Optimal design of periodic structures using evolutionary topology optimization

This paper presents a method for topology optimization of periodic structures using the bi-directional evolutionary structural optimization (BESO) technique. To satisfy the periodic constraint, the designable domain is divided into a certain number of identical unit cells. The optimal topology of the unit cell is determined by gradually removing and adding material based on a sensitivity analysis. Sensitivity numbers that consider the periodic constraint for the repetitive elements are developed. To demonstrate the capability and effectiveness of the proposed approach, topology design problems of 2D and 3D periodic structures are investigated. The results indicate that the optimal topology depends, to a great extent, on the defined unit cells and on the relative strength of other non-designable part, such as the skins of sandwich structures.     

Multi-objective concurrent topology optimization of thermoelastic structures composed of homogeneous porous material

The present paper studies multi-objective design of lightweight thermoelastic structure composed of homogeneous porous material. The concurrent optimization model is applied to design the topologies of light weight structures and of the material microstructure. The multi-objective optimization formulation attempts to find minimum structural compliance under only mechanical loads and minimum thermal expansion of the surfaces we are interested in under only thermo loads. The proposed optimization model is applied to a sandwich elliptically curved shell structure, an axisymmetric structure and a 3D structure. The advantage of the concurrent optimization model to single scale topology optimization model in improving the multi-objective performances of the thermoelastic structures is investigated. The influences of available material volume fraction and weighting coefficients are also discussed. Numerical examples demonstrate that the porous material is conducive to enhance the multi-objective performance of the thermoelastic structures in some cases, especially when lightweight structure is emphasized. An “optimal” material volume fraction is observed in some numerical examples.     

Convergence of topological patterns of optimal periodic structures under multiple scales

The current techniques for topology optimization of material microstructure are typically based on infinitely small and periodically repeating base cells. These base cells have no actual size. It is uncertain whether the topology of the microstructure obtained from such a material design approach could be translated into real structures of macroscale. In this work we have carried out a first systematic study on the convergence of topological patterns of optimal periodic structures, the extreme case of which is a material microstructure with infinitesimal base cells. In a series of numerical experiments, periodic structures under various loading and boundary conditions are optimized for stiffness and frequency. By increasing the number of unit cells, we have found that the topologies of the unit cells converge rapidly to certain patterns. It is envisaged that if we continue to increase the number of unit cells and thus reduce the size of each unit cell until it becomes the infinitesimal material base cell, the optimal topology of the unit cell would remain the same. The finding from this work is of significant practical importance and theoretical implication because the same topological pattern designed for given loading and boundary conditions could be used as the optimal solution for the periodic structure of vastly different scales, from a structure with a few (e.g. 20) repetitive modules to a material microstructure with an infinite number of base cells.     

Topological design of modular structures under arbitrary loading

A numerical method for the topological design of periodic continuous domains under general loading is presented. Both the analysis and the design are defined over a single cell. Confining the analysis to the repetitive unit is obtained by the representative cell method which by means of the discrete Fourier transform reduces the original problem to a boundary value problem defined over one module, the representative cell. The repeating module is then meshed into a dense grid of finite elements and solved by finite element analysis. The technique is combined with topology optimization of infinite spatially periodic structures under arbitrary static loading. Minimum compliance structures under a constant volume of material are obtained by using the densities of material as design variables and by satisfying a classical optimality criterion which is generalized to encompass periodic structures. The method is illustrated with the design of an infinite strip possessing 1D translational symmetry and a cyclic structure under a tangential point force. A parametric study presents the evolution of the solution as a function of the aspect ratio of the representative cell.     

Optimal design of two-dimensional band-gap materials for uni-directional wave propagation

New configurations and new properties of materials used in engineering components can be developed by introducing band-gap materials for which a two-dimensional design domain is optimized for uni-directional wave propagation. In this paper, uni-directional band-gap materials with periodic two-dimensional appearances are designed for in-plane waves. By using the Finite Element Method to solve the dynamic behavior of the representative unit cell, the dispersion relation of wave propagation is built up based on Floquet-Bloch theory. With the goal of maximizing the width of band gaps for a certain direction, the distribution of two material phases in a two-dimensional unit cell is determined by a gradient-based topology optimization method. The numerical results show that the proposed periodic hierarchical laminates and the corresponding periodic layered material with homogenized anisotropic layers exhibit wider band gaps than multilayered materials optimized by a one-dimensional design domain. Meanwhile, the influence of the anisotropic property of combined layers and homogenized layers on the band-gap characteristics is analyzed.     

Design of band-gap grid structures

This paper discusses issues related to designing band-gaps in periodic plane grid structures. Finite element analysis is used to solve the dynamic behavior of a representative unit cell and Bloch–Floquet theory is used to extend the results to the infinite structure. Particular attention is given to the addition of non-structural masses that are introduced as design variables. These are used to create desirable features in the dispersion diagram. Physical insight is presented into the optimal choice of locations where masses should be added and the results of several numerical examples are provided to highlight this and other features of how band-gaps can be created and located at desired frequency ranges. The effect of the skew angle of the underlying grid structure is also explored, as are mathematical refinements of the modelling of the beam elements and the rotational inertia of the added masses. A scaling feature between the size of the reducible and the irreducible reference cell is exploited and the manner in which this can simplify optimization approaches is discussed.     

Optimum design of honeycomb sandwich constructions with buckling constraints

Optimum design of honeycomb sandwich constructions with buckling constraints is treated in this paper. Four modes of instability for honeycomb sandwich structures are considered in buckling constraints, including overall buckling, core shear instability, face wrinkling, and monocell buckling. The face thicknesses, core depth, cell wall thickness, and diameter of an inscribed circle in a honeycomb cell are taken as design variables. Eight-nodal quadrilateral honeycomb sandwich isoparametric shell elements and hybrid approximation techniques in combination with the dual solution are used. Some comparisons are also made between the cases with and without buckling constraints. Numerical results are given for four examples.     

Topology optimization of periodic layouts of dielectric materials

A topology optimization method is used to design two dimensional periodic structures with desirable transmission properties by distributing two materials of different permittivity over a rectangular representative cell. A plane wave expansion of the electric field at the input and output boundaries is used in the analysis. This allows non-homogeneous material distributions near the boundaries. Numerical examples are used to verify the robustness of the method and to investigate the importance of retaining higher modes in the expansions. It is found that the optimization problem typically admits possibly many local optima and the relevance of higher modes depends on the nature of the solution found. In some instances, higher modes play an important role and using only the dominant mode in the analysis is shown to result in errors in the evaluation of the performance of the design.     

Damping optimization of viscoelastic laminated sandwich composite structures

Recent developments on the optimization of passive damping for vibration reduction in sandwich structures are presented in this paper, showing the importance of appropriate finite element models associated with gradient based optimizers for computationally efficient damping maximization programs. A new finite element model for anisotropic laminated plate structures with viscoelastic core and laminated anisotropic face layers has been formulated, using a mixed layerwise approach. The complex modulus approach is used for the viscoelastic material behavior, and the dynamic problem is solved in the frequency domain. Constrained optimization is conducted for the maximization of modal loss factors, using gradient based optimization associated with the developed model, and single and multiobjective optimization based on genetic algorithms using an alternative ABAQUS finite element model. The model has been applied successfully and comparative optimal design applications in sandwich structures are presented and discussed.     

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.     

An efficient optimization method for periodic lattice cellular structure design based on the K-fold SVR model

The design optimization of periodic lattice cellular structure relying exclusively on the computational simulation model is a time-consuming, even computationally prohibitive process. To relieve the computational burden, an efficient optimization method for periodic lattice cellular structure design based on the K-fold support vector regression model (K-SVR) is proposed in this paper. First, based on the loading experiments, the most promising unit cell of periodic lattice cellular structure is selected from five typical unit cells. Second, an initial SVR model is constructed to replace the simulation model of the periodic lattice cellular structure, and the K-fold cross-validation approach is used to extract the error information from the SVR model at the sample points. According to the error information, the sample points are sorted and classified into several sub-sets. Then, a global K-SVR model is re-constructed by aggregating each SVR model under each sub-set. Third, considering that there exists prediction errors between the K-SVR model and the simulation model, which may lead to infeasible optimal solutions, an uncertainty quantification approach is developed to ensure the feasibility of the optimal solution for the periodic lattice cellular structure design. Finally, the effectiveness and merits of the proposed approach are demonstrated on the design optimization of the A-pillar and seat-bottom frame.

     

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.     

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.     

Reliability-based vibro-acoustic microstructural topology optimization

A reliability-based vibro-acoustic microstructural topology optimization (RBVAMTO) model taking into consideration the uncertainty of several design-independent parameters, such as the direction of the load, the excitation frequency, or their combinations is presented. The design objective is to minimize the sound power of the vibrating composite macrostructure that is assumed to be constructed by periodic micro unit cell filled up by two kinds of prescribed isotropic materials. The SIMP based bi-material interpolation model is employed at the micro-scale and the design variable is the relative material volume density of the micro unit cell. A design process consisting of the uncertainty analysis and vibro-acoustic microstructural topology optimization is proposed and implemented. The influences of different combination values of the normalized variables corresponding to the uncertainty parameters on the design results are investigated. Numerical examples show that in the vibro-acoustic microstructural design the uncertainty of the excitation frequency may play a very important role. It is also shown that when the normalized variable corresponding to the random excitation frequency takes the higher value, the optimum microstructural topology may not be so sensitive to perturbation of the loading direction. Monte Carlo simulation results demonstrate that the RBVAMTO designs normally lead to the optimum results that are more robust for perturbations of both the excitation frequency and the loading direction than the deterministic design.     

Optimum design of sandwich constructions

Finite element analysis and the optimization problem of sandwich constructions are treated. The thicknesses of the face plates and the core are used as design variables. The hybrid approximation technique in combination with the dual method from mathematical programming is used. Three examples are solved using six-nodal triangular and eight-nodal quadrilateral sandwich shell elements.     

Size-dependent optimal microstructure design based on couple-stress theory

The purpose of this paper is to propose a size-dependent topology optimization formulation of periodic cellular material microstructures, based on the effective couple-stress continuum model. The present formulation consists of finding the optimal layout of material that minimizes the mean compliance of the macrostructure subject to the constraint of permitted material volume fraction. We determine the effective macroscopic couple-stress constitutive constants by analyzing a unit cell with specified boundary conditions with the representative volume element (RVE) method, based on equivalence of strain energy. The computational model is established by the finite element (FE) method, and the design density and FE stiffness of the RVE are related by the solid isotropic material with penalization power (SIMP) law. The required sensitivity formulation for gradient-based optimization algorithm is also derived. Numerical examples demonstrate that this present formulation can express the size effect during the optimization procedure and provide precise topologies without increase in computational cost.     

Concurrent finite element analysis of periodic boundary value problems

A parallel finite element approach for analyzing micromechanical problems with periodic unit cells is discussed. The method uses a direct solution strategy so that general periodic boundary conditions can be treated using a two-step domain decomposition strategy. The speedup results show a good performance of the method on coarse-grained problems, i.e. for cases where the computational work done on the substructures that are treated in parallel is relatively large compared to the total amount of computational work. Application examples using crystal-plasticity on an array of planar crystals and a metal matrix composite are used to show that the overall response of these materials is rather strongly dependent on the constraint imposed on the unit cell so that a correct treatment of the periodic boundary conditions is required to accurately predict the macroscopic response of a periodic material even though a unit cell with a large number of grains or fibers is used.     

Microstructural topology optimization of structural-acoustic coupled systems for minimizing sound pressure level

The aim of this paper is to present a microstructural topology optimization methodology for the structural-acoustic coupled system. In the structural-acoustic system, the structure is considered to be a thin composite plate composed of periodic uniform microstructures. The discrete design variables are used in the microstructural topology optimization, and the constitutive matrix is interpolated by the power-law scheme at the micro scale. The equivalent macro material properties of the microstructure are computed through the homogenization method. The design objective is to minimize the sound pressure level (SPL) in an interior acoustic medium. The sensitivities of the SPL with respect to design variables are derived. The bi-directional evolutionary structural optimization (BESO) method is extended to solve the structural-acoustic coupled optimization problem to find the optimal material distribution of the microstructure. Numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented microstructural topology optimization method.     

Multiobjective evolutionary optimization of periodic layered materials for desired wave dispersion characteristics

An important dispersion-related characteristic of wave propagation through periodic materials is the existence of frequency bands. A medium effectively attenuates all incident waves within stopbands and allows propagation within passbands. The widths and locations of these bands in the frequency domain depend on the layout of contrasting materials and the ratio of their properties. Using a multiobjective genetic algorithm, the topologies of one-dimensional periodic unit cells are designed for target frequency band structures characterizing longitudinal wave motion. The decision variables are the number of layers in the unit cell and the thickness of each layer. Binary and mixed formulations are developed for the treatment of the optimization problems. Designs are generated for the following novel objectives: (1) maximum attenuation of time harmonic waves, (2) maximum isolation of general broadband pulses, and (3) filtering signals at predetermined frequency windows. The saturation of performance with the number of unit-cell layers is shown for the first two cases. In the filtering application, the trade-off between the simultaneous realization of passband and stopband targets is analyzed. It is shown that it is more difficult to design for passbands than it is to design for stopbands. The design approach presented has potential use in the development of vibration and shock isolation structures, sound isolation pads/partitions, and multiple band frequency filters, among other applications.     

Optimal design of periodic functionally graded composites with prescribed properties

The computational design of a composite where the properties of its constituents change gradually within a unit cell can be successfully achieved by means of a material design method that combines topology optimization with homogenization. This is an iterative numerical method, which leads to changes in the composite material unit cell until desired properties (or performance) are obtained. Such method has been applied to several types of materials in the last few years. In this work, the objective is to extend the material design method to obtain functionally graded material architectures, i.e. materials that are graded at the local level (e.g. microstructural level). Consistent with this goal, a continuum distribution of the design variable inside the finite element domain is considered to represent a fully continuous material variation during the design process. Thus the topology optimization naturally leads to a smoothly graded material system. To illustrate the theoretical and numerical approaches, numerical examples are provided. The homogenization method is verified by considering one-dimensional material gradation profiles for which analytical solutions for the effective elastic properties are available. The verification of the homogenization method is extended to two dimensions considering a trigonometric material gradation, and a material variation with discontinuous derivatives. These are also used as benchmark examples to verify the optimization method for functionally graded material cell design. Finally the influence of material gradation on extreme materials is investigated, which includes materials with near-zero shear modulus, and materials with negative Poisson’s ratio.