Design of laminated piezocomposite shell transducers with arbitrary fiber orientation using topology optimization approach

Sensor and actuator based on laminated piezocomposite shells have shown increasing demand in the field of smart structures. The distribution of piezoelectric material within material layers affects the performance of these structures; therefore, its amount, shape, size, placement, and polarization should be simultaneously considered in an optimization problem. In addition, previous works suggest the concept of laminated piezocomposite structure that includes fiber‐reinforced composite layer can increase the performance of these piezoelectric transducers; however, the design optimization of these devices has not been fully explored yet. Thus, this work aims the development of a methodology using topology optimization techniques for static design of laminated piezocomposite shell structures by considering the optimization of piezoelectric material and polarization distributions together with the optimization of the fiber angle of the composite orthotropic layers, which is free to assume different values along the same composite layer. The finite element model is based on the laminated piezoelectric shell theory, using the degenerate three‐dimensional solid approach and first‐order shell theory kinematics that accounts for the transverse shear deformation and rotary inertia effects. The topology optimization formulation is implemented by combining the piezoelectric material with penalization and polarization model and the discrete material optimization, where the design variables describe the amount of piezoelectric material and polarization sign at each finite element, with the fiber angles, respectively. Three different objective functions are formulated for the design of actuators, sensors, and energy harvesters. Results of laminated piezocomposite shell transducers are presented to illustrate the method. Copyright © 2012 John Wiley & Sons, Ltd.     

Composite material design of two‐dimensional structures using the homogenization design method

Composite materials of two‐dimensional structures are designed using the homogenization design method. The composite material is made of two or three different material phases. Designing the composite material consists of finding a distribution of material phases that minimizes the mean compliance of the macrostructure subject to volume fraction constraints of the constituent phases, within a unit cell of periodic microstructures. At the start of the computational solution, the material distribution of the microstructure is represented as a pure mixture of the constituent phases. As the iteration procedure unfolds, the component phases separate themselves out to form distinctive interfaces. The effective material properties of the artificially mixed materials are defined by the interpolation of the constituents. The optimization problem is solved using the sequential linear programming method. Both the macrostructure and the microstructures are analysed using the finite element method in each iteration step. Several examples of optimal topology design of composite material are presented to demonstrate the validity of the present numerical algorithm. Copyright © 2001 John Wiley & Sons, Ltd.     

Isogeometric topology optimization of anisotropic metamaterials for controlling high-frequency electromagnetic wave

This study presents a level set–based topology optimization with isogeometric analysis (IGA) for controlling high-frequency electromagnetic wave propagation in a domain with periodic microstructures (unit cells). The high-frequency homogenization method is applied to characterize the macroscopic high-frequency waves in periodic heterogeneous media whose wavelength is comparative to or smaller than the representative length of a unit cell. B-spline basis functions are employed for the IGA discretization procedure to improve the performance of electromagnetic wave analysis in a unit cell and topology optimization. Also, to keep the same order of continuity on the periodic boundaries as on other element edges in the domain, we propose the extended domain approach, while incorporating Floquet periodic boundary condition (FPBC). Two types of optimization problems are taken as examples to demonstrate the effectiveness of the proposed method in comparison with the standard finite element analysis (FEA). The optimization results provide optimized topologies of unit cells qualified as anisotropic metamaterials with hyperbolic and bidirectional dispersion properties at the macroscale.     

零膨胀材料设计与模拟验证

零膨胀材料对提高航空航天结构和电子设备等的热几何稳定性有重要意义。采用拓扑优化技术设计各相材料在单胞域的分布形式, 以获得零膨胀材料的微结构形式。给出了由二相实体材料和空心构成的各向同性零膨胀材料的设计方案, 讨论了初始设计依赖性问题, 分析了该依赖性的存在原因。采用有限元技术代替实际测试, 分析了所设计材料的试件在均匀温度变化下的变形, 验证了所设计材料的零膨胀(低膨胀) 性质, 说明通过拓扑优化技术设计材料的微结构是设计零膨胀材料的有效方法。     

Optimal design of a multilayered piezoelectric transducer based on a special unit cell homogenization method

Piezoelectric transducers with high directional-dependent response are of interest in several applications because of their ability to sense and actuate along specific directions where they can distinguish individual principal strain components. This paper presents an improved unidirectional sensor obtained using a multi-laminate piezocomposite that maximizes the electromechanical coupling factor or the piezoelectric strain constant in one direction with respect to the other directions. Furthermore, multi-laminate structures provide significant design potential by the variation of the orientation and stacking sequence of fibers to obtain the desired properties. The effective properties depend on the number of layers, the fibers orientation as well as the thickness of each layer, and they are estimated by the variational asymptotic method for unit cell homogenization. A design that guarantees maximum directional dependence in terms of piezoelectric strain constant is determined by a global optimization technique using a genetic algorithm. Layered transducers composed of several orthotropic passive layers and a single active layer are considered.     

尺度关联的微结构构型与排布的材料/结构动力学设计

考虑微观尺度和宏观尺度的关联, 将微结构胞元设计和多尺度均匀化设计结合, 建立了由相同尺寸和内部构型的微结构胞元组成的复合材料结构的材料/结构一体化动力学优化设计方法, 给出了相应的算例。方法中引入了微观和宏观两个尺度上的独立密度变量, 采用材料属性的合理近似模型对密度进行惩罚, 利用有限元超单元技术建立材料与结构的联系。基于算例获得了超单元尺寸与微观、宏观材料用量对结构拓扑构型的影响规律。将算例结果与已有方法的结果比较, 表明本文方法具有合理有效性, 可作为对轻质结构进行动力学设计的一种新方法。      

Level-set method for design of multi-phase elastic and thermoelastic materials

The problem of designing composite materials with desired mechanical properties is to specify the materials microstructures in terms of the topology and distribution of their constituent material phases within a unit cell of periodic microstructures. In this paper we present an approach based on a multi-phase level-set model for the geometric and material representation and for numerical solution of a least squares optimization problem. The level-set model precisely specifies the material regions and their sharp boundaries in contrast to a raster discretization of the conventional homogenization-based approaches. Combined with the classical shape derivatives, the level-set method yields a computational system of partial differential equations. In using the Eulerian computation scheme with a fixed rectilinear grid and a fixed mesh in the unit cell, the gradient descent solution of the optimization captures the interfacial boundaries naturally and performs topological changes accurately. The proposed method is illustrated with several 2D examples for the synthesis of heterogeneous microstructures of elastic and/or thermoelastic composites composed of two and three material phases.     

Analysis and optimization of heterogeneous materials using the variational asymptotic method for unit cell homogenization

The variational asymptotic method for unit cell homogenization is used to find the sensitivity of the effective properties of periodically heterogeneous materials, within a periodic base-cell. The sensitivities are found by the direct differentiation of the variational asymptotic method for unit cell homogenization (VAMUCH) and by the method of adjoint variables. This sensitivity theory is implemented using the finite element method and the engineering program VAMUCH. The methodology is used to design the periodic microstructure of a material that allows obtaining prescribed constitutive properties. The microstructure is modeled as a 2D periodic structure, but a complete set of 3D material properties are obtained. Furthermore, the present methodology can be used to perform the micromechanical analysis and related sensitivity analysis of heterogeneous materials that have 3D periodic structures. The effective material properties of the artificially mixed materials of the microstructure are obtained by the density approach, in which the solid material and void are mixed artificially.     

一种新型的压电材料参数的精确表征方法

针对压电材料精确表征问题,本文提出了一种包含材料损耗特性,基于模拟退火(Simulated Annealing,SA)优化算法的压电材料新型表征方法.并采用提出的新型表征方法对厚度振动模式下的高损耗偏铌酸铅压电陶瓷和l-3型压电复合材料样品的材料特性进行了研究,样品电子阻抗共振特性的理论优化结果与实际测量值准确拟合,表明该方法能够精确表征具有损耗特性压电材料参数.     

Multiscale topology optimization for frequency domain response with bi-material interpolation schemes

In areas that require high performance components, such as the automotive, aeronautics and aerospace industries, optimization of the dynamic behavior of structures is sought through different approaches, such as the design of materials specific to the application, for instance through structural topology optimization. The bi-directional evolutionary structural optimization (BESO) method, in particular, has been used for the simultaneous design of hierarchical structures, which means that the structural domain consists not only of the macrostructure but also of the microstructural topology of the materials employed. The purpose of this work is to apply the BESO method to solve two-dimensional multiscale problems in order to minimize the response of structures subjected to forced vibrations in a given frequency range. The homogenization method is applied to integrate the different scales of the problem. In particular, the material interpolation model for two materials is used. The BESO method is applied to different cases of optimization, in macroscale, microscale, and multiscale structural domains. Numerical examples are presented to validate the optimization and demonstrate the potential of this approach. The numerical examples show that the multiscale bi-material topology optimization method implemented here is able to produce structures and microstructures for optimization of the frequency domain response, satisfying prescribed volume constraints.

     

Topological design of microstructures of cellular materials for maximum bulk or shear modulus

This paper presents a new approach to designing periodic microstructures of cellular materials. The method is based on the bidirectional evolutionary structural optimization (BESO) technique. The optimization problem is formulated as finding a micro-structural topology with the maximum bulk or shear modulus under a prescribed volume constraint. Using the homogenization theory and finite element analysis within a periodic base cell (PBC), elemental sensitivity numbers are established for gradually removing and adding elements in PBC. Numerical examples in 2D and 3D demonstrate the effectiveness of the proposed method for achieving convergent microstructures of cellular materials with maximum bulk or shear modulus. Some interesting topological patterns have been found for guiding the cellular material design.     

Topological optimization for the design of microstructures of isotropic cellular materials

The aim of this study was to design isotropic periodic microstructures of cellular materials using the bidirectional evolutionary structural optimization (BESO) technique. The goal was to determine the optimal distribution of material phase within the periodic base cell. Maximizing bulk modulus or shear modulus was selected as the objective of the material design subject to an isotropy constraint and a volume constraint. The effective properties of the material were found using the homogenization method based on finite element analyses of the base cell. The proposed BESO procedure utilizes the gradient-based sensitivity method to impose the isotropy constraint and gradually evolve the microstructures of cellular materials to an optimum. Numerical examples show the computational efficiency of the approach. A series of new and interesting microstructures of isotropic cellular materials that maximize the bulk or shear modulus have been found and presented. The methodology can be extended to incorporate other material properties of interest such as designing isotropic cellular materials with negative Poisson's ratio.     

Scale‐related topology optimization of cellular materials and structures

The integrated optimization of lightweight cellular materials and structures are discussed in this paper. By analysing the basic features of such a two‐scale problem, it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell (MUC), i.e. the so‐called representative volume element (RVE). However, with the asymptotic homogenization method used widely in actual topology optimization procedure, effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases, properties and spatial distribution of constituents in the microstructure regardless of scale effect. From this consideration, we propose the design element (DE) concept being able to deal with conventional designs of materials and structures in a unified way. By changing the scale and aspect ratio of the DE, scale‐related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns. To illustrate the proposed approach, numerical design problems of 2D layered structures with cellular core are investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal design of laminated piezocomposite energy harvesting devices considering stress constraints

Energy harvesting devices are smart structures capable of converting the mechanical energy (generally, in the form of vibrations) that would be wasted otherwise in the environment into usable electrical energy. Laminated piezoelectric plate and shell structures have been largely used in the design of these devices because of their large generation areas. The design of energy harvesting devices is complex, and they can be efficiently designed by using topology optimization methods (TOM). In this work, the design of laminated piezocomposite energy harvesting devices has been studied using TOM. The energy harvesting performance is improved by maximizing the effective electric power generated by the piezoelectric material, measured at a coupled electric resistor, when subjected to a harmonic excitation. However, harmonic vibrations generate mechanical stress distribution that, depending on the frequency and the amplitude of vibration, may lead to piezoceramic failure. This study advocates using a global stress constraint, which accounts for different failure criteria for different types of materials (isotropic, piezoelectric, and orthotropic). Thus, the electric power is maximized by optimally distributing piezoelectric material, by choosing its polarization sign, and by properly choosing the fiber angles of composite materials to satisfy the global stress constraint. In the TOM formulation, the Piezoelectric Material with Penalization and Polarization material model is applied to distribute piezoelectric material and to choose its polarization sign, and the Discrete Material Optimization method is applied to optimize the composite fiber orientation. The finite element method is adopted to model the structure with a piezoelectric multilayered shell element. Numerical examples are presented to illustrate the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust topology optimization of phononic crystals with random field uncertainty

The uncertain spatial variation of material properties can remarkably affect the band gap characteristics of phononic crystals (PnCs). It is necessary to consider this issue when designing and manufacturing PnC materials/structures. This paper investigates a robust topology optimization method for designing the microstructures of PnCs by considering random‐field material properties. Herein, the spatial distribution of the material properties is first represented by a random field and then discretized into uncorrelated stochastic variables with the expansion optimal linear estimation method; stochastic band gap analysis is then conducted with polynomial chaos expansion. Furthermore, a robust topology optimization formulation of PnCs is proposed on the basis of the relative elemental density, where a weighted objective function handles the compromise of the mean value and standard deviation of the PnC band gap. The band gap response is analyzed, employing the finite element method for each sample of polynomial chaos expansion. In this context, the sensitivities of the stochastic band gap behaviors to the design variables are also derived. Numerical examples demonstrate that the proposed method can generate meaningful optimal topologies of PnCs with a relatively large width and less sensitive band gap. Additionally, the effects of the weight factors in the objective function and the variation coefficient of material properties are discussed.     

基于单胞解析模型的复合材料层合板渐进损伤数值分析

基于单胞解析模型,建立一种从复合材料细观组分到宏观层合板的渐进损伤分析模型。根据连续介质力学和均匀化方法构建细-宏观关联矩阵,通过该矩阵将细观组分材料的弹性和损伤性能传递到宏观复合材料中。该模型只需给出纤维和基体的材料属性及纤维体积含量无需层合板的弹性和强度参数,通过组分材料的损伤失效判据确定其是否损伤,如果发生损伤则用损伤因子折算成相应的刚度衰减。通过用户材料子程序UMAT 及VUMAT将单胞解析模型以及损伤理论嵌入到有限元软件ABAQUS 中,对开孔复合材料层合板的渐进损伤过程进行模拟,预测了层合板强度。结果表明:预报的强度与试验值吻合较好,验证了该方法的有效性。     

特定弹性性能材料的细观结构设计优化

针对具有特定弹性性质的两相复合材料, 研究了特定性能材料优化设计问题的数学模型,提出了基于形状优化的材料设计方法。该方法利用形状优化技术, 设计两相复合材料的细观结构形式, 以使复合材料具有特定的弹性性质。材料的宏观性质由均匀化方法确定。最后给出了零泊松比材料的设计过程和结果。      

Metamaterial topology optimization of nonpneumatic tires with stress and buckling constraints

This article presents the design of a metamaterial for the shear layer of a nonpneumatic tire using topology optimization, under stress and buckling constraints. These constraints are implemented for a smooth maximum function using global aggregation. A linear elastic finite element model is used, implementing solid isotropic material with penalization. Design sensitivities are determined by the adjoint method. The method of moving asymptotes is used to solve the numerical optimization problem. Two different optimization statements are used. Each requires a compliance limit and some aspect of continuation. The buckling analysis is linear, considering the generalized eigenvalue problem of the conventional and stress stiffness matrices. Various symmetries, base materials, and starting geometries are considered. This leads to novel topologies that all achieve the target effective shear modulus of 10 MPa, while staying within the stress constraint. The stress-only designs generally were susceptible to buckling failure. A family of designs (columnar, noninterconnected representative unit cells) that emerge in this study appears to exhibit favorable properties for this application.     

Thermo-electro-mechanical response of 1–3–2 piezoelectric composites: effect of fiber orientations

A micromechanics-based analytical model is developed to evaluate the performance of 1–3–2 piezoelectric composite where both matrix and fiber materials are piezoelectrically active. A parametric study is conducted to investigate the effects of variations in the poling characteristics of the fiber phase on the overall thermo-electro-mechanical behavior of a 1–3–2 piezocomposite. The performance of the 1–3–2 composite as a transducer for underwater and biomedical imaging applications is analyzed. The proposed model is capable of predicting the effective properties of the composite subjected to thermo-electro-mechanical loading conditions. The predicted variations in the effective elastic, piezoelectric and dielectric material constants with fiber volume fraction are nonlinear in nature. It is observed that the influence of thermal effects on effective properties of the composite also induces polarization in the composite. The analytical results show that an appropriate selection of the poling characteristics of the individual fiber and matrix phases could lead to the development of a piezocomposite with significant effective properties.     

Tailoring materials with prescribed elastic properties

This paper describes a method to design the periodic microstructure of a material to obtain prescribed constitutive properties. The microstructure is modelled as a truss or thin frame structure in 2 and 3 dimensions. The problem of finding the simplest possible microstructure with the prescribed elastic properties can be called an inverse homogenization problem, and is formulated as an optimization problem of finding a microstructure with the lowest possible weight which fulfils the specified behavioral requirements. A full ground structure known from topology optimization of trusses is used as starting guess for the optimization algorithm. This implies that the optimal microstructure of a base cell is found from a truss or frame structure with 120 possible members in the 2-dimensional case and 2016 possible members in the 3-dimensional case. The material parameters are found by a numerical homogenization method, using Finite-Elements to model the representative base cell, and the optimization problem is solved by an optimality criteria method.

Numerical examples in two and three dimensions show that it is possible to design materials with many different properties using base cells modelled as truss or frame works. Hereunder is shown that it is possible to tailor extreme materials, such as isotropic materials with Poisson's ratio close to − 1, 0 and 0.5, by the proposed method. Some of the proposed materials have been tested as macro models which demonstrate the expected behaviour.