass of three 24 × 24 m grid shells with different boundary conditions. Analysis of the three case studies clearly indicates the benefits of the coupled form-finding and grid configuration optimization approach: material mass reduction of up to 50% is achieved.

Bi-directional Evolutionary Structural Optimization (BESO) is a well-established topology optimization technique. This method is used in this paper to optimize the shape of a passive energy dissipater designed for earthquake risk mitigation. A previously proposed shape design of a steel slit damper (SSD) device is taken as the initial design and its shape is optimized using a slightly modified BESO algorithm. Some restrictions are imposed to maintain simplicity and to reduce fabrication cost. The optimized shape shows increased energy dissipation capacity and even stress distribution. Experimental verification has been carried out and proved that the optimized shape is more resistant to low-cycle fatigue.

A topology optimization based approach is proposed to study the optimal configuration of stiffeners for the interior sound reduction. Since our design target is aimed at reducing the low frequency noise, a coupled acoustic-structural conservative system without damping effect is considered. Modal analysis method is used to evaluate the interior sound level for this coupled system. To formulate the topology optimization problem, a recently intro- duced Microstructure-based Design Domain Method (MDDM) is employed. Using the MDDM, the optimal stiffener configurations problem is treated as a material distribution problem and sensitivity analysis of the coupled system is derived analytically. The norm of acoustic excitation is used as the indicator of the interior sound level. The optimal stiff- ener design is obtained by solving this topology optimization problem using a sequential convex approximation method. Examples of acoustic box under single frequency excita- tion and a band of low frequency excitations are presented and discussed. @DOI: 10.1115/1.1569512#

Optimization algorithms based on convex separable approximations for optimal structural design often use reciprocal-like approximations in a dual setting; CONLIN and the method of moving asymptotes (MMA) are well-known examples of such sequential convex programming (SCP) algorithms. We have previously demonstrated that replacement of these nonlinear (reciprocal) approximations by their own second order Taylor series expansion provides a powerful new algorithmic option within the SCP class of algorithms. This note shows that the quadratic treatment of the original nonlinear approximations also enables the restatement of the SCP as a series of Lagrange-Newton QP subproblems. This results in a diagonal trust-region SQP type of algorithm, in which the second order diagonal terms are estimated from the nonlinear (reciprocal) intervening variables, rather than from historic information using an exact or a quasi-Newton Hessian approach. The QP formulation seems particularly attractive for problems with far more constraints than variables (when pure dual methods are at a disadvantage), or when both the number of design variables and the number of (active) constraints is very large.

A T junction in a photonic crystal waveguide is designed with the topology-optimization method. The gradient-based optimization tool is used to modify the material distribution in the junction area so that the power transmission in the output ports is maximized. To obtain high transmission in a large frequency range, we use an active-set strategy by using a number of target frequencies that are updated repeatedly in the optimization procedure. We apply a continuation method based on artificial damping to avoid undesired local maxima and also introduce artificial damping in a penalization scheme to avoid nondiscrete properties in the design domain.

This paper deals with topological design optimization of vibrating bi-material elastic structures placed in an acoustic medium.
The structural vibrations are excited by a time-harmonic external mechanical surface loading with prescribed excitation frequency,
amplitude and spatial distribution. The design objective is minimization of the sound pressure generated by the vibrating
structures on a prescribed reference plane or surface in the acoustic medium. The design variables are the volumetric densities
of material in the admissible design domain for the structure. A high frequency boundary integral equation is employed to
calculate the sound pressure in the acoustic field. This way the acoustic analysis and the corresponding sensitivity analysis
can be carried out in a very efficient manner. The structural damping is considered as Rayleigh damping. Penalization models
with respect to the acoustic transformation matrix and/or the damping matrix are proposed in order to eliminate intermediate
material volume densities, which have been found to appear obstinately in some of the high frequency designs. The influences
of the excitation frequency and the structural damping on optimum topologies are investigated by numerical examples. Also,
the problem of maximizing (rather than minimizing) sound pressures in points on a reference plane in the acoustic medium is
treated. Many interesting features of the examples are revealed and discussed.
KeywordsStructural topology optimization-Harmonic dynamic loading-Minimization and maximization of sound pressure -Acoustic medium-High frequency approximation-Acoustic transformation matrix-Structural damping-Penalization model

This paper addresses a novel method of topology and shape optimization. The basic idea is the iterative positioning of new holes (so-called bubbles) into the present structure of the component. This concept is therefore called the bubble method. The iterative positioning of new bubbles is carried out by means of different methods, among others by solving a variational problem. The insertion of a new bubble leads to a change of the class of topology. For these different classes of topology, hierarchically structured shape optimizations that determine the optimal shape of the current bubble, as well as the other variable boundaries, are carried out.

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.

Based on the modal truncation method and some approximate treatments, a structural mean square dynamic response expression under narrow-band random excitations as well as their sensitivities are derived. According to engineering practical requirements, a set of topology optimization formula and its procedure, with static loads and narrowband random excitations on the structure, are built. Numerical examples show that the method proposed is of good rationality and efficiency in engineering application.

The evolutionary structural optimization (ESO) and basic ESO (BESO) processes and given various illustrative examples are described. Such processes are based on the concept of slowly removing inefficient materials from a structures so that the residual structure evolves towards the optimum. It is shown that the simple ESO and BESO algorithms are capable of solving a wide range of shape and topology optimization problems.

In the past, the possibilities of structural optimization were restricted to an optimal choice of profiles and shape. Further improvement can be obtained by selecting appropriate advanced materials and by optimising the topology, i.e. finding the best position and arrangement of structural elements within a construction. The optimization of structural topology permits the use of optimization algorithms at a very early stage of the design process. The method presented in this book has been developed by Martin Bendsoe in co-operation with other researchers and can be considered as one of the most effective approaches to the optimization of layout and material design.

Aim of this work is the maximization of the fundamental eigenfrequency of 2D bodies made of micropolar (or Cosserat) materials using a topology optimization approach. A classical SIMP–like model is used to approximate the constitutive parameters of the micropolar medium. A suitable penalization is introduced for both the linear and the spin inertia of the material, to avoid the occurrence of undesired local modes. The robustness of the proposed procedure is investigated through numerical examples; the influence of the material parameters on the optimal material layouts is also discussed. The optimal layouts for Cosserat solids may differ significantly from the truss–like solutions typical of Cauchy solids, as the intrinsic flexural stiffness of the material can lead to curved beam-like material distributions. The numerical simulations show that the results are quite sensitive to the material characteristic length and the spin inertia.

This paper implements an evolutionary operations based global optimization algorithm for the minimum cost design of a two span continuous prestressed concrete (PC) I-girder bridge structure. Continuity is achieved by applying additional deck slab reinforcement in negative flexure zone. The minimum cost design problem of the bridge is characterized by having a nonlinear constrained objective function, and a combination of continuous, discrete and integer design variables. A global optimization algorithm called EVolutionary OPeration (EVOP), is used which can efficiently solve the presented constrained minimization problem. Minimum cost design is achieved by determining the optimum values of 13 numbers of design variables. All the design constraints for optimization belong to AASHTO Standard Specifications. The paper concludes that the robust search capability of EVOP algorithm has efficiently solved the presented structural optimization problem with relatively small number of objective function evaluation. Minimum design achieved by application of this optimization approach to a practical design example leads to around 36% savings in cost.

It is often hard to optimise constrained layer damping (CLD) for structures more complicated than simple beams and plates as its performance depends on its location, the shape of the applied patch, the mode shapes of the structure and the material properties. This paper considers the use of cellular automata (CA) in conjunction with finite element analysis to obtain an efficient coverage of CLD on structures. The effectiveness of several different sets of local rules governing the CA are compared against each other for a structure with known optimum coverage---namely a plate. The algorithm which attempts to replicate most closely known optimal configurations is considered the most successful. This algorithm is then used to generate an efficient CLD treatment that targets several modes of a curved composite panel. To validate the modelling approaches used, results are also presented of a comparison between theoretical and experimentally obtained modal properties of the damped curved panel.

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.

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.

We study the ‘classical’ discrete, solid-void or black-and-white topology optimization problem, in which minimum compliance is sought, subject to constraints on the available material resource. We assume that this problem is solved using methods that relax the discreteness requirements during intermediate steps, and that the associated programming problems are solved using sequential approximate optimization (SAO) algorithms based on duality. More specifically, we assume that the advantages of the well-known Falk dual are exploited. Such algorithms represent the state-of-the-art in (large-scale) topology optimization when multiple constraints are present; an important example being the method of moving asymptotes (MMA).We depart by noting that the aforementioned SAO algorithms are invariably formulated using strictly convex subproblems. We then numerically illustrate that strictly concave constraint functions, like those present in volumetric penalization, as recently proposed by Bruns and co-workers, may increase the difficulty of the topology optimization problem when strictly convex approximations are used in the SAO algorithm. In turn, volumetric penalization methods are of notable importance, since they seem to hold much promise for generating predominantly solid-void or discrete designs.We then argue that the nonconvex problems we study may in some instances efficiently be solved using dual SAO methods based on nonconvex (strictly concave) approximations which exhibit monotonicity with respect to the design variables.Indeed, for the topology problem resulting from SIMP-like volumetric penalization, we show explicitly that convex approximations are not necessary. Even though the volumetric penalization constraint is strictly concave, the maximum of the resulting dual subproblem still corresponds to the optimum of the original primal approximate subproblem.

In practice, a continuum structure is usually designed to carry the traction applied to the boundary of the structure subject to prescribed displacements imposed on its boundary. The design domains of practical structures are often limited and significantly affect the final optimal design of the structures. Structural boundaries under traction and prescribed displacements should be treated as a zero level set in the level set methods. Firstly, to overcome the limitations of current level set methods and the stopping issue of structural boundary movements, this paper presents a set of new level set based optimization formulae for the optimal design of continuum structures with bounded design domains. A set of new normal velocities required by level set movements is given. A level set based optimization algorithm is developed and implemented with several robust and efficient numerical techniques, which include the level set regularization algorithm, gradient projection algorithm, nonlinear velocity mapping algorithm and return mapping algorithm. Secondly, in order to overcome the difficulty in nucleating holes in the design domain in the level set based optimization method, this paper introduces a new optimization strategy with a small possibility random topology mutations and crossovers. A mixed topology optimization algorithm is implemented and presented for the compliance minimization problems of continuum structures with material volume constraints. The validity and effectiveness of the proposed method are demonstrated with two examples.

Eigenvalue sensitivity can be analytically computed using Fox and Kapoor's formula. However, if the eigenvalues are not distinct, the calculation of eigenvalue sensitivities for the repeated eigenvalues using Fox's formula sometimes does not yield correct answers unless the eigenvectors associated with the repeated eigenvalues are properly defined. It has been found in this paper that for two types of frequently designed structures, although the repeated eigenvalues exist, their sensitivities with respect to specific design variables can be found without forming an auxiliary eigenequation as proposed by earlier researchers. For one type of structure, the design variable has equal dynamic influence in two coordinate directions, and for the other type, the design variable affects the dynamic behavior in only one coordinate direction. Two numerical examples representing these two types of structures are given to demonstrate this finding.

In this paper, a topology optimization method mixed structures with continuum and discrete elements is proposed. First, based on the stress and displacement formulations of the finite element analysis, a set of stress sensitivity numbers for beam and plane-stress structures are derived. Then, relative difference quotients for topology and size variables of mixed continuum structures with several different types of components are formulated. The optimization criteria, which incorporate the element stress level and relative different quotients, are developed for the topology optimization of continuum domains and the sizing optimization of discrete elements. A new optimization procedure based on the bi-directional evolutionary structural optimization method is proposed. Two examples are provided to demonstrate the validity and effectiveness of the proposed method.

This paper presents topology optimization of geometrically and materially nonlinear structures under displacement loading. A revised bi-directional evolutionary optimization (BESO) method is used. The objective of the optimization problem is to maximize the structural stiffness within the limit of prescribed design displacement. The corresponding sensitivity number is derived using the adjoint method. The original BESO technique has been extended and modified to improve the robustness of the method. The revised BESO method includes a filter scheme, an improved sensitivity analysis using the sensitivity history and a new procedure for removing and adding material. The results show that the developed BESO method provides convergent and mesh-independent solutions for linear optimization problems. When the BESO method is applied to nonlinear structures, much improved designs can be efficiently obtained although the solution may oscillate between designs of two different deformation modes. Detailed comparison shows that the nonlinear designs are always better than the linear ones in terms of total energy. The optimization method proposed in this paper can be directly applied to the design of energy absorption devices and structures.

This paper introduces a method to determine strut-and-tie models in reinforced concrete (RC) structures using evolutionary structural optimization (ESO). Even though strut-and-tie models are broadly adapted in the design of reinforced concrete members subjected to shear and torsion, conventional methods can hardly offer useful models of RC members subjected to complex loadings and geometry conditions. In this paper, the basic idea of the ESO method is used to determine more rational strut-and-tie models. Since an optimum topology of structures, finally obtained by the ESO method, usually represents a truss-like structure, the ESO method using truss elements can effectively be used in finding the best strut-and-tie models in RC structures. To prevent the structural instability that may occur during the evolutionary optimization process, a brick element composed of six truss elements is designed as a basic element unit. Systematic removal of each brick element that has the least virtual strain energy follows, and the optimal load transfer mechanism of an RC structure, which is equivalent to the optimal topology of the strut-and-tie model, is finally characterized on the basis of an optimization criterion of minimizing the total elastic strain energy of the structure. Several RC structures are used as examples to demonstrate the capability of the proposed method in finding the best strut-and-tie model of each RC structure and to verify its efficiency in application to real design problems.

When elastic structures are subjected to dynamic loads, a propagation problem is considered to predict structural transient response. To achieve better dynamic performance, it is important to establish an optimum structural design method. Previous work focused on minimizing the structural weight subject to dynamic constraints on displacement, stress, frequency, and member size. Even though these methods made it possible to obtain the optimal size and shape of a structure, it is necessary to obtain an optimal topology for a truly optimal design. In this paper, the homogenization design method is utilized to generate the optimal topology for structures and an explicit direct integration scheme is employed to solve the linear transient problems. The optimization problem is formulated to find the best configuration of structures that minimizes the dynamic compliance within a specified time interval. Examples demonstrate that the homogenization design method can be extended to the optimal topology design method of structures under impact loads.

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

In this paper, an improved method for evolutionary structural optimisation against buckling is proposed for maximising the critical buckling load of a structure of constant weight. First, based on the formulations of derivatives for eigenvalues, the sensitivity numbers of the first eigenvalue or the first multiple eigenvalues (for closely spaced and repeated eigenvalues) are derived by performing a variation operation. In order to effectively increase the buckling load factor, a set of optimum criteria for closely-spaced eigenvalues and repeated eigenvalues are established, based on the sensitivity numbers of the first multiple eigenvalues. Several examples are provided to demonstrate the validity and effectiveness of the proposed method.

We present a method for topology optimization of an acoustic horn with the aim of radiating sound as efficiently as possible. Using a strategy commonly employed for topology optimization of elastic structures, we optimize over a scalar function indicating presence of material. The Helmholtz equation modeling the wave propagation is solved using the finite element method and the associated adjoint equation provides the required gradients. Numerical experiments validates that the result of the optimization provides horns with the desired acoustical properties. The resulting horns are very efficient in the frequency span subject to optimization.

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.

The present paper is concerned with the layout optimization of resonating actuators using topology optimization techniques. The goal of the optimization is a maximization of the magnitude of steady-state vibrations for a given excitation frequency. The problem formulation includes an external viscous damper at the output port which models a working load on the structure. Copyright © 2002 John Wiley & Sons, Ltd.

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.

A new and powerful mathematical programming method is described, which is capable of solving a broad class of structural optimization problems. The method employs mixed direct/reciprocal design variables in order to get conservative, first-order approximations to the objective function and to the constraints. By this approach the primary optimization problem is replaced with a sequence of explicit subproblems. Each subproblem being convex and separable, it can be efficiently solved by using a dual formulation. An attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs. Examples of application to real-life aerospace structures are offered to demonstrate the power and generality of the approach presented.

This paper deals with vibro-acoustic optimization of laminated composite plates. The vibration of the laminated plate is excited by time-harmonic external mechanical loading with prescribed frequency and amplitude, and the design objective is to minimize the total sound power radiated from the surface of the laminated plate to the surrounding acoustic medium. Instead of solving the Helmholtz equation for evaluation of the sound power, advantage is taken of the fact that the surface of the laminated plate is flat, which implies that Rayleigh’s integral approximation can be used to evaluate the sound power radiated from the surface of the plate. The novel Discrete Material Optimization (DMO) formulation has been applied to achieve the design optimization of fiber angles, stacking sequence and selection of material for laminated composite plates. Several numerical examples are presented in order to illustrate this approach.

The frequency averaged transverse vibration levels of a plate with a harmonic excitation is minimized by optimizing the position and shape of attached passive constrained layer damping. A modified gradient method is used in the finite-element context to successively add pieces of constrained damping layers at the elemental positions showing the steepest gradient of the goal function as a result of the treatment. The coding is done in MATLAB and different stop conditions can be included so as to set limits for the cost or weight that can be spent on the treatment of the structure. It is demonstrated that for a square plate, only a few iterations are needed to reduce the average vibration level with up to 18 dB by covering less than 30 percent of the surface with a sandwich type applied damping material. For an industrial example, measurements show that the solution proposed by the optimization procedure will decrease the vibration levels for the two dominant modes of vibrations with 3–4 dB, by covering 3.4 percent of the surface with a single-sided constraining layer type applied damping material.

A solution strategy to find the shape and topology of structures that maximize a natural frequency is presented. The methodology is based on a homogenization method and the representation of the shape of the structure as a material property. The problem is formulated as a reinforcement problem in which a given structure is reinforced using a prescribed amount of material. Two dimensional, plane elasticity problems are considered. Examples are presented for illustration.

The paper presents a gradient-based topology optimization formulation that allows to solve acoustic–structure (vibro-acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface-coupling integrals, however, such a formulation does not allow for free material re-distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u/p-formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two-dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

A new method for non-linear programming in general and structural optimization in particular is presented. In each step of the iterative process, a strictly convex approximating subproblem is generated and solved. The generation of these subproblems is controlled by so called ‘moving asymptotes’, which may both stabilize and speed up the convergence of the general process.

This paper deals with topological design optimization of vibrating bi-material elastic structures of given volume, domain
and boundary conditions, with the objective of minimizing the sound power radiated from the structural surfaces into a surrounding
acoustic medium. The structural vibrations are excited by a time-harmonic mechanical loading with prescribed forcing frequency
and amplitude, and structural damping is not considered. It is assumed that air is the acoustic medium and that a feedback
coupling between the acoustic medium and the structure can be neglected. Certain conditions are assumed, where the sound power
radiated from the structural surface can be estimated by using a simplified approach instead of solving the Helmholz integral
equation. This implies that the computational cost of the structural-acoustical analysis can be considerably reduced. Numerical
results are presented for plate and pipe-like structures with different sets of boundary conditions.

A concept of independent continuous topological variables was proposed to establish its corresponding smooth model of structural topological optimization. In order to use efficient differential optimization algorithms, the topological variables should be handed as a continuous instead of the discrete one. Therefore, a mapping and its inverse between original discrete topology problem and corresponding continuous one were proposed in modelling and optimization. The method can overcome difficulties that are encountered in conventional models and algorithms for the optimization of the structural topology. Its application to truss topological optimization with stress and displacement constraints is satisfactory, with convergence faster than that of sectional optimization.

In density-based topological design, 0/1 solutions are often sought, that is, one expects that the final design includes either elements with full material or no material, excluding grey areas. The accepted technique for achieving binary values for the densities is to use a solid isotropic microstructure with penalization (SIMP) material for which Young’s modulus is a polynomial function of the otherwise continuous relative densities. This approach indeed enhances 0/1 solutions in a significant manner and as such it has achieved prominent status in topological design. Nevertheless, this paper proposes a possible alternative to the SIMP methodology for generating 0/1 structures. The design variables are still the densities of the finite elements but Young’s modulus is a linear function of these densities (in some sense, a SIMP material without penalty). In order to drive the solution to a 0/1 layout a new constraint, labeled the sum of the reciprocal variables (SRV), is introduced. The constraint stipulates that the SRV must be larger or equal to its value at a discrete design for a specified amount of material. It is understood that this implies a minimum gage on the design variables, a provision which is also present in the standard fixed-grid formulation to avoid singular stiffness matrices. The technique turned out to be very effective in conjunction with the method of moving asymptotes (MMA) when using topological design methods for finding optimal layouts of patches of piezo-electric (PZT) material in order to minimize the mechanical noise emanating from vibrating surfaces. It also performed satisfactorily in classical structural topological design instances, as can be seen in the numerical examples that illustrate this work.

The aim of this article is to evaluate and compare established numerical methods of structural topology optimization that
have reached the stage of application in industrial software. It is hoped that our text will spark off a fruitful and constructive
debate on this important topic.