Topology Optimisation Methods for Satellite Structures: Comparison
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Please note this is a comparison between Version 2 by Peter Tang and Version 1 by Arturo Benjamin Hurtado-Perez.

Launching satellites into the Earth’s orbit is a critical area of research, and very demanding satellite services increase exponentially as modern society takes shape. At the same time, the costs of developing and launching satellite missions with shorter development times increase the requirements of novel approaches in the several engineering areas required to build, test, launch, and operate satellites in the Earth’s orbit, as well as in orbits around other celestial bodies. One area with the potential to save launching costs is that of the structural integrity of satellites, particularly in the launching phase where the largest vibrations due to the rocket motion and subsequent stresses could impact the survival ability of the satellite. To address this problem, two important areas of engineering join together to provide novel, complete, and competitive solutions: topology optimisation methods and additive manufacturing. Topology optimisation methods are mathematical methods that allow iteratively optimising structures (usually by decreasing mass) while improving some structural properties depending on the application (load capacity, for instance), through the maximisation or minimisation of a uni- or multi-objective function and multiple types of algorithms. This area has been widely active in general and has two main core types of algorithms: continuum methods that modify continuous parameters such as density, and discrete methods that work by adding and deleting material elements in a meshing context. 

  • structural topology optimisation
  • continuum topology
  • density methods
  • aerospace satellite structure

1. Introduction

Within the realm of structural mechanics, there exist diverse methods for optimising mechanical designs. In contrast to size optimisation and shape optimisation, topology optimisation (TO) stands apart. It does not require a nearly optimal starting design to properly work and has the capability to generate optimal solutions when conventional design strategies prove ineffective [19][1]. This might happen due to complex inter-relationships between design parameters and structural responses [9][2]. Applying TO can result in much greater savings of material than size optimisation or shape optimisation [19,20,21,22][1][3][4][5].
TO addresses the question—how can we strategically distribute a specified amount of material within a predefined design space to achieve optimal structural performance, considering defined loads and constraints?
The answer is characterised by its broad applicability, encompassing scenarios involving singular or multiple materials with isotropic or anisotropic behaviour. It extends to both two-dimensional and three-dimensional spaces and involves considerations related to weight, displacement, and stress restrictions, combined with multiple loads of diverse physical nature. Moreover, it encompasses a range of physical phenomena, including mechanical aspects such as stiffness, natural frequency, buckling, and impact, as well as thermo-mechanical and fluid interaction effects. These varied conditions have led to the emergence of a multitude of methods aiming to address as many combinations as possible. The underlying objective of these methods is to provide optimal solutions while maintaining efficiency and computational affordability.
The proposed classification can be observed in Table 1, where the Metaheuristic Algorithms and Metaheuristic Hybrid Algorithms have their own sub-classification, which can be found in Table 2 and Table 3, respectively.
Table 1.
Topology optimisation approaches.

Topology Optimisation Approaches
Metaheuristic Algorithms
in
Table 1
.

Metaheuristic Algorithms

Continuum Approaches
 
Table 3.
Topology optimisation approaches—subclassification within
Metaheuristic Hybrid Algorithms
in
Table 1
.

Metaheuristic Hybrid Algorithms

Discrete Approaches

Diverse Approaches

Ant Colony Algorithm

Electro-Search Algorithm

Simulated Annealing (SA)

Artificial Immune System Algorithm

ESO with GA

BESO with HS

BESO with SA

Homogenisation Method

Evolutionary Structural Optimisation (ESO)

Firefly Algorithm

Machine Learning Applied to TO

BESO with GA

BESO with PSO

Symbiotic Organisms Search Algorithm

SIMP with a GA

Level-Set Method (LSM)

Bacterial Chemotaxis

Moving Morphable Voids (MMV)

Multi-Physics TO

Genetic Algorithms (GAs)

Plasma Generation Algorithm

Phase-Field Method

Soft-Kill Bidirectional ESO (SBESO)

Bat Algorithm

Multi-Scale TO

Solid Isotropic Material with Penalty (SIMP)

Metaheuristic Algorithms

  

Topology Derivative Method

Metaheuristic Hybrid Algorithms

  
Table 2.
Topology optimisation approaches—subclassification within

Harmony Search (HS)

 

Differential Evolution

Particle Swarm (PSO)

2. Topology Optimisation Continuum Approaches

2.1. Beginnings

The first formal paper to address the problem is probably proposed by Michell back in 1904 [23][6]. In this paper, he derived optimal criteria for the minimum weight layout of 2D truss structures.
Then, after some decades, the first general theory of TO was given by Prager and Rozvany in 1977 [24][7] in the form of their “optimal layout theory”. In this work, the optimisation analysis is obtained analytically for grid-type structures.
Exact analytical solutions for TO problems are very difficult to obtain. Even today, there are very few papers that address the problem from such a point of view; the reader might find a thorough way to address the derivation of analytical solutions for TO problems in the recompilation of lectures given at the International Centre for Mechanical Sciences (CISM) [25][8].

2.2. Homogenisation Method

The transition from truss structures to continuum structures undergoing TO began with the seminal paper of Bendsøe and Kikuchi in 1988 [26][9]. Originally, TO was perceived as a challenge involving discrete 0-1 variables, implying a binary design configuration. This particular configuration, often associated with structural compliance, is recognised as being ill-conditioned in nature [27,28,29][10][11][12]. To tackle this issue, Bendsøe and Kikuchi introduced a solution. They proposed re-framing the problem by introducing intricate porous microstructures at a smaller separate scale. This is performed using the homogenisation theory, thereby transforming the problem into one that is mathematically well-defined.
The homogenisation method aims to optimise through density variables that are associated with a specific microstructure model at a separate lower scale. TO is achieved by iteratively modifying the corresponding size variables of each unit cell. It is a difficult method to implement due to mathematical complications, particularly in 3D extensions of method [30][13]. This preseaperrch meant the birth of practical (numerical) TO methods [16,20,31[3][14][15][16],32], and since their first appearance, their use has been extensive. This surge in numerical investigations coincided with revolutionary advances in computing capabilities and the progress made in numerical simulation techniques [21][4].
The original paper of Bendsøe and Kikuchi meant the birth of practical (numerical) TO methods. Since its introduction, the concept of continuum TO has evolved in various ways, giving rise to a wide array of methods that have emerged and progressed over time.

2.3. Solid Isotropic Material with Penalty Method

To circumvent the mathematical complications inherent to homogenisation theory, Bendsøe [33][17] proposed Solid Isotropic Material with Penalty (SIMP) in 1989, another density-based method with a simplified material assumption. By penalising exponentially the isotropic material density in terms of element density variables, it is possible to manipulate structural properties related to density (like structural stiffness). The design domain is discretised by using elastic linear isotropic continuum finite elements (FE), and then the finite element method (FEM) is used to obtain the structural response; analytical sensitivities are then calculated with an adjoint method (numerically solved via the Method of Moving Asymptotes (MMA) [34][18]). A filter is then applied to smooth sensitivities [35][19]. The process occurs in a loop until the required volume constraint is satisfied. The name (SIMP) was first suggested by Zhou and Rozvany in 1991 [36][20]. For its simple conceptualisation and numerical implementation, SIMP is one of the most popular and widespread TO methods [5,9,22,30][2][5][13][21].
Until today, SIMP has maintained its status and continues to develop. Recent advances include SIMP combined with phase-field method [37][22], multi-material TO [38,39,40,41,42,43,44,45][23][24][25][26][27][28][29][30], anisotropic material behaviour [46[31][32],47], dynamic performance and fatigue TO [48[33][34][35][36][37],49,50,51,52], large elastic deformation TO [53][38], additive manufacturing (AM) constrained TO [54[39][40][41][42][43][44],55,56,57,58,59], subtractive manufacture (SM) constrained TO [60][45], and casting constrained TO [61][46]. Some approaches lead to self-weight TO [62][47], nonlinear load cases [63[48][49],64], stress constrained TO [65][50], surface corrosion TO [66][51], buckling considerations [67[52][53],68], non-linear heat conduction [69][54], thermal dissipation [70][55].
Some interesting applications include flexoelectric structures (nano-structures) [71][56] and shape-memory alloys [72][57]. Recent approaches use Iso-Geometric Analysis (IGA) [73,74][58][59]. Others eliminate mesh of the design [75][60].

2.4. Topology Derivative

Another approach that emerged in the 1990s is the utilisation of topology derivatives in the fields of topology and shape optimisation, pioneered by Eschenauer [76][61] in 1994 through the introduction of the bubble-method. The fundamental concept underlying topology derivatives involves predicting the impact of introducing an infinitesimally small hole at any location within the design domain. The predictive information obtained is then employed to create holes. Topology derivative is used in conjunction with a shape optimisation method acting as the mechanism to place new holes that are then manipulated by the latter method. A more recent development is brought by Amstutz [77][62], demonstrates that for some choices of material properties, the topology gradients used in minimising compliance correspond to the standard density gradients as used in the SIMP scheme [16][14]. Amstutz’s research highlighted that both methods yield equivalent outcomes under certain conditions (when the penalisation exponent is set to 𝑝=3, specifically in the context of 2D elastic problems with a Poisson’s ratio of 1/3). An additional aspect is that deriving topology derivatives demands the application of intricate mathematical concepts, as emphasised by Sokołowski and Zochowski [78][63], a point later reiterated in [79][64]; this feature has the effect of prevent widespread of the method and practical applications, instead, one can find the topology derivative method implicit within level-set methods [79,80,81,82,83,84][64][65][66][67][68][69].

2.5. Level-Set Method

Methods up to now track changes in topology in an explicit way. level-set Method (LSM) is a TO method that defines boundary design implicitly. When dealing with the problem of TO, the groundwork for the LSM is initially laid out by Osher and Sethian [85,86][70][71]. This foundational mathematical framework subsequently found practical utility by Sethian [87][72], Wang [88[73][74],89], Allaire [90][75], and Jouve [91][76].
LSM is originally developed to model moving boundaries. Initially, LSM is primarily employed to depict the progression of interfaces in scenarios like multi-phase flows and image segmentation. However, the application of LSM to TO commenced in the year 2000 with Sethian’s work and has advanced rapidly since then.
The core principle of the LSM involves expressing the interface (the demarcation between material and void within a design space) as the level set of a continuous function. Defining the interface enables the explicit formulation of objectives and restrictions pertaining to the interface while also facilitating the description of boundary conditions at that interface. Normally, the outcomes obtained through the utilisation of LSM are heavily reliant on the initial solution guess [16,92][14][77]. A notable limitation of this approach is its tendency to experience sub-optimal convergence rates, often requiring a substantial number of iterations during the optimisation process [93][78].
A comprehensive examination of the utilisation of LSM in TO is provided by van Dijk in a detailed review [93][78], whereas a more recent review by Gibou [94][79] shows the capacities of this method, which has not stopped evolving.
Recent works on LSM are: LSM combined with BESO [95][80], multi-material approaches [96[81][82],97], multi-component TO [98][83], micro-architectured materials [99][84], reliability-based TO [100[85][86][87][88][89],101,102,103,104], dynamic TO [105[90][91],106], buckling [107][92], design-dependent loads [108][93], heat conduction TO [109][94], AM constrained TO [110[95][96][97][98],111,112,113], SM constrained TO [114][99], parallel LSM proposals [115[100][101][102][103],116,117,118], and Iso-Geometric Analysis (IGA) with LSM [119,120,121][104][105][106].
Additionally, a level-set band approach has been introduced with the aim of enhancing the smoothness of objective and constraint functions. This method integrates the benefits of both the level-set method (LSM) and density-based techniques by introducing a single parameter, referred to as the ’level-set band’, to seamlessly merge the strengths of both methods [122][107].

2.6. Phase-Field Method

The phase-field method for TO is initially developed as a means to depict the surface dynamics of phase-transition phenomena, particularly in scenarios involving transitions between solid and liquid states [123,124][108][109]. These methods have found application in various simulations of surface dynamics, particularly in materials science. They have been employed for tasks such as modelling diffusion, multi-phase flow, crack propagation, solidification, and phase transitions [22][5]. The phase-field method for TO is first introduced by Bourdin in 2003 [125,126][110][111] and then subsequently extended by Wang [127][112].
Phase-field TO employs a penalty approach to approximate the perimeter of the interface. Through the selection of a very small positive penalty parameter, it becomes possible to achieve a penalty as a well-defined interface zone that separates solid regions from voids [128][113]. In contrast to LSM, the equations governing phase transition are solved across the entire design domain without any pre-existing knowledge regarding the whereabouts of the phase interface. However, there exist some problems; for example, the correct selection of the parameters is an important factor in ensuring the convergence behaviour of the numerical solution (affecting the number of iterations needed to obtain a solution). Unfortunately, the selection of parameters is case dependent and requires a set of tests [129][114]. Additionally, there is a high computational cost of solving the underlying fourth order Cahn–Hilliard Equation [92][77]. These are some of the reasons that have prevented the spread of the method. Despite not being a popular method, advances can be further studied in [128,129,130,131,132][113][114][115][116][117].

3. Topology Optimisation Discrete Approaches

In TO discrete approaches, the treatment of individuals is twofold: one option consists of treating every finite element (FE) of a unique design as an individual of a population; then, least fit individuals of that population are removed as the optimisation process advances. This approach is common in Evolutionary Structural Optimisation methods. Specifics of formulation and implementation vary depending on each method. On the other side, the second option is to treat a design space (and its TO solution) as an individual; then, one wants to generate a large number of solutions (population) and apply some technique to generate the best individual. This approach is typical of genetic algorithms, for instance.

3.1. Evolutionary Structural Optimisation Methods

Evolutionary Structural Optimisation (ESO) is a discrete approach to the TO problem. Although in density approaches the discrete design space is relaxed, offering a continuous space design, in discrete methods, the design space is not relaxed and thus the design variable is completely discrete (0,1). The most prominent family of discrete methods is the ESO method, the first one being proposed by Xie and Steven in 1993 [133,134][118][119], which is grounded in the straightforward notion that a structure progresses towards an optimal state by progressively eliminating material experiencing low stress levels, while adhering to specified material volume criteria. However, achieving effective solutions can prove challenging under certain circumstances. In order to address its initial shortcomings, this method has undergone modifications throughout the years. First changes were aimed at enabling the restoration of removed elements, particularly those adjacent to heavily stressed components.
Originally, ESO could only remove low-stress elements, preventing finding optimal solutions. To improve ESO capabilities, bi-directional approaches allow previously eliminated individuals to reappear as part of the design. Early versions of the Bi-Directional ESO (BESO) method were developed by Querin [135,136][120][121] and Yang [137][122]. However, these methods relied heavily on heuristic concepts and lacked strong theoretical foundations. Additionally, they were susceptible to issues such as mesh dependency and the emergence of undesirable chequerboard patterns [32][16]. An improved BESO method was delivered by Huang and Xie [138,139][123][124], and, almost at the end of the decade a soft-kill BESO (SBESO) [140][125] version of their method was proposed.
In SBESO, standard adjoint gradient analysis is incorporated, introducing a higher level of theoretical rigour to the approach. Additionally, filtering techniques similar to those employed in density-based methods are integrated to stabilise the algorithm and mitigate issues like chequerboard patterns. Various benchmark examples for BESO and SBESO are fully reported in the book by Huang and Xie [141][126]. For the final SBESO version, the design domain is discretised through a finite element mesh, and initial property values are assigned to construct an initial design. Subsequently, a finite element analysis is conducted and nodal sensitivity numbers are computed using an adequate compliance equation. Notably, these sensitivity numbers are directly derived from nodal values to avoid the need for elemental calculations. The implementation includes a mesh independence filter step that refines the sensitivity values by considering neighbouring nodes and employing linear weight factors within a specified radius. Furthermore, the process involves averaging the current sensitivity numbers with historical data to ensure continuity and readiness for the subsequent iteration. The determination of the target volume for upcoming iterations is based on an evolutionary volume ratio concept, wherein the structure’s volume undergoes adjustment until a predefined constraint is met. The optimisation process dynamically modifies the design configuration by introducing or removing elements according to specific conditions tied to sensitivity thresholds. Importantly, the iteration cycle, encompassing steps 2 to 6, persists until the prescribed constraint volume is attained and optimisation objectives are successfully achieved, all while maintaining convergence through a pre-established threshold.
A multitude of recent works were also found: LSM with BESO [142][127], phase-field method with BESO [143][128], nonlinear structures under dynamic loads [144][129], non-linear reliability TO [145][130], buckling TO [146][131], fatigue considerations [147][132], crashworthiness TO with BESO [148][133], frequency optimisation [149[134][135],150], BESO with casting constraints [61][46], and BESO with AM constraints [151,152,153,154,155,156][136][137][138][139][140][141]. Other interesting works are the Iso-Geometric Analysis (IGA) approach with BESO [157][142], a meshless BESO technique [158][143], and a parallel framework for BESO [159][144].

3.2. Moving Morphable Voids Method

On 2014, Guo proposed Moving Morphable Components (MMC) [160][145]. In this method, a set of MMC are utilised as fundamental elements for TO. The process involves enhancing the arrangement, positioning, and configuration of these components to achieve an optimal structural layout; its ability to significantly decrease the quantity of design variables required in the problem definition makes this method appealing. This reduction proves advantageous in cutting down the required computational time for numerical optimisation procedures.
Based on Guo’s work, Zhang proposed Moving Morphable Voids (MMV) in 2017 [161][146], where a set of MMV are adopted as basic building blocks for TO. This method decreases the number of design variables, the number of degrees of freedom involved in the FEA solution process, and it can also be further extended to 3D optimisation. Examples of use can be further explored in [162,163[147][148][149][150],164,165], whereas explicit use of Iso-Geometric Analysis (IGA) has also been explored [166,167,168,169][151][152][153][154].

3.3. Metaheuristic Topology Optimisation Methods

Although SIMP is a method exclusively centred around gradients, involving calculating derivatives of objective functions, metaheuristic algorithms prioritise sensitivity numbers by evaluating their magnitudes. As a result, this enables a more adaptable objective function and simplifies the implementation process, leading to significant advancements in the utilisation of TO within engineering disciplines [170][155]. In what follows, wthe shresearchers shall revise some advancements in the application of metaheuristic algorithms to TO.

Genetic Algorithms

Over the past thirty years, efforts have been devoted to applying genetic algorithms (GAs) to address challenges in structural optimisation problems [171,172,173][156][157][158]. GAs function by working with a population of potential solutions, as opposed to enhancing a singular solution. Because of this characteristic, a typical structural optimisation problem tackled with GAs has to contend with a significantly larger set of design variables compared to conventional mathematical programming methods. As a result, there is a notable drawback in terms of computational burden [174][159]. Research integrating GAs into structural optimisation has focused on relatively small problem sizes. The implementation of GAs in TO has been found utilising different approaches: we can find pure GAs applied to solve the TO problem for simple compliance optimisation [175,176,177][160][161][162]. In this type of approach, the entire design space is considered as a component of the chromosome of a single individual; the idea is to generate hundreds or thousands of designs which evolve towards the optimal solution. For a structure to effectively withstand a mechanical load, it must possess a strong level of connectivity.
However, ensuring adequate structural connectivity through the use of GAs is challenging due to their inherently stochastic nature. This presents a significant limitation when applying GAs to TO, and finding a viable solution to this issue proves to be exceedingly difficult. Jackiela [178,179][163][164] suggests altering solid elements that lack connectivity to the seed elements in order to be emptied; ref. [180,181][165][166] suggests enhancing connectivity by incorporating a chromosome mask to selectively filter out chromosome segments originating from disconnected positions; Wang [177,182][162][167] suggests the incorporation of image processing techniques. Those approaches failed to effectively reduce randomly disconnected topologies, or are excessively cumbersome. However according to Li and other scholars, GAs might have better probabilities to converge towards global solutions [183,184,185,186,187,188][168][169][170][171][172][173].
For some successful applications of TO using GAs, it is appropriate to review the work in [182,189,190][167][174][175].

Other Metaheuristic Approaches

Other metaheuristic techniques have been applied without hybridising. Such are the cases of artificial immune algorithms [191,192[176][177][178],193], ant colonies [194[179][180][181][182],195,196,197], particle swarms [198[183][184][185],199,200], simulated annealing [201[186][187][188][189],202,203,204], harmony search [205[190][191][192][193],206,207,208], and differential evolution [209,210][194][195].
There exist a plethora of other proposals aiming to apply metaheuristic approaches with TO, but they have not experienced further development. The Weresearchers observed a multi-objective TO based on bacterial chemotaxis [211][196], a joint firefly and particle swarm optimisation (PSO) algorithm [212][197], a symbiotic organisms search algorithm [213][198], an improved electro-search algorithm [214][199], a plasma generation optimisation [215][200], a binary bat algorithm [216][201], an enhanced binary GA along with morphological reconstruction [217][202], a semi-definite programming-based approach combined with a GA [218][203], as well as a ground structure approach based on curved elements [219][204]. A constant in metaheuristic methods is the large quantity of iterations needed to converge which makes them difficult to extend to 3D design spaces where the computational burden increases considerably.
There is another approach in which the idea is to combine established TO algorithms with metaheuristic algorithms. Such is the case of Garcia and Silva, [220][205] who combined SIMP with simulated annealing, or Xue [221][206] who combined SIMP with a GA. In this sense of hybridisation, there is a tendency to combine ESO and BESO methods with other metaheuristic algorithms. This is mainly due to the discrete nature of ESO algorithms, which makes hybridisation a relatively straightforward process.
Far from the approaches used in the application of pure metaheuristics, hybrid approaches do not work over populations of individual solutions; instead, they try to assist the base algorithm to improve individual solutions. Such is the case of Liu [222,223][207][208]; the proposed approach treats each element within a structural domain as an individual member of a genetic algorithm (GA) population, in contrast to creating multiple entire structural designs to compose the population. In this method, the traditional ESO sensitivity number is employed as a fitness function, and a natural selection process is employed to eliminate less-fit elements, mimicking the elimination of less-fit individuals in traditional GA design problems. Cui [224][209] presents a work following Liu’s ideas. A similar work is presented by Zuo [225][210] to generate a genetic version of BESO. Yhunzhen [226][211] applies Zuo’s work to architectural design optimisation. Other works hybridising BESO with GAs might be consulted [227,228][212][213]. In a similar way, it is possible to find works hybridising BESO with other metaheuristics; such is the case of swarm optimisation algorithm [229][214], simulated annealing [230][215], and harmony search [231][216].

4. Diverse Approaches

Multi-scale, multi-physics, and machine learning applied to TO constitute a series of problems that may be solved by applying TO techniques that use either continuum, discrete, or combined approaches. These novel approaches have seen important development in the last 10 to 20 years, positively expanding the realm of methods and techniques available to solve a multitude of TO problems.

4.1. Multi-Scale Topology Optimisation Methods

Over the past three decades, TO methods have predominantly focused on the macro-scale aspects of design. However, there has been a growing interest in the development of optimisation methods that can enhance the overall performance of structures by incorporating the concept of multi-scale structural TO.
Multi-scale TO offers a novel way to optimise structures. Taking into account that the structure itself may be composed of small unit cells with a certain degree of porosity, this characteristic gives multi-scale methods the ability to obtain comparably lighter structures than using full-scale TO methods.
Features of multi-scale, such as high level of porosity, may appear when using full-scale TO methods, small details on the mesoscale (ranging from 0.1 to 10 mm) may appear if the FE mesh is fine enough to capture such details. As a result, structures that resemble grid-like patterns filling the overall macro-scale structure are obtained. Such a full-scale approach is computationally intensive as the resources needed to solve 3D problems increase with mesh density. As an option to alleviate the problem, multi-scale approaches work in two or more levels (sometimes based on homogenisation theory) of optimisation; in one level, a unit cell is proposed to fill the design volume delimited by the macro optimisation level that resembles the usual solution obtained in full-scale approaches. The main difference is that lighter structures can be obtained. Periodic patterns using TO may be imposed by subdividing the design volume and then assigning a repetitive pattern on each subdomain. Such an approach allows the obtaining of Lattice structures, and if the density (size) of unit cells is allowed to change along the design, we are then speaking of functionally graded materials.
Multi-scale structures offer the potential to attain exceptional performance characteristics, all while possessing inherent qualities of being lightweight and versatile. However, the actual advantages of these innovative multi-scale structures must be verified through numerical simulations and experimental validations as investigation research matures. Multi-scale topology approaches have been extensively reviewed by Wu et al. [17][217].
Many full-scale TO methods can be used to obtain multi-scale lattice structures and functionally graded materials; notable examples are:

4.2. Multi-Physics Topology Optimisation Approaches

TO is usually applied to mechanical problems were elasticity is the most relevant phenomenon at hand. However, there are problems where the interaction of multiple physical phenomena needs to be accounted for; such problems are categorised as “multi-physics” [271][257].
It is possible, in general, to find TO applied to problems where thermal boundary conditions impose temperature gradients, yielding thermo-mechanical stresses acting on the structure. It is possible to find in the literature implementations that are either uni-objective or multi-objective, and that provide options on how to tackle the problem at hand. Coupled thermo-mechanical TO has found application on heat dissipation structure design [272,273,274,275,276][258][259][260][261][262].
TO has also found application in electromagnetic actuators, the work in [277][263] considers the coupled interaction between structural and electromagnetic phenomena, whereas the research in [278][264] considers thermal and electromagnetic interaction, both employing multi-objective TO.
Furthermore, other less-studied multi-physics TO problems include fluid–structure interaction, which is usually relevant in aeronautics for the design and optimisation of airfoils, wings, and compressor blades. Relevant applications for the purpose of this preseaperrch are found in selected works where interaction between molecular atmosphere and satellite structure is relevant to optimise orbit lifetime [279,280,281][265][266][267]. As interaction occurs in the surface of the structures, only shape optimisation is needed to solve the problem.
Finally, layout optimisation deserves to be mentioned, although it does not necessarily refer to TO. Layout optimisation refers to the optimisation of the relative position of objects inside the satellite with the intention of adjusting the satellite centre of mass. Such an optimisation might be of great help to improve stability characteristics in orbit. However, its integration with structural TO might prove useful for structure redesign in the late development stages. Satellite layout optimisation can be found in [282,283,284,285,286][268][269][270][271][272].

4.3. Machine Learning Applied to Topology Optimisation

Research in this area has focused on data-driven TO, which means using a traditional TO method to gather a database of possible solutions for a problem [287,288,289][273][274][275]. Then, aiming to improve the process of obtaining optimal structures, different machine learning techniques are applied, like supervised, unsupervised, as well as reinforcement learning. These techniques serve several purposes, including speeding up the design iteration [289,290,291,292,293,294,295][275][276][277][278][279][280][281].
Apart from data-driven TO, some works try to achieve non-iterative optimisation [296[282][283][284][285][286][287][288][289][290][291][292][293][294][295][296][297][298][299],297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313], or intend to replace the typical FEM process, creating meta-models [314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332][300][301][302][303][304][305][306][307][308][309][310][311][312][313][314][315][316][317][318]. Other approaches try to reduce the dimensionality of the design space [333,334,335][319][320][321].
Such approaches have effectively accelerated the optimisation process and decreased computational burden. Despite the progress in this field, there remain challenges in many studies. These include addressing the low resolution of generated designs, enhancing the performance of machine learning methods, adapting the methodologies to different design domains (three-dimensional spaces), grappling with the costs associated with gathering data and managing the high computational burden of both FEA and the machine learning approach [336][322].

5. Cases of Study for TO Methods Applied to Satellite Structures

In the extension of the preseaper, werch, the researchers were able to provide brief explanations of TO methods applied to satellite structures. A diverse distribution of applications across various TO techniques is observed, with a considerable number of them reaching the manufacturing stage. For such instances, AM is employed for fabrication. It is evident that each application shows optimised weight and safety margins appropriate for their intended use, which is sometimes demonstrated by applying verification techniques designed to validate satellite structures. Some applications are then successfully integrated into operational service.

Reviewed papers report a substantial reduction in mass. According to the articles reviewed in this preseaperrch, mass reductions ranging from 8% to 55% have been observed in satellite structure components where TO has been applied, with a trend in the majority of the studies towards a 50% mass reduction. This wide variation is primarily attributed to differences in the studied satellite components as well as in the safety margins allowed. A relevant work that demonstrates the clear potential of the combination of TO and AM is was presented, the application of TO resulted in cost savings of 15,600 USD just by decreasing slightly more than one kilogram of mass from a satellite component, the main reason for cost savings is mass reduction as launch costs are directly related to the total mass of the satellite. It is noteworthy that reporting monetary savings is unusual in the majority of papers.

Despite success, authors report challenges present in the development and fabrication of satellite structures, of which the following stand out:

  • There is little support from FEA available software for the implementation of customised TO algorithms.
  • FEA's available software for TO tools does not report implemented algorithms and/or parameters to the final user, hindering the possibility of proper comparisons between scholars.
  • There is a need for support structures in metal AM processes such as SLM to print overhangs and bridges. This adds mass and requires post-processing to remove supports.
  • Achieving the necessary surface finishes for satellite components can pose a challenge due to the roughness resulting from AM. Typically, extra finishing processes are needed.
  • Lack of design standards for AM satellite components, especially for structural parts and mechanisms. Current international regulations have not been aimed at encompassing all the AM methods (ASTM-529002021, ASTM-2792). Moreover, in the absence of robust AM regulations, the oversight of international regulations within the aerospace field has received even less attention with only a few standards for certain methods ISO-52942. Even though, NASA has published some standards for Laser Powder Bed Fusion methods (MSFC-STD-3716, MSFC-SPEC-3717).
  • It is essential to recognise and certify novel AM alloys and methods for diverse satellite uses, as material characteristics may differ among various AM machines and methods.

References

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