Topology optimization of three-dimensional structures using optimum microstructures

In this paper, optimum three-dimensional microstructures derived in explicit analytical form by Gibianski and Cherkaev (1987) are used for topology optimization of linearly elastic three-dimensional continuum structures subjected to a single case of static loading. For prescribed loading and boundary conditions, and subject to a specified amount of structural material within a given three-dimensional design domain, the optimum structural topology is determined from the condition of maximum integral stiffness, which is equivalent to minimum elastic complicance or minimum total elastic energy at equilibrium.The use of optimum microstructures in the present work renders the local topology optimization problem convex, and the fact that local optima are avoided implies that we can develop and present a simple sensitivity based numerical method of mathematical programming for solution of the complete optimization problem.Several examples of optimum topology designs of three-dimensional structures are presented at the end of the paper. These examples include some illustrative full three-dimensional layout and topology optimization problems for plate-like structures. The solutions to these problems are compared to results obtained earlier in the literature by application of usual two-dimensional plate theories, and clearly illustrate the advantage of the full three-dimensional approach.     

Topology optimization of energy harvesting devices using piezoelectric materials

Energy harvesting devices based on the piezoelectric effect that converts ambient energy to electric energy is a very attractive energy source for remote sensors and embedded devices. Although topology optimization has been applied to the design of piezoelectric transducers, the locations of piezoelectric materials are predefined and only the optimal layout of elastic materials is considered. In this paper, both elastic materials as well as piezoelectric materials are considered for the design of energy harvesting devices under the topology optimization formulation. The objective function for this study is to maximize the energy conversion factor. The sensitivities of both stored strain energy and electrical energy are derived by the adjoint method. Examples of energy harvesting devices are presented and discussed using the proposed method.     

Optimal topology design for stress-isolation of soft hyperelastic composite structures under imposed boundary displacements

Soft hyperelastic composite structures that integrate soft hyperelastic material and linear elastic hard material can undergo large deformations while isolating high strain in specified locations to avoid failure. This paper presents an effective topology optimization-based methodology for seeking the optimal united layout of hyperelastic composite structures with prescribed boundary displacements and stress constraints. The optimization problem is modeled based on the power-law interpolation scheme for two candidate materials (one is soft hyperelastic material and the other is linear elastic material). The ?-relaxation technique and the enhanced aggregation method are employed to avoid stress singularity and improve the computational efficiency. Then, the topology optimization problem can be readily solved by a gradient-based mathematical programming algorithm using the adjoint variable sensitivity information. Numerical examples are given to show the importance of considering prescribed boundary displacements in the design of hyperelastic composite structures. Moreover, numerical solutions demonstrate the validity of the present model for the optimal topology design with a stress-isolated region.     

Design of phononic crystals for self-collimation of elastic waves using topology optimization method

Self-collimating phononic crystals (PCs) are periodic structures that enable self-collimation of waves. While various design parameters such as material property, period, lattice symmetry, and material distribution in a unit cell affect wave scattering inside a PC, this work aims to find an optimal material distribution in a unit cell that exhibits the desired self-collimation properties. While earlier studies were mainly focused on the arrangement of self-collimating PCs or shape changes of inclusions in a unit cell having a specific topological layout, we present a topology optimization formulation to find a desired material distribution. Specifically, a finite element based formulation is set up to find the matrix and inclusion material distribution that can make elastic shear-horizontal bulk waves propagate along a desired target direction. The proposed topology optimization formulation newly employs the geometric properties of equi-frequency contours (EFCs) in the wave vector space as essential elements in forming objective and constraint functions. The sensitivities of these functions with respect to design variables are explicitly derived to utilize a gradient-based optimizer. To show the effectiveness of the formulation, several case studies are considered.     

有限弹性体中声表面波波速的分析

声表面波在无限大弹性体中以瑞利波(Rayleigh wave)的形式传播,弹性体中声表面波波速的求解最早也是由Rayleigh从各向同性材料在半无限空间求解开始的.以后的研究工作都沿用半无限空间模型,但在实际工程问题中,声表面波必定是在有限弹性体中传播的.这就要求我们必须研究在有限弹性体中声表面波传播的问题.在声表面波的分析中我们利用Matlab软件进行计算分析.求解了声表面波在带有电极层的各向同性材料弹性体中的波速.此外还推导并求解了各向异性材料弹性体中声表面波的波速方程..     

Design of piezocomposite materials and piezoelectric transducers using topology optimization— Part III

Summary Following the topic introduced in [1, 2] this paper discusses the design of piezoelectric transducers used in applications such as acoustic wave generation and resonators. These applications require goals in the transducer design such as high electromechanical energy conversion for a certain transducer vibration mode and narrowband or broadband response. The development of these devices has been based on the use of simple analytical models, test of prototypes, and analysis by the finite element method. However, in all cases the design is limited to a parametric optimization where only some dimensions of a chosen transducer configuration are optimized. By changing the topology of these devices or their components, we may obtain devices with better performance since the design space of solutions is enlarged. Based on this idea, we have proposed the use of topology optimization for designing these devices. This method consists of finding the distribution of the material and void phases in the design domain that optimizes a defined objective function. The optimized solution is obtained using Sequential Linear Programming (SLP). Considering acoustic wave generation and resonator applications, three kinds of objective functions were defined: maximize the energy conversion for a specific mode or a set of modes; design a transducer with specified frequencies; and design a transducer with narrowband or broadband response. Although only two-dimensional plane strain transducer topologies have been considered in order to illustrate the implementation of the method, it can be extended to three-dimensional topologies. Transducer designs were obtained that conform to the desired design requirements and have better performance characteristics than other common designs.     

Topology optimization of periodic microstructures for enhanced dynamic properties of viscoelastic composite materials

We present a topology optimization method for the design of periodic composites with dissipative materials for maximizing the loss/attenuation of propagating waves. The computational model is based on a finite element discretization of the periodic unit cell and a complex eigenvalue problem with a prescribed wave frequency. The attenuation in the material is described by its complex wavenumber, and we demonstrate in several examples optimized distributions of a stiff low loss and a soft lossy material in order to maximize the attenuation. In the examples we cover different frequency ranges and relate the results to previous studies on composites with high damping and stiffness based on quasi-static conditions for low frequencies and the bandgap phenomenon for high frequencies. Additionally, we consider the issues of stiffness and connectivity constraints and finally present optimized composites with direction dependent loss properties.     

轨道扣件弹性垫板结构优化设计

采用OptiStruct对某型铁路扣件轨下垫板进行拓扑优化和自由形状优化,实现垫板结构轻量化设计.根据优化空间最大化原则,对垂向载荷工况进行拓扑优化、对极限载荷工况进行自由形状优化.使用Abaqus极限载荷扣件系统仿真模型对优化后垫板结构性能进行验证评价.结果表明:优化后的垫板性能基本不变,质量减少约10%,刚度满足设计要求.优化结果可为垫板轻量化设计提供参考.     

Optimizing inclusion shapes and patterns in periodic materials using Discrete Object Projection

Current topology optimization methodologies assume a monolithic, free form approach to design. Many engineered materials and structures, however, are composed of discrete, non-overlapping objects such as fiber or particle-based materials. Application of the topology optimization methodology to these types of materials therefore requires controlling the shape and interaction of designed features to ensure solutions are meaningful and physically realizable. Achieving such control on continuum domains is challenging as features form via the union of elements of like phase. A topology optimization approach is proposed herein for optimizing the size, shape, and layout of inclusion-like features in a continuum domain. The technique is based on the Heaviside Projection Method and uses multiple regularized Heaviside functions whose interaction is tailored so that the designer may restrict the minimum and maximum length scale of inclusions, and minimum spacing between inclusions. The technique is demonstrated on the design of material microstructures with enhanced elastic stiffness.     

Minimization of sound radiation from vibrating bi-material structures using topology optimization

Up to now, work on topological design optimization of vibrating structures against noise radiation has mainly addressed the maximization of eigenfrequencies and gaps between consecutive eigenfrequencies of free vibration, and minimization of the dynamic compliance subject to harmonic loading on the structure. In this paper, we deal with topology optimization problems formulated directly with the design objective of minimizing the sound power radiated from the structural surface(s) into a surrounding acoustic medium. Bi-material elastic continuum structures without material damping are considered. The structural vibrations are excited by time-harmonic external mechanical loading with prescribed frequency and amplitude. It is assumed that air is the acoustic medium and that a feedback coupling to the structure can be neglected. Certain conditions are assumed that imply that the sound power emission from the structural surface can be obtained in a simpler way than by solving Helmholz’ integral equation. Hereby, the computational cost of the structural-acoustical analysis is substantially reduced. Several numerical results are presented and discussed for plate- and pipe-like structures with different sets of boundary and loading conditions.     

Piezoresistive sensor design using topology optimization

Piezoresistive materials, materials whose resistivity properties change when subjected to mechanical stresses, are widely utilized in many industries as sensors, including pressure sensors, accelerometers, inclinometers, and load cells. Basic piezoresistive sensors consist of piezoresistive devices bonded to a flexible structure, such as a cantilever or a membrane, where the flexible structure transmits pressure, force, or inertial force due to acceleration, thereby causing a stress that changes the resistivity of the piezoresistive devices. By applying a voltage to a piezoresistive device, its resistivity can be measured and correlated with the amplitude of an applied pressure or force. The performance of a piezoresistive sensor is closely related to the design of its flexible structure. In this research, we propose a generic topology optimization formulation for the design of piezoresistive sensors where the primary aim is high response. First, the concept of topology optimization is briefly discussed. Next, design requirements are clarified, and corresponding objective functions and the optimization problem are formulated. An optimization algorithm is constructed based on these formulations. Finally, several design examples of piezoresistive sensors are presented to confirm the usefulness of the proposed method.     

Topology optimization of dielectric substrates for filters and antennas using SIMP

Summary In this paper a novel design procedure based on the integration of full wave Finite Element Analysis (FEA) and a topology design method employing Sequential Linear Programming (SLP) is introduced. The employed design method is the Solid Isotropic Material with Penalization (SIMP) technique formulated as a general non-linear optimization problem. SLP is used to solve the optimization problem with the sensitivity analysis based on the adjoint variable method for complex variables. A key aspect of the proposed design method is the integration of optimization tools with a fast simulator based on the finite element-boundary integral (FE-BI) method. The capability of the design method is demonstrated by two design examples. First, we developed a metamaterial substrate with arbitrary material composition and subject to a pre-specified antenna bandwidth enhancement. The design is verified and its performance is evaluated via measurements and simulation. As a second example, the material distribution for a Thermo-Photovoltaic (TPV) filter subject to pre-specified bandwidth and compactness criteria is designed. Results show that the proposed design method is capable of designing full three-dimensional volumetric material textures and printed conductor topologies for filters and patch antennas with enhanced performance.     

Topology optimization of microstructures of viscoelastic damping materials for a prescribed shear modulus

Damping performance of a passive constrained layer damping (PCLD) structure mainly depends on the geometric layout and physical properties of the viscoelastic damping material. Properties such as the shear modulus of the damping material need to be tailored for improving the damping of the structures. This paper presents a topology optimization method for designing the microstructures in 2D, i.e., the structure of the periodic unit cell (PUC), of cellular viscoelastic materials with a prescribed shear modulus. The effective behavior of viscoelastic materials is derived through the use of a finite element based homogenization method. Only isotropic matrix material was considered and under such assumption it is found that the effective loss factor of viscoelastic material is independent of the geometrical configuration of the PUC. Based upon the idea of a Solid Isotropic Material with Penalization (SIMP) method of topology optimization, the relative material densities of the elements of the PUC are considered as the design variables. The topology optimization problem of viscoelastic cellular material with a prescribed property and with constraints on the isotropy and volume fraction is established. The optimization problem is solved using the sequential linear programming (SLP) method. Several examples of the design optimization of viscoelastic cellular materials are presented to demonstrate the validity of the method. The effectiveness of the design method is illustrated by comparing a solid and an optimized cellular viscoelastic material as applied to a cantilever beam with the passive constrained layer damping treatment.     

Design of periodic elastoplastic energy dissipating microstructures

The design of periodic elastoplastic microstructures for maximum energy dissipation is carried out using topology optimization. While the topology optimization of elastic microstructures has been performed in numerous studies, microstructural design considering inelastic behavior is relatively untouched due to a number of reasons which are addressed in this study. An RVE-based multiscale model is employed for computational homogenization with periodic boundary constraints, satisfying the Hill-Mandel principle. The plastic anisotropy which may be prevalent in materials fabricated through additive manufacturing processes is considered by modeling the constitutive behavior at the microscale with Hoffman plasticity. Discretization is done using enhanced assumed strain elements to avoid locking from incompressible plastic flow under plane strain conditions and a Lagrange multiplier approach is used to enforce periodic boundary constraints in the discrete system. The design problem is formulated using a density-based parameterization in conjunction with a SIMP-like material interpolation scheme. Attention is devoted to issues such as dependence on initial design and enforcement of microstructural connectivity, and a number of optimized microstructural designs are obtained under different prescribed deformation modes.

     

Topology optimization of fluid flow channel in cold plate for active phased array antenna

This article presents a topology optimization method for the design of the fluid flow channel of cold plate, aiming to solve the problem of heat dissipation for power devices in active phased array antenna (APAA). The density-based topology optimization method is used in the topology optimization design of flow path, in which the conjugate heat transfer analysis is performed. In the numerical experiment, we use the central plane of the cold plate to make a two-dimensional topology optimization, and then the result of two-dimensional topology optimization was used to build the three-dimensional flow channel of cold plate. The results of simulation show that the optimized flow channel has a better ability of heat dissipation compared with the traditional S style flow channel, which provides important reference for engineering application.     

Applications of one-shot methods in PDEs constrained shape optimization

The paper deals with applications of numerical methods for optimal shape design of composite materials structures and devices. We consider two different physical models described by specific partial differential equations (PDEs) for real-life problems. The first application relates microstructural biomorphic ceramic materials for which the homogenization approach is invoked to formulate the macroscopic problem. The obtained homogenized equation in the macroscale domain is involved as an equality constraint in the optimization task. The second application is connected to active microfluidic biochips based on piezoelectrically actuated surface acoustic waves (SAWs). Our purpose is to find the best material-and-shape combination in order to achieve the optimal performance of the materials structures and, respectively, an improved design of the novel nanotechnological devices. In general, the PDEs constrained optimization routine gives rise to a large-scale nonlinear programming problem. For the numerical solution of this problem we use one-shot methods with proper optimization algorithms and inexact Newton solvers. Computational results for both applications are presented and discussed.     

Topology optimization of continuum structures subjected to pressure loading

This paper presents a generalization of topology optimization of linearly elastic continuum structures to problems involving loadings that depend on the design. Minimum compliance is chosen as the design objective, assuming the boundary conditions and the total volume within the admissible design domain to be given. The topology optimization is based on the usage of a SIMP material model. The type of loading considered in this paper occurs if free structural surface domains are subjected to static pressure, in which case both the direction and location of the loading change with the structural design. The presentation of the material is given in a 2D context, but an extension to 3D is straightforward. The robustness of the optimization method is illustrated by some numerical examples in the end of the paper. Received August 3, 1999     

Topology optimization of three-dimensional non-centrosymmetric micropolar bodies

The topology optimization problem for linearly elastic micropolar solids is dealt with. The constituent materials are supposed to lack in general of centro-symmetry, which means that force stresses and microcurvatures are coupled, and so are couple stresses and micropolar strains. The maximum global stiffness is taken as objective function. According to the SIMP model, the constitutive tensors are assumed to be smooth functions of the design variable, that is, the material density. Optimal material distributions are obtained for several significant three-dimensional cases. The differences respect to the optimal configurations obtained with classical Cauchy materials and centrosymmetric materials are pointed out. The influence of the constants defining the non-centrosymmetric behaviour on the optimal configurations is discussed.     

Material nonlinear topology optimization using the ground structure method with a discrete filtering scheme

Topology optimization of truss lattices, using the ground structure method, is a practical engineering tool that allows for improved structural designs. However, in general, the final topology consists of a large number of undesirable thin bars that may add artificial stiffness and degenerate the condition of the system of equations, sometimes even leading to an invalid structural system. Moreover, most work in this field has been restricted to linear material behavior, yet real materials generally display nonlinear behavior. To address these issues, we present an efficient filtering scheme, with reduced-order modeling, and demonstrate its application to two- and three-dimensional topology optimization of truss networks considering multiple load cases and nonlinear constitutive behavior. The proposed scheme accounts for proper load levels during the optimization process, yielding the displacement field without artificial stiffness by simply using the truss members that actually exist in the structure (spurious members are removed), and improving convergence performance. The nonlinear solution scheme is based on a Newton-Raphson approach with line search, which is essential for convergence. In addition, the use of reduced-order information significantly reduces the size of the structural and optimization problems within a few iterations, leading to drastically improved computational performance. For instance, the application of our method to a problem with approximately 1 million design variables shows that the proposed filter algorithm, while offering almost the same optimized structure, is more than 40 times faster than the standard ground structure method.     

Long-wave interfacial instabilities in a thin electrolyte film undergoing coupled electrokinetic flows: a nonlinear analysis

Spatiotemporal deformations of the free charged surface of a thin electrolyte film undergoing a coupled electrokinetic flow composed of an electroosmotic flow (EOF) on a charged solid substrate and an electrophoretic flow (EPF) at its free surface are explored through linear stability analysis and the long-wave nonlinear simulations. The nonlinear evolution equation for the deforming surface is derived by considering both the Maxwell’s stresses and the hydrodynamic stresses. The electric potential across the film is obtained from the Poisson–Boltzmann equation under the Debye–Hückel approximation. The simulations show that at the charged electrolyte–air interface, the applied electric field generates an EPF similar to that of a large charged particle. The EOF near the solid–electrolyte interface and the EPF at the electrolyte–air interface are in the same (or opposite) directions when the zeta potentials at the two interfaces are of the opposite (or same) signs. The linear and nonlinear analyses of the evolution equation predict the presence of travelling waves, which is strongly modulated by the applied electric field and the magnitude/sign of the interface zeta potentials. The time and length scales of the unstable modes reduce as the sign of zeta potential at the two interfaces is varied from being opposite to same and also with the increasing applied electric field. The increased destabilization is caused by a reverse EPF near the free surface when the interfaces bear the same sign of zeta potentials. Flow reversal by EPF at the free surface occurs at smaller zeta potential of the free surface when the film is thicker because of less influence of the EOF arising at the solid–electrolyte boundary. The amplitude of the surface waves is found to be smaller when the unstable waves travel at a faster speed. The films can undergo pseudo-dewetting when the free surface is almost stationary under the combined influences of EPF and EOF. The free surface instability of the coupled EOF and EPF has some interesting implications in the development of micro/nano fluidic devices involving a free surface.