In topology optimization of structures, materials and mechanisms, parametrization of geometry is often performed by a grey-scale density-like interpolation function. In this paper we analyze and compare the various approaches to this concept in the light of variational bounds on effective properties of composite materials. This allows us to derive simple necessary conditions for the possible realization of grey-scale via composites, leading to a physical interpretation of all feasible designs as well as the optimal design. Thus it is shown that the so-called artificial interpolation model in many circumstances actually falls within the framework of microstructurally based models. Single material and multi-material structural design in elasticity as well as in multi-physics problems is discussed.

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Material interpolation schemes in topology optimization

This review paper provides an overview of different level-set methods for structural topology optimization. Level-set methods can be categorized with respect to the level-set-function parameterization, the geometry mapping, the physical/mechanical model, the information and the procedure to update the design and the applied regularization. Different approaches for each of these interlinked components are outlined and compared. Based on this categorization, the convergence behavior of the optimization process is discussed, as well as control over the slope and smoothness of the level-set function, hole nucleation and the relation of level-set methods to other topology optimization methods. The importance of numerical consistency for understanding and studying the behavior of proposed methods is highlighted. This review concludes with recommendations for future research.

Level-set methods for structural topology optimization: a review

Design of multiphysics actuators using topology optimization - Part I: One-material structures

Manufacturing-oriented topology optimization has been extensively studied the past two decades, in particular for the conventional manufacturing methods, for example, machining and injection molding or casting. Both design and manufacturing engineers have benefited from these efforts because of the close-to-optimal and friendly-to-manufacture design solutions. Recently, additive manufacturing (AM) has received significant attention from both academia and industry. AM is characterized by producing geometrically complex components layer-by-layer, and greatly reduces the geometric complexity restrictions imposed on topology optimization by conventional manufacturing. In other words, AM can make near-full use of the freeform structural evolution of topology optimization. Even so, new rules and restrictions emerge due to the diverse and intricate AM processes, which should be carefully addressed when developing the AM-specific topology optimization algorithms. Therefore, the motivation of this perspective paper is to summarize the state-of-art topology optimization methods for a variety of AM topics. At the same time, this paper also expresses the authors' perspectives on the challenges and opportunities in these topics. The hope is to inspire both researchers and engineers to meet these challenges with innovative solutions.

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Current and future trends in topology optimization for additive manufacturing

The paper deals with topology optimization of structures undergoing large deformations. The geometrically nonlinear behaviour of the structures are modelled using a total Lagrangian finite element formulation and the equilibrium is found using a Newton-Raphson iterative scheme. The sensitivities of the objective functions are found with the adjoint method and the optimization problem is solved using the Method of Moving Asymptotes. A filtering scheme is used to obtain checkerboard-free and mesh-independent designs and a continuation approach improves convergence to efficient designs.

Different objective functions are tested. Minimizing compliance for a fixed load results in degenerated topologies which are very inefficient for smaller or larger loads. The problem of obtaining degenerated "optimal" topologies which only can support the design load is even more pronounced than for structures with linear response. The problem is circumvented by optimizing the structures for multiple loading conditions or by minimizing the complementary elastic work. Examples show that differences in stiffnesses of structures optimized using linear and nonlinear modelling are generally small but they can be large in certain cases involving buckling or snap-through effects.

Stiffness design of geometrically nonlinear structures using topology optimization

This work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization. In the proposed framework, alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain. Since two or more materials are permitted to simultaneously occupy local regions of the design domain, small-strain integration algorithms for general two-material mixtures of solids are developed for the Voigt (isostrain) and Reuss (isostress) assumptions, and hybrid combinations thereof. Structural topology optimization problems involving non-linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis (DSA) of energy type functionals are presented. The consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials, elastoplastic materials, and viscoelastic materials. The good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures. It is demonstrated through the second example that structures optimized for maximal strength can be significantly different than those optimized for minimal elastic compliance. © 1997 John Wiley & Sons, Ltd.

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Voigt-reuss topology optimization for structures with nonlinear material behaviors

The desired results of variable topology material layout computations are stable and discrete material distributions that optimize the performance of structural systems. To achieve such material layout designs a continuous topology design framework based on hybrid combinations of classical Reuss (compliant) and Voigt (sti) mixing rules is investigated. To avoid checkerboarding instabilities, the continuous topology optimization formulation is coupled with a novel spatial ltering procedure. The issue of obtaining globally optimal discrete layout designs with the proposed formulation is investigated using a continuation method which gradually transitions from the sti Voigt formulation to the compliant Reuss formulation. The very good performance of the proposed methods is demonstrated on four structural topology design optimization problems from the literature. ? 1997 by John Wiley & Sons, Ltd.

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Voigt–Reuss topology optimization for structures with linear elastic material behaviours

http://user.engineering.uiowa.edu/~swan/struct_opt/topo_papers/cs99_symmetry_topology.pdf

A symmetry reduction method for continuum structural topology optimization

Discrete ground structure topology optimization design methods have to date received considerable attention in structural engineering. An alternative class of structural topology optimization methods, which have not yet received much attention in structural engineering, but which have undergone considerable development in the past decade, are the so-called continuum formulations. In this work, a continuum structural topology optimization formulation is presented and applied to the concept design optimization of structural bracing systems that are needed to stiffen tall structures against sidesway under lateral-wind and seismic-type loading. Although demonstrated here in the context of these specific design examples, continuum structural topology optimization methods are believed to hold potential as a design tool for a wide range of civil engineering type structures. A variety of continuum topology design formulations, including static compliance minimization and eigenvalue optimization, are explored, and solution parameters are varied to show that a number of design possibilities can be realized as solutions.

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Iku Kosaka

Papers

Voigt-reuss topology optimization for structures with nonlinear material behaviors

Voigt–Reuss topology optimization for structures with linear elastic material behaviours

A symmetry reduction method for continuum structural topology optimization

Continuum Topology Optimization for Concept Design of Frame Bracing Systems

Homogenization-based analysis and design of composites