Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsoe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

Topology optimization approaches: A comparative review

Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method's focuses on new developments and novel applications.

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A survey of structural and multidisciplinary continuum topology optimization: post 2000

Recent advancements in computational inverse-design approaches — algorithmic techniques for discovering optical structures based on desired functional characteristics — have begun to reshape the landscape of structures available to nanophotonics. Here, we outline a cross-section of key developments in this emerging field of photonic optimization: moving from a recap of foundational results to motivation of applications in nonlinear, topological, near-field and on-chip optics. Starting with a desired optical output it is possible to use computational algorithms to inverse design devices. The approach is reviewed here with an emphasis on nanophotonics.

Inverse design in nanophotonics

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

Topology optimization has become an effective tool for least-weight and performance design, especially in aeronautics and aerospace engineering. The purpose of this paper is to survey recent advances of topology optimization techniques applied in aircraft and aerospace structures design. This paper firstly reviews several existing applications: (1) standard material layout design for airframe structures, (2) layout design of stiffener ribs for aircraft panels, (3) multi-component layout design for aerospace structural systems, (4) multi-fasteners design for assembled aircraft structures. Secondly, potential applications of topology optimization in dynamic responses design, shape preserving design, smart structures design, structural features design and additive manufacturing are introduced to provide a forward-looking perspective.

Topology Optimization in Aircraft and Aerospace Structures Design

SUMMARY
This paper presents a level-set-based topology optimization method based on numerically consistent sensitivity analysis. The proposed method uses a direct steepest-descent update of the design variables in a level-set method; the level-set nodal values. An exact Heaviside formulation is used to relate the level-set function to element densities. The level-set function is not required to be a signed-distance function, and reinitialization is not necessary. Using this approach, level-set-based topology optimization problems can be solved consistently and multiple constraints treated simultaneously.

The proposed method leads to more insight in the nature of level-set-based topology optimization problems. The level-set-based design parametrization can describe gray areas and numerical hinges. Consistency causes results to contain these numerical artifacts. We demonstrate that alternative parameterizations, level-set-based or density-based regularization can be used to avoid artifacts in the final results.

The effectiveness of the proposed method is demonstrated using several benchmark problems. The capability to treat multiple constraints shows the potential of the method. Furthermore, due to the consistency, the optimizer can run into local minima; a fundamental difficulty of level-set-based topology optimization. More advanced optimization strategies and more efficient optimizers may increase the performance in the future. Copyright © 2012 John Wiley & Sons, Ltd.

Explicit level-set-based topology optimization using an exact Heaviside function and consistent sensitivity analysis

A phase-field model for strength and fracture analyses of fiber-reinforced composites

In these papers we study networks of queues with general routing- and service-characteristics.

Networks of queues

Geometrically nonlinear structural analyses in conventional density-based Topology Optimization (TO) may fail due to excessive deformation, concerning in particular compression in low stiffness parts (void) of the domain. This limits the application of TO in the field of realistic large deflection mechanisms, actuators and multi-stable structures.

This paper investigates the source of the instabilities that may be encountered using the conventional strategy to scale the stiffness of finite elements using the density (e.g. SIMP). Based on the findings, we propose a new design interpolation, called Element Deformation Scaling (EDS), to obtain more robust structural analyses for geometrically nonlinear TO. Instead of scaling the stiffness, EDS scales the local internal displacements, and therefore, the deformation, in a low-density finite element. This ensures that, even for extremely deformed finite elements, the internal displacements remain in the range of applicability of the material model and finite element description.

The effectiveness of the proposed method is compared with the conventional approach (e.g. SIMP) and the Element Connectivity Parameterization (ECP) method using several numerical experiments using path-following techniques. The proposed method, EDS, is demonstrated to lead to more robust structural analyses than the other approaches. However, EDS still has limitations. These limitations are discussed in detail.

Nico P. van Dijk

Papers

Level-set methods for structural topology optimization: a review

Explicit level-set-based topology optimization using an exact Heaviside function and consistent sensitivity analysis

A phase-field model for strength and fracture analyses of fiber-reinforced composites

Networks of queues

Element deformation scaling for robust geometrically nonlinear analyses in topology optimization