A review of homogenization and topology optimization III-topology optimization using optimality criteria

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.     

Linear free flexural vibration of cracked functionally graded plates in thermal environment

In this paper, the linear free flexural vibrations of functionally graded material plates with a through center crack is studied using an 8-noded shear flexible element. The material properties are assumed to be temperature dependent and graded in the thickness direction. The effective material properties are estimated using the Mori–Tanaka homogenization scheme. The formulation is developed based on first-order shear deformation theory. The shear correction factors are evaluated employing the energy equivalence principle. The variation of the plates natural frequency is studied considering various parameters such as the crack length, plate aspect ratio, skew angle, temperature, thickness and boundary conditions. The results obtained here reveal that the natural frequency of the plate decreases with increase in temperature gradient, crack length and gradient index.     

Computational networks and systems – homogenization of variational problems on micro-architectured networks and devices

Networked materials and micro-architectured systems gain increasingly importance in multi-scale physics and engineering sciences. Typically, computational intractable microscopic models have to be applied to capture the physical processes and numerous transmission conditions at singularities, interfaces and borders. The topology of the periodic microstructure governs the effective behaviour of such networked systems. A mathematical concept for the analysis of microscopic models on extremely large periodic networks is developed. We consider microscopic models for diffusion–advection–reaction systems in variational form on periodic manifolds. The global characteristics are identified by a homogenization approach for singularly perturbed networks with a periodic topology. We prove that the solutions of the variational models on varying networks converge to a two-scale limit function. In addition, the corresponding tangential gradients converge to a two-scale limit function for vanishing lengths of branches. We identify the variational homogenized model. Complex network models, previously considered as completely intractable, can now be solved by standard PDE-solvers in nearly no time. Furthermore, the homogenized coefficients provide an effective characterization of the global behaviour of the variational system.     

Elastic-plastic dynamic analysis of anisotropic laminated plates

This paper presents the application of a refined finite element model to the elastic and elastic-plastic dynamic analysis of anisotropic laminated plates. Dynamic analysis is based on Newmark's algorithm used in conjunction with the Hughes and Liu predictor-corrector scheme resulting in an ‘effective static problem’ which is solved using a Newton-Raphson-type process. Flow theory is used in the inelastic range and the Huber-Mises yielding surface extended by Hill for anisotropic materials is adopted. Numerical results obtained for two categories of anisotropic structures, namely cross-ply laminated plates and angle-ply laminated plates, are presented and the effects of anisotropy and bending/ stretching coupling on the dynamic elastic and elastic-plastic responses are discussed. The effects of lamina stack sequences and lamina angle sequences on the dynamic responses are also considered.     

局部各向异性的薄壳收缩变形

基于位置动力学提出局部各向异性的薄壳收缩变形方法.首先针对基于位置动力学变形模拟方法的材质局限性的不足,提出薄壳收缩变形的弹性变形能,实现了多材质的弹性收缩变形.其次,针对薄壳收缩变形过程中的抖动问题,给出适当的弯曲能系数,实现了稳定的收缩变形.第三,针对薄壳局部类球面结构收缩变形缓慢且细微的不足,定义了局部各向异性ARAP变形能等,实现了薄壳局部类球结构的快速、显著、稳定的收缩变形.最后以轴向平行包围盒与非渗透滤波器作为碰撞检测的预处理,剔除不可能发生碰撞的图元对,提高了收缩变形过程中的碰撞检测效率.相关实验结果表明,提出的薄壳收缩变形算法适用于多种材质模型以及多种各向异性能量,且有效地解决了抖动及局部类球结构收缩变形缓慢且不显著等问题.     

Reduced basis technique for evaluating the sensitivity of the nonlinear vibrational response of composite plates

A reduced basis technique and a computational procedure are presented for generating the nonlinear vibrational response, and evaluating the first-order sensitivity coefficients of composite plates (derivatives of the nonlinear frequency with respect to material and geometric parameters of the plate). The analytical formulation is based on a form of the geometrically nonlinear shallow shell theory with the effects of transverse shear deformation, rotatory inertia and anisotropic material behavior included. The plate is discretized by using mixed finite element models with the fundamental unknowns consisting of both the nodal displacements and the stress-resultant parameters of the plate. The computational procedure can be conveniently divided into three distinct steps. The first step involves the generation of various-order perturbation vectors, and their derivatives with respect to the material and lamination parameters of the plate, using Linstedt-Poincaré perturbation technique. The second step consists of using the perturbation vectors as basis vectors, computing the amplitudes of these vectors and the nonlinear frequency of vibration, via a direct variational procedure. The third step consists of using the perturbation vectors, and their derivatives, as basis vectors and computing the sensitivity coefficients of the nonlinear frequency via a second application of the direct variational procedure. The effectiveness of the proposed technique is demonstrated by means of numerical examples of composite plates.     

Transverse shear effect on vibration of laminated plate using higher-order plate element

An approach using a higher-order plate element to include the effect of transverse shear deformation on free vibration of laminated plate is presented. The total displacement of the element is expressed as the sum of the displacement due to bending and that due to shear deformation. The double-sized stiffness and mass matrices due to the separation of bending and shear displacements are then reduced to the size as if only the total deflection was considered. Numerical results for natural frequencies for a range of different isotropic and anisotropic plates with various thickness-to-length ratios are obtained and compared with solutions available in the literature. The effect of transverse shear deformation on natural frequencies of higher modes of laminated plates is also discussed.     

Sensitivity strategies in modelling heterogeneous media undergoing finite deformation

This paper deals with homogenization of microscopically heterogeneous media which are subjected to finite deformations. The updated Lagrangian scheme is applied to obtain linear subproblems which can be homogenized using the two-scale convergence. Microscopic equations and homogenized stiffness coefficients are derived for the hyperelastic material with incompressible inclusions. A sensitivity analysis of homogenized coefficients is proposed to study their dependence on local deformations of the microstructure. This approach can assist in reducing the number of the local microscopic equations that have to be solved in each iteration of macroscopic problems.     

On the modelling of ribbed plates for shape optimization

The development of a homogenized plate model suitable for shape optimization is presented. The development is based on a homogenization method for layered materials with a periodic microstructure. A particular advantage of the approach is that it accommodates without significant changes the modelling of ribbed, honeycomb and perforated plates. The model is compared with others that have appeared in the literature and that are also useful in the context of the optimization of the shape and layout of plates and plate-like structures. The results indicate that the model presented here is useful in the optimization of both thick and thin plates.     

基于修正变分法开孔板和加强板结构的自由振动分析

为研究船舶开孔板和加强板结构的振动特性,用1阶剪切变形板理论描述各向同性板的位移场,并采用修正变分原理和区域分解方法建立板的离散动力学模型.每一块子域板的位移和转角分量通过第一类切比雪夫正交多项式展开.针对加强板模型,将该方法获得的结果与已经发表的文献和有限元商用软件计算结果进行对比,验证该方法的收敛性和正确性.基于修正变分法探讨多种开孔和加强板模型的自由振动特性,充分说明该数理模型和半解析方法是一种适合处理复杂板结构问题的数值工具.     

Anisotropic mesh adaptation for the generalized Ambrosio–Tortorelli functional with application to brittle fracture

Quasi-static crack propagation in brittle materials is modeled via the Ambrosio–Tortorelli approximation. The crack is modeled by a smooth phase-field, defined on the whole computational domain. Since the crack is confined to a thin layer, the employment of anisotropic adapted grids is shown to be a really effective tool in containing computational costs. We extend the anisotropic error analysis, applied to the classical Ambrosio–Tortorelli approximation by Artina et al., to the generalized Ambrosio–Tortorelli functional, where a unified framework for several elasticity laws is dealt with as well as a non-convex fracture energy can be accommodated. After deriving an anisotropic a posteriori error estimator, we devise an algorithm which alternates optimization and mesh adaptation. Both anti-plane and plane-strain configurations are numerically checked.     

An autowave based methodology for deformable object simulation

This paper presents a new methodology for deformable object simulation by drawing an analogy between autowaves and elastic deformation. The potential energy stored in an elastic body as a result of a deformation caused by an external force is propagated among mass points by non-linear autowaves. The novelty of the methodology is that autowave techniques are established to describe the potential energy distribution of a deformation for extrapolating internal elastic forces, and non-linear material properties are modelled with non-linear autowaves other than geometric non-linearity. A haptic virtual reality system has been developed for deformation simulation with force feedback. The proposed methodology not only deals with large-range deformations, but also accommodates isotropic, anisotropic and inhomogeneous materials by simply modifying diffusion coefficients.     

On recent developments in the homogenization theory of elastic plates and their application to optimal design: Part I

The paper presents an up-to-date review of results concerning the evaluation of the stiffnesses of elastic plates with periodic structure. The homogenization formulae of Kohn and Vogelius (1984) (modela = 1) are viewed as the formulae of reference. The paper discusses the range of applicability of the homogenization formulae for the two-dimensional plate models of Kirchhoff and Reissner-Hencky. Transversely symmetric as well as asymmetric plates are considered. Particular attention is paid to plates stiffened asymmetrically with ribs in one direction. For this case, the closed formulae for effective membrane, reciprocal and bending stiffnesses are derived. The homogenization results overviewed are a prerequisite for the discussion of the methods of regularization of optimum design plate problems presented in Part II of the paper.Revised version of a paper presented at NATO ASI Optimization of Large Structural Systems, Berchtesgaden, Germany, Sept. 23 – Oct. 4, 1991     

Optimization of perforated domains through homogenization

We begin by explaining briefly why some shape/topology optimization problems need to be relaxed through homogenization. In Section 2 we present, from a mechanical viewpoint, the formula for the homogenized coefficients for a periodic infinitesimal perforation, and then briefly discuss the locally periodic ones (Section 3). Sections 4–6 describe a program which minimizes a certain functional over the set of model holes, and then its integration into a larger program, intended to treat topology and shape optimization problems. Numerical results are presented.     

A mixed shear flexible finite element for the analysis of laminated plates

A mixed shear flexible finite element based on the Hencky-Mindlin type shear deformation theory of laminated plates is presented and their behavior in bending is investigated. The element consists of three displacements, two rotations, and three moments as the generalized degrees of freedom per node. The numerical convergence and accuracy characteristics of the element are investigated by comparing the finite element solutions with the exact solutions. The present study shows that reduced-order integration of the stiffness coefficients due to shear is necessary to obtain accurate results for thin plates.     

Elasto-plastic shear buckling of square plates with circular holes

With in-plane stresses calculated by finite element analysis, critical loads are obtained by the Rayleigh-Ritz method for a square plate subjected to uniform edge shear stress and containing centrally located circular holes. Elastic and elasto-plastic buckling is examined for clamped and simply supported plates, and results are compared with previous analyses and experiments for various sized holes. The range of hole sizes considered is extended to include larger holes than previously examined, and for small holes, the results suggest that the critical stress is higher than previously thought. For elasto-plastic buckling, critical shear stresses are given for the full range of appropriate slenderness. Experimental results for the cases of simply supported plates support the analytical results, whereas verification for clamped plates remains inconclusive on account of limited reliable test data.     

Computation of capacitance coefficients of rectangular thick patches on multilayered substrates

This article presents an alternative method to compute the capacitance coefficients of a system of rectangular thick patches on a multilayered substrate. Following the M-strips, previously developed to analyze infinite length transmission lines, the conductors are modeled by a set of zero-thickness plates at the same potential. The capacitance coefficients of the model structure are obtained after applying the Galerkin's method in the spatial domain. A suitable approach for computing the static spatial Green's function is also presented. © 1996 John Wiley & Sons, Inc.     

Topological sensitivity derivative and finite topology modifications: application to optimization of plates in bending

The concept of topological sensitivity derivative is introduced and applied to study the problem of optimal design of structures. It is assumed, that virtual topology variation is described by topological parameters. The topological derivative provides the gradients of objective functional and constraints with respect to these parameters. This derivative enables formulation of the conditions of topology transformation. In this paper formulas for the topological sensitivity derivative for bending plates are derived. Next, the topological derivative is used in the optimization process in order to formulate conditions of finite topology modifications and in order to localize positions of the modifications. In the case of plates they are related to introduction of holes and introduction of stiffeners. The theoretical considerations are illustrated by some numerical examples.     

点阵多孔金属夹芯板振动特性分析及优化设计

采用空间桁架结构分析的网架结构连续化方法对点阵多孔金属夹芯板的夹芯层桁架进行了连续化处理,即,将夹芯层桁架等效为连续介质,并分别推导了金字塔型、四面体型、kagome型和4杆型点阵多孔金属夹芯板的抗弯刚度和等效剪切刚度．然后,应用分解刚度法推导了四边简支条件下点阵多孔金属夹芯板的固有振动频率公式,并与有限元计算结果进行了对比,表明所得公式具有较高的精度．最后,研究了夹芯板单胞结构尺寸对固有振动频率的影响,以夹芯层高度和桁架杆截面尺寸为设计变量,以第一阶频率最大化为目标对夹芯板进行了优化,优化后的夹芯板振动频率得到了明显提高．