Introduction

In this Introduction, the background and motivations of the book are presented in section I.1. A literature review on related subjects, including topology optimization methods, material design and multiscale optimization, and fracture resistance design, is presented in section I.2. The outline of the book is presented in section I.3.

I.1. Background and motivations

Topology optimization has been an active research topic in the last decades and has become a subject of major importance with the growing development of additive manufacturing processes, which allow the fabrication of workpieces such as lattice structures with arbitrary geometrical details. In this context, topology optimization (Bendsøe and Kikuchi 1988, Allaire 2012) aims to define the optimal structural or material geometry with regard to specific objectives (e.g. maximal stiffness, minimal mass or maximizing other physical/mechanical properties) under mechanical constraints such as equilibrium and boundary conditions. The key merit of topology optimization over conventional size and shape optimization is that the former can provide more design freedom, consequently leading to the creation of novel and highly efficient designs. With the topology optimization technique, designers can make the best use of limited materials and guide the concept design of various practical structures, especially in automotive and aerospace engineering.

In recent years, there has been an increase in the use of high-performance heterogeneous materials such as fibrous composite, concrete materials and 3D printed materials. Mechanical and physical properties of complex heterogeneous materials are determined, on the one hand, by the composition of their constituents but can, on the other hand, be drastically modified at a constant volume fraction of heterogeneities, by their geometrical shape and by the presence of interfaces. Topology optimization of microstructures can help design materials with higher effective properties while maintaining the volume fraction of constituents or obtaining new properties which are not naturally available (metamaterials). Recently, the development of 3D printing techniques and additive manufacturing processes has made it possible to directly manufacture designed materials from a numerical file, opening routes for new designs, as shown in Figure I.1. It is no exaggeration to say that "additive manufacturing" and "topology optimization" are the best couple made for each other. To this end, systematic and comprehensive research on the topological design of complex heterogeneous materials is of great significance for academic research and engineering applications.

Figure I.1. 3D printed lattice materials: (a) cubic and (b) cylindrical configurations (Mohammed et al. 2017)

However, in topology optimization of material modeling, the scale separation is often assumed. This assumption states that the characteristic length of microstructural details is much smaller than the dimensions of the structure, or that the characteristic wavelength of the applied load is much larger than that of the local fluctuation of mechanical fields (Geers et al. 2010). In additive manufacturing of architectured materials such as lattice structures, the manufacturing process may induce limitations on the size of local details, which can lead to a violation of scale separation when the characteristic size of the periodic unit cell within the lattice is not much smaller than that of the structure. In such a case, classical homogenization methods may lead to inaccurate description of the effective behavior as non-local effects, or strain-gradient effects may occur within the structure. On the other hand, using a fully detailed description of the lattice structure in an optimization framework could be computationally expensive. One objective of this book is to develop multiscale topology optimization procedures not only for heterogeneous materials but also for mesoscopic structures in the context of non-separated scales.

Figure I.2. Damage phenomena in engineering: (a) macroscopic structure; (b) cracks (Nguyen 2015)

On the other hand, fatigue or failure characteristics of engineering structures are another subject of great concern, as shown in Figure I.2. Microcracking is known to be a significant factor affecting the mechanical properties and the long-term behavior of engineering facilities. The accurate modeling of these phenomena, as well as their coupled effects have received special attention. In addition, topology optimization design of composite materials accounting for fracture resistance is a rather challenging task. It is necessary to improve the fracture resistance of heterogeneous materials in terms of the required mechanical work, through an optimal placement of the inclusion phase, taking into account the crack nucleation, propagation and interaction. However, this research remains relatively unexplored so far due to the following reasons. First, there has been a lack of robust numerical methods for fracture propagation in the presence of complex heterogeneous media until recently, especially when interface effects are presented. Second, these numerical simulation models should be formulated in a context compatible with the topology optimization scheme. For these reasons, there has been very limited research in the literature on topology optimization for maximizing the fracture resistance of heterogeneous materials before the recent works from the author and his coworkers (Xia et al. 2018a, Da et al. 2018a).

I.2. Literature review

In the following, section I.2.1 provides a brief literature review on the development of topology optimization methods. Section I.2.2 reviews material microstructure design and extension to multiscale topology optimization with or without scale separation. Section I.2.3 presents the newly proposed fracture resistance design framework, by combining the phase field method to take into account the heterogeneities and their interfaces in the material.

I.2.1. Topology optimization methods

Over the past decades, topology optimization has undergone a tremendous development since the seminal paper by Bendsøe and Kikuchi (1988). The key merit of topology optimization over conventional size and shape optimization is that the former can provide more design freedom, consequently leading to the creation of novel and highly efficient designs. Various topology optimization methods have been proposed so far, for example density-based methods (Bendsøe 1989, Zhou and Rozvany 1991, Bendsøe and Sigmund 2004), evolutionary procedures (Xie and Steven 1993, 1997), level-set method (LSM) (Sethian and Wiegmann 2000, Wang et al. 2003, Allaire et al. 2004), hybrid cellular automaton (Tovar et al. 2004) and phase field method (Bourdin and Chambolle 2003). All of these methods are based on finite element analysis (FEA) where the design domain is discretized into a number of finite elements. With such a setting, the optimization procedure is then to determine which points of the design domain should be full of material (solid elements) and which void (soft elements), as shown in Figure I.3. According to the update algorithm, these methods can be generally categorized into two groups: density variation and shape/boundary variation. The topology optimization technique has already become an effective tool for both academic research and engineering applications. A general review of various methods and their applications was presented by Deaton and Grandhi (2014). Regarding their strengths, weaknesses, similarities and dissimilarities, a critical review and comparison on different approaches was also given by Sigmund and Maute (2013).

Figure I.3. Illustration for structure topology optimization

Level-set method (LSM) is a typical shape/boundary variation approach that maintains the capability of topological change. It describes the structural topology implicitly by the iso-contours of a level-set function. Using the LSM, a fixed rectilinear spatial grid and a finite element mesh of a given design domain can be constructed separately, which allows the separation of the topological description from the physical model. With the merits of the flexibility in handling complex topological changes and the smoothness of boundary representation, the LSM has been successfully applied to an increasing variety of design problems, involving, for example, multi-phase materials (Wang and Wang 2004), shell structures (Park and Youn 2008), geometric nonlinearities (Luo and Tong 2008), stress minimization (Allaire and Jouve 2008) and contact problems (Myslinski 2008). The reader can refer to the comprehensive review in van Dijk et al. (2013) for more theoretical details of different LSMs for structural topology optimization.

Density-based methods are the most commonly used topology optimization approaches, such as the popular solid with isotropic material with penalization (SIMP) method. The SIMP method uses continuous design variables for topology optimization, which can be interpreted as material pseudo densities (Bendsøe 1989, Zhou and Rozvany 1991, Mlejnek 1992). The physical justification of the SIMP method was provided by Bendsøe and Sigmund (1999). A popular 99-line topology optimization Matlab code using the SIMP method was developed by Sigmund (2001) for education purposes. As...