Multiobjective design of actively controlled structures using a hybrid optimization method

A multiobjective approach to the combined structure and control optimization problem for flexible space structures is presented. The proposed formulation addresses robustness considerations for controller design, as well as a simultaneous determination of optimum actuator locations. The structural weight, controlled system energy, stability robustness index and damping augmentation provided by the active controller are considered as objective functions of the multiobjective problem which is solved using a cooperative game-theoretic approach. The actuator locations and the cross-sectional areas of structural members are treated as design variables. Since the actuator locations are spatially discrete, whereas the cross-sectional areas are continuous, the optimization problem has mixed discrete-continuous design variables. A solution approach to this problem based on a hybrid optimization scheme is presented. The hybrid optimizer is a synergetic blend of artificial genetic search and gradient-based search techniques. The computational procedure is demonstrated through the design of an ACOSS-FOUR space structure. The optimum solutions obtained using the hybrid optimizer are shown to outperform the optimum results obtained using gradient-based search techniques.     

A consistent grayscale‐free topology optimization method using the level‐set method and zero‐level boundary tracking mesh

This paper proposes a level‐set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming. Because the boundary tracking mesh is always conformed to the structural boundary, good approximation to the boundary is maintained during optimization; therefore, structural design problems are solved completely without grayscale material. Previously, we introduced the boundary tracking mesh generating method into level‐set based topology optimization and updated the design variables by solving the level‐set equation. In order to adapt our previous method to general structural optimization frameworks, the incorporation of the method with nonlinear programming is investigated in this paper. To successfully incorporate nonlinear programming, the optimization problem is regularized using a double‐well potential. Furthermore, the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming. We expect the investigation to open up a new class of grayscale‐free topology optimization. The usefulness of the proposed method is demonstrated using several numerical examples targeting two‐dimensional compliant mechanism and metallic waveguide design problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A scaled boundary finite element based explicit topology optimization approach for three-dimensional structures

This article proposes an efficient approach for solving three-dimensional (3D) topology optimization problem. In this approach, the number of design variables in optimization as well as the number of degrees of freedom in structural response analysis can be reduced significantly. This is accomplished through the use of scaled boundary finite element method (SBFEM) for structural analysis under the moving morphable component (MMC)-based topology optimization framework. In the proposed method, accurate response analysis in the boundary region dictates the accuracy of the entire analysis. In this regard, an adaptive refinement scheme is developed where the refined mesh is only used in the boundary region while relating coarse mesh is used away from the boundary. Numerical examples demonstrate that the computational efficiency of 3D topology optimization can be improved effectively by the proposed approach.     

Optimality criteria-based topology optimization of a bi-material model for acoustic–structural coupled systems

This article investigates topology optimization of a bi-material model for acoustic–structural coupled systems. The design variables are volume fractions of inclusion material in a bi-material model constructed by the microstructure-based design domain method (MDDM). The design objective is the minimization of sound pressure level (SPL) in an interior acoustic medium. Sensitivities of SPL with respect to topological design variables are derived concretely by the adjoint method. A relaxed form of optimality criteria (OC) is developed for solving the acoustic–structural coupled optimization problem to find the optimum bi-material distribution. Based on OC and the adjoint method, a topology optimization method to deal with large calculations in acoustic–structural coupled problems is proposed. Numerical examples are given to illustrate the applications of topology optimization for a bi-material plate under a low single-frequency excitation and an aerospace structure under a low frequency-band excitation, and to prove the efficiency of the adjoint method and the relaxed form of OC.     

Nonlinear response topology optimization using equivalent static loads—case studies

Nonlinear structural optimization is fairly expensive and difficult, because a large number of nonlinear analyses is required due to the large number of design variables involved in topology optimization. In element density based topology optimization, the low density elements create mesh distortion and the updating of finite element material with low density elements has a severe effect on the optimization results in the next cycles. In order to overcome these difficulties, the equivalent static loads method for nonlinear response structural optimization (ESLSO) primarily used for size and shape optimization has been applied to topology optimization. The nonlinear analysis is performed with the given loading conditions to calculate equivalent static loads (ESLs) and these ESLs are used to perform linear response optimization. In this paper, the authors have presented the results of five case studies with material, geometric and contact nonlinearities showing good agreement and providing justification of the proposed method.     

Topology optimization of composite material plate with respect to sound radiation

In this paper, topology optimization of composite material plate with respect to minimization of the sound power radiation has been studied. A new low noise design method based on topology optimization is proposed, which provides great guidance for acoustic designers. The structural vibrations are excited by external harmonic mechanical load with prescribed frequency and amplitude. The sound power is calculated using boundary element method. An extended solid isotropic material with penalization (SIMP) model is introduced for acoustic design sensitivity analysis in topology optimization, where the same penalization is applied for the stiffness and mass of the structural volume elements. Volumetric densities of stiffer material are chosen as design variables. Finally, taking a simple supported thin plate as a simulation example, the sound power radiation from structures subjected to forced vibration can be considerably reduced, leading to a reduction of 20 dB. It is shown that the optimal topology is easy to manufacture at low frequency, while as the loading frequency increases, the optimal topology shows a more and more complicated periodicity which makes it difficult to manufacture.     

Achieving minimum length scale in topology optimization using nodal design variables and projection functions

A methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described. Nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology. The projection is made via mesh independent functions that are based upon the minimum length scale. A simple linear projection scheme and a non‐linear scheme using a regularized Heaviside step function to achieve nearly 0–1 solutions are examined. The new approach is demonstrated on the minimum compliance problem and the popular SIMP method is used to penalize the stiffness of intermediate volume fraction elements. Solutions are shown to meet user‐defined length scale criterion without additional constraints, penalty functions or sensitivity filters. No instances of mesh dependence or checkerboard patterns have been observed. Copyright © 2004 John Wiley & Sons, Ltd.     

Shape morphing of laminated composite structures with photostrictive actuators via topology optimization

In this paper, a new design method is presented for achieving remote wireless shape morphing of laminated composite structures using topology optimization methods. A recently emerging family of smart materials, photostrictive materials, is introduced as the actuation discipline to implement the active control of optical structures by utilizing the photostriction mechanism, which arises from the superposition of photovoltaic effect and converse piezoelectric effect when exposed to the illumination of near ultraviolet light. In terms of the Mindlin plate theory of first-order shear deformation, a finite element formulation including multiphysics effects of photovoltaic, pyroelectric and thermal expansion is developed to model composite structures of ferroelectric materials polarized in 0–1 and 0–3 directions, respectively. The design is formulated as a multi-constrained optimization problem with a least square objective function to minimize structural shape errors. The topology optimization method is used as a systematic design approach to seek the optimal topologies of material layouts for both the photostrictive and host layers as well as the actuator light distribution. In terms of design sensitivity analysis, many gradient-based optimization algorithms can be applied to solve the problem effectively. Numerical examples are presented to demonstrate the effectiveness of this method in the field of active photonic control of laminated composite structures.     

A moving superimposed finite element method for structural topology optimization

Level set methods are becoming an attractive design tool in shape and topology optimization for obtaining efficient and lighter structures. In this paper, a dynamic implicit boundary‐based moving superimposed finite element method (s‐version FEM or S‐FEM) is developed for structural topology optimization using the level set methods, in which the variational interior and exterior boundaries are represented by the zero level set. Both a global mesh and an overlaying local mesh are integrated into the moving S‐FEM analysis model. A relatively coarse fixed Eulerian mesh consisting of bilinear rectangular elements is used as a global mesh. The local mesh consisting of flexible linear triangular elements is constructed to match the dynamic implicit boundary captured from nodal values of the implicit level set function. In numerical integration using the Gauss quadrature rule, the practical difficulty due to the discontinuities is overcome by the coincidence of the global and local meshes. A double mapping technique is developed to perform the numerical integration for the global and coupling matrices of the overlapped elements with two different co‐ordinate systems. An element killing strategy is presented to reduce the total number of degrees of freedom to improve the computational efficiency. A simple constraint handling approach is proposed to perform minimum compliance design with a volume constraint. A physically meaningful and numerically efficient velocity extension method is developed to avoid the complicated PDE solving procedure. The proposed moving S‐FEM is applied to structural topology optimization using the level set methods as an effective tool for the numerical analysis of the linear elasticity topology optimization problems. For the classical elasticity problems in the literature, the present S‐FEM can achieve numerical results in good agreement with those from the theoretical solutions and/or numerical results from the standard FEM. For the minimum compliance topology optimization problems in structural optimization, the present approach significantly outperforms the well‐recognized ‘ersatz material’ approach as expected in the accuracy of the strain field, numerical stability, and representation fidelity at the expense of increased computational time. It is also shown that the present approach is able to produce structures near the theoretical optimum. It is suggested that the present S‐FEM can be a promising tool for shape and topology optimization using the level set methods. Copyright © 2005 John Wiley & Sons, Ltd.     

结构主动控制的一体化多目标优化研究

基于Pareto多目标遗传算法提出了结构主动控制系统的一体化多目标优化设计方法,对作动器位置与主动控制器进行同步优化设计.外界激励采用平稳过滤白噪声来模拟,在状态空间下通过求解Lyapunov方程,得到结构响应和主动控制力的均方值.主动控制器采用LQG控制算法来进行设计.以结构位移和加速度均方值最大值与相应无控响应均方值的最大值之比,以及所需控制力均方值之和作为多目标同步优化的目标函数.优化过程还考虑了结构与激励参数对优化结果的影响.最后以某6层平面框架有限元模型为例进行了计算机仿真分析,结果表明所提出的主动控制系统多目标一体化优化方法简单,高效,实用,具有较好的普适性.     

考虑作动器联接方式的结构形状控制优化

以压电材料梁式作动器控制复合材料层合板形状的设计问题为对象,研究有限个独立控制参数条件下的形状最优控制问题。研究了作动器与信号发生器(独立控制参数)联接关系的参数化描述方式,建立了作动器联接方式与控制参数协同设计的问题提法;针对优化问题中离散变量(联接方式描述参数)和连续变量(控制参数)共存的特点,建立了遗传算法和线性最小二乘(Linear Least Square,LLS)方法相结合的求解策略和方法;在响应分析所采用的有限元模型中,采用粘结层单元描述本体结构与作动器之间的连接。复合材料层合板形状控制设计的实例,验证了该文中建立的问题提法、优化模型和求解策略的有效性。     

Integrated layout design of multi‐component system

A new integrated layout optimization method is proposed here for the design of multi‐component systems. By introducing movable components into the design domain, the components layout and the supporting structural topology are optimized simultaneously. The developed design procedure mainly consists of three parts: (i) Introduction of non‐overlap constraints between components. The finite circle method (FCM) is used to avoid the components overlaps and also overlaps between components and the design domain boundaries. (ii) Layout optimization of the components and supporting structure. Locations and orientations of the components are assumed as geometrical design variables for the optimal placement while topology design variables of the supporting structure are defined by the density points. Meanwhile, embedded meshing techniques are developed to take into account the finite element mesh change caused by the component movements. (iii) Consistent material interpolation scheme between element stiffness and inertial load. The commonly used solid isotropic material with penalization model is improved to avoid the singularity of localized deformation in the presence of design dependent loading when the element stiffness and the involved inertial load are weakened by the element material removal. Finally, to validate the proposed design procedure, a variety of multi‐component system layout design problems are tested and solved on account of inertia loads and gravity center position constraint. Solutions are compared with traditional topology designs without component. Copyright © 2008 John Wiley & Sons, Ltd.     

An adaptive mesh-adjustment strategy for continuum topology optimization to achieve manufacturable structural layout

This paper presents an adaptive mesh adjustment algorithm for continuum topology optimization method to describe the structural boundary using nonuniform isoparametric element. A criterion on the basis of the node movement is proposed; herein, the densities and coordinates of the nodes are defined to instruct the deformation of finite elements in subsequent optimization iterations. With such a scheme, the topology optimization can start from a regular mesh discretization then gradually yields an optimal design with clear structural boundaries. The element in the transition along the boundary is refined; on the contrary, the pure solid or void element is coarsen. The contribution of this work is to improve the resolution of the structural boundaries and decrease the percentage of transitional regions with the invariant design variable. Several 2D and 3D numerical examples indicate the effectiveness of our proposed method. Seen from the examples, the structural boundary become smoother and the intermediate densities have been reduced up to 70%. In addition, a design process based on the presented method is proposed to make the optimum solutions be fabricated conveniently and accurately by linking it with the 3D design software, ie, SolidWorks, which is also demonstrated in the numerical examples.     

A hierarchical method for discrete structural topology design problems with local stress and displacement constraints

In this paper, we present a hierarchical optimization method for finding feasible true 0–1 solutions to finite‐element‐based topology design problems. The topology design problems are initially modelled as non‐convex mixed 0–1 programs. The hierarchical optimization method is applied to the problem of minimizing the weight of a structure subject to displacement and local design‐dependent stress constraints. The method iteratively treats a sequence of problems of increasing size of the same type as the original problem. The problems are defined on a design mesh which is initially coarse and then successively refined as needed. At each level of design mesh refinement, a neighbourhood optimization method is used to treat the problem considered. The non‐convex topology design problems are equivalently reformulated as convex all‐quadratic mixed 0–1 programs. This reformulation enables the use of methods from global optimization, which have only recently become available, for solving the problems in the sequence. Numerical examples of topology design problems of continuum structures with local stress and displacement constraints are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Simultaneous partial topology and size optimization of a wing structure using ant colony and gradient based methods

This article presents a methodology and process for a combined wing configuration partial topology and structure size optimization. It is aimed at achieving a minimum structural weight by optimizing the structure layout and structural component size simultaneously. This design optimization process contains two types of design variables and hence was divided into two sub-problems. One is structure layout topology to obtain an optimal number and location of spars with discrete integer design variables. Another is component size optimization with continuous design variables in the structure FE model. A multi city-layer ant colony optimization (MCLACO) method is proposed and applied to the topology sub-problem. A gradient based optimization method (GBOM) built in the MSC.NASTRAN SOL-200 module was employed in the component size optimization sub-problem. For each selected layout of the wing structure, a size optimization process is performed to obtain the optimum result and feedback to the layout topology process. The numerical example shows that the proposed MCLACO method and a combination with the GBOM are effective for solving such a wing structure optimization problem. The results also indicate that significant structural weight saving can be achieved.     

Hierarchical remeshing strategies with mesh mapping for topology optimisation

This work investigates the use of hierarchical mesh decomposition strategies for topology optimisation using bi‐directional evolutionary structural optimisation algorithm. The proposed method uses a dual mesh system that decouples the design variables from the finite element analysis mesh. The investigation focuses on previously unexplored areas of these techniques to investigate the effect of five meshing parameters on the analysis solving time (i.e. computational effort) and the analysis quality (i.e. solution optimality). The foreground mesh parameters, including adjacency ratio and minimum and maximum element size, were varied independently across solid and void domain regions. Within the topology optimisation, strategies for controlling the mesh parameters were investigated. The differing effects of these parameters on the efficiency and efficacy of the analysis and optimisation stages are discussed, and recommendations are made for parameter combinations. Some of the key findings were that increasing the adjacency ratio increased the efficiency only modestly – the largest effect was for the minimum and maximum element size parameters – and that the most dramatic reduction in solve time can be achieved by not setting the minimum element size too low, assuming mapping onto a background mesh with a minimum element size of 1. © 2016 The Authors. International Journal for Numerical Methods in Engineering Published by John Wiley & Sons, Ltd.     

压电智能连续体结构/控制一体化广义拓扑优化

对于以往较少涉及到的同时考虑结构拓扑、作动器位置与数目和控制器参数等多种优化设计变量参与的压电智能板结构的一体化优化设计问题,研究了结构/控制一体化广义拓扑优化设计的方法。提出采用基于耦合模态空间的二次型最优控制系统设计与基于遗传算法和数学形态学处理的策略进行一体化拓扑优化设计实现。数值算例的结果表明,所提方法合理、有效,能够得到清晰的结构拓扑和良好的可控性。     

Enhanced damping of lightweight structures by semi-active joints

Summary Lightweight structures typically have low inherent structural damping. Effective vibration suppression is required, for example, in certain applications involving precision positioning. The present approach is based on friction damping in semi-active joints which allow relative sliding between the connected parts. The energy dissipation due to interfacial slip in the friction joints can be controlled by varying the normal pressure in the contact area using a piezo-stack actuator. This paper focuses on the optimal placement of semi-active joints for vibration suppression. The proposed method uses optimality criteria for actuator and sensor locations based on eigenvalues of the controllability and observability gramians. Optimal sensor/actuator placement is stated as a nonlinear multicriteria optimization problem with discrete variables and is solved by a stochastic search algorithm. At optimal locations, conventional rigid connections of a large truss structure are replaced by semi-active friction joints. Two different concepts for the control of the normal forces in the friction interfaces are implemented. In the first approach, each semi-active joint has its own local feedback controller, whereas the second concept uses a global, clipped-optimal controller. Simulation results for a 10-bay truss structure show the potential of the proposed semi-active concept. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

An immersed boundary element method for level‐set based topology optimization

In this paper, we propose a new BEM for level‐set based topology optimization. In the proposed BEM, the nodal coordinates of the boundary element are replaced with the nodal level‐set function and the nodal coordinates of the Eulerian mesh that maintains the level‐set function. Because this replacement causes the nodal coordinates of the boundary element to disappear, the boundary element mesh appears to be immersed in the Eulerian mesh. Therefore, we call the proposed BEM an immersed BEM. The relationship between the nodal coordinates of the boundary element and the nodal level‐set function of the Eulerian mesh is clearly represented, and therefore, the sensitivities with respect to the nodal level‐set function are strictly derived in the immersed BEM. Furthermore, the immersed BEM completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization. By using the immersed BEM, we construct a concrete topology optimization method for solving the minimum compliance problem. We provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results. Copyright © 2012 John Wiley & Sons, Ltd.     

A simultaneous approach for compliance minimization and piezoelectric actuator design considering the polarization profile

This article addresses the compliance problem along with the piezoelectric actuator design for active vibration control. The topology structural design is obtained by solving a compliance minimization problem with volume constraint, whereas the actuator design is carried out by the maximization of a control performance index written in terms of the controllability Gramian. This measure describes the ability of the actuator to move the structure from an initial condition to a desired final state, at rest for instance, in a finite time interval. The actuator design is also characterized by the polarization profile, which is defined according to the distribution of an additional design variable. Therefore, the actuators can yield both tensile and compressing fields at different points of the structure using the same applied control voltage. To achieve this goal, a material interpolation scheme based on the solid isotropic material with penalization and the piezoelectric material with penalization and polarization (PEMAP-P) models is employed, and both the optimum structure/actuator layout and polarization profile are obtained simultaneously. The sensitivities with respect to the polarization and design variables are calculated analytically. Numerical examples are presented considering the design and vibration control for a cantilever beam, a beam fixed at both ends, and an L-bracket structure to show the efficiency of the proposed formulation. The control performance of the designed structures are analyzed employing a linear-quadratic regulator simulation, and these results are compared to verify the influence of the optimized polarization profiles.