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... Chu et al. [27] used stress penalty to deal with localized problems of stress constraints. Wang et al. [28] introduced von Mises stress into fail-safe optimization, this method eliminated unnecessary failure working conditions and get good achievement. Meng et al. [29] used P-norms to convert a large number of stress constraints into a global constraint. ...

... To assess the merit of the proposed fail-safe models than other fail-safe structures. Let compare Example 5 with literature [28] chapter 5.2. Literature [28]'s arithmetic is based on the von mises stress and used the KS function to approximate the max-operator. ...

... Let compare Example 5 with literature [28] chapter 5.2. Literature [28]'s arithmetic is based on the von mises stress and used the KS function to approximate the max-operator. Fig. 39 is the topology results for literature [28]. ...

... In this work, the optimization problem was the mass minimization of lattice structures subjected to stress constraints. More recently, and following the idea of reducing the number of involved failures scenarios, Wang et al. [22] introduced the von Mises stress into fail-safe topology optimization to instruct the measure of local failure. By defining the failure coefficient, material property of the patch was regarded as degenerated if its von Mises stress exceeded the allowable stress. ...

... For that reason, stochastic collocation methods represent an attractive tool for uncertainty propagation tasks when these take advantage of High Performance Computing (HPC). Task-level parallelism is used for implementing the resolution of the direct and adjoint state equations in (22) at each stochastic node in clustering. This is done adopting a Master-Worker Pattern, which permits to perform simultaneous processing across multiple machines or processes via a Master and multiple Workers. ...

... 2. Define the target volume V = L|D| (with L the target volume fraction in (6)), an evolutionary volume ratio (ER) and the allowable volume change ∆V for stabilization purposes. 3. Calculate the distribution of the optimality criterion Φ(χ O , x) (i) of (22) using an X-FEM integration scheme for solving the state and adjoint state equations at each stochastic collocation point at the current i-th iteration of the optimization approach. Project this field from the FE integration points to the nodes. ...

This paper presents a novel probabilistic approach for fail-safe robust topology optimization with the following novelties: (1) the probability for failure to occur at a specified location is considered; (2) the possibility for random failure size is incorporated; (3) a multi-objective problem is pursued encompassing both the expected value of the structural performance and its variance as a robustness criterion. Compared against alternative worst-case-based formulations, the probabilistic framework employed allows designers to assume certain level of risk, avoiding undesirable increments in structural performance due to low probability damage configurations; (4) alternatively to most existing works within fail-safe topology optimization, considering density-based methods, this paper pursues for the first time an optimization technique where the structural boundary is represented implicitly by an iso-level of an optimality criterion field, which is gradually evolved using a bisection method. A key advantage of this technique is that it provides optimized solutions for different volume fractions during the optimization process, allowing to efficiently find a trade-off between structural performance, cost and robustness. Finally, numerical results are included demonstrating the ability of the proposed formulation to provide smooth and clearly defined structural boundaries and to enhance structural robustness with respect to conventional deterministic designs.

... References [42,113] consider the material degradation, where stress exceeds a given stress, in the density-based topology optimization. Consequently, fail-safe design [59,61,65,115] which incorporates the failure uncertainties e.g. unknown various failure shapes, sizes, and locations into the topology optimization has been developed to obtain still a good design while damage happens. ...

... Ref [3] proposed the unified framework on worst-case in parametric and shape optimization under the uncertainties of geometry, elastic modulus, applied body force. A fail-safe design is proposed [115] where the damaged compliance for the worst failure case is set as the optimization objective. However, it is not easy to define the worst-case within hybrid uncertainties, and multi-physics loading. ...

Topology optimization is a systemic design that requires simulation and optimization of a system for a single or multiple physics coupling processes. However, it is short of the engineering sense regarding the absence of uncertainties and limitations on applied monophase material. The foundation of this dissertation is to combine homogenization and stochastic processing into topology optimization to formulate a robust multiscale topology optimization approach. Accordingly, this Ph.D. dissertation concerns (1) the multiscale and multiphysics performance of heterogeneous materials/structures embedded with microstructures material, taking into account the uncertainties, (2) for further optimizing the heterogeneous structure at different scales to satisfy target performance. These microstructures may arise from the processing of biological materials, or from dedicated engineered materials, e.g., aerogels, foams, composites, acoustics metamaterials, etc. We parametrize architecture material; study the performances of the microstructure at the macroscopic scale by homogenization method. Then, the homogenization model can be considered a stochastic model with presented uncertainties exhibited in the unit cell. It can be built from a polynomial chaos development. In addition, these parametrized micro geometry features can be mapped into homogenized properties space, which can be utilized as design variables to control the macrostructure performance. Afterward, we combined the topology optimization, homogenization, and uncertainties qualification to (1) design macro topology and micro material distribution to maximum structure stiffness (2) reduce the structure sensitivity to presented uncertainties (e.g., loading and material properties). This proposed general framework has the advance and compatibility ability in solving optimization problems considering the (1) multiple parametrized architectures cells, (2) complex loading problem, (3) hybrid uncertified, etc., with an affordable computation manner.

... Long et al. [22] studied the robust topology optimization based on the reciprocal SIMP method, taking into account local failure and load uncertainty. Wang et al. [23] eliminated some unnecessary failure cases by introducing the von Mises stress into the fail-safe optimization design. ...

... The fail-safe topology optimization of continuum structures mentioned above [18][19][20][21][22][23] is only confined to the field of statics. The vibration problem must be considered in the practical engineering application. ...

It is an important topic to improve the redundancy of optimized configuration to resist the local failure in topology optimization of continuum structures. Such a fail-safe topology optimization problem has been solved effectively in the field of statics. In this paper, the fail-safe topology optimization problem is extended to the field of frequency topology optimization. Based on the independent continuous mapping (ICM) method, the model of fail-safe topology optimization is established with the objective of minimal weight integrating with the discrete condition of topological variables and the constraint of the fundamental frequency. The fail-safe optimization model established above is substituted by a sequence of subproblems in the form of the quadratic program with exact second-order information and solved efficiently by the dual sequence quadratic programming (DSQP) algorithm. The numerical result reveals that the optimized fail-safe structure has more complex configuration and preserved materials than the structure obtained from the traditional frequency topology optimization, which means that the optimized fail-safe structure has higher redundancy. Moreover, the optimized fail-safe structure guarantees that the natural frequency meets the constraint of fundamental frequency when the local failure occurs, which can avoid the structural frequency to be sensitive to local failure. The fail-safe optimization topology model is proved effective and feasible by four numerical examples.

... While this method proved to be conservative, the number of analyses could only be reduced by about 40% for the examples considered. Wang et al. [39] followed the same idea and disregarded damage scenarios based on the stress state of the intact structure inside each damage shape and reported a similar reduction in computational cost. Thus, the achievable savings for both strategies are again not high enough. ...

Structural optimization has become an increasingly important part of product development, especially in the aerospace industry, where weight savings due to lightweight design have a particularly strong impact on efficiency and thus economy and environmental compatibility. One area of structural optimization is topology optimization, which offers maximum design freedom and thus enables the greatest improvements. However, load-adapted designs obtained by topology optimization are usually highly sensitive to an unpredictable local loss of stiffness, like e.g. for the case of randomly inflicted damage to individual load paths of the structure. Therefore, these designs are not considered fail-safe.
This thesis presents a two-stage procedure for density-based optimization towards a fail-safe design. Existing approaches are either computationally extremely expensive or do not explicitly consider fail-safe requirements in the optimization. The presented method trades off both aspects by employing a two-stage optimization approach to provide redundant designs that offer robustness to the failure of single load paths. In the first stage, a topology optimization with local volume constraints is performed. The second stage is referred to as "density-based shape optimization" since it only alters the outline of the structure while still acting on a fixed voxel-type finite element mesh with pseudo-densities assigned to each element. The performance gain and computational efficiency of the proposed method are demonstrated by application to various 2D and 3D examples. The results show, that the presented method can be carried out with reasonable computational effort, in contrast to existing approaches with explicit consideration of fail-safety in topology optimization. For the 2D examples considered, the number of analyses for a fail-safe optimization is reduced by three orders of magnitude compared to existing methods and is at most 5.6 times higher than for a standard topology optimization. Consequently, the proposed method is also applicable for large-scale models in an industrial context.
With the possibility to compute and manufacture single optimized components, the question of how to optimize the connections between different components in an assembly arises. This thesis therefore also provides a method for the simultaneous optimization of the topology of components and their corresponding joint locations in an assembly. Therein, the joint locations are not discrete and predefined, but continuously movable. The underlying coupling equations allow for connecting dissimilar meshes and avoid the need for remeshing when joint locations change. The presented method models the force transfer at a joint location not only by using single spring elements but accounts for the size and type of the joints. When considering e.g. riveted or bolted joints, the local part geometry at the joint location consists of matching holes that are surrounded by material. For spot welds, the joint locations are filled with material and may be smaller than for bolts. The presented method incorporates these material and clearance zones into the simultaneously running topology optimization of the components. Furthermore, failure of joints may be taken into account at the optimization stage, yielding assemblies connected in a fail-safe manner. Finally, by embedding the above-mentioned efficient method for fail-safe optimization of single components in the presented assembly optimization framework, damage tolerant assemblies can be obtained that are robust to the failure of joints and single load paths of each component.

... In general, fail-safe topology optimization approaches seek to provide optimized structures that are tolerant to any imaginable local failure, which in turn may occur at any position in the design space (see, for example, Refs. [32][33][34][35][36][37][38][39][40][41]). In the proposed formulation for predetermined breaking points, on the other hand, the position of the damage region is predefined, that is, it is chosen before solving the optimization problem; the optimization problem is thus formulated in such a way to ensure that the first structural member that experiences failure is within the predefined damage region. ...

This paper addresses the concept of predetermined breaking points in topology optimization. The aim is to propose and investigate a novel formulation to design optimized topologies in which one can control where failure will occur first in case of overload; in addition, the optimized topology must withstand the design load after the damaged part is removed. In order to achieve this goal, a stress-constrained formulation based on two realizations of material distributions is proposed: one realization represents the nominal design, without damage, and the other represents the damaged design. In the nominal design, the predetermined damage region is defined, which is the region where failure is programmed to occur first in case of overload. The design constraints are defined in a way that ensures that a structural member is formed within the predetermined damage region and that the maximum von Mises equivalent stress of this member is slightly larger than the maximum von Mises stress in the rest of the structure. After failure has occurred, stress constraints are employed to ensure that the resulting design without the damaged part still resists the applied load. Two design problems with several variants are addressed: the L-shaped and the MBB beam problems. Numerical investigations demonstrate that: (1) the conventional design is extremely sensitive to localized damage of structural members and, moreover, its almost fully stressed configuration does not allow to predict where failure will occur first in case of overload; (2) the proposed formulation for predetermined breaking points is able to provide optimized structures where one knows in advance the region where failure is expected to occur first; in addition, the structure remains safe after the damaged part is removed.

... Further damage scenario reduction strategies for continuum structures include an adaptive stress criterion [16] in which only damage patches above a stress threshold are evaluated and a nonlinear mapping of principal stress to approximate regions representing members and nodes [17,18]. ...

To investigate the characteristics of optimal fail-safe structures subjected to single and multi-member damage scenarios, we consider a pin-jointed cantilever truss with all members directly connected from the load point to the boundary. Two problem formulations are considered-minimizing the compliance with a volume constraint and minimizing the volume with stress constraints. Whilst these formulations produce equivalent structures for traditional truss design problems, we find that this is not always the case in the fail-safe setting. Analytical solutions are developed for a three-bar truss under both problem formulations. Damage is modelled as the complete removal of any one member and a minmax problem is constructed to minimize the compliance or volume of the structure for the worst-case damage scenario. These new analytical solutions provide much needed benchmarks for numerical fail-safe methods. The problems are extended to n-bar systems with damage to multiple members. Results show that as the structural complexity (number of members in a system) increases, the optimum fail-safe structure tends towards a variation of the nominal 2-bar design with overlapping members. From these observations, we then approach the idea of full redundancy through the introduction of parallel substructures into a more complex truss design. We compare our fully redundant truss design to a benchmark fail-safe solution and show that the fully redundant design has significantly better performance and with fewer members. Practically, this suggests that fully redundant structural designs are highly efficient and have the +61 3 9925 3655 additional benefit of only requiring the computation of the nominal solution.

... Several investigations have analyzed the local failure in topology optimization for truss structures [1][2][3][4][5][6], where local failure can be modeled straightforwardly by removing one bar from the truss, since there is a clear definition of structural members. In contrast, other authors [7][8][9][10][11][12] addressed local failure in continuum topology optimization, where there is no real notion of a structural member and holes of varying shape and size can occur. In these approaches, damage was limited to a small rectangular subset of the design domain where all material was removed. ...

The aerospace industry demands designs capable of withstanding the simultaneous collapse of several structural elements. Examples of this failure scenario are propeller blade or uncontained engine rotor failures, where the structural integrity of the aircraft may be compromised by flying debris. To assess aircraft survivability, reliable simulation of accidental damage scenarios using physics-based models is required. Due to the complexity of characterizing these events, practitioners and researchers have traditionally assumed conservative damage envelopes, restraining potential design improvements. This research presents an object-oriented framework to automatically generate any number of damaged aircraft meshes in a realistic manner, taking into account the randomness of the event. Due to the flexibility in the software design, both random and deterministic input parameters are allowed, such as debris origin, impact orientation, number of impacts, debris size, debris velocity, spread angles and ballistic penetration equations. The tool is applied to a commercial narrow-body aircraft in which real failure scenarios are simulated.

... As a result, the computational cost was reduced significantly. Wang et al. [42] proposed an efficient optimization strategy to obtain the designs which are insensitive to the occurrence of local failure, introducing the von Mises failure criterion to evaluate the patches to be damaged or undamaged. By defining the failure coefficient, the material properties of a given patch was regarded as degenerated if its von Mises stress exceeded an allowable stress. ...

This research proposes a new formulation for fail-safe size optimization, considering the probability of occurrence of each failure scenario and the random structural parameters as sources of uncertainty. Essentially, the fail-safe reliability-based design optimization is reformulated, where the term “damaged structure” coalesces information of the whole set of damaged configurations. Thus, a single random reliability index is defined, representing the reliability of a limit-state of the damaged structure, which accounts for the safety level of the entire set of damaged configurations. The method provides the optimum design for which the reliability indices of the damaged structure are achieved at the confidence level the designer demands. The first application example corresponds to an academic analytical problem. The second and third application examples correspond to practical engineering cases: a 2D truss structure with stress constraints as well as the tail section of an aircraft fuselage with stress and buckling constraints. Results show a considerable reduction of the objective function compared to the fail-safe RBDO, which could lead to oversized designs. In this sense, mass savings up to 13.6% are achieved for the industrial-like application example.

... Models and algorithms for fail-safe topology optimization through density based approaches and material interpolation schemes are presented in e.g. [23,51,33,28,19,15,6,40,50]. Fail-safe optimal design in the framework of fracture mechanics and uncertainty analysis is studied in [39]. ...

In conventional fail-safe optimization of frame structures the damage is usually modelled as complete removal of one or more members. We propose and incorporate two additional types of damage models into the fail-safe design problem. The first describes thickness degradation caused e.g. by corrosion. The second describes severe local damage by removal of a part of a member that causes a gap and free ends in the member. The latter damage model can cause undesirable local vibration modes. By combining the two damage models, local thickness degradation in a part of a member can be modelled. The considered design problem minimizes structural mass and includes local stress constraints and limits on eigenfrequencies. Besides the new damage models, a working-set algorithm is applied on the fail-safe optimization problem to reduce the computational cost. Numerical experiments on two-dimensional frame structures illustrate that the working-set algorithm can effectively handle the relatively large number of constraints and damage scenarios in fail-safe optimization.

... Since the computationally cost is so high, Zhou and Fleury (2016) proposed to only use as much failure patches as needed to cover the design domain with no gap and no overlap. To further decrease the computational cost, Wang et al. (2020) selected the active failure patches based on a von Mises stress criterion and used a stabilized optimality criterion update scheme. Ambrozkiewicz and Kriegesmann (2018) chose to use actual load paths as failure patches instead of a regular grid, which significantly reduces computational costs. ...

Explicitly considering fail-safety within design optimization is computationally very expensive, since every possible failure has to be considered. This requires solving one finite element model per failure and iteration. In topology optimization, one cannot identify potentially failing structural members at the beginning of the optimization. Hence, a generic failure shape is applied to every possible location inside the design domain. In the current paper, the maximum stress is considered as optimization objective to be minimized, since failure is typically driven by the occurring stresses and thus of more practical relevance than the compliance. Due to the local nature of stresses, it is presumed that the optimization is more sensitive to the choice of the failure shape than compliance-based optimization. Therefore, various failure shapes, sizes and different numbers of failure cases are investigated and compared on the basis of a general load-path-based evaluation scheme. Instead of explicitly considering fail-safety, redundant structures are obtained at much less computational cost by controlling the maximum length scale. A common and easy to implement maximum length scale approach is employed and fail-safe properties are determined and compared against the explicit fail-safe approach.

... The present article focuses on the second criteria (for work on the first criteria, see, e.g. Wang et al. (2020)). ...

Designs obtained with topology optimization (TO) are usually not safe against damage. In this paper, density-based TO is combined with a moving morphable component (MMC) representation of structural damage in an optimization problem for fail-safe designs. Damage is inflicted on the structure by an MMC which removes material, and the goal of the design problem is to minimize the compliance for the worst possible damage. The worst damage is sought by optimizing the position of the MMC to maximize the compliance for a given design. This non-convex problem is treated using a gradient-based solver by initializing the MMC at multiple locations and taking the maximum of the compliances obtained. The use of MMCs to model damage gives a finite element-mesh-independent method, and by allowing the components to move rather than remain at fixed locations, more robust structures are obtained. Numerical examples show that the proposed method can produce fail-safe designs with reasonable computational cost.

... In this work, the optimization problem was the mass minimization of lattice structures subjected to stress constraints. More recently, and following the idea of reducing the number of involved failures scenarios, [37] introduced the von Mises stress into fail-safe topology optimization to instruct the measure of local failure. By defining the failure coefficient, the material properties of a given patch was regarded as degenerated if its von Mises stress exceeded an allowable stress. ...

This paper presents a risk-averse approach in the context of fail-safe topology optimization. The main novelty is the minimization of two risk functions quantifying the costs inherent to partial or full collapses, whose occurrence is considered as a source of uncertainty. This provides the designer with the flexibility to explicitly incorporate probabilistic information of occurrence of different structural failures, in contrast to the worst case approach, that penalizes all the damage configurations regardless their probability of occurrence. For the first time in the context of fail-safe topology optimization, a level-set method is employed. The level-set function is updated by means of a reaction–diffusion equation incorporating the topological derivative of the two risk-averse functions considered. Finally, the numerical experiments reveal the capability of the proposed formulations to yield redundant structures less sensitive to inherent losses of stiffness resulting from possible failures, whilst allowing designers to assume an acceptable level of risk. The benefits and drawbacks of the formulations proposed are compared against deterministic and fail-safe worst-case formulations.

... Several topology optimisation methods, for example, the solid isotropic material with penalization (SIMP) (Bendsøe 1989;Sigmund and Petersson 1998;Rozvany et al. 1992), rational material with penalization (RAMP) (Stolpe and Svanberg 2001), evolutionary structural optimisation (ESO) (Xie and Steven 1993), bi-directional evolutionary structural optimisation (BESO) (Yang et al. 1999;Querin et al. 2000;Xie 2007, 2009), level-set method (Wang et al. 2003;Allaire et al. 2004;Xia et al. 2014 Guo et al. 2016;Zhang et al. 2017;Zhang and Zhou 2018), and moving morphable void (MMV)-based method (Zhang et al. 2018a(Zhang et al. ,b, 2020 have been developed during the past few decades. In recent years, these topology optimisation approaches have been applied in a wide range of distinct engineering problems, including frequency responses (Ma et al. 1993;Zuo et al. 2012), stress problems De Leon et al. 2015), convection problems (Alexandersen et al. 2014(Alexandersen et al. , 2016Asmussen et al. 2019), structural failure problems (Nabaki et al. 2019;Wang et al. 2020d), large-scale problems (Pollini et al. 2020;Aage and Lazarov 2013;Wang et al. 2020b,c), nanophotonics design (Jensen and Sigmund 2011), metamaterial design (Chen and Huang 2019), and structural design for aircraft and aerospace (Aage et al. 2017;Orme et al. 2017Orme et al. , 2018. Some well-established optimisation algorithms that are used to update design variables for topology optimisation are the optimality criteria (OC) method (Kohn and Strang 1986), sequential quadratic programming (SQP) (Wilson 1963), method of moving asymptotes (MMA) (Svanberg 1987), and globally convergent method of moving asymptotes (GCMMA) (Svanberg 2002). ...

Background: Additive manufacturing (AM) oriented topology optimisation has become one of the most important branches in Design for Additive Manufacturing (DfAM). Traditional topology optimisation algorithms are finite element (FE) based which result in mesh-dependent zigzag or blurry boundaries, requiring extra effort to obtain accurate boundary information for manufacturing. Currently, introducing additional support structures and designing self-supporting structures are two effective ways to avoid collapse during fabrication. Self-supporting design is becoming preferable as it can reduce the use of materials and avoid the extra efforts of removing support structures after fabrication. Therefore, smooth design of self-supporting topologies is a promising research field in terms of generating print-ready geometries for 3D printing machines.
Purpose: The objective of this thesis is to explore smooth self-supporting topologies to obtain print-ready designs without needing post-processing methods for smoothing boundaries before fabrication and adding extra support structures during fabrication.
Approach: An element-based topology optimisation algorithm named Smooth-Edged Material Distribution for Optimising Topology (SEMDOT) is developed through introducing extra grid points to each element and using elemental volume fractions in Finite Element Analysis (FEA). In SEMDOT, multiple (dual) filtering steps are used instead of the single filtering step used in general element-based algorithms. The combination of SEMDOT and Langelaar's AM filter is used. Manufacturability experiments are set up in two typical AM technologies: Fused Deposition modelling (FDM) and Selective Laser Melting (SLM).
Findings: The proposed SEMDOT algorithm is capable of forming smooth topologies. A lower penalty coefficient can be used in SEMDOT, meaning that the optimisation problem is much closer to a convex problem. SEMDOT is capable of obtaining topological designs comparable or better than standard element-based algorithms. The use of multiple filtering steps enhances the ﬂexibility of SEMDOT in exploring better performance and different topological designs. Incorporating Langelaar's AM filter enables SEMDOT to generate convergent self-supporting topologies which have been demonstrated to be printable using both FDM and SLM. Experimental results show that the conservative overhang angle criterion of 45 degrees would sacrifice more performance than is necessary to achieve the self-supporting goal. Topological designs obtained by SEMDOT can be directly manufactured by 3D printing machines without requiring redesign and post-processing.
Novelty: Accurate boundary information can be obtained using SEMDOT. Multiple filtering steps are used to obtain better performance. The Heaviside smooth function is used to implement the solid/void design of grid points and simultaneously mitigate numerical instabilities. The mathematical model of combining SEMDOT and Langelaar's AM filter is established.

... The remarkable thing is that for model 2 (honeycomb structure), the stress distribution is not as severe as other models. This means that it might exhibit low von Mises stress [11][12][13][14][15][16][17] compared to other models. Therefore, von Mises stress is calculated for each model. ...

This study presents the characteristics of the eleven commonly used porous structures. The structures are designed using ten different unit cells. Some of the unit cells consist of free-form surfaces (e.g., triply periodic minimal surface). Some of them are straightforward in design (e.g., honeycomb structure). Some of them have a hybrid structure. The 3D CAD models of the structures are created using commercially available CAD software. The finite element analysis is conducted for each structure to know how it behaves under a static load. The structures are also manufactured using a 3D printer to confirm the manufacturability of them. It is found that some of the structures are easy to manufacture, and some are not. Particularly, metal-alloy-printed structures need a minimal thickness. However, the structures’ printed or virtual models are evaluated by determining their respective mass, production cost, production time, Mises stress, and surface area. Using the values of mass, production time and cost, Mises stress, and surface area, the optimal structure is identified. Thus, the outcomes of this study can help identify the optimal porous structure for a given purpose.

... Models and algorithms for fail-safe topology optimization through density based approaches and material interpolation schemes are presented in e.g. [23,51,33,28,19,15,6,40,50]. Fail-safe optimal design in the framework of fracture mechanics and uncertainty analysis is studied in [39]. ...

In conventional fail-safe optimization of frame structures, the damage is usually modeled as complete removal of one or more members. We propose and incorporate two additional types of damage models into the fail-safe design problem. The first describes thickness degradation caused e.g. by corrosion. The second describes severe local damage by removal of a part of a member that causes a gap and free ends in the member. The latter damage model can cause undesirable local vibration modes. By combining the two damage models, local thickness degradation in a part of a member can be modeled. The considered design problem minimizes structural mass and includes local stress constraints and limits on eigenfrequencies. Besides the new damage models, a working-set algorithm is applied to the fail-safe optimization problem to reduce the computational cost. Numerical experiments on two-dimensional frame structures illustrate that the working-set algorithm can effectively handle the relatively large number of constraints and damage scenarios in fail-safe optimization.

... Hz). It is verified that the DAR and NF are efficiently improved by the Taguchi [36]. Therefore, the smaller the stress is preferred when designing the PA. ...

The slope stability is generally analyzed by a 2D method, ignoring the spatial effect (SE), which no longer meets the evaluation accuracy requirement. This problem is particularly prominent for the embankment built on the double V-shaped gully (E-DVSG), which is a typical but unique topography in the mountainous area. Moreover, the technical requirements for engineers will increase sharply if a 3D slope stability calculation method is adopted. To tackle this issue, pragmatically, a dimension-reduced slope stability analysis method (DR-SSAM) considering the SE of the E-DVSG was proposed and verified by a case study. Specifically, the DR-SSAM obtained the 3D safety factor of the E-DVSG by using the spatial effect curves and the safety factor from the 2D model. The results showed that the method was rational and efficient for engineering applications. Besides, the SE generation mechanism and condition along with its influences resulted from the major factor were addressed thoroughly.

... As a result, the computational cost was reduced significantly. Wang et al. [42] proposed an efficient optimization strategy to obtain the designs which are insensitive to the occurrence of local failure, introducing the von Mises failure criterion to evaluate the patches to be damaged or undamaged. By defining the failure coefficient, the material properties of a given patch was regarded as degenerated if its von Mises stress exceeded an allowable stress. ...

The fail-safe design philosophy aims to achieve safe designs under the different accidental scenarios that structures might undergo throughout their lifespan. The Probability-Damage approach for Fail-Safe Optimization (PDFSO) led to an improvement over the traditional multi-model optimization strategy, since it takes into account the available data on the probability of occurrence of each partial collapse and allows the designer to assume certain risk over the damaged configurations less likely to occur. However, that methodology can be improved considering the inherent uncertainty in some random parameters such as loads or material properties. The objective of this research is to formulate a PDFSO approach including this randomness and therefore, obtain more reliable designs. Two application examples were considered: a 2D truss structure with stress constraints as well as the tail section of an aircraft fuselage with stress and buckling constraints.

... The work by [8] uses the same approach as [7], but instead minimizes the volume subject to displacement constraints for each of the damage scenarios, and thus it is not a min-max problem. The recent and interesting work by [9] also subdivides the design region in patches as in [7], but it incorporates stress constraints for the damage scenarios (i.e., for the patches). To achieve this, the stiffness of the patch is degraded as a function of the ratio of the maximum stress in the patch to the specified stress limit, and the structural performance is subsequently given by the compliance of the structure (this is essentially akin to the approach advanced by [10] for stress-based topology optimization). ...

This work introduces a topology optimization technique for the design of fail-safe structures made of geometric components. Specifically, the structure is made of the union of bar or plate geometric primitives. The geometry projection method is employed to smoothly map the geometric parameters that describe the primitives onto a continuous density field on the design region. As in conventional topology optimization techniques, this density field is subsequently used to define an ersatz material and perform the structural and sensitivity analyses on a non-body fitted mesh, thus circumventing the need for re-meshing with a body-fitted mesh upon design changes. In the proposed fail-safe design methodology, the performance of the structure is evaluated upon removal of each individual geometric component. Since the number of analyses required is proportional to the number of geometric components and independent from the mesh, the proposed methodology is significantly more efficient than density-based techniques for fail-safe design. Numerical examples of minimization of the maximum compliance for all component removal scenarios are presented to demonstrate the method.

... Hz). It is verified that the DAR and NF are efficiently improved by the Taguchi [36]. Therefore, the smaller the stress is preferred when designing the PA. ...

The Taguchi method (TM) and TM-combined methods (e.g., TM fuzzy method (TM-Fuzzy), TM data envelopment analysis (TM-DEA), TM coupled with grey relational analysis (TM-GRA), etc.) have been proven to be effective to achieve robust design performance. However, the efficient strategy to identify the significant design parameters, and the link between different quality characteristics and design parameters have not been fully studied. To fill the gaps, a combined method, i.e., TM-Response Surface Methodology (RSM)-Desirability Function (DF), (TM-RSM-DF), was proposed. The significance of the TM-RSM-DF method is able to address the relationship between design parameters and quality characteristics, which facilitated to sort out the most significant design parameters precisely; Besides, it also provided a robust strategy to optimize the multiple quality characteristics of the precision product. A design process of a precision amplification (PA) was taken as an example to prove the effectiveness of the TM-RSM-DF. The results showed that the TM-RSM-DF method improved the estimation performance of displacement amplification ratio (DAR) and natural frequency (NF) by 6.148% and 0.537% compared with the initial desired design. Besides, compared with others, the TM-RSM-DF method reduced the stress amount about 16% and had the lowest DAR and NF errors of 2.882% and 1.305%, respectively. Overall, the proposed TM-RSM-DF method outperformed the TM-DEA, TM-GRA, and TM-Fuzzy in the robustness of the PA design.

... During the model analysis, three-dimensional stresses and strains are developed in several directions on each part of the robot. A usual way of expressing these multidirectional tensions is to summarize them to an equivalent tension, also called Von-Mises stress [10], as shown in figure 3. ...

... A number of topology optimization algorithms have been proposed based on different strategies: homogenization of microstructures [4], using elemental densities as design variables [5], evolutionary approaches [6], topological derivative [7], level-set (LS) [8,9], phase field [10], moving morphable component (MMC) [11], moving morphable void (MMV) [12], ele-mental volume fractions [13], and using the floating projection [14,15]. In recent years, these topology optimization approaches have been applied in a wide range of distinct engineering problems, including frequency responses [16,17], stress problems [18,19], convection problems [20][21][22], structural failure problems [23,24], large-scale problems [25][26][27], nanophotonics [28], metamaterial design [29], and manufacturing oriented methods [30][31][32][33][34] have been presented in recent years. ...

Element-based topology optimization algorithms capable of generating smooth boundaries have drawn serious attention given the significance of accurate boundary information in engineering applications. The basic framework of a new element-based continuum algorithm is proposed in this paper. This algorithm is based on a smooth-edged material distribution strategy that uses solid/void grid points assigned to each element. Named Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT), the algorithm uses elemental volume fractions which depend on the densities of grid points in the Finite Element Analysis (FEA) model rather than elemental densities. Several numerical examples are studied to demonstrate the application and effectiveness of SEMDOT. In these examples, SEMDOT proved to be capable of obtaining optimized topologies with smooth and clear boundaries showing better or comparable performance compared to other topology optimization methods. Through these examples, first, the advantages of using the Heaviside smooth function are discussed in comparison to the Heaviside step function. Then, the benefits of introducing multiple filtering steps in this algorithm are shown. Finally, comparisons are conducted to exhibit the differences between SEMDOT and some well-established element-based algorithms. The validation of the sensitivity analysis method adopted in SEMDOT is conducted using a typical compliant mechanism design case. In addition, this paper provides the Matlab code of SEMDOT for educational and academic purposes.

... Stolpe (2019) focused on fail-safe truss topology optimization, where the problem is modeled as convex conic optimization problems by enumerating all possible damage scenarios. Wang et al. (2020) proposed efficient robust fail-safe topological designs based on the von Mises stress. ...

Fail-safe design often leads to oversized structures as the design must be able to behave properly for the different accidental situations that could undergo throughout their lifespan. The existing research about structural optimization applied to fail-safe design does not contemplate the idea of having accidental situations with different probabilities of occurrence. This means that the structure must satisfy the design constraints in the damaged structure regardless of whether one accidental situation is much more likely than another. The objective of this research is to formulate a new optimization strategy to get fail-safe structures with minimum weight taking into account the available data about the probability of occurrence associated to each partial collapse. A multi-model probabilistic optimization problem is defined and applied to two structures: a 2D truss structure with stress constraints as well as the tail section of an aircraft fuselage with stress and buckling constraints.

... A number of topology optimization algorithms have been proposed based on different strategies: homogenization of microstructures [4], using elemental densities as design variables [5], evolutionary approaches [6], topological derivative [7], level-set (LS) [8,9], phase field [10], moving morphable component (MMC) [11], moving morphable void (MMV) [12], elemental volume fractions [13], and using the floating projection [14,15]. In recent years, these topology optimization approaches have been applied in a wide range of distinct engineering problems, including frequency responses [16,17], stress problems [18,19], convection problems [20,21,22], structural failure problems [23,24], large-scale problems [25,26,27], nanophotonics [28], metamaterial design [29], and manufacturing oriented methods [30,31,32,33,34] have been presented in recent years. ...

Element-based topology optimization algorithms capable of generating smooth boundaries have drawn serious attention given the significance of accurate boundary information in engineering applications. The basic framework of a new element-based continuum algorithm is proposed in this paper. This algorithm is based on a smooth-edged material distribution strategy that uses solid/void grid points assigned to each element. Named Smooth-Edged Material Distribution for Optimizing Topology (SEMDOT), the algorithm uses elemental volume fractions which depend on the densities of grid points in the Finite Element Analysis (FEA) model rather than elemental densities. Several numerical examples are studied to demonstrate the application and effectiveness of SEMDOT. In these examples, SEMDOT proved to be capable of obtaining optimized topologies with smooth and clear boundaries showing better or comparable performance compared to other topology optimization methods. Through these examples, first, the advantages of using the Heaviside smooth function are discussed in comparison to the Heaviside step function. Then, the benefits of introducing multiple filtering steps in this algorithm are shown. Finally, comparisons are conducted to exhibit the differences between SEMDOT and some well-established element-based algorithms. The validation of the sensitivity analysis method adopted in SEMDOT is conducted using a typical compliant mechanism design case. In addition, this paper provides the Matlab code of SEMDOT for educational and academic purposes.

This paper presents a methodology to optimize the cable system in cable-stayed bridges, whose main novelty is to take into account the accidental breakage of one cable within the design process. To this end, a multi-model optimization strategy is proposed by establishing design constraints on both the intact and damaged models. The dynamic effect of cable breakage is accounted for in the damaged models by the application of impact loads at the tower and deck anchorages. The objective function is to minimize the steel volume in the cable system by varying the cable anchor positions on the deck, the number of cables, the cross sectional areas and prestressing forces. This approach is applied to the Queensferry Crossing Bridge, the longest three-tower cable-stayed bridge in the world and also the largest with crossing cables in the central spans. The fail-safe optimization of the cable system leads to a different layout than the optimum design without considering cable breakage, with more cables and smaller areas, having a minimum penalty in steel volume.

The design philosophy of fail-safe structures was first proposed in the aerospace industry to provide redundant load paths as back-ups when local damage happens. Most fail-safe topology optimization methods paid more attention to minimizing compliance of the worst failure case. However, the stress concentration due to local failure may lead to secondary damage and further destroy the structure. In the current work, the von Mises stress of damaged structures is considered as the optimization objective, to alleviate the stress concentration caused by possible local failures. Two sorts of topology optimization objectives are investigated: (1) the worst-case formulation; (2) the mean-performance formulation. To avoid the ‘singularity’ problem, the stress is penalized through the RAMP interpolation scheme. The Kreisselmeier-Steinhauser (KS) aggregation function is used to approximate the global stress level. Concerning the highly nonlinear stress behavior, the Method of Moving Asymptotes (MMA) solver is adopted. Finally, the benefits and drawbacks of these two objective functions are systematically compared and discussed through several numerical examples.

The aim of this paper is to investigate the mechanical effects of three different core designs of solid plate, honeycomb, and dimple surface sandwich panel subjected to static and fatigue loading conditions. The performance of sandwich panel is important and dependent on the main core material and design. The higher density reduction on core structure of sandwich panel, made it vulnerable to the catastrophic failure such as delamination and reduced its bonding strength under varies cyclic loading condition. Four points bending simulation was carried out to obtain the mechanical behaviour under static condition in terms of von Mises stress, total deformation, and shear stress. Under fatigue condition, the Gerber mean stress theory has been used with a load ratio of − 1 to investigate the potential of debonding. The results indicate that the mechanical performance and bonding strength of sandwich panel with dimpled surface core achieved better results in terms of stress distribution and permanent deformation under static and fatigue condition. Moreover, the uses of magnesium alloy also showed good potential for its use as a core material and alternative candidate of common core material such as aluminium alloy. The modification on the surface of core panel improved the bonding strength and provide good future in critical defence application such as protective panel for armour vehicle. In addition, this simplified finite element analysis provides promising technique to estimate the delamination phenomena in terms of stress distribution on bonding area of sandwich panel without using complex fracture modelling.

In the current work, a fail-safe optimization of beam structures is carried out. This approach may provide an insight into the robustness of lattice structures. The use of beam elements allows a commonly used engineering approach for obtaining a fail-safe design to be applied. This consists of removing one beam element at a time and optimizing the remaining structure. At the end of the process, the maximum beam radii are used for the final design. This approach is computationally extremely expensive for lattice structures, as it requires one optimization per removed beam. In our contribution, we show that the design obtained from this approach does not actually achieve the desired fail-safe behaviour. We therefore apply a multi-model approach in which the fail-safe requirement is an optimization constraint. This is still computationally demanding and therefore, methods for reducing the number of failure cases to be considered within the optimization are discussed. Furthermore, the p-norm is applied to the stress constraints to reduce the computational effort for the gradient calculation. Reduction of failure cases and use of the p-norm have opposite effects on the conservatism of the result and therefore compensate each other to some extent.
Highlights Stress constrained optimization of beam structures. Evaluation of different approaches for fail-safe design. Strategies for efficiency improvement (reducing failure cases/p-norm).

This paper develops a non-probabilistic reliability-based topology optimization (NRBTO) framework for continuum structures under multi-dimensional convex uncertainties. Combined with the solid isotropic material with penalization (SIMP) model and the set-theoretical convex method, the uncertainty quantification (UQ) analysis is firstly conducted to obtain mathematical approximations and boundary laws of considered displacement responses. By normalization treatment of the limit-state function, a new quantified measure of the non-probabilistic reliability is then defined and further deduced by the principle of the hyper-volume ratio. For circumventing optimization difficulties arising from large-scale design variables, the adjoint vector scheme for sensitivity analysis of the reliability index with respect to design variables are discussed as well. Numerical applications eventually illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques.

As a typical form of material imperfection, cracks generally cannot be avoided and are critical for load bearing capability and integrity of engineering structures. This paper presents a topology optimization method for generating structural layouts that are insensitive/sensitive as required to initial cracks at specified locations. Based on the linear elastic fracture mechanics model (LEFM), the stress intensity of initial cracks in the structure is analyzed by using singularity finite elements positioned at the crack tip to describe the near-tip stress field. In the topology optimization formulation, the J integral, as a criterion for predicting crack opening under certain loading and boundary conditions, is introduced into the objective function to be minimized or maximized. In this context, the adjoint variable sensitivity analysis scheme is derived, which enables the optimization problem to be solved with a gradient-based algorithm. Numerical examples are given to demonstrate effectiveness of the proposed method on generating structures with desired overall stiffness and fracture strength property. This method provides an applicable framework incorporating linear fracture mechanics criteria into topology optimization for conceptual design of crack insensitive or easily detachable structures for particular applications.

Fail-safe robustness of critical load carrying structures is an important design philosophy for aerospace industry. The basic idea is that a structure should be designed to survive normal loading conditions when partial damage occurred. Such damage is quantified as complete failure of a structural member, or a partial damage of a larger structural part. In the context of topology optimization fail-safe consideration was first proposed by Jansen et al. Struct Multidiscip Optim 49(4):657–666, (2014). While their approach captures the essence of fail-safe requirement, it has two major shortcomings: (1) it involves analysis of a very large number of FEA models at the scale equal to the number of elements; (2) failure was introduced in generic terms and therefore the fundamental aspects of failure test of discrete members was not discussed. This paper aims at establishing a rigorous framework for fail-safe topology optimization of general 3D structures, with the goal to develop a computationally viable solution for industrial applications. We demonstrate the effectiveness of the proposed approach on several examples including a 3D example with over three hundred thousand elements.

In this paper, we propose a new method for topology optimization with local stress constraints. In this method, material in which a stress constraint is violated is considered as damaged. Since damaged material will contribute less to the overall performance of the structure, the optimizer will promote a design with a minimal amount of damaged material. We tested the method on several benchmark problems, and the results show that the method is a viable alternative for conventional stress-based approaches based on constraint relaxation followed by constraint aggregation.

This paper presents a new topology optimization approach based on the so-called Moving Morphable Components (MMC) solution framework. The proposed method improves several weaknesses of the previous approach (e.g., Guo et al. in J Appl Mech 81:081009, 2014a) in the sense that it can not only allow for components with variable thicknesses but also enhance the numerical solution efficiency substantially. This is achieved by constructing the topological description functions of the components appropriately, and utilizing the ersatz material model through projecting the topological description functions of the components. Numerical examples demonstrate the effectiveness of the proposed approach. In order to help readers understand the essential features of this approach, a 188 line Matlab implementation of this approach is also provided.

This paper reports an efficient approach for uncertain topology optimization in which the uncertain optimization problem is equivalent to that of solving a deterministic topology optimization problem with multiple load cases. Probabilistic and fuzzy property of the directional uncertainty of the applied loads is considered in the topology optimization; the cloud model is employed to describe that property which can also take the correlations of the probability and fuzziness into account. Convergent and mesh-independent bi-directional evolutionary structural optimization (BESO) algorithms are utilized to obtain the final optimal solution. The proposed method is suitable for linear elastic problems with uncertain applied loads, subject to volume constraint. Several numerical examples are presented to demonstrate the capability and effectiveness of the proposed approach. In-depth discussions are also given on the effects of considering the probability and fuzziness of the directions of the applied loads on the final layout.

This paper presents a 100-line Python code for general 3D topology optimization. The code adopts the Abaqus Scripting Interface that provides convenient access to advanced finite element analysis (FEA). It is developed for the compliance minimization with a volume constraint using the Bi-directional Evolutionary Structural Optimization (BESO) method. The source code is composed of a main program controlling the iterative procedure and five independent functions realising input model preparation, FEA, mesh-independent filter and BESO algorithm. The code reads the initial design from a model database (.cae file) that can be of arbitrary 3D geometries generated in Abaqus/CAE or converted from various widely used CAD modelling packages. This well-structured code can be conveniently extended to various other topology optimization problems. As examples of easy modifications to the code, extensions to multiple load cases and nonlinearities are presented. This code is intended for educational purposes and would be useful for researchers and students in the topology optimization field. With further extensions, the code could solve sophisticated 3D conceptual design problems in structural engineering, mechanical engineering and architecture practice. The complete code is given in the appendix section and can also be downloaded from the website: www.rmit.edu.au/research/cism/.

There is a general interest to consider stress constraints in topology optimization of continuum structures. By their very nature stress constraints are local constraints which result in large scale optimization problems that are often expensive to solve. Here in order to reduce the computing effort we explore an alternative technique based on equivalent global (that is integrated) constraints. We define two global stress constraints based on the p-norm and p-mean of the e-relaxed overall stress criteria in the finite elements. We present a new formulation of the e-relaxation technique which is better suited to topology optimization of continuum structures and which makes the relaxation process automatic. The "p-mean" and "p-norm" functions bound by lower and upper value the maximum value of the e-relaxed overall stress criterion. Based on numerical experiments this study compares the global and the local constraint formulations. Even if the use of integrated constraints leads a reduction of the computing time by one or two orders of magnitude, they definitely give a weaker control of local stress level. This sometimes can lead to solutions that are a bit different.

Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

The paper presents a compact Matlab implementation of a topology optimization code for compliance minimization of statically
loaded structures. The total number of Matlab input lines is 99 including optimizer and Finite Element subroutine. The 99
lines are divided into 36 lines for the main program, 12 lines for the Optimality Criteria based optimizer, 16 lines for a
mesh-independency filter and 35 lines for the finite element code. In fact, excluding comment lines and lines associated with
output and finite element analysis, it is shown that only 49 Matlab input lines are required for solving a well-posed topology
optimization problem. By adding three additional lines, the program can solve problems with multiple load cases. The code
is intended for educational purposes. The complete Matlab code is given in the Appendix and can be down-loaded from the web-site
http://www.topopt.dtu.dk.

Previous research on topology optimization focussed primarily on global structural behaviour such as stiffness and frequencies. However, to obtain a true optimum design of a vehicle structure, stresses must be considered. The major difficulties in stress based topology optimization problems are two-fold. First, a large number of constraints must be considered, since unlike stiffness, stress is a local quantity. This problem increases the computational complexity of both the optimization and sensitivity analysis associated with the conventional topology optimization problem. The other difficulty is that since stress is highly nonlinear with respect to design variables, the move limit is essential for convergence in the optimization process. In this research, global stress functions are used to approximate local stresses. The density method is employed for solving the topology optimization problems. Three numerical examples are used for this investigation. The results show that a minimum stress design can be achieved and that a maximum stiffness design is not necessarily equivalent to a minimum stress design.

This paper presents an improved algorithm for the bi-directional evolutionary structural optimization (BESO) method for topology optimization problems. The elemental sensitivity numbers are calculated from finite element analysis and then converted to the nodal sensitivity numbers in the design domain. A mesh-independency filter using nodal variables is introduced to determine the addition of elements and eliminate unnecessary structural details below a certain length scale in the design. To further enhance the convergence of the optimization process, the accuracy of elemental sensitivity numbers is improved by its historical information. The new approach is demonstrated by solving several compliance minimization problems and compared with the solid isotropic material with penalization (SIMP) method. Results show the effectiveness of the new BESO method in obtaining convergent and mesh-independent solutions.

Aimed at the topology optimization of continuum structures considering the fail-safe principle, for the purpose of overcoming the shortcomings of the topologies obtained by the traditional topology optimization being too sensitive to local damages for the lack of reasonable redundancy components, the fail-safe design is achieved. At first, four concepts are clarified: the structural local failure mode, the structural local failure region, structural failure case, and the pre-estimation distribution of structural failure cases. Secondly, based on the ICM (independent continuous mapping) method, a minimizing structural volume model with structural performance constraints is established for the fail-safe topology optimization problems of continuum structures. While establishing the objective function, minimizing the maximum of structural volumes of all structural failure cases is converted into minimizing the structural volume of the ground structure without failure regions. Therefore, the difficulty of dealing with multi-objective optimization is avoided. While establishing the approximation functions of constraints, mechanical property constraints of all of the structural failure cases are taken into account. The problems with a single load case or multi-load cases can be solved by the presented model. At last, optimization problems with displacement constraints are taken as examples. The optimization model is established and the solution method is also presented. Some examples with displacement constraints under a single load case or multi-load case are presented to verify the validity of this method. The results show that the optimal topologies obtained by this method are more complex and has a greater volume than that obtained by the topology optimization without fail-safe. Namely, optimal topologies have more redundancy, which is the result of considering the fail-safe principle. The proposed research is an important progress for the design of vehicles serving for aviation, aerospace, water or land fields and other engineering structures undergoing accident damages, war wounds or terrorist attacks. © 2018, Editorial Office of Chinese Journal of Theoretical and Applied Mechanics. All right reserved.

Considering fail-safe requirements in a topology optimization, where the location of the damage is unknown in advance, leads to a high number of potential damage scenarios to be calculated. Since this is driving the overall calculation time of the optimization, a reduction of the considered damage cases is desirable. In this paper, two strategies to achieve a significant reduction of damage cases are shown: An active-set strategy as an extension to the simplified local damage model first introduced by Jansen et al. as well as a newly developed load-path based algorithm for the placement of damage zones. Available at: https://link.springer.com/chapter/10.1007%2F978-3-319-97773-7_19

This paper presents an evolutionary structural optimization method of designing continuum structures with clear and smooth boundaries for stress minimization. The stress optimization problem is formulated with the P-norm function based on the distortion strain energy density, so as to avoid the stress relaxation and the local nature of the maximum stress. In order to obtain a smooth topology, the surrogate design variables on the volume fraction of elements are defined based on the proportion of solid and void points within an element. Based on sensitivity analysis, topology optimization evolves the structure by gradually decreasing the structural volume to the optimized one with the prescribed volume. 2D and 3D numerical examples are presented and discussed to demonstrate the effectiveness of the proposed topology optimization method for minimizing the maximum stress of continuum structures and alleviating stress concentration.

Compliant mechanism-based microdevices with multiple inputs and multiple outputs have a wide range of applications in precision mechanics, e.g., cell manipulation, electronic microscopy and MEMS (Micro-Electro-Mechanical Systems). In designing this kind of microdevice, the movement coupling among the microdevices becomes critical because many inputs and outputs are involved. This paper presents a systematic method for designing fully decoupled compliant mechanisms with multiple degrees of freedom by using topology optimization. An optimization formulation is posed by considering both output coupling and input coupling issues to achieve fully decoupled motion. The SIMP (Solid Isotropic Material with Penalization) and MMA (Method of Moving Asymptotes) methods are adopted to identify the optimized material distribution in the design domain. Several numerical examples are presented to demonstrate the validity of the proposed method.

This paper proposes two effective constraint schemes to address the stress-constrained topology optimization of continuum structures. By considering the maximum stress measure in the global and local forms, respectively, the STM (stability transformation method)-based stress correction scheme and the violated set enhanced stress measure are developed to tackle the challenging issues from numerous local stress constraints and highly nonlinear stress behavior. Particularly, a stress aggregation function is involved in the design sensitivity analysis. Moreover, the nodal variable based SIMP method and adjoint sensitivity analysis are employed to solve the optimum topological design problems with two different optimization formulations. Finally, several representative examples demonstrate the validity of the present approach. It is also indicated that the numerical performance of the stress aggregation function is closely related to the problem formulation of topology optimization. The STM-based stress correction scheme is appropriate to the material volume minimization design, while the violated set enhanced stress measure is suitable for the mean compliance minimization design. Meanwhile, the proposed optimization approach can handle the stress-constrained topology optimization with easy implementation, low computational cost and stable convergence.

This paper presents a compact and efficient 88-line MATLAB code for the parameterized level set method based topology optimization using radial basis functions (RBFs), which is applied to minimize the compliance of a two-dimensional linear elastic structure. This parameterized level set method using radial basis functions can maintain a relatively smooth level set function with an approximate re-initialization scheme during the optimization process. It also has less dependency on initial designs due to its capability in nucleation of new holes inside the material domain. The MATLAB code and simple modifications are explained in detail with numerical examples. The 88-line code included in the appendix is intended for educational purposes.

Topology optimization considering stress constraints has received ever-increasing attention in recent years for both of its academic challenges and great potential in real-world engineering applications. Traditionally, stress-constrained topology optimization problems are solved with approaches where structural geometry/topology is represented in an implicit way. This treatment, however, would lead to problems such as the existence of singular optima, the risk of low accuracy of stress computation, and the lack of direct link between optimized results and computer-aided design/engineering (CAD/CAE) systems. With the aim of resolving the aforementioned issues straightforwardly, a Moving Morphable Void (MMV)-based approach is proposed in the present study. Compared with existing approaches, the distinctive advantage of the proposed approach is that the structural geometry/topology is described in a completely explicit way. This feature provides the possibility of obtaining optimized designs with crisp and explicitly parameterized boundaries using much fewer numbers of degrees of freedom for finite element analysis and design variables for optimization, respectively. Several numerical examples provided demonstrate the effectiveness and advantages of the proposed approach.

This work proposes an evolutionary topology optimization method for stress minimization design using the bi-directional evolutionary structural optimization (BESO) method. The discrete nature of the BESO method avoids naturally the well-known "singularity" problem in density-based methods with degenerated materials. The p-norm stress aggregation scheme is adopted for the measure of global stress level. A computationally efficient sensitivity number formulation is derived from the adjoint sensitivity of the global stress measure. With regard to the highly nonlinear stress behavior, both sensitivity numbers and topology variables are filtered to stabilize the optimization procedure; meanwhile, the filtered sensitivity numbers are further stabilized with their historical information. The method has been shown efficient, practical and easy-to-implement through a series of 2D and 3D benchmark designs.

Few researches have paid attention to designing structures in consideration of the uncertainties in
the loading locations, which may significantly influence the structural performances. In this work,
the cloud models are employed to depict the uncertainties in the loading locations. A robust
algorithm is developed in context of minimizing the expectation of the structural compliance,
while conforming to a material volume constraint. To guarantee optimal solutions, sufficient cloud
drops are used which in turn leads to the low efficiency. An innovative strategy is then
implemented to enormously improve the computational efficiency. A modified soft-kill
Bi-directional Evolutionary Structural Optimization (BESO) method using the derived sensitivity
numbers is employed to output the robust novel configurations. Several numerical examples are
presented to demonstrate the effectiveness and the efficiency of the proposed algorithm.

Three dimensional (3D) topology optimization problems always involve huge numbers of Degrees of Freedom (DOFs) in finite element analysis (FEA) and design variables in numerical optimization, respectively. This will inevitably lead to large computational efforts in the solution process. In the present paper, an efficient and explicit topology optimization approach which can reduce not only the number of design variables but also the number of degrees of freedom in FEA is proposed based on the Moving Morphable Voids (MMVs) solution framework. This is achieved by introducing a set of geometry parameters (e.g., control points of B-spline surfaces) to describe the boundary of a structure explicitly and removing the unnecessary DOFs from the FE model at every step of numerical optimization. Numerical examples demonstrate that the proposed approach does can overcome the bottleneck problems associated with a 3D topology optimization problem in a straightforward way and enhance the solution efficiency significantly.

Thin membrane structures would experience wrinkling due to local buckling deformation when compressive stresses are induced in some regions. Using the stress criterion for membranes in wrinkled and taut states, this paper proposed a new stress-based topology optimization methodology to seek the optimal wrinkle-free design of macro-scale thin membrane structures under stretching. Based on the continuum model and linearly elastic assumption in the taut state, the optimization problem is defined as to maximize the structural stiffness under membrane area and principal stress constraints. In order to make the problem computationally tractable, the stress constraints are reformulated into equivalent ones and relaxed by a cosine-type relaxation scheme. The reformulated optimization problem is solved by a standard gradient-based algorithm with the adjoint-variable sensitivity analysis. Several examples with post-bulking simulations and experimental tests are given to demonstrate the effectiveness of the proposed optimization model for eliminating stress-related wrinkles in the novel design of thin membrane structures.

A moving bounds strategy is proposed to implement simultaneous shape optimization of curved shell structures and openings. Design variables related to the hole shape are constrained in a planar reference domain by the moving bounds whose values are adaptively updated as functions of design variables related to the surface by an arc-length rule. It is shown that this strategy is essential not only to ensure the geometric consistence in the simultaneous design process but also to hold the shape-preserving of the mapped FE mesh from reference domains. Numerical results are presented to validate the proposed method.

This paper proposes a new topology optimization algorithm based on the bi-directional evolutionary structural optimization (BESO) method to design photonic crystals with broad all-angle negative refraction (AANR) frequency range. The photonic crystals are assumed to be two-dimensional periodical structures, which consist of dielectric materials and air. The conditions for the occurrence of AANR are identified and the design objective is to enlarge the AANR frequency range. The BESO algorithm is proposed based on finite element analysis for band diagrams of photonic crystals and the derived sensitivity numbers. Starting from a simple initial design without any AANR, BESO gradually re-distributes the dielectric materials within the periodical unit cell so that the AANR property emerges and its frequency range is enlarged accordingly. The numerical results show that the proposed BESO algorithm can effectively obtain AANR photonic crystals with novel patterns. The effects of dielectric permittivity contrast of two constituent materials, mesh-refinement and filter are discussed.

Stress-related problems have not been given the same attention as the minimum compliance topological optimization
problem in the literature. Continuum structural topological optimization with stress constraints is
of wide engineering application prospect, in which there still are many problems to solve, such as the stress
concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization
method of continuum structures with the stress constraints and the objective function being the
structural volume has been presented in this paper. To solve the stress concentration issue, an approximate
stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient
constraint is constructed for the optimized structure under the r�th load case. To obtain stable convergent
series solutions and enhance the control on the stress level, two p-norm global stress constraint functions
with different indexes are adopted, and some weighting p-norm global stress constraint functions are introduced
for any load case. And an equivalent topological optimization model with reduced stress constraints is
constructed,being incorporated with the rational approximation for material properties, an active constraint
technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve
the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed,
based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level
optimization problem with artificial variables and many possible non-active design variables is proposed by
adopting an inequality constrained nonlinear programming method with simple trust regions, based on the
primal-dual theory, in which the non-smooth expressions of the design variable solutions are reformulated
as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a twolevel
optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed
to deal with the aggregation constraints that always are of loose constraint (non active constraint) features
in the conventional structural optimization method. A novel structural topological optimization method with
stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed
method is feasible and very effective.

In the present work, we intend to demonstrate how to do topology optimization in an explicit and geometrical way. To this end, a new computational framework for structural topology optimization based on the concept of moving morphable components is proposed. Compared with the traditional pixel or node point-based solution framework, the proposed solution paradigm can incorporate more geometry and mechanical information into topology optimization directly and therefore render the solution process more flexibility. It also has the great potential to reduce the computational burden associated with topology optimization substantially. Some representative examples are presented to illustrate the effectiveness of the proposed approach.

Topology optimization of mechanical structures often leads to efficient designs which resemble statically determinate structures. These economical structures are especially vulnerable to local loss of stiffness due to material failure. This paper therefore addresses local failure of continuum structures in topology optimization in order to design fail-safe structures which remain operable in a damaged state.
A simplified model for local failure in continuum structures is adopted in the robust approach. The complex phenomenon of local failure is modeled by removal of material stiffness in patches with a fixed shape. The damage scenarios are taken into account by means of a minimax formulation of the optimization problem which minimizes the worst case performance.
The detrimental influence of local failure on the nominal design is demonstrated in two representative examples: a cantilever beam optimized for minimum compliance and a compliant mechanism. The robust approach is applied successfully in the design of fail-safe alternatives for the structures in these examples.

The basic idea presented herein is that a systematic method for optimal structural design can be developed that accounts for probable future damage to the structure. A general mathematical model for the damage-tolerant structure design problem is defined. Design sensitivity analysis and procedures for treatment of a large number of constraints for the problem are presented. As a practical design example, damage-tolerant design of an open truss helicopter tail-boom structure is considered. Optimal designs for several cases of the tail-boom are presented. It is shown that considerable design variation is necessary to achieve the damage-tolerant design objective. It is also shown that if the structure is optimized without consideration of damage, the structure will fail catastrophically when any damage occurs to the structure. © 1980 American Institute of Aeronautics and Astronautics, Inc., All rights reserved.

The effectiveness and efficiency of the Bi-directional Evolutionary Structural Optimization (BESO) method has been demonstrated on the minimization compliance problem with fixed external loads. This paper considers the minimization of mean compliance for continuum structure subjected to design-dependent self-weight loads. Due to the non-monotonous behaviour for this type of the optimization problems, the extended BESO method using discrete design variables has its difficulty to obtain convergent solutions for such problems. In this paper, a new BESO method is developed based on the sensitivity number computation utilizing an alternative material interpolation scheme. A number of examples are presented to demonstrate the capabilities of the proposed method for achieving convergent optimal solutions for structures including self-weight loads.

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.

A simple evolutionary procedure is proposed for shape and layout optimization of structures. During the evolution process low stressed material is progressively eliminated from the structure. Various examples are presented to illustrate the optimum structural shapes and layouts achieved by such a procedure.

Iterative and gradient methods used in constrained optimization problems can lead to local minima which can not be improved. The reasons for this anomaly are investigated and methods leading to the global minimum are suggested.

We introduce an extension of current technologies for topology optimization of continuum structures which allows for treating local stress criteria. We first consider relevant stress criteria for porous composite materials, initially by studying the stress states of the so-called rank 2 layered materials. Then, on the basis of the theoretical study of the rank 2 microstructures, we propose an empirical model that extends the power penalized stiffness model (also called SIMP for Solid Isotropic Microstructure with Penalization for inter-mediate densities). In a second part, solution aspects of topology problems are considered. To deal with the so-called ‘singularity’ phenomenon of stress constraints in topology design, an ϵ-constraint relaxation of the stress constraints is used. We describe the mathematical programming approach that is used to solve the numerical optimization problems, and show results for a number of example applications. © 1998 John Wiley & Sons, Ltd.

This paper presents a so-called -relaxed approach for structural topology optimization problems of discrete structures. The distinctive feature of this new approach is that unlike the typical treatment of topology optimization problems based on the ground structure approach, we eliminate the singular optima from the problem formulation and thus unify the sizing and topology optimization within the same framework. As a result, numerical methods developed for sizing optimization problems can be applied directly to the solution of topology optimization problems without any further treatment. The application of the proposed approach and its effectiveness are illustrated with several numerical examples.

The paper deals with the imposition of local stress constraints in topology optimization. The aim of the work is to analyze
the performances of an alternative methodology to the ε-relaxation introduced in Cheng and Guo (Struct Optim 13:258–266, 1997), which handles the well-known stress singularity problem. The proposed methodology consists in introducing, in the SIMP
law used to apply stress constraints, suitable penalty exponents that are different from those that interpolate stiffness
parameters. The approach is similar to the classical one because its main effect is to produce a relaxation of the stress
constraints, but it is different in terms of convergence features. The technique is compared with the classical one in the
context of stress-constrained minimum-weight topology optimization. Firstly, the problem is studied in a modified truss design
framework, where the arising of the singularity phenomenon can be easily shown analytically. Afterwards, the analysis is extended
to its natural context of topology bidimensional problems.

Structural optimization for damage tolerance under various unforeseen damage scenarios is computationally challenging. It
couples non-linear progressive failure analysis with sampling based stochastic analysis of random damages. This work shows
that analysis of damage tolerance depends on specification of damages, and optimizing a structure under one damage specification
can be sensitive to other damages not considered. This work demonstrates the importance of understanding the underlying mechanics
that provide damage tolerance in order to develop computationally efficient methods for optimization. Understanding features
of load distributions in damage tolerant structures can result in efficient methods for optimization. To understand and identify
these features, one compared and contrasted designs with varying degree of damage tolerance. A method to describe load distributions
based on principal component analysis is presented. It is found that the number of dominant eigenvalues of principal components
in a structure correlates with the number of alternate paths.
KeywordsDamage tolerant–Truss structure–Load redistribution–Load path

This paper presents a new approach to structural topology optimization. We represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension. Such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure. Furthermore, a well-founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints. The result is a 3D topology optimization technique that demonstrates outstanding flexibility of handling topological changes, fidelity of boundary representation and degree of automation. We have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods. The benefit and the advantages of the proposed method are illustrated with several 2D examples that are widely used in the recent literature of topology optimization, especially in the homogenization based methods.

Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, isotropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

The need for structural safety under a variety of loading and accident conditions has focused attention on redundancy, ductility and reliability of structural systems. The concepts of component reserve strength and system residual strength, system reliability and system residual reliability and their application are described. Several different structural configuration examples are illustrated in which component sizes are optimized. Design models for extreme loading and accident conditions for both brittle and ductile models are developed. System design methods are recommended.

- W Zhang
- J Yuan
- J Zhang
- X Guo

W. Zhang, J. Yuan, J. Zhang, X. Guo, A new topology optimization approach based
on Moving Morphable Components (MMC) and the ersatz material model, Struct.
Multidiscip. Optim. 53 (6) (2016) 1243-1260.

A 99 line topology optimization code written in Matlab

- Sigmund

Optimal shape design as a material distribution problem

- Bendsøe