Julien Yvonnet
email: julien.yvonnet@univ-eiffel.fr

Daicong Da
email: daicongda@boisestate.edu

Topology optimization to fracture resistance: a review and recent developments

Keywords: Topology optimization, fracture, structures, materials

Topology Optimization (TO) methods for fracture resistance offer new possibilities for designing stronger structures or materials with lower masses than conventional designs. This article presents an overview of topology optimization techniques for fracture resistance, from pioneering works to the most recent developments at the time of writing. We first review stress-based methods, which were the forerunners of crack resistance methods, producing optimal designs that prevent any damage or crack initiation. Other works followed, taking into account the presence of defects or cracks in structures, but using classical approaches aimed at minimizing compliance in an elastic framework. TO methods for fatigue damage are also an important branch of these approaches and are reviewed. We then present more recent methodologies, including non-linear effects in structural design, such as plasticity and damage. Finally, we describe the latest methods of TO design for fracture resistance, including an explicit description of crack propagation during loading, from initiation to failure of structures and materials. In particular, the design of two-phase materials that are more resistant to cracking and that can be manufactured by 3D printing is discussed. The article concludes with some challenges and promising avenues for the coming years in this field.

Introduction

Topology optimization (TO) is a well-known computational method for designing structures with optimal geometries in order to optimize an objective under possible constraints. Since the pioneering work of Bendsøe and Kikuchi in the late 1980s, the technique has been widely developed and has enjoyed considerable success in the engineering community. Initially applied to mass minimization under compliance constraints or, conversely, to compliance minimization under volume constraints, TO has since been extended and applied to a wide variety of problems such as mechanical and thermal loads on structures, fluid flow, waste management, structural mechanics and thermics, fluid flow, dynamics, acoustics and biomechanics, among others. (see review articles in [START_REF] Eschenauer | Topology optimization of continuum structures: a review[END_REF][START_REF] Deaton | A survey of structural and multidisciplinary continuum topology optimization: post 2000[END_REF]).

TO can be categorized into several popular approaches: Solid Isotropic Material with Penalization (SIMP) method [START_REF] Bendsøe | Optimal shape design as a material distribution problem[END_REF][START_REF] Rozvany | Generalized shape optimization without homogenization[END_REF][START_REF] Bendsøe | Material interpolation schemes in topology optimization[END_REF], Bidirectional Evolutionary Structural Optimization (BESO) method [START_REF] Querin | Computational efficiency and validation of bi-directional evolutionary structural optimisation[END_REF][START_REF] Li | Evolutionary topology optimization for structures made of multiple materials with different properties in tension and compression[END_REF], Level-Set Method (LSM) [START_REF] Wang | A level set method for structural topology optimization[END_REF][START_REF] Allaire | Structural optimization using sensitivity analysis and a level-set method[END_REF], Phase Field method [START_REF] Takezawa | Shape and topology optimization based on the phase field method and sensitivity analysis[END_REF], Moving Morphable Components (MMC) [START_REF] Guo | Doing topology optimization explicitly and geometrically-a new moving morphable components based framework[END_REF][START_REF] Liu | Optimal design of shell-gradedinfill structures by a hybrid mmc-mmv approach[END_REF], and their combinations [START_REF] Da | Evolutionary topology optimization of continuum Springer Nature 2021 L A T E X template structures with smooth boundary representation[END_REF]. Review of these methods can be found in state-of-the-art papers [START_REF] Deaton | A survey of structural and multidisciplinary continuum topology optimization: post 2000[END_REF][START_REF] Rozvany | A critical review of established methods of structural topology optimization[END_REF][START_REF] Van Dijk | Level-set methods for structural topology optimization: a review[END_REF][START_REF] Zhu | Topology optimization in aircraft and aerospace structures design[END_REF][START_REF] Wang | A comprehensive review of educational articles on structural and multidisciplinary optimization[END_REF][START_REF] Wang | Open-source codes of topology optimization: A summary for beginners to start their research[END_REF]. An interesting comparison review on these techniques, with advantages and drawbacks, can be found in [START_REF] Sigmund | Topology optimization approaches: A comparative review[END_REF].

Recently, an exciting new branch of TO has emerged, whose aim is not just to minimize the mass/stiffness ratio, but to take account of damage during service loading. In many engineering applications, damage or cracks may occur inevitably, and designing structures or materials able to continue working in damaged conditions may be crucial for safety in a variety of applications from the design of airframes to civil infrastructures [START_REF] James | Topology optimization of structures under variable loading using a damage superposition approach[END_REF]. Another important feature is energy dissipation during a static or impact load, where it might be desirable to authorize some damage/plasticity/cracks to dissipate energy and preserve other critical locations within the structure. Structures of this type are needed in a wide variety of fields including crashworthiness designs, seismic applications, blast protection, personal safety and sport equipment, among others [START_REF] Lu | Energy absorption of structures and materials[END_REF]. Finally, resisting to fatigue by accurately designing the geometry is also critical for engineering performances. These different features can be obtained by optimizing the structural geometry, or the microstructure of a material itself, thanks to recent progresses in 3D printing [START_REF] Ngo | Additive manufacturing (3D printing): A review of materials, methods, applications and challenges[END_REF][START_REF] Wang | 3D printing of polymer matrix composites: A review and prospective[END_REF][START_REF] Kao | Loading-unloading cycles of 3D-printing built bi-material structures with ceramic and elastomer[END_REF][START_REF] Amin | Dynamic response of 3D-printed bi-material structure using drop weight impact test[END_REF]. Globally, these different applications can be handled by Topology Optimization to fracture resistance and are of industrial and technological critical importance, for applications in aircraft, automotive or biomechanics, among many others.

TO capable of taking crack propagation into account poses enormous challenges, as it can combine TO with fully nonlinear path-dependent simulations. This is in contrast with previous linear approaches. The first challenge is then the computational cost. The second issue is the need for a robust and efficient crack propagation simulation method, compatible with TO. Usually, TO requires fixed meshes for computational efficiency. This makes classical FEM-based crack propagation methods with remeshing (see e.g. [START_REF] Cai | Numerical simulation of the Brazilian test and the tensile strength of anisotropic rocks and rocks with pre-existing cracks[END_REF][START_REF] Ludwig | Simulation of anisotropic crack growth behavior of nickel base alloys under thermomechanical fatigue[END_REF]) not well adapted to TO. The eXtended Finite Element Method (XFEM) [START_REF] Belytschko | Elastic crack growth in finite elements with minimal remeshing[END_REF][START_REF] Moës | A finite element method for crack growth without remeshing[END_REF][START_REF] Daux | Arbitrary branched and intersecting cracks with the extended finite element method[END_REF][START_REF] Gravouil | Non-planar 3D crack growth by the extended finite element and level setspartii: level set update[END_REF] offers the possibility to simulate cracks over fixed meshes. However, as being based on linear fracture mechanics, it can not handle crack initiation. In addition, the simulation of complex and multiple cracks in XFEM induces an enormous level of complexity, which limits its application in TO, even if there are a few examples, as presented in this review article. The phase field method to fracture, developed by Marigo, Bourdin, Francfort and Miehe [START_REF] Francfort | Revisiting brittle fracture as an energy minimization problem[END_REF][START_REF] Bourdin | Numerical experiments in revisited brittle fracture[END_REF][START_REF] Amor | Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments[END_REF][START_REF] Miehe | A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits[END_REF] has many advantages for being easily combined with TO: first, it can operate on fixed meshes, as based on a regularized gradient damage model; then, it can easily handle initiation, propagation and merging of cracks with arbitrary geometries. For these reasons, its combination with TO, initiated in the seminal paper by Xia et al. [START_REF] Xia | Topology optimization for maximizing the fracture resistance of quasi-brittle composites[END_REF], has opened the way to a large number of new methodologies for complex applications, from structural ones to material engineering.

In this article, we present an overview of these methods, combining TO and fracture propagation, but also all the previous and related approaches that have led to these techniques. Note that we focus on Topology Optimization, and do not review other optimization techniques like parameterized geometries (see e.g. [START_REF] Desmorat | Topology optimization in damage governed low cycle fatigue[END_REF][START_REF] Lew | A deep learning augmented genetic algorithm approach to polycrystalline 2D material fracture discovery and design[END_REF]) for fracture resistance, or related works like optimal control for cracks [START_REF] Khimin | Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems[END_REF], even though these studies share common objectives with TO for fracture resistance. In section 2, we first review stress-based methods, which were the forerunners of crack resistance methods by producing optimal designs that prevents any initiation of damage or cracking. In section 3, works taking into account the presence of defects or cracks in structures, but following classical approaches aimed at minimizing compliance in an elastic framework are presented. In section 4, TO methods for fatigue damage are reviewed. In section 5, we then describe more recent approaches, including non-linear effects in the design of structures, such as plasticity and damage. Finally, we provide in section 6 a recent state-of-the-art of most recent TO design methods to fracture resistance, including an explicit description of crack propagation during loading, from initiation to failure of structures and materials. In particular, the design of two-phase materials that are more resistant to cracking and can be manufactured by 3D printing is discussed. The article concludes with some challenges and promising avenues for the coming years in this field.

Stress-based approaches

Initially flourishing in compliance optimization, stress-based design is another prominent and active topic within the TO community. The first work on TO for eliminating stress concentration can be traced back to [START_REF] Yang | Stress-based topology optimization[END_REF]. Another pioneering work, recognized in the field, employs the SIMP approach to address local stress constraints [START_REF] Duysinx | Topology optimization of continuum structures with local stress constraints[END_REF] by Duysinx and Bendsøe. Currently, stress-based design problems have extensively explored various TO approaches mentioned earlier. One commonly studied benchmark is the L-shape beam, which aims to eliminate stress concentrations in the right-angle area (a brief summary is provided in Fig. 1). The single-constrained stress-based optimizations can be categorized into two types: mass minimization under maximal stress constraints [START_REF] Holmberg | Stress constrained topology optimization[END_REF][START_REF] Emmendoerfer | A level set approach for topology optimization with local stress constraints[END_REF][START_REF] Jeong | Development of a novel phasefield method for local stress-based shape and topology optimization[END_REF][START_REF] Bruggi | On an alternative approach to stress constraints relaxation in topology optimization[END_REF][START_REF] Luo | Topology optimization of continuum structures with Drucker-Prager yield stress constraints[END_REF][START_REF] Verbart | Damage approach: A new method for topology optimization with local stress Springer Nature 2021 L A T E X template constraints[END_REF][START_REF] Long | Stressconstrained topology optimization of continuum structures subjected to harmonic force excitation using sequential quadratic programming[END_REF][START_REF] Paulino | Polystress: a Matlab implementation for local stress-constrained topology optimization using the augmented Lagrangian method[END_REF][START_REF] Nguyen | Stressconstrained shape and topology optimization with the level set method using trimmed hexahedral meshes[END_REF][START_REF] Mcbane | Stress-constrained topology optimization of lattice-like structures using component-wise reduced order models[END_REF][START_REF] Kundu | Multimaterial stress-constrained topology optimization with multiple distinct yield criteria[END_REF][START_REF] Norato | A maximum-rectifierfunction approach to stress-constrained topology optimization[END_REF], or conversely, maximal stress minimization under volume constraints [START_REF] Allaire | Topology optimization for minimum stress design with the homogenization method[END_REF][START_REF] Allaire | Minimum stress optimal design with the level set method[END_REF][START_REF] Xia | A level set solution to the stress-based structural shape and topology optimization[END_REF][START_REF] Cai | Stress constrained shape and topology optimization with fixed mesh: A b-spline finite cell method combined with level set function[END_REF][START_REF] Lian | Combined shape and topology optimization for minimization of maximal von mises stress[END_REF][START_REF] Picelli | Stress-based shape and topology optimization with the level set method[END_REF][START_REF] Xia | Stress-based topology optimization using bi-directional evolutionary structural optimization method[END_REF].

Stress-based TO poses greater challenges compared to compliance-based TO due to several factors: (1) stress is inherently a local characteristic, (2) it involves non-self-adjoint optimization, and (3) the imposition of numerous stress constraints is required. In order to address these challenges, aggregation functions such as p-norm [START_REF] Le | Stress-based topology optimization for continua[END_REF] and Kreisselmeier-Steinhauser (KS) functions [START_REF] Yang | Stress-based topology optimization[END_REF] have been widely adopted to represent the stress constraint corresponding to the number of finite elements or Gauss points. However, achieving a balance between accurately predicting the maximum local stress and employing a gradientbased robust algorithm for highly nonlinear stress optimization is crucial. This balance heavily relies on the choice of aggregation degree, such as the p-value in the p-norm function. Recently, the application of Heaviside projection-based aggregation [START_REF] Qian | Undercut and overhang angle control in topology optimization: a density gradient based integral approach[END_REF] in stress-based TO [START_REF] Wang | Heaviside projectionbased aggregation in stress-constrained topology optimization[END_REF] has enabled precise predictions of maximal stress while significantly reducing computational complexity. To tackle large-scale design problems, augmented Lagrangian approaches have been proposed [START_REF] Pereira | Topology optimization of continuum structures with material failure constraints[END_REF][START_REF] Silva | Three-dimensional manufacturing tolerant topology optimization with hundreds of millions of local stress constraints[END_REF][START_REF] Paulino | A unified approach for topology optimization with local stress constraints considering various failure criteria: von Mises[END_REF][START_REF] Giraldo-Londoño | Local stress constraints in topology optimization of structures subjected to arbitrary dynamic loads: a stress aggregationfree approach[END_REF], where solutions involving over 135 million elements and 505 million stress constraints have been achieved. Similar strategies have been employed in the context of reliability-based topology optimization (RBTO) [START_REF] Silva | Reliabilitybased topology optimization of continuum structures subject to local stress constraints[END_REF] and in considering manufacturing uncertainties [START_REF] Silva | Stress-constrained topology optimization considering uniform manufacturing uncertainties[END_REF], respectively. Additionally, a comparison of local and global stress constraint strategies in TO is presented in [START_REF] Silva | Local versus global stress constraint strategies in topology optimization: a comparative study[END_REF].

The stress-based design approach was initially combined with classical compliance topology optimization in [START_REF] Yang | Stress-based topology optimization[END_REF], where the simultaneous minimization of stress and compliance was carried out under a mass constraint. Subsequent studies, such as [START_REF] Bruggi | A mixed FEM approach to stress-constrained topology optimization[END_REF][START_REF] Sharma | Stress-based topology optimization using spatial gradient stabilized XFEM[END_REF][START_REF] Yang | Stress-constrained topology optimization based on maximum stress measures[END_REF][START_REF] Granlund | Stress-constrained topology optimization of structures subjected to nonproportional loading[END_REF][START_REF] Ogawa | Stress constraint topology optimization of coupled thermomechanical problems using the temperature dependence of allowable stress[END_REF] and [START_REF] Bruggi | Topology optimization for minimum weight with compliance and stress constraints[END_REF][START_REF] Suresh | Stressconstrained topology optimization: a topological level-set approach[END_REF], focused on formulating compliance minimization with stress and mass constraints, as well as achieving minimum weight while considering both compliance and stress constraints, respectively. In [START_REF] Boissier | Elastoplastic topology optimization of cyclically loaded structures via direct methods for shakedown[END_REF], Boissier et al. used a Level-set TO to solve cyclic elastoplastic problems based on the Melans theorem, in which only the existence of certain residual stress fields based on elastic solutions is required to guarantee asymptotic elastoplastic behavior (shakedown). In [START_REF] Bruggi | A mixed FEM approach to stress-constrained topology optimization[END_REF], the optimization of elastoplastic and incompressible media was made possible by employing a mixed finite-element discretization scheme. Recently, the same optimization formulations were solved using the BESO method for linearly elastic materials in [START_REF] Fan | Evolutionary topology optimization of continuum structures with stress constraints[END_REF][START_REF] Chen | Controlling the maximum first principal stress in topology optimization[END_REF] and the LSM for shell structures in [START_REF] Ho-Nguyen-Tan | Level set-based topology optimization for compliance and stress minimization of shell structures using trimmed quadrilateral shell meshes[END_REF]. Notably, [START_REF] Da | Simple strategy toward tailoring fracture properties of brittle architected materials[END_REF] by Da and Chen represents the first work that systematically analyzes the effects of stress-based designs on fracture resistance properties (see Fig. 2). The authors found that minimizing the critical maximum stress1 can delay the onset of cracking, and when combined with compliance optimization, it leads to significant enhancements in both peak load and toughness. Furthermore, the authors employed a data-driven strategy in [START_REF] Da | Data-driven and topological design of structural metamaterials for fracture resistance[END_REF] to minimize multiscale homogenized stress, indirectly enhancing the toughness of heterogeneous structures. The study emphasizes the significant size effects observed in multiscale stress-oriented irregular architected materials (see Fig. 3). [START_REF] Da | Simple strategy toward tailoring fracture properties of brittle architected materials[END_REF], and (b-e) the following approaches: (b) SIMP [START_REF] Le | Stress-based topology optimization for continua[END_REF], (c) BESO [START_REF] Xia | Stress-based topology optimization using bi-directional evolutionary structural optimization method[END_REF], (d) LSM [START_REF] Guo | Stress-related topology optimization via level set approach[END_REF], and (e) MMC [START_REF] Zhang | A moving morphable void (MMV)-based explicit approach for topology optimization considering stress constraints[END_REF]. Reproduced figures used with permission. 3 Linear elastic structures containing virtual or fixed pre-existing cracks

One of the first reported work introducing the notion of crack in TO is by Challis et al. [START_REF] Challis | Fracture resistance via topology optimization[END_REF]. In their study, an objective function defining a virtual crack extension was proposed on the boundary nodes of a structure discretized by Finite Elements. The framework only involved linear elastic structures. A series of other works have investigated the TO of structures embedding pre-existing cracks or defects within a structure. In these problems, a linear elastic structure was usually considered, and the TO was carried out with respect to compliance volume, or toughness, taking into account the presence of fixed cracks. In an early work [START_REF] Jansen | Topology optimization of failsafe structures using a simplified local damage model[END_REF], Jansen et al. defined a TO framework for minimizing compliance of structures such that they were less sensitive to defects. For this purpose, several "worst case" scenarios were defined, where material was totally damaged in a square patch. The compliance was then minimized for a given number of damage scenarios. As a result, the optimized structure was assumed to be insensitive to cracks in the patch zones.

In [START_REF] Shobeiri | The topology optimization design for cracked structures[END_REF], Shobeiri investigated TO of linear elastic structures containing fixed cracks by combining the Element Free Galerkin Method (EFG) [START_REF] Belytschko | Element-free Galerkin methods[END_REF] and BESO. Differences between EFG and FEM were reviewed in this context.

An innovative contribution was proposed by Buehler's group in [START_REF] Gu | Optimization of composite fracture properties: method, validation, and applications[END_REF], where a TO framework was introduced to optimize for the first time the toughness of a bi-material sample. In their work, the TO of a notched plate made of soft and stiff phases distributed on a regular grid was carried out to improve the toughness under mode-I failure. A greedy algorithm was proposed to obtain the topology of the second phase, and finite elements were used to evaluate the toughness using linear analysis. The toughness was here defined as the area under the force/displacement curve at failure strain. In a follow-up work [START_REF] Gu | Algorithm-driven design of fracture resistant composite materials realized through additive manufacturing[END_REF], the same authors used the optimized topologies to 3D-printed samples made of soft and hard phases (see Figs. 4 and5). In these studies, no crack propagation was considered in the simulations and thus in the TO process, but the samples were experimentally cracked to evaluate the toughness improvements (see Fig. 5). It was shown that a redistribution of soft and stiff phases could lead to a clear improvement of toughness and strength as compared to other benchmark designs.

In [START_REF] Kang | Topology optimization considering fracture mechanics behaviors at specified locations[END_REF], Kang et al. evaluated the J-integral around predefined cracks at specific locations and used it in the objective function to optimize the topology of elastic structures. In [START_REF] Banh | Multi-material topology optimization design for continuum structures with crack patterns[END_REF] Banh and Lee used SIMP and a multi-material TO to minimize compliance in presence of fixed cracks, where the discontinuities and singularities around the crack tip were modelled by XFEM [START_REF] Belytschko | Elastic crack growth in finite elements with minimal remeshing[END_REF][START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]. The same authors extended this approach to Functionally Graded Materials (FGM) in [START_REF] Banh | A nonhomogeneous multi-material topology optimization approach for functionally graded structures with cracks[END_REF]. A similar procedure was developed by combining XFEM and BESO in [START_REF] Hu | Fracture strength topology optimization of structural specific position using a bi-directional evolutionary structural optimization method[END_REF] (see Fig. 6), as well as in [START_REF] Klarbring | Minimizing crack energy release rate by topology optimization[END_REF] to define the topology of a second material playing the role of patches near existing cracks to minimize the energy release rate within a BESO framework. Mechanical ice protection systems was investigated in [START_REF] Marboeuf | Improving mechanical ice protection systems with topology optimization[END_REF] using topology optimization to enhance ice shedding through fracture.

Finally, peridynamics [START_REF] Silling | Reformulation of elasticity theory for discontinuities and long-range forces[END_REF] was investigated for compliance optimization of linearly elastic cracked structures in [START_REF] Kefal | Topology optimization of cracked structures using peridynamics[END_REF][START_REF] Sohouli | Continuous density-based topology optimization of cracked structures using peridynamics[END_REF][START_REF] Motlagh | Comparative study of peridynamics and finite element method for practical modeling of cracks in topology optimization[END_REF][START_REF] Kendibilir | Peridynamics topology optimization of threedimensional structures with surface cracks for additive manufacturing[END_REF]. A comparison between peridynamics and FEM in this context was discussed in [START_REF] Motlagh | Comparative study of peridynamics and finite element method for practical modeling of cracks in topology optimization[END_REF]. In [START_REF] Kendibilir | Peridynamics topology optimization of threedimensional structures with surface cracks for additive manufacturing[END_REF], Kendibilir et al. fabricated optimized structures by 3D printing and tested it experimentally.

Fatigue

Fatigue resistance TO is primarily stress-based and can be enhanced by controlling the stress level. It typically requires ensuring that the stress level of each finite element remains within the safe region, i.e., below the yield stress, after a certain number of stress cycles. The adoption of stress aggregation functions, such as the p-norm, allows constraining the maximum accumulated "damage" to maintain it at a tolerable level even after a significant number of stress cycles [START_REF] Chen | Fatigue-resistance topology optimization of continuum structure by penalizing the cumulative fatigue damage[END_REF]. Fatigue resistance in TO can be categorized based on the number of cycles as low-cycle fatigue (LCF) or high-cycle fatigue (HCF), with a dividing line typically set around 10 4 cycles [START_REF] Hou | Topology optimization of the multifasteners jointed structure considering fatigue constraints[END_REF]. Due to the complexity of LCF, which is primarily governed by plasticity, the majority of research in fatigue TO has focused on HCF. One notable example of TO work addressing LCF design is found in [START_REF] Desmorat | Topology optimization in damage governed low cycle fatigue[END_REF] by Desmorat and Desmorat. In their study, fatigue life was maximized by optimizing the topology of a 3D double-clamped cantilever beam, considering cyclic plasticity in conjunction with the Lemaitre damage law [START_REF] Lemaitre | Mechanics of solid materials[END_REF][START_REF] Lemaitre | Engineering damage mechanics: ductile, creep, fatigue and brittle failures[END_REF].

In the context of HCF designs, early studies such as [START_REF] Mrzyglod | Parametric structural optimization with respect to the multiaxial high-cycle fatigue criterion[END_REF] employed three different HCF criteria (Sines [START_REF] Sines | Behavior of metals under complex static and alternating stresses[END_REF], Crossland [START_REF] Crossland | Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel[END_REF], and Dang Van [111]) within a parametric structural optimization framework, solved using a gradient-free evolutionary algorithm. This work was followed by a TO framework by the same author in [START_REF] Mrzyg | Two-stage optimization method with fatigue constraints for thinwalled structures[END_REF], which demonstrated that TO can effectively reduce the mass of a structure while maintaining its durability at an established level. In the same year, Sherif et al. [START_REF] Sherif | Efficient topology optimization of large dynamic finite element systems using fatigue[END_REF] investigated dynamic response TO, where fatigue played a crucial role in determining the optimization process, and proposed a criterion for meaningful equivalent static loads (ESL) [START_REF] Choi | Transformation of dynamic loads into equivalent static loads based on modal analysis[END_REF]. Subsequently, Yoon's group comprehensively incorporated dynamic fatigue and static failure criteria into TO in their works [START_REF] Jeong | Fatigue and static failure considerations using a topology optimization method[END_REF][START_REF] Lee | Topology optimization considering fatigue life in the frequency domain[END_REF][START_REF] Jeong | Topology optimization considering the fatigue constraint of variable amplitude load based on the equivalent static load approach[END_REF], utilizing Miner's cumulative damage rule to calculate the total damage at each spatial element.

In [START_REF] Holmberg | Fatigue constrained topology optimization[END_REF] and [START_REF] Oest | Topology optimization with finite-life fatigue constraints[END_REF], the same cumulative damage rule was employed, with the principal stress and Sines damage criterion being constrained in TO, respectively. The use of the clustered approach and Rainflow-counting method in these works simplifies the optimization problem, making it as convenient as the classical von Mises stress-based TO framework. However, both studies show that the optimized topologies obtained through fatiguebased TO exhibit similarities with stress-based designs. In contrast, [START_REF] Suresh | Topology optimization using a continuous-time highcycle fatigue model[END_REF] adopts a continuous time approach for fatigue, where the evolution of damage is governed by a system of ordinary differential equations. This approach does not rely on cycle-counting techniques such as Rainflowcounting and integrates fatigue damage over the entire stress history. Consequently, it can handle more general load histories, including those with Fig. 5 Stress-strain relations. (a) Both optimized materials fail at a higher stress and strain compared to all benchmark samples. (b) Stress-strain relations for the softer materials that exhibit higher strain and lower failure stress. (c) Toughness amplification. The two optimized geometries have on average a more than 20-fold increase in toughness modulus compared to the stiff homogenous sample. (d) In terms of strength, optimized geometries (see Fig. 4) have on average a more than two-fold increase in strength compared to stiff and random geometries. Reproduced from [START_REF] Gu | Algorithm-driven design of fracture resistant composite materials realized through additive manufacturing[END_REF] with permission.

non-proportional loads. The computational acceleration of the proposed framework is addressed in [START_REF] Suresh | Acceleration of continuoustime, high-cycle fatigue constrained problems in topology optimization[END_REF], which presents HCF-constrained TO for a 3D bracket subjected to periodic load history. Furthermore, the investigation of fatigue-based TO under non-proportional loads has been recently explored in [START_REF] Zhang | Fatigue-based topology optimization with non-proportional loads[END_REF], where different objective functions, including maximizing fatigue life under volume constraints, were studied.

Zhao et al. [START_REF] Zhao | Structural topological optimization with dynamic fatigue constraints subject to dynamic random loads[END_REF] addressed the same finitelife local fatigue under random vibrations using a density-based SIMP-classified method, similar to the other reviewed works. This problem was recently solved using the BESO method [START_REF] Nabaki | Evolutionary topology optimization of continuum structures considering fatigue failure[END_REF] and unstructured polygonal meshes [START_REF] Teng | Structural topology optimization with local finite-life fatigue constraints[END_REF]. In contrast, Collet et al. [START_REF] Collet | Topology optimization for minimum weight with compliance and simplified nominal stress constraints for fatigue resistance[END_REF] proposed infinite-life fatigue TO, aiming to minimize mass while subject to compliance and simplified nominal stress constraints. They adopted a modified Goodman failure criterion based on the Sines method.

Due to the prevalence of fatigue failure as a leading cause of rupture in fastened parts, fatigue TO has gained significant attention in real-world engineering applications. This is particularly evident with the rise of additive manufacturing, which has led to numerous engineering examples and experimental investigations (see Fig. 7). A recent study by Olesen et al. [START_REF] Olesen | Simultaneous optimization of topology and print orientation for transversely isotropic fatigue[END_REF] explored the simultaneous optimization of topology and print orientation for transversely isotropic fatigue. Suresh et al. [START_REF] Suresh | Topology optimization for transversely isotropic materials with highcycle fatigue as a constraint[END_REF] investigated TO for additive manufactured components and transversely isotropic materials, focusing on HCF as a constraint. Trudel et al. [START_REF] Trudel | Penalization techniques for fatigue-based topology optimizations of structures with embedded functionally graded lattice materials[END_REF] proposed penalization techniques for fatigue TO of structures incorporating embedded functionally graded lattice materials. Correspondingly, several studies have examined fatigue TO for real engineering and medical parts, such as the Helicopter Tail Rotor Pitch Arm [START_REF] Demir | Fatigue damage-based topology optimization of helicopter tail rotor pitch arm[END_REF], Aerospace Bracket [START_REF] Niutta | Defect-driven topology optimization for fatigue design of additive manufacturing structures: Application on a real industrial aerospace component[END_REF], and Temporomandibular Joint Prosthesis [START_REF] Al-Ali | Topology optimization and fatigue analysis of temporomandibular joint prosthesis[END_REF] (see Fig. 8). Additionally, Dagkolu et al. [START_REF] Dagkolu | Design and additive manufacturing of a fatigue-critical aerospace part using topology optimization and l-pbf process[END_REF] reported the design of a fatigue-critical aerospace part using TO, which was manufactured using laser power bed fusion (L-PBF). Lastly, Liu et al. [START_REF] Liu | Enhanced fatigue characteristics of a topologyoptimized porous titanium structure produced by selective laser melting[END_REF] achieved enhanced fatigue characteristics in a topology-optimized porous titanium structure produced via selective laser melting.

Path-dependent approaches: plasticity and damage

Stress-based approaches have limitations to correctly address situations where damage has already formed, and to correctly measure failure of a structure. Therefore, the stress-based design procedures may produce structures that Fig. 8 Fatigue TO of actual engineering components of (a) Helicopter Tail Rotor Pitch Arm [START_REF] Demir | Fatigue damage-based topology optimization of helicopter tail rotor pitch arm[END_REF], (b) aerospace bracket [START_REF] Niutta | Defect-driven topology optimization for fatigue design of additive manufacturing structures: Application on a real industrial aerospace component[END_REF]; and (c) Temporomandibular joint prosthesis [START_REF] Al-Ali | Topology optimization and fatigue analysis of temporomandibular joint prosthesis[END_REF]. Reproduced from [START_REF] Demir | Fatigue damage-based topology optimization of helicopter tail rotor pitch arm[END_REF][START_REF] Niutta | Defect-driven topology optimization for fatigue design of additive manufacturing structures: Application on a real industrial aerospace component[END_REF][START_REF] Al-Ali | Topology optimization and fatigue analysis of temporomandibular joint prosthesis[END_REF] with permission.

lack robustness in such scenarios [START_REF] James | Topology optimization of structures under variable loading using a damage superposition approach[END_REF]. More specifically, for materials having highly nonlinear behavior, such as elastoplastic and strain-softening, once a threshold for stress or strain is reached, the material loses part of its stiffness but may experience large deformations before actually failing. For these materials, failure cannot easily be expressed in terms of a maximal stress or strain. By constraining the stress to be lower than the yield stress, the optimized designs will be structurally safe but overly conservative since the behavior beyond yielding or damage initiation is not considered [START_REF] Barbier | Topology optimization of damage-resistant structures with a predefined load-bearing capacity[END_REF]. To overcome these limitations, authors have introduced TO approaches including fully nonlinear and path-dependent simulations. Early approaches have first included plasticity in TO approaches, then elastic damage models, and later models coupling damage and plasticity.

Plasticity

Plasticity can be the onset of failure in ductile materials. A first contribution to TO in pathdependent problems with plasticity was proposed by Maute et al. [START_REF] Maute | Adaptive topology optimization of elastoplastic structures[END_REF][START_REF] Schwarz | Topology and shape optimization for elastoplastic structural response[END_REF], where an elastoplastic structure was considered in the simulations. In their work, the structural ductility, defined as the integral of the strain energy over a given range of a prescribed displacement, was maximized under mass constraint. In [START_REF] Kato | Analytical sensitivity in topology optimization for elastoplastic composites[END_REF], Kato et al. proposed a SIMP TO in composites considering elastoplastic deformations to maximize the energy absorption capacity (defined as the external work) of a structure under a prescribed material volume and a whole incremental simulation. Analytical sensitivities were derived for the path-dependent problem.

In [START_REF] Nakshatrala | Topology optimization for effective energy propagation in rate-independent elastoplastic material systems[END_REF], Nakshatrala and Tortorelli introduced the TO of dynamically loaded structured with rate-independent elastoplastic material behavior with applications to impact mitigation. An extension of TO for elastoplastic structures at finite strains was performed by Wallin et al. in [START_REF] Wallin | Topology optimization based on finite strain plasticity[END_REF]. Hencky's law of plasticity with regularization was investigated within the Level-set TO method in [START_REF] Maury | Elastoplastic shape optimization using the level set method[END_REF].

Recently, Jia et al. [START_REF] Jia | Crashworthiness design of periodic cellular structures using topology optimization[END_REF], investigated material design by performing TO in cellular periodic structures considering plasticity and dynamics, showing that energy absorption could be effectively improved as compared to classical microstructures such as honeycombs (see Fig. 9).

Damage

Pioneer works combining topology optimization and damage may be attributed to Achtziger and Bendsøe [START_REF] Achtziger | Design for maximal flexibility as a simple computational model of damage[END_REF], who designed truss structures with a damage model. In a next study, Bendsøe and Diaz [START_REF] Bendsøe | A method for treating damage related criteria in optimal topology design of continuum structures[END_REF] used an elastic-damage model for designing structures with damage constraints. In [START_REF] Desmorat | Topology optimization in damage governed low cycle fatigue[END_REF], Desmorat and Desmorat combined TO with continuum damage to maximize the lifetime of structures subject to fatigue damage. In [START_REF] Amir | A topology optimization procedure for reinforced concrete structures[END_REF], Amir included discrete steel reinforcements embedded into a concrete continuum, which was subject to damage. In [START_REF] Amir | Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization[END_REF] Amir and Sigmund optimized reinforcements in concrete structures, where the concrete was described by a gradient-enhanced damage model with strain softening, and reinforcements modeled by elastic bars embedded in the matrix. In this work, final compliance was minimized but after solving a fully nonlinear incremental path-dependent problem (see Fig. 10).

In [START_REF] James | Failure mitigation in optimal topology design using a coupled nonlinear continuum damage model[END_REF], James and Waisman used TO to generate minimum-weight structural layouts, subject to a compliance constraint. Failure mitigation was achieved by constraining the maximum local damage. The damage was modeled using a continuum damage approach in which the nonlocal damage evolution law was coupled to a finite element model. In a follow-up study [START_REF] James | Topology optimization of structures under variable loading using a damage superposition approach[END_REF], the same authors performed the TO of structures able to sustain material damage and subjected to multiple loading cases, using a coupled nonlinear brittle damage model. A nonlocal damage model was developed by Noël et al. in [START_REF] Noël | Level set topology optimization considering damage[END_REF] in the framework of a level-set TO. More recently, Barbier et al. [START_REF] Barbier | Topology optimization of damage-resistant structures with a predefined load-bearing capacity[END_REF] considered minimization of volume in a structure by solving an elastic problem with nonlinear damage model [START_REF] Peerlings | Gradientenhanced damage for quasi-brittle materials[END_REF] under a maximum loading capacity, including a fully nonlinear strategy and a simplified strategy, where damage was computed into the structure in only one step.

Plasticity and damage

The aforementioned approaches only included damage in the nonlinear analysis. Other works coupled damage and plasticity in TO pathdependent strategies to maximize the load bearing capacity of elastoplastic structures with damage.

In [START_REF] Alberdi | Topology optimization of pressure dependent elastoplastic energy absorbing structures with material damage constraints[END_REF] Alberdi and Khandelwalb optimized the volume of elastoplastic-damageable structures under maximum plastic work and maximum damage constraints. In [START_REF] Li | Design of fracture resistant energy absorbing structures using elastoplastic topology optimization[END_REF], Li and Khandelwal developed the optimization of structures resistant to ductile fracture by using elastoplasticdamage path-dependent simulations for minimizing the structural volume under maximum plastic work and maximum damage constraints. Similar approaches were developed by the same authors in [START_REF] Li | Topology optimization of energy absorbing structures with maximum damage constraint[END_REF] where the plastic work was maximized under damage and volume constraints, and in [START_REF] Li | Failure resistant topology optimization of structures using nonlocal elastoplastic-damage model[END_REF] using a nonlocal damage model. More recently, Zhang and Khandelwal [START_REF] Zhang | Gursontvergaard-needleman model guided fracture-resistant structural designs under finite deformations[END_REF] extended such strategy to a finite strains Gurson-Tvergaard-Needleman plasticity model within a SIMP TO framework, where the plastic work was maximized under damage and volume constraints (see Fig. 11)

Path-dependent approaches: crack propagation

A new step in the development of TO to fracture resistance was accomplished by incorporating an explicit description of crack propagation in pathdependent TO approaches. To our best knowledge, this has been firstly proposed in Zhang et al. [START_REF] Zhang | Topological design of allceramic dental bridges for enhancing fracture resistance[END_REF], where an XFEM fracture propagation was combined with a BESO TO method. The emergence of such approach has been accelerated by the recent development of the phase field method to fracture (see e.g. [32-35, 157, 158]), which has considerably simplified the analysis of fracture propagation in engineering problems. The first combination of the phase field method to fracture with TO was achieved by Xia et al. in [START_REF] Xia | Topology optimization for maximizing the fracture resistance of quasi-brittle composites[END_REF] and Da et al. in [START_REF] Da | Topology optimization of particle-matrix composites for optimal fracture resistance taking into account interfacial damage[END_REF] in the framework of bi-materials, and then followed by several groups both in the contexts of single and bi-material structures with different variants regarding the definition of the objective function and the TO technique. The different methods are descried in the following.

Single material structures

In [START_REF] Russ | Topology optimization for brittle fracture resistance[END_REF], Russ and Waisman developed a SIMP TO framework with path-dependent phase field simulations. In their work, the total mass of the structure was minimized under maximum compliance and fracture energy constraints during fracture propagation simulations. A careful analysis of the effects of phase field model features, like degradation function and strain splits was carried out to investigate their effects on the final topologies. It was evidenced that including the fracture energy as a constraint could improve the fracture resistance as compared to including the compliance constraint only (see Fig. 12). In [START_REF] Russ | A novel topology optimization formulation for enhancing fracture resistance with a single quasi-brittle material[END_REF], a threshold formulation within phase field was investigated by the same authors, and a different objective function combining mas and external work under constraint of maximal cumulated external work was analyzed. These works have been followed by similar studies with different variants in the choice of TO methods or definition of the optimization problem. In [START_REF] Wu | A path-dependent level set topology optimization with fracture criterion[END_REF], Wu et al. used a level-set TO combined with phase field fracture and a p-norm function formulation to aggregate the phase field variables into a single constraint. The authors evidenced the practical interest of such approaches in dental bridge applications. In [START_REF] Desai | Topology optimization of structures undergoing brittle fracture[END_REF], Desai et al. also developed a Level-set TO combined with fracture simulations to minimize total external work during fracture simulations of structures and conducted advanced 3D applications (see Fig. 13). In [START_REF] Jia | Controlling the fracture response of structures via topology optimization: From delaying fracture nucleation to maximizing toughness[END_REF], Jia et al. combined a SIMP TO including phase field fracture and used a multi-objective function that allowed optimizing in a weighted manner: (i) the initial stiffness of the structure; (ii) the first instance at which fracture nucleates and (iii) the energy dissipated by fracture propagation once fracture has occurred (see Fig. 14).

All the above-mentioned works only considered static, quasi-brittle structures. In [START_REF] Li | Improved fracture resistance of 3D-printed elastoplastic structures with respect to their topology and orientation of deposited layers[END_REF], Li et al. developed a ductile fracture behavior combined within a BESO TO formulation. Another novelty of this work was the introduction of the anisotropy in the fracture behavior, to account for orientation in 3D printed fabrication process (see Figs. 1516). In a recent paper, [START_REF] Noii | Level-set topology optimization for ductile and brittle fracture resistance using the phase-field method[END_REF], Noii et al. also investigated TO with ductile fracture within a Level-set TO. In their work, the total work was maximized while accounting from structural damage and mass constraints and the level-set surface used within TO was updated by a reaction-diffusion algorithm.

Finally, another recent and promising advance is the consideration of dynamic fracture within TO. The first related study and the only one so far to our best knowledge has been conducted by Wu et al. in [START_REF] Wu | Topology optimization for enhanced dynamic fracture resistance of structures[END_REF], where SIMP TO was combined with dynamic phase field propagation analysis. In this work, fracture energy was minimized during a given time load after impact on the structure and with volume constraints. Introducing dynamic effects allowed to take into account the effects of loading rates (see Fig. 17).

Fig. 12 Force-displacement (a) and fracture energy functional (b) curves using the principal split of the elastic energy and the quadratic degradation function for three topologies. Reproduced from [START_REF] Russ | Topology optimization for brittle fracture resistance[END_REF] with permission.

Heterogeneous structures and materials

The structural design discussed above is primarily derived from the pioneering work by Xia et al. [START_REF] Xia | Topology optimization for maximizing the fracture resistance of quasi-brittle composites[END_REF] and Da et al. [START_REF] Da | Topology optimization of particle-matrix composites for optimal fracture resistance taking into account interfacial damage[END_REF], with a specific emphasis on two-phase quasi-brittle composite structures.

In their study [START_REF] Xia | Topology optimization for maximizing the fracture resistance of quasi-brittle composites[END_REF], the authors employed TO to optimize the distribution of the stiffer phase in order to maximize the absorbed energy throughout the fracture process, encompassing damage, crack initiation, propagation, and ultimate failure. Furthermore, they perform a comprehensive sensitivity analysis of the objective function with respect to the topological design variable. The paper showcased numerical examples in both 2D and 3D settings, illustrating the enhanced fracture resistance achieved when compared to classical linearly elastic compliance-based TO approaches. Da et al. [START_REF] Da | Topology optimization of particle-matrix composites for optimal fracture resistance taking into account interfacial damage[END_REF] presented a significant advancement by addressing the widespread issue of interfacial damage and its implications. The authors introduced a novel approach to optimize the topology of reinforced inclusion phases by modifying the geometry of the interface. Their objective function was designed to maximize toughness considering entire reaction force and load-point displacement curves, with a constraint on the inclusion volume. The study investigated scenarios with and without initial cracks, shedding light on the interfacial effects. By employing a highly nonlinear cohesive zone model [START_REF] Verhoosel | A phasefield model for cohesive fracture[END_REF][START_REF] Nguyen | A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography[END_REF], the researchers observed that cracks tend to initiate at the optimized interface in the absence of predetermined or initial cracks. Through the optimization process, the study reveals the emergence of concave interfaces, effectively preventing inclusion bypass and promoting crack nucleation on the opposite side of the inclusion. Moreover, the concave interfaces exhibit twisting behavior and elongate the path of interfacial crack propagation, consequently demanding elevated fracture energies to fracture the component. These insights have a direct impact on fracture resistance, offering an effective mechanism for toughening through crack bridging and twisting.

Li et al. [START_REF] Li | A SIMPphase field topology optimization framework to maximize quasi-brittle fracture resistance of 2D and 3D composites[END_REF] and Wu et al. [START_REF] Wu | Level-set topology optimization for maximizing fracture resistance of brittle materials using phasefield fracture model[END_REF] have both employed a similar framework, utilizing different TO methods such as SIMP and LSM, respectively. Li et al. [START_REF] Li | A SIMPphase field topology optimization framework to maximize quasi-brittle fracture resistance of 2D and 3D composites[END_REF] compared the BESO and SIMP methods by analyzing various initial guess designs. In their study, Wu et al. presented a practical example of an all-ceramic dental bridge, demonstrating the improved fracture toughness achieved by combining LSM with phase-field fracture modelling. Expanding upon this framework, Da and Yvonnet [START_REF] Da | Topology optimization for maximizing the fracture resistance of periodic quasi-brittle composites structures[END_REF] incorporated periodic composite structures and multiple objectives into their work. They enforced the periodicity of the unit cell geometry while performing TO for a given load applied across the entire structure. The authors investigated size effects, which exhibited similar trends to classical fracture experiments in quasibrittle structures (refer to Fig. 18). Additionally, they explored enhancing the fracture resistance of a unit cell associated with a material under multiple loads, resulting in a complex optimized Reproduced from [START_REF] Desai | Topology optimization of structures undergoing brittle fracture[END_REF] with permission. microstructure (refer to Fig. 19). The aforementioned studies [START_REF] Xia | Topology optimization for maximizing the fracture resistance of quasi-brittle composites[END_REF][START_REF] Da | Topology optimization of particle-matrix composites for optimal fracture resistance taking into account interfacial damage[END_REF][START_REF] Da | Topology optimization for maximizing the fracture resistance of periodic quasi-brittle composites structures[END_REF] comprise the second segment of Da's book [START_REF] Da | Topology optimization design of heterogeneous materials and structures[END_REF] in TO of heterogeneous materials and structures.

In [START_REF] Da | Fracture resistance design through biomimicry and topology optimization[END_REF], a departure from linear analysis in assessing the toughness of bio-inspired bi-material samples, as seen in [START_REF] Gu | Algorithm-driven design of fracture resistant composite materials realized through additive manufacturing[END_REF], was made by introducing phase field modeling. The primary focus of the study is the analysis of the "brick-and-mortar" biological composites and its interlock and noninterlock architectural associations. Through this investigation, the researchers identified two distinct fracture stages that align closely with experimental observations [START_REF] Liu | Combination of stiffness, strength, and toughness in 3D printed interlocking nacre-like composites[END_REF]. To further enhance fracture performance, the phase field-based TO framework aiming to maximize toughness while Load-displacement curves for an L-shaped bracket (optimized design), showing the effects of 3D printing orientation angles θ, and the transition from quasibrittle to ductile fracture regimes. Reproduced from [START_REF] Li | A SIMPphase field topology optimization framework to maximize quasi-brittle fracture resistance of 2D and 3D composites[END_REF] with permission.

accounting for the mass constraint of the soft phase was proposed (refer to Figs. 20 (a-f)). Through strategic modifications to the constitutive material distribution, the authors have successfully eliminated the significant stress concentrations located ahead of the crack tip (refer to Figs. 20 (g) and (h)). As a result, the energy dissipation capability is significantly enhanced, leading to a notable improvement in fracture performance.

In a recent study by Da [START_REF] Da | Model reduction on 3D fracture resistance design[END_REF], the author investigated similar soft and stiff bio-inspired bimaterials and compared them with interpenetrating phase composites, focusing on more advanced 3D cases. The research included model reduction techniques applied to 3D fracture resistance design, demonstrating significant improvements in critical fracture resistance properties while achieving computational efficiency in inverse design. Moreover, the author and co-workers successfully fabricated bi-material topology-optimized structures using 3D printing. Experimental tests, such as the model-I test conducted by Da et al. [START_REF] Da | Enhancing fracture resistance in architected bi-material structures: Design optimization, experimental validation, and mechanical analysis[END_REF], revealed that the toughness of the bi-material structures in brittle fracture can be enhanced by more than 100 times compared to pure homogeneous structures with a 100% volume fraction of constitutive stiff material. Further investigations on design optimization of inclusion phases in bi-material composites against failure can be found in the works of Singh et al. [START_REF] Singh | Material optimization to enhance delamination resistance of composite structures using viscous regularization[END_REF][START_REF] Singh | Robust design optimization for enhancing delamination resistance of composites[END_REF][START_REF] Singh | On optimization of heterogeneous materials for enhanced resistance to bulk fracture[END_REF]. These studies focus on achieving delamination resistance in composite structures through variations in the size, rotation, and shifting positions of the inclusions within the context of periodic microstructure optimization. The similar crack bridging mechanism as in [START_REF] Da | Topology optimization of particle-matrix composites for optimal fracture resistance taking into account interfacial damage[END_REF] offers valuable insights into diverse toughening mechanisms, encompassing crack nucleation, growth, and ultimate structural failure.

Conclusion and future challenges

In this article, we have presented the state of the art of Topology Optimization (TO) for fracture resistance, including: (i) stress-based approaches, (ii) TO for linear structures containing initial nonevolving cracks, (iii) TO for fatigue resistance, (iv) TO taking into account path-dependent nonlinear damage and plasticity and finally (v) TO incorporating crack propagation in structures and materials.

Stress-based TO methods offer valuable insights for mitigating stress concentrations in linear structures, which is essential for ensuring structural integrity. However, they often have limitations when it comes to predicting plasticity, damage, or the behavior of cracks in scenarios where they cannot be entirely avoided. Fatigue TO approaches, on the other hand, predominantly rely on stress-based methodologies and are particularly useful for ensuring that stresses remain within safe limits over a specified number of load cycles. Nevertheless, there have been advancements within this framework to account for damage and plasticity. TO models incorporating pre-existing cracks primarily revolve around compliance optimization, wherein existing cracks are modeled as discontinuities in the structural design. These methods are well-suited for addressing structures with pre-defined flaws, offering insights into optimizing their performance.

TO techniques explicitly describing the development of plasticity, damage or crack propagation in load path scenarios add a strong layer of complexity, as they require the solution of a pathdependent nonlinear problem (usually with finite elements) at each iteration of the TO scheme. Various commonly used TO methodologies have been employed in these endeavors. The last-mentioned techniques represent the most recent contributions to TO for fracture resistance, and the various related studies have shown promising gains in terms of fracture toughness relative to mass, compared with previous design techniques. In addition, they also make it possible to develop new features, such as the ability to maintain structural integrity or to dissipate energy in the most efficient way in the case of unavoidable damage situations. However, the development of these techniques for practical engineering applications still faces a number of technical and scientific challenges. The first of these is computational cost. Unlike linear approaches, each iteration of the TO process usually involves a full non-linear simulation. Advanced computational methods are therefore needed to drastically reduce these costs for engineering applications. In this respect, model reduction or machine learning can be promising ways of solving this problem.

Another important future challenge will be to develop multi-scale approaches to more accurately take into account the effects of microstructures or architected materials in optimized designs capable of withstanding fracture. Finally, many very recent aspects of fracture resistance, including Reproduced from [START_REF] Da | Topology optimization for maximizing the fracture resistance of periodic quasi-brittle composites structures[END_REF] with permission.

anisotropic effects, dynamic fracture, and composite material design, are still under development and will have to meet the challenges mentioned above, integrating more levels of complexity.

Fig.

  Fig. Stress distribution of the optimal L-shaped beams achieved using various stress-based TO methods: (a) compliance TO with stress concentration [71], and (b-e) the following approaches: (b) SIMP[START_REF] Le | Stress-based topology optimization for continua[END_REF], (c) BESO[START_REF] Xia | Stress-based topology optimization using bi-directional evolutionary structural optimization method[END_REF], (d) LSM[START_REF] Guo | Stress-related topology optimization via level set approach[END_REF], and (e) MMC[START_REF] Zhang | A moving morphable void (MMV)-based explicit approach for topology optimization considering stress constraints[END_REF]. Reproduced figures used with permission.

Fig.

  Fig. Effects of stress-optimized designs on fracture resistance properties [71]: (a) diverse optimized topologies for various scenarios; (b) stress distribution highlighting the maximum von Mises stress; (c) fracture patterns displaying normalized toughness ( T ) and peak load ( P ) values. Reproduced from [71] with permission.

Fig.

  Fig. Size effects of the global heterostructure: (a) final crack patterns of the data-driven designed structures; (b) corresponding nominal stressstrain curves. Reproduced from [85] with permission.

Fig. 4

 4 Fig. 4 Bi-material notched plate (a) Several designs with different distributions of the soft and stiff phases. (b) 3D printed samples. Reproduced from [91] with permission.

Fig. 6 L

 6 Fig.6L-bracket optimization: von Mises stress contour in the optimized grometry for several initial cracks with different lengths. Reproduced from[START_REF] Hu | Fracture strength topology optimization of structural specific position using a bi-directional evolutionary structural optimization method[END_REF] with permission.

Fig. 7

 7 Fig. 7 Fatigue TO and experiments of a 3D-Printed metal aerospace bracket [127]: (a) assembly of the original bracket and the fatigue testing platform; (b) topology of the optimized bracket; (c) stress analysis of the new bracket; (d) 3D printed sample; and (e) experimental fatigue test. Reproduced from [127] with permission.

Fig. 9

 9 Fig. 9 Iteration of cellular structure with 8×4 periodic unit cells: (a) iterative histories of mean energy absorption and volume fraction constraint; (b) force-displacement curves of different iterative steps. Reproduced from [143] with permission.

Fig. 10

 10 Fig. 10 Optimized reinforcement layouts and response of a 2-D structure taking into account damage. Reproduced from [147] with permission.

Fig. 11

 11 Fig. 11 Optimized L-bracket taking into account path-dependent finite strains Gurson-Tvergaard-Needleman plasticity model and damage. Reproduced from [155] with permission.

Fig. 13

 13 Fig. 13 Damage variable at the final time plotted for several optimization iterations (black regions indicate cracks).Reproduced from[START_REF] Desai | Topology optimization of structures undergoing brittle fracture[END_REF] with permission.

Fig. 14

 14 Fig. 14 Design setup and results for the optimization of a disk under rotational displacement. (a) Design domain and boundary conditions (all length units are in mm). (b) Optimized designs for two different choices of weighting factors defining importance of initial stiffness, onset of fracture and energy dissipation. (c) Force-displacement curves for the optimized designs and corresponding radar chart comparing their normalized structural performance. The classical stiffness design obtained from conventional stiffness maximization is included for reference. Reproduced from [164] with permission.

Fig. 15

 15 Fig. 15 Optimized designs of an L-shape structure with anisotropic ductile fracture behavior arising from 3D-printing process. The values α, d and p denote interfacial damage, bulk damage and equivalent plastic strain, respectively. The topologies and fields are depicted for 3D printing orientations θ: (a) θ = 0 • ; (b) θ = 90 • ; θ = 120• . Reproduced from[START_REF] Li | Improved fracture resistance of 3D-printed elastoplastic structures with respect to their topology and orientation of deposited layers[END_REF] with permission.

Fig. 16

 16 Fig. 16Load-displacement curves for an L-shaped bracket (optimized design), showing the effects of 3D printing orientation angles θ, and the transition from quasibrittle to ductile fracture regimes. Reproduced from[START_REF] Li | A SIMPphase field topology optimization framework to maximize quasi-brittle fracture resistance of 2D and 3D composites[END_REF] with permission.

Fig. 17

 17 Fig. 17Optimized structures for dynamic fracture resistance. Columns correspond to different loading rates; (a) Final design taking into account Dynamic fracture; (b) Final crack pattern; (c) design obtained by static analysis for comparison.Reproduced from[START_REF] Wu | Topology optimization for enhanced dynamic fracture resistance of structures[END_REF] with permission.

  Fig. 17Optimized structures for dynamic fracture resistance. Columns correspond to different loading rates; (a) Final design taking into account Dynamic fracture; (b) Final crack pattern; (c) design obtained by static analysis for comparison.Reproduced from[START_REF] Wu | Topology optimization for enhanced dynamic fracture resistance of structures[END_REF] with permission.

Fig. 18

 18 Fig.18TO of a bi-material structure; black phases denote stiff inclusions, grey phase denote brittle matrix: (a) fracture (in red) in 3-point bending in the initial design; (b) fracture in the optimized design; (c) comparison between the response of initial and optimized designs. Reproduced from[START_REF] Da | Topology optimization for maximizing the fracture resistance of periodic quasi-brittle composites structures[END_REF] with permission.

Fig. 19

 19 Fig. 19 Unit cell of a material containing randomly distributed square inclusions: (a) initial design and load case 1 geometry; (b) load case 2; (c) load case 3; (d) final design for load cases 1 and 2; (e) final design for load cases 1, 2, and 3.Reproduced from[START_REF] Da | Topology optimization for maximizing the fracture resistance of periodic quasi-brittle composites structures[END_REF] with permission.

Fig. 20

 20 Fig.20Comparing final crack patterns of the structure (a-e) with the optimized design structure (f). Additionally, Figures (g) and (h) showcase the topology of an alternative designed structure along with its in-plane stress distribution. Reproduced from[START_REF] Da | Fracture resistance design through biomimicry and topology optimization[END_REF] with permission.

By critical, we mean the stress concentration directly contributing to crack initiation.

Conflict of interest statement

On behalf of all authors, the corresponding authors state that there is no conflict of interest.


S Hebib

Member, IEEE N Raveu

Cantor Spiral Array for the Design of Thinned Arrays

Keywords: Fractal, spiral array, thinned array

This letter is focused on the design of thinned planar arrays achieving simultaneously the two following requirements: 1) a given beamwidth in the broadside direction and 2) a given peak side-lobe level in a specified sub-domain of the visible region. It is seen that Cantor spiral arrays are excellent candidates. Peak sidelobes of the order of 20 dB and a beamwidth of 0.6 are obtained with only 200 radiating elements.

I. INTRODUCTION

T HINNED planar antenna arrays are of great practical im- portance, especially for space applications where the minimization of the number of radiating elements in satellites is crucial. This letter is focused on the design of thinned arrays achieving simultaneously the two following requirements: 1) a given beamwidth in the broadside direction and, 2) a given peak side-lobe level in a specified subdomain of the visible range.

Among the many thinning strategies reported in the literature, Cantor Ring Arrays are promising solutions [START_REF] Jaggard | Cantor ring arrays[END_REF]. Such arrays consist in a uniform distribution of radiating elements on circles whose diameters are derived from a symmetric finite-stage polyadic Cantor set. Instead of adopting uniform distributions of elements on the circles, we propose here to locate elements at the intersection of the Cantor rings and the radials. We show that such configuration allows the design of Cantor-based arrays that reduce the number of radiating elements compared to classical ring arrays [START_REF] Sherman | Thinning planar arrays antennas with ring arrays[END_REF] composed of concentric rings with equal spacing between the rings. This is the first result reported here. Next, in order to reduce the number of radiating elements but keeping the beamwidth and peak side-lobe level relatively unchanged, we break the alignment of radiating elements in each radial by constituting spiral arms. Such "ring rotation technique" has been advantageously applied in [START_REF] Sherman | Thinning planar arrays antennas with ring arrays[END_REF] for reducing the peak side-lobe levels in arrays composed concentric rings with equal spacing between the rings.

We show here that this technique allows the design of Cantor spiral arrays that present the same performances than the original Cantor-based arrays in terms of beamwidth and side-lobe level, but with a significant reduction of 10% in the number of radiating elements. This is the second major result of the research work reported here. Let us determine the Cantor-based ring array with a minimum number of radiating elements that satisfies the two following requirements:

II. THE CANTOR SPIRAL ARRAY

1)

a beamwidth of 0.6 in the broadside direction; 2) a peak side-lobe level of the order of in the range around the broadside direction. Genetic Algorithms (GAs) are used [START_REF] Haupt | Thinned arrays using genetic algorithms[END_REF]- [START_REF] Marcano | Synthesis of antenna arrays using genetic algorithms[END_REF] to derive such thinned array. GAs have been chosen for their ability to converge toward a global solution, in the case of complex functions [START_REF] Haupt | Practical Genetic Algorithms[END_REF], [START_REF] Haupt | An introduction to genetic algorithms for electromagnetics[END_REF]. The fitness function is here given by peak side-lobe level (dB)

A random crossover point is applied with a probability of 0.7.

A random mutation on one bit is performed with a probability of 0.1 (the mutation allows to avoid local minima). The GA reaches a solution for Cantor-based ring array with 224 sources, , , and , where designates the free-space wavelength. The resulting array factor is given in Fig. 3 ( and traduce the polar into cartesian coordinates normalized by 90 ). For comparison, classical ring arrays composed of concentric rings with equal spacing between the rings [START_REF] Sherman | Thinning planar arrays antennas with ring arrays[END_REF], with the same number of radiating elements (224) and the same beamwidth (0.6 ) presents a peak side-lobe level of the order of , that is, 8 dB higher than one of the Cantor-based ring array.

Next, in order to reduce the number of radiating elements but keeping the beamwidth and peak side-lobe level relatively unchanged, we break the alignment of radiating elements in each radial. As shown in the insert of Fig. 4, elements of the original Cantor-based ring arrays are shifted by a constant angle so as to generate spiral arms (see Fig. 5). The peak sidelobe levels and beamwidth are determined for several normalized ratio

, where stands for the constant angle between two radials in the original Cantor-based ring array (see Fig. 2). In order to keep a relatively constant beamwidth, the parameters and have to be slightly tuned for each value of . As an example, consider the 224 radiating elements of the Cantor-based ring array shown in the insert of Fig. 3 and determine the variation of the peak side-lobe level versus in a Cantor spiral array (see Fig. 4). We observe that the Cantor spiral array such that present a slight improvement in term of side-lobe level compared to the original Cantor-based ring array ( instead of

). Moreover we derive from GA that the Cantor spiral array with a beamwidth of 0.6 and a peak side-lobe level of the order of presents only 200 elements. This interesting thinned array, shown in Fig. 5, has the following parameters: , , , and . Consequently, in terms of beamwidth and peak side-lobe level, this Cantor spiral array present the same performances than the original Cantor-based ring array, but with a significant reduction of 10% in the number of radiating elements. The normalized array factor of this thinned array is shown in Fig. 6 in the range of interest around the broadside direction.

III. CONCLUSION

For a given number of radiating elements and beamwidth in the broadside direction, Cantor-based ring arrays may present lower peak side-lobe level than one of the classical ring arrays composed of concentric rings with equal spacing between the rings. Moreover,in terms of beamwidth and peakside-lobelevel, Cantor spiral arrays present the same performances than Cantor-based ring arrays with alignment of radiating elements, but with a significant reduction of 10% in the number of elements.

  Throughout this paper the arrays are composed of isotropic elements and equal amplitude signals are assumed at each el-Manuscript received October 28, 2005; revised January 17, 2006. The authors are with the Ecole Nationale supérieure d'Electrotechnique, d'Electronique, d'Informatique et d'Hydraulique et des Télécommunications de Toulouse, 31071 Toulouse, France (e-mail: raveu@enseeiht.fr).

Fig. 1 .

 1 Fig. 1. (a) Symmetric polyadic Cantor set at stage of growth S = 1; 2; 3; and 4. (b) Resulting concentric rings at stage S = 4.

Fig. 2 .

 2 Fig. 2. Cantor-based ring array with elements located at the intersection of the Cantor rings and the equiangular radials.

Fig. 3 .

 3 Fig. 3. Normalized array factor of the Cantor-based ring array shown in the insert of (a) with N = 1, = 0:4367, a = 56:23 and S = 4: (a) side view and (b) top view.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Sidelobe level versus relative misalignment.

Fig. 6 .

 6 Fig. 6. Normalized array factor of the Cantor spiral array shown in the insert of (a) with N = 1, S = 4, = 0:43, a = 55:7 and 1 =1 = 0:13: (a) side view and (b) top view.