Topology optimization of material‐nonlinear continuum structures by the element connectivity parameterization

The application of the element density‐based topology optimization method to nonlinear continuum structures is limited to relatively simple problems such as bilinear elastoplastic material problems. Furthermore, it is very difficult to use analytic sensitivity when a commercial nonlinear finite element code is used. As an alternative to the element density formulation, the element connectivity parameterization (ECP) formulation is developed for the topology optimization of isotropic‐hardening elastoplastic or hyperelastic continua by using commercial software. ECP varies the stiffness of zero‐length linear elastic links that connect design domain‐discretizing finite elements. Unloading was not considered. But the advantages of ECP in material‐nonlinear problems were demonstrated: considerably simple analytic sensitivity calculation using a commercial code and simple link stiffness penalization regardless of nonlinear material behaviour. Copyright © 2006 John Wiley & Sons, Ltd.     

基于Preisach理论的形状记忆合金温度-位移迟滞仿真研究

摘 要：智能材料如形状记忆合金(Shape Memory Alloy,SMA)已经广泛应用于驱动器和传感器的设计,实现定位和主动控制目的。然而,受迟滞影响,SMA驱动器的工作精度大大降低,限制了其应用。多数智能材料中,选择Preisach理论成为迟滞建模工具,近年来,也涉及到SMA材料系统。本文,讨论运用Preisach模型描述SMA驱动器系统的迟滞行为,尤其针对驱动器系统的模型建立过程,修正经典Preisach模型的几何解释和数值实现方法。最后,引入Gobert给出的Preisach平面的辨识函数执行仿真计算,数值结果表明该模型能够很好地描述SMA驱动器的迟滞行为。     

Simultaneous optimal design of structural topology, actuator locations and control parameters for a plate structure

 Simultaneous optimization with respect to the structural topology, actuator locations and control parameters of an actively controlled plate structure is investigated in this paper. The system consists of a clamped-free plate, a H ₂ controller and four surface-bonded piezoelectric actuators utilized for suppressing the bending and torsional vibrations induced by external disturbances. The plate is represented by a rectangular design domain which is discretized by a regular finite element mesh and for each element the parameter indicating the presence or absence of material is used as a topology design variable. Due to the unavailability of large-scale 0–1 optimization algorithms, the binary variables of the original topology design problem are relaxed so that they can take all values between 0 and 1. The popular techniques in the topology optimization area including penalization, filtering and perimeter restriction are also used to suppress numerical problems such as intermediate thickness, checkerboards, and mesh dependence. Moreover, since it is not efficient to treat the structural and control design variables equally within the same framework, a nested solving approach is adopted in which the controller syntheses are considered as sub processes included in the main optimization process dealing with the structural topology and actuator locations. The structural and actuator variables are solved in the main optimization by the method of moving asymptotes, while the control parameters are designed in the sub optimization processes by solving the Ricatti equations. Numerical examples show that the approach used in this paper can produce systems with clear structural topology and high control performance. Received 16 November 2001 / Accepted 26 February 2002     

Material nonlinear topology optimization considering the von Mises criterion through an asymptotic approach: Max strain energy and max load factor formulations

This paper addresses material nonlinear topology optimization considering the von Mises criterion by means of an asymptotic analysis using a fictitious nonlinear elastic model. In this context, we consider the topology optimization problem subjected to prescribed energy, which leads to robust convergence in nonlinear problems. Two nested formulations are considered. In the first, the objective is to maximize the strain energy of the system in equilibrium, and in the second, the objective is to maximize the load factor. In both cases, a volume constraint is imposed. The sensitivity analysis is quite effective and efficient in the sense that there is no extra adjoint equation. In addition, the nonlinear structural equilibrium problem is solved using direct minimization of the structural strain energy using Newton's method with an inexact line-search strategy. Four numerical examples demonstrate the features of the proposed material nonlinear topology optimization framework for approximating standard von Mises plasticity.     

Topology optimization of multiscale elastoviscoplastic structures

This paper extends current concepts of topology optimization to the design of structures made of nonlinear microheterogeneous materials. The objective is to maximize the macroscopic structural stiffness for a prescribed material volume usage while accounting for the nonlinearity and the microstructure of the material. The resulting design problem considers two scales: the macroscopic scale at which the optimization is performed and the microscopic scale at which the material heterogeneities and the nonlinearities are observed. The topology optimization at the macroscopic scale is performed by means of the bi‐directional evolutionary structural optimization method. The solution of the macroscopic boundary value problem requires as inputs the effective constitutive response with full consideration of the microstructure. While computational homogenization methods such as the FE² method could be used to solve the nonlinear multiscale problem, the associated numerical expense (CPU time and memory) is highly unacceptable. In order to regain the computational feasibility of the computational scale transition, a recent model reduction technique of the authors is employed: the potential‐based reduced basis model order reduction with graphics processing unit acceleration. Numerical examples show the efficiency of the resulting nonlinear two‐scale designs. The impact of different load amplitudes on the design is examined. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

A three-dimensional shape memory alloy/elastomer actuator

This paper presents a mathematical model for analysis of a thermally-driven shape memory alloy (SMA)/elastomer actuator under arbitrary loading and boundary conditions. The actuator is a three-dimensional laminated composite box beam that consists of SMA and elastomer layers with a uniform rectangular cross section. The thermomechanical behavior of SMA layers is modeled utilizing Tanaka and Nagaki's constitutive equation and linear phase transformation kinetics. The behavior of the elastomer layers is assumed to be thermoelastic, in which the elastic modulus is considered to be temperature dependent. The classical laminated beam theory is employed to obtain the force-deformation relationships. The analysis provides explicit solutions to the structural response of the actuator, including strain and curvature of actuator's midplane. A numerical example for a cantilever box beam with uniform square cross-section subjected to a transverse concentrated load is presented. Results demonstrate that significant changes occur in the actuator's responses during phase transformation due to the strain recovery.     

Conceptual design of reinforced concrete structures using topology optimization with elastoplastic material modeling

Design of reinforced concrete structures is governed by the nonlinear behavior of concrete and by its different strengths in tension and compression. The purpose of this article is to present a computational procedure for optimal conceptual design of reinforced concrete structures on the basis of topology optimization with elastoplastic material modeling. Concrete and steel are both considered as elastoplastic materials, including the appropriate yield criteria and post‐yielding response. The same approach can be applied also for topology optimization of other material compositions where nonlinear response must be considered. Optimized distribution of materials is achieved by introducing interpolation rules for both elastic and plastic material properties. Several numerical examples illustrate the capability and potential of the proposed procedure. Copyright © 2012 John Wiley & Sons, Ltd.     

Material modeling for parametric,anisotropic finite strain hyperelasticity based on machine learning with application in optimization of metamaterials

Mechanical metamaterials such as open- and closed-cell lattice structures, foams, composites, and so forth can often be parametrized in terms of their microstructural properties, for example, relative densities, aspect ratios, material, shape, or topological parameters. To model the effective constitutive behavior and facilitate efficient multiscale simulation, design, and optimization of such parametric metamaterials in the finite deformation regime, a machine learning-based constitutive model is presented in this work. The approach is demonstrated in application to elastic beam lattices with cubic anisotropy, which exhibit highly nonlinear effective behaviors due to microstructural instabilities and topology variations. Based on microstructure simulations, the relevant material and topology parameters of selected cubic lattice cells are determined and training data with homogenized stress-deformation responses is generated for varying parameters. Then, a parametric, hyperelastic, anisotropic constitutive model is formulated as an artificial neural network, extending a recent work of the author extending a recent work of the author, Comput Mech., 2021;67(2):653-677. The machine learning model is calibrated with the simulation data of the parametric unit cell. The authors offer public access to the simulation data through the GitHub repository https://github.com/CPShub/sim-data . For the calibration of the model, a dedicated sample weighting strategy is developed to equally consider compliant and stiff cells and deformation scenarios in the objective function. It is demonstrated that this machine learning model is able to represent and predict the effective constitutive behavior of parametric lattices well across several orders of magnitude. Furthermore, the usability of the approach is showcased by two examples for material and topology optimization of the parametric lattice cell.     

Forced response of cylindrical shells coupled with nonlinear shape-memory-alloy (SMA) actuators regulated by sinusoidal and saw-tooth temperature profiles

Dynamic responses of cylindrical shells coupled with shape-memory alloy (SMA) ring segments placed at critical locations are investigated. Since the SMA actuators are highly nonlinear and governed by the temperate-dependent stiffness and martensite/austenite fraction, input shaping and phase shift of temperature profiles are incorporated to control the shell vibrations. Open-loop equations of an elastic cylindrical shell panel are defined first and then used with assumed mode-shape functions satisfying specified boundary conditions in the free-vibration analysis. Modal-analysis data are used to determine spatial strain distributions of natural modes. Distributed modal-signal characteristics suggest optimal placements of SMA actuator segment(s) for each given mnth mode. Based on the modal-expansion method, the open-loop control force induced by the SMA ring segments applied to a simply supported cylindrical shell panel is formulated. This formulation indicates that only the odd modes in the circumferential direction can be controlled. Longitudinal modes are controlled via placing specific number, depending on the mode, of actuator segments along the panel length. To predict control effects of the nonlinear SMA ring segments, the modal participation factor response is determined for an external harmonic excitation applied to the shell along with induced SMA control forces, via sinusoidal and saw-tooth temperature profiles to generate desirable control forces and to eliminate the unwanted effects. Analysis results suggest that with proper choice of temperature waveform function to the SMA ring segments and minor modifications to frequency and phase, the SMA ring segments can attenuate unwanted external vibrations of cylindrical shells.     

Design optimization of dome actuators

This paper reports the conceptual design, analysis, and modeling of the electromechanical behavior of dome actuators. The geometric parameters of the actuator (dome thickness, width, radius, and depth), poling direction, electric field, and material properties (elastic compliance, piezoelectric constants, and dielectric permittivity) have been taken into account in the modeling work. The results of the analysis indicate that a dome actuator with a tangentially alternating poling direction and electric field (Case C) exhibits much larger displacement and force responses than dome actuators with other poling directions and electric fields. The first mode of natural frequency of the Case C dome actuator also was investigated, and its predicted performance was compared with that of moonie and rainbow actuators. The findings of this research clearly demonstrate the merit of design optimization of electromechanical devices.     

正交异性膜材大变形行为的有限质点法求解

建筑薄膜具有正交异性和拉伸非线性的力学特性,其本构关系的表征和大变形行为的描述都较为复杂,具有很强的几何、材料双重非线性特征。有限质点法是一种新颖的结构数值分析方法,它将传统分析力学方法中复杂的函数连续体模型用清晰的离散质点物理模型取代,通过质点的运动描述结构的行为。该文根据途径单元的基本概念直接在质点内力计算过程中引入不同的膜材本构,将有限质点法拓展应用于正交异性薄膜结构的几何与材料非线性大变形分析。为了准确表征膜材力学特性,根据复合材料本构理论分别建立了适用于有限质点法的正交异性膜材的线性与非线性拉伸本构模型,并通过若干算例探讨了该文方法和程序的适用性和正确性。     

Level set topology optimization of structural problems with interface cohesion

This paper presents a finite element topology optimization framework for the design of two‐phase structural systems considering contact and cohesion phenomena along the interface. The geometry of the material interface is described by an explicit level set method, and the structural response is predicted by the extended finite element method. In this work, the interface condition is described by a bilinear cohesive zone model on the basis of the traction‐separation constitutive relation. The non‐penetration condition in the presence of compressive interface forces is enforced by a stabilized Lagrange multiplier method. The mechanical model assumes a linear elastic isotropic material, infinitesimal strain theory, and a quasi‐static response. The optimization problem is solved by a nonlinear programming method, and the design sensitivities are computed by the adjoint method. The performance of the presented method is evaluated by 2D and 3D numerical examples. The results obtained from topology optimization reveal distinct design characteristics for the various interface phenomena considered. In addition, 3D examples demonstrate optimal geometries that cannot be fully captured by reduced dimensionality. The optimization framework presented is limited to two‐phase structural systems where the material interface is coincident in the undeformed configuration, and to structural responses that remain valid considering small strain kinematics. Copyright © 2017 John Wiley & Sons, Ltd.     

Smart soft composite actuator with shape retention capability using embedded fusible alloy structures

This work presents a new kind of shape memory alloy (SMA) based composite actuators that can retain its shape in multiple configurations without continuous energy consumption by changing locally between a high-stiffness and a low-stiffness state. This was accomplished by embedding fusible alloy (FA) material, Ni-chrome (Ni–Cr) wires and SMA wires in a smart soft composite (SSC) structure. The soft morphing capability of SMA-based SSC structures allows the actuator to produce a smooth continuous deformation. The stiffness variation of the actuator was accomplished by melting the embedded FA structures using Ni–Cr wires embedded in the FA structure. First, the design and manufacturing method of the actuator are described. Then, the stiffness of the structure in the low and high-stiffness states of the actuator were measured for different applied currents and heating durations of the FA structure and results show that the highest stiffness of the actuator is more than eight times that of its lowest stiffness. The different shape retention capability of the actuator were tested using actuators with one or two segments and these were compared with a numerical model.     

Shape Memory Alloy (SMA)‐Based Microscale Actuators with 60% Deformation Rate and 1.6 kHz Actuation Speed

Shape memory alloys (SMAs) are widely utilized as an actuation source in microscale devices, since they have a simple actuation mechanism and high‐power density. However, they have limitations in terms of strain range and actuation speed. High‐speed microscale SMA actuators are developed having diamond‐shaped frame structures with a diameter of 25 µm. These structures allow for a large elongation range compared with bulk SMA materials, with the aid of spring‐like behavior under tensile deformation. These actuators are validated in terms of their applicability as an artificial muscle in microscale by investigating their behavior under mechanical deformation and changes in thermal conditions. The shape memory effect is triggered by delivering thermal energy with a laser. The fast heating and cooling phenomenon caused by the scale effect allows high‐speed actuation up to 1600 Hz. It is expected that the proposed actuators will contribute to the development of soft robots and biomedical devices.     

结构地震响应控制的优化方法

针对大跨空间结构地震响应主动控制,以超磁致伸缩材料为核心元件设计了一种可以应用于大跨空间结构振动主动控制中的超磁致伸缩作动器,制作出了作动器的原型并对其进行了输出性能测试。使用遗传算法对大跨空间结构主动控制作动器的布置位置进行了优化设计,最后进行了优化效果的数值模拟分析。以此验证了超磁致伸缩作动器具有良好的作动效应,利用遗传算法在大大提高结构主动控制优化设计效率时,可以保证实现对结构的整体优化以及作动器能高效、经济地实现对结构进行主动控制的目的。     

Fault tolerance of magnetic bearings by generalized bias currentlinearization

The mathematical basis for bias linearization of quadratic magnetic actuators, as typified by magnetic bearings, is developed. The approach generalizes prior ad hoc methods of linearizing the relationship between actuator force and electromagnet current, obviating the earlier assumptions of stator symmetry. This relationship is fundamentally quadratic in the regime where the magnetic material is unsaturated and flux is essentially proportional to magnet current. Growing from the properties of a fundamental representation for the current-force relationships in magnetic bearings, conditions are determined under which linearization may be possible. A numerical optimization problem is posed whose solution provides a linearization scheme which maximizes the available force capacity of the actuator. A corollary result is a method for obtaining coil-fault tolerance in magnetic bearings without adding coils to existing actuators. Several paper examples are presented to illustrate linearization of asymmetric actuators, actuators with failed coils, and force/moment actuators     

Electromechanical coupling and output efficiency of piezoelectric bending actuators

Electromechanical coupling mechanisms in piezoelectric bending actuators are discussed in this paper based on the constitutive equations of cantilever bimorph and unimorph actuators. Three actuator characteristic parameters, (e.g., electromechanical coupling coefficient, maximum energy transmission coefficient, and maximum mechanical output energy) are discussed for cantilever bimorph and unimorph actuators. In the case of the bimorph actuator, if the effect of the bonding layer is negligible, these parameters are directly related to the transverse coupling factor lest. In the case of the unimorph actuator, these parameters also depend on the Young's modulus and the thickness of the elastic layer. Maximum values for these parameters can be obtained by choosing proper thickness ratio and Young's modulus ratio of elastic and piezoelectric layers. Calculation results on four unimorph actuators indicate that the use of stiffer elastic material is preferred to increase electromechanical coupling and output mechanical energy in unimorph actuators.     

Towards the stabilization of the low density elements in topology optimization with large deformation

This work addresses the treatment of lower density regions of structures undergoing large deformations during the design process by the topology optimization method (TOM) based on the finite element method. During the design process the nonlinear elastic behavior of the structure is based on exact kinematics. The material model applied in the TOM is based on the solid isotropic microstructure with penalization approach. No void elements are deleted and all internal forces of the nodes surrounding the void elements are considered during the nonlinear equilibrium solution. The distribution of design variables is solved through the method of moving asymptotes, in which the sensitivity of the objective function is obtained directly. In addition, a continuation function and a nonlinear projection function are invoked to obtain a checkerboard free and mesh independent design. 2D examples with both plane strain and plane stress conditions hypothesis are presented and compared. The problem of instability is overcome by adopting a polyconvex constitutive model in conjunction with a suggested relaxation function to stabilize the excessive distorted elements. The exact tangent stiffness matrix is used. The optimal topology results are compared to the results obtained by using the classical Saint Venant–Kirchhoff constitutive law, and strong differences are found.     

A nonlinear plate finite element formulation for shape memory alloy applications

The aim of the present work is to develop a new finite element model for the finite strain analysis of plate structures constituted of shape memory alloy (SMA) material. A three‐dimensional constitutive model for shape memory alloys able to reproduce the special thermomechanical behavior of SMA characterized by pseudoelasticity and shape memory effects is adopted. The finite strain constitutive model is thermodynamically consistent and is completely formulated in the reference configuration. A two‐dimensional plate theory is proposed based on a tensor element shape function formulation. The displacement field is expressed in terms of increasing powers of the transverse coordinate. The equilibrium statement is formulated on the basis of the virtual displacement principle in a total Lagrangian format. The proposed displacement formulation is particularly suitable for the simple derivation of high‐order finite elements. Numerical applications are performed to assess the efficiency and locking performance of the proposed plate finite element. Some additional numerical examples are carried out to study the accuracy and robustness of the proposed computational technique and its capability of describing the structural response of SMA devices. Copyright © 2011 John Wiley & Sons, Ltd.