*2.4. Fluid–Structure Interaction*

In this section, the advances within topology optimisation of fluid–structure interaction (FSI) problems are discussed.

Figure 3 shows the types of design modifications possible for FSI problems. Dry optimisation only changes the internal structure, keeping the solid–fluid interface constant. Wet optimisation modifies the solid–fluid interface and topology. Since this review paper is focused on the optimisation of the fluid flow, only works where the wet surface and topology is allowed to change significantly are included (wet and wet+dry). A whole range of works describe the topology optimisation of structural parts subjected to fluid loads, where the deformation of the structure may or may not be taken into account when computing the fluid induced loads. However, because they do not modify the wet surface and topology, they are not considered herein.

**Figure 3.** Description of different degrees of design modification for fluid–structure interaction (FSI) problems.

Yoon [161] can be considered the seminal paper on topology optimisation for FSI. A unified density-based formulation of the elastic Navier–Cauchy equations assuming small strains and the incompressible Navier–Stokes equation is obtained by converting the interface condition to a volumetric integral representation, previously used for acoustic–structure interaction [162]. The fluid stress in the interaction is slightly simplified and a pressure filter function determines where the fluid pressure applies. The fluid problem is solved in the deformed mesh and a full coupling is modelled; however, the obtained deformations of the solid domain are extremely small and the two-way coupling is not really active. Another approach was taken by Kreissl et al. [163], where micro-fluidic devices are optimised subject to external mechanical actuation. A one-way coupling from structure to fluid is used to deform the fluid domain. The backward fluid-structural coupling is assumed negligible and hence ignored. Yoon [164] extends the previous work [161] to cover electro-fluid-thermal-compliant actuators, including two additional physical fields, electric and thermal, in the coupling. Planar multiphysics MEMS devices are optimised with electrical and thermal response being computed in the reference mesh. Subsequently, Yoon extended the framework to minimise the structural mass subject to stress constraints [165], and also applied the framework to the optimisation of a compliant flapper valve [166].

Jenkins and Maute [167] presented a coupled level set based framework utilising X-FEM and deformed meshes, demonstrating generalised shape optimisation of a bio-prosthetic heart valve and topology optimisation of the wall example of Yoon [161]. Munk et al. [168] present a simplified model for designing baffle plates, with a one-way pressure coupling omitting fluid shear stress. The fluid is modelled with LBM and the loads are mapped to the structural model in the reference configuration. The Bi-directional Evolutionary Structural Optimisation (BESO) method is used in a soft-kill version, but the sensitivities of changing the fluid flow are neglected. Similarly, Picelli et al. [169] also neglect the sensitivities of changing the fluid flow when updating the wet surface when applying a hard-kill BESO method to design various FSI problems.

Yoon [170] presented an extension of previous work [165], where a material failure criteria is applied to design the material distribution. Lundgaard et al. [171] revisited the unified density-based formulation of Yoon [161], however, solving the fluid in the reference mesh under the assumption of small deformations. Multiple objective functions and design problems are reviewed and thorough discussions of current limitations, artefacts and future extensions for density-based topology optimisation of FSI problems are given. Munk et al. [172] compared the previous formulation [168] to level set and density-based methods for the case of minimising the compliance of a fluid-loaded baffle plate. Subsequently, Munk et al. [173] ported the work to graphics processing unit (GPU) architecture in order to reduce the high computational time for the LBM model. Feppon et al. [174] used a level set-based framework to explicitly track and advance the interface using the Hamilton-Jacobi equation. The meshes are iteratively updated based on the convected level set by local operations, with the physics being weakly coupled, modeled in referenced configuration and solved using a staggered procedure.

#### *2.5. Microstructure and Porous Media*

In relation to the origin of topology optimisation, namely the homogenisation approach, there are a range of studies that consider the optimisation of material microstructures. Typically a unit-cell, or representative volume element (RVE), is subjected to periodic boundary conditions and the effective parameters are obtained by imposing a set of volumetric loads. This approach can be utilised in an inverse manner to optimise the material design and the corresponding effective parameters. For solids, this is related to the effective stiffness tensor, while for fluids this is naturally related to the permeability. For fluid–structure interaction in a porous medium, a pressure coupling term can also be obtained by homogenisation.

#### 2.5.1. Material Microstructures

Guest and Prévost [175] maximised the permeability of a porous material microstructures using a Darcy–Stokes interpolation [11] subject to isotropic symmetry constraints. The work is extended in Guest and Prévost [176] to optimise microstructures for combined maximum stiffness and permeability. Bones and tissue contain porous materials and Hollister and Lin [177] optimised tissue engineering scaffolds using a hybrid stiffness and permeability optimisation routine. Xu and Cheng [178] proposed a multiscale optimisation problem, where the macroscopic elastic compliance is minimised subject to a flow constraint ensuring a permeable microstructure. Physically related to this, Andreasen and Sigmund [179] optimised the microstructure of a poroelastic material for maximum poroelastic coupling during pressurisation subject to permeability constraints. Chen et al. [180] studied the optimisation of bio-scaffolds using homogenisation for tissue regeneration including permeability considerations. Chen et al. [181] extended the work to consider shear induced wall erosion. Goncalves Coelho et al. [182] introduced permeability constraints in an extensive multiscale optimisation framework for the multiscale topology optimisation of trabecular bone. For most material properties, certain bounds apply in property space and Challis et al. [183] investigated the cross-property bounds between stiffness and permeability by exploiting the Pareto-front using a level-set based approach [25].

#### 2.5.2. Porous Media

In this subsection, works where the final design is supposed to be a porous structure i.e., intermediate design variables, are reviewed. This can be in terms of multiscale problems obtained e.g., by two-scale asymptotic expansion. A different take on FSI problems was presented by Andreasen and Sigmund [184] for optimisation of material design for poroelastic actuators and in Andreasen and Sigmund [185] for impact energy absorption in porous structures. Furthermore, in the context of macroscale problems, Youssef et al. [186] optimised a porous scaffold with macroscale flow channels to control the internal shear stress in a bioreactor. Ha et al. [187] used the Darcy–Stokes interpolation method [175] to maximise the permeability of three-dimensional woven materials. A multimaterial approach is taken by Wein et al. [188], where a highly nonlinear saturated porous model with multiple materials is applied to design diapers that quickly transports fluid away from the surface to capture it in the interior. Takezawa et al. [189] used a Brinkman–Forchheimer macro model to optimise material microstructures for minimum flow resistance considering the trade-off between permeability and form-drag. Lurie et al. [190] optimised the distribution of the wick (porous media used to transport condensate to the evaporator due to capillary effects) in a heat pipe. Takezawa et al. [191] applied a multiscale method to the thermofluid problem of metal printed lattice design. Effective parameters for permeability, form drag and conductivity are obtained for a generic orthogonal truss microstructure and used in a macroscopic material distribution method based on the Brinkman–Forchheimer and a convection–diffusion equation. Takezawa et al. [192] later extended this to the fluid-thermo-elastic problem of metal printed heat sinks.

#### **3. Quantitative Analysis**

In this section, a quantitative study of the referenced papers is carried out. In some cases, the method details might be unclear or not mentioned, excluding the paper from the statistics.

#### *3.1. Total Publications*

Figure 4 shows the number of papers published per year and the total accumulated number of publications over time since the inaugural paper by Borrvall and Petersson [7] in 2003. The year of publication is here taken as the year of the final journal issue.

**Figure 4.** Number of papers published per year and total accumulated publications over time.

From 2003 to 2015, a slow increase in the number of papers per year is observed, reaching an average of 10 per year for the period 2012–2015. In 2016 and 2017, the number of papers per year almost doubled to 18 and 16, respectively, bringing the total number of publications to 100 after 2016. In 2018 and 2019, the number of papers per year again almost doubled to 31 and 36, respectively. The total number of papers by the end of 2019 reached 182. This approximate doubling behaviour shows itself as an exponential-like increase in the number of total papers. In 2020, covering only the month of January, there have been five publications so far. This amounts to a total of 186 papers covered by this review.

#### *3.2. Design Representations*

As discussed in Section 1, several different design representations exist for topology optimisation. The papers are grouped into three groups: "density" covering interpolation and homogenisation approaches; "level set" covering Ersatz-material, adapted (here refering to adapted meshes and/or ignoring the solid elements in computations.) and surface-capturing level set approaches; "other" covering anything else, e.g., BESO.

Figure 5 shows how the included papers are distributed among the two main design representations, density-based approaches and level set approaches. Firstly, the use of density-based approaches vastly outnumbers any other methods with 144 papers or 77%. This reflects the general tendency within the topology optimisation community to prefer density-based methods [2,3]. Secondly, the approaches relying on an implicit level set description of the geometry are also numerous at 31 papers or 17%. Of these 31 papers, 11 use an Ersatz-material, 11 use an adapted approach and only nine use a surface-capturing discretisation method. Lastly, the rest of the papers are distributed as follows: 5 using BESO [168,169,172,173,177]; 2 using phase field [42,59]; 1 using a discrete surface representation [63]; 1 using a geometry-projection method [138]; and 2 utilising the topological gradient [46,74].

#### *3.3. Discretisation Methods*

For discretising the design and physics, a variety of methods are used in the included papers: finite element method (FEM); finite volume method (FVM); lattice Boltzmann method (LBM) including Boltzmann equation related schemes; particle-based methods (PM).

**Figure 5.** Distribution of papers in overall design representation type.

Figure 6 shows the distribution of papers in these overall methods. It is clear that FEM is the most widely used discretisation method with 134 papers or 76%. The next most used method is LBM at 28 papers [14,20,22,24,26,36–38,44,55,65,69,75,77,78,84,97,108,113,118,119,126,131,163,168,172,173,186] or 16%. PM is the least used method with only a single paper [79]. Surprisingly, FVM is the second least used method with only 12 papers [21,47,52,80,82,115,129,130,137,144,159,188] or 7%, despite the fact that FVM for many years has been the preferred discretisation method for computational fluid dynamics. This can probably be explained by several factors: topology optimisation originates from solid mechanics where FEM is the preferred method; discrete adjoint approaches are easier using FEM than FVM; stabilised FEM has grown to be a mature and accurate method [193,194].

**Figure 6.** Distribution of papers in overall discretisation method: FEM = finite element methods; FVM = finite volume methods; LBM = lattice Boltzmann methods; PM = particle-based methods.

#### *3.4. Problem Types*

The main problem types treated in this review is: pure fluid (PF); species transport (ST); conjugate heat transfer (CHT); fluid–structure interaction (FSI); microstructure and porous media (MP).

Figure 7 shows the distribution of the included papers in these problem types. It is clearly seen that the largest number of papers deal with purely fluid flow problems, namely 82 papers or 44%. The second largest group deals with conjugate heat transfer covering 55 papers or 30%. Species transport covers 19 papers or 10%, with FSI covering 15 papers or 8%, and porous media covering 14 papers or 8%.

**Figure 7.** Distribution of papers in overall problem type: PF = pure fluid; ST = species transport; CHT = conjugate heat transfer; FSI = fluid–structure interaction; MP = microstructure and porous media.

#### *3.5. Flow Types*

In this review, the fluid model type can be boiled down to four categories: steady-state laminar flow (SS); transient laminar flow (TR); steady-state turbulent flow (TU); and Non-Newtonian fluid (NN).

Figure 8 shows the distribution of papers for fluid model type, both for (a) all papers and (b) papers treating only fluid flow. Analysing all papers, the vast majority use a steady-state laminar flow model with 158 papers or 85% as seen in Figure 8a. Only 13 papers or 7% consider a transient laminar flow model [72–79,145,150,152,158,188], with a meager six papers or 3% treating turbulent flow [21,80–83,129]. In the case of time-dependent problems, this is most likely due to the vast increase in computational cost related to simulation of transient flow problems, where all temporal details must be resolved sufficiently and all temporal solutions saved in memory (or recomputed) for the adjoint solve. Likewise, turbulent flow also carries an increase in computational cost with it, since turbulence models with additional degrees-of-freedom are used and fine meshes are needed to resolve the turbulent boundary layers.

All of the above use a Newtonian fluid model, but nine papers or 5% use a Non-Newtonian model [84–91,139].

Looking only at papers treating fluid flow only, Figure 8b shows that the percentage of the more complex flow models increases. This indicates that more work has been done on treating transient, turbulent and non-Newtonian models for fluid flow only compared to overall for fluid-based problems. This makes sense since pure fluid flow is the obvious place to start working with and tackling the large computational cost associated with the more complex flow models.

**Figure 8.** Distribution of papers for fluid model type: SS = steady-state laminar flow; TR = transient laminar flow; TU = turbulent flow; NN = Non-Newtonian fluid.

#### *3.6. Three-Dimensional Problems*

A paper is classified as treating three-dimensional problems only if at least one example uses a three-dimensional model in the optimisation process. Therefore, two-dimensional results that are extruded and post-analysed in three dimensions are not included.

Figure 9 shows the distribution of papers treating two-dimensional and three-dimensional problems.

**Figure 9.** Distribution of papers for two-dimensional (2D) and three-dimensional (3D) problems.

Figure 9a shows that a significant share of the included papers contain three-dimensional results, namely 31% or 58 out of 186 papers. For the pure fluid flow, there are 15 papers for steady laminar flow [14,15,21,23,25,32–34,36,46,55,56,61,66,70], 3 for unsteady flow [74,76,77], 2 for turbulent flow [80,82] and none for non-Newtonian fluids. For species transport, there are four papers [93,98,105,107]. For conjugate heat transfer, there are eight in forced convection [110,117,119,122,126,129,132,135] and six in natural convection [150,151,153–155,160]. The fluid–structure interaction category counts four papers [164,168,172,173], but it must be noted that, common for all, the three-dimensional design freedom is severely limited, as the design domain is restricted in the third dimension. Finally, a very large share of the three-dimensional results belong in the category of microstructure and porous media problems with 14 papers [175–177,179–184,187–189,191,192]. Since, in material design, it is natural to consider

both stiffness and permeability, the need for three-dimensional design freedom is obvious, as two-dimensional models would have either the fluid or the solid phase disconnected.

Figure 9b shows the yearly publication count for two- and three-dimensional problems. It can be seen that they follow the same trend, which is reflected in the fact that the percentage of three-dimensional papers has been close to constant since 2010 at approximately 30%. In January 2020, there has been a single three-dimensional paper [160] out of 5, making it 20% so far.

#### **4. Recommendations**

Based on the extensive literature review, analysis of the methods used, as well as the authors' personal experience and opinions, some recommendations are made to the research community in order to help with moving forward.

#### *4.1. Optimisation Methods*

Ninety-eight percent of the included papers use gradient-based optimisation approaches, covering amongst others nonlinear programming algorithms, velocity-based level set updates and discrete BESO updates. Most of these use first-order methods, with notable exceptions being the work of Evgrafov [34,41] using higher-order schemes. Only three papers use gradient-free optimisation approaches, consisting of genetic algorithms [47,186] and neural networks [71]. As pointed out by Sigmund et al. [195], gradient-free approaches seldomly make sense for topology optimisation, due to the high dimensional problems for increasing design resolutions. This is perfectly illustrated in the work of Gaymann and Montomoli [71], where the design resolution is absurdly coarse and useless in practise. However, gradient-free approaches can be useful when gradient information is not available, like when using a commercial solver as a black-box, or when dealing with discontinuous functions with non-well-defined gradients. However, even in the case of black-box solvers, gradients can easily be approximated using finite differences at a fraction of the cost of most genetic algorithms. Furthermore, gradient-free methods may have advantages for multi-objective problems, although these can also be included in gradient-based approaches, e.g., [125].

However, gradient-free methods should in general be avoided for topology optimisation of fluid-based problems, and the recommendations of Sigmund [195] should be followed.

#### *4.2. Density-Based Approaches*

For density-based approaches, the interpolation of material properties between solid and fluid is of utmost importance to ensure final designs without intermediate design variables. Especially when moving to multiple physics, with an increasing number of material properties, the complexity of choosing the correct form of interpolation increases substantially, see, e.g., [145]. It is often not easy to intuitively choose the various interpolation functions to provide a correct relation between the material properties for intermediate design field values. Thus, it is necessary to either perform analytical derivations (often not possible) or numerical experiments. It is important to investigate the behaviour of the chosen objective functional with respect to the design field to ensure the interpolation functions provides well-scaled and monotonic behaviour [171,196]. Furthermore, it is extremely important to rigorously validate adjoint sensitivities with other methods, such as the complex step method or finite difference approximations as discussed by Lundgaard et al. [145].

Relying on Brinkman penalisation to model an immersed solid geometry in a unified domain has its drawbacks. Due to the nature of the penalisation, there will always exist fluid flow inside the solid. The penalisation factor must be large enough to ensure this flow is negligible inside the solid domain, but small enough to ensure numerical stability of the solution and optimisation algorithms. Generally, this is not observed to be an issue in general, except for a few very specific problems, where pressure diffusion through the solid domains are problematic [30,66]. However, when the pressure field is of direct interest, the Brinkman penalisation must be significantly higher to ensure an accurate evaluation [171].

For density-based methods and Ersatz-based level set methods on regular meshes, a smooth transition region, as ensured by e.g., density filtering, provides proper convergence of the design description with mesh refinement. It might not be an advantage to have a fully discrete 0–1 design field, since this will lead to staircase-like descriptions of the fluid–solid interface. For coarse meshes, this may lead to flow instabilities near the interface and thus a poor description of the boundary layer. For finer meshes, this is not as big of an issue.

#### *4.3. Level Set-Based Approaches*

The level set method is often praised for its accurate description of the geometric interface between solid and fluid. However, if this accurate description of the interface is not transferred to the simulation model, then nothing is gained. Therefore, in order to exploit the full potential of a level set design description, it becomes essential to use surface-capturing schemes, such as e.g., X-FEM [30,150], CutFEM [76], or adaptive body-fitted meshes [89,174]. This will allow for increased accuracy of the boundary layer, which becomes increasingly important when moving to more complex fluid problems, such as turbulent flow discussed in Section 4.7. Therefore, it is recommended that future work using the level set design representation should focus on applying surface-capturing schemes with local refinement of the boundary layer regions.

#### *4.4. Steady-State Laminar Incompressible Flow*

A steady-state incompressible laminar flow model is used for the vast majority of the work on topology optimisation of fluid-based problems, with 85% of all papers and 74% of fluid flow only papers. With 61 papers treating steady-state incompressible fluid flow only, it is proposed that the community not spend more time on this, especially for minimum dissipated energy and pressure drop. A large range of methods have been applied to these energy-based functionals, with only a minority of papers treating more complex objective functionals such as flow distribution and uniformity [33,50], diodicity [43,53,64,68,70,91], minimum drag and maximum lift [29,59].

Future papers treating only steady-state incompressible fluid flow should either present novel objective functionals, constraints or applications. This should preferably be in the context of application to practical engineering applications, since the treatment of steady-state incompressible laminar flow is already rather mature.

#### *4.5. Benchmarking*

Future papers should build and improve upon the current literature, not reproduce it. During the development and testing phases of research, already published examples should absolutely be used as benchmarks. However, merely reproducing old examples using a new method does not represent a scientific contribution. Future papers proposing new methodologies should show clear improvements compared to the old, focusing on the extension of applicability rather than reproducing old examples. Therefore, if a new method is not or can not be shown to provide a clear improvement in one or more of the following, the work does not warrant publication:


If the above is not shown, then the work should not be submitted by the authors and should be rejected by reviewers. Works reinventing the wheel with a new methodology without showing *clear* advantages, only serves to clog up the cogs of scientific progress.

In extension of the above, when comparing methodologies in order to show a clear improvement, it is pertinent to use the exact problem setup of the previously published works. There is a tendency to

change dimensions or physical settings slightly from paper to paper, which reduces the weight carried by the comparison.

#### *4.6. Time-Dependent Problems*

Only 13 papers in total treat time-dependent problems [72–79,145,150,152,158,188], eight of which are for fluid flow only. Since most realistic flow applications exhibit some form of time-dependent behaviour through either time-dependent boundary conditions or flow instabilities, it is strongly recommended that more community effort is dedicated to expanding the research on applying topology optimisation to time-dependent fluid-based problems. Due to the iterative nature of topology optimisation often requiring hundreds or thousands of simulations, the computational cost of a single time-dependent simulation becomes a significant bottleneck. The topology optimisation of time-dependent problems is therefore seen as the next frontier, requiring research into high-performance computing, efficient numerical methods and time integration and storage reduction methods. Novel ways to treat transient optimisation problems, such as the work by Chen et al. [77], can also aid in this progress. The topology optimisation community should draw inspiration from other fields, such as computational science and mathematics, and collaborate with researchers from those fields.

#### *4.7. Turbulent Flow*

Most industrial flow applications are turbulent, rather than laminar. Turbulence is inherently time-dependent and this research area goes hand in hand with the above. Current works on topology optimisation of turbulent flow only amount to six papers [21,80–83,129] and they all consider a steady-state time-averaged approximation of turbulence, namely the Reynolds-Averaged Navier–Stokes (RANS) equations. This is a natural starting point and there is certainly still room for research to be done at this level of turbulence modelling for topology optimisation.

Capturing the turbulent boundary layers is a significant challenge in topology optimisation, since the solid–fluid interface is not known a priori and, thus, local boundary layer mesh refinement is not easily applied. This can potentially be a significant problem for density-based methods with a gradual transition from solid to fluid or with a staircase description of the boundary. Surface-capturing level set methods can potentially deliver significant benefits to this type of problems, as well as adaptive body-conforming meshes [63,174] or local mesh refinement [61]. However, despite the attractive properties of accurate boundary identification, to date, only density-based methods for turbulent flow have been presented.

As will be discussed in Section 4.11, the introduction of approximate models as a surrogate for full-blown turbulent models may also be a viable way to treat very complex flow problems in the context of topology optimisation.

## *4.8. Compressible Flow*

To the knowledge of the authors, no works treating fully compressible flows have been published as to the date of submission of this review paper. There are a few papers treating slightly compressible fluids, but only as an approximation of fully incompressible fluids. For this type of problem, local conservation properties may well prove important when introducing a varying design representation and, thus, methods such as FVM or Discontinuous Galerkin (DG-)FEM might be necessary to ensure conservation of mass.

#### *4.9. Fluid–Structure Interaction*

The efforts within wet topology optimisation of FSI problems only cover 15 papers [161,163–174], and these all remain restricted to small deformations and steady-state. Thus, the solution of the problems in the deformed state is either negligible or of minor importance to the optimisation procedure, at least if the design objective is minimum compliance. There seems to be a large

potential in extending the methodology to transient problems exhibiting large deformations, e.g., in a biomechanical context.

#### *4.10. Three-Dimensional Problems*

As for time-dependent and turbulent problems, three-dimensionality is present in most industrial applications. Therefore, it is important for the community to focus on large scale three-dimensional problems. While 31% of papers treat three-dimensional problems, many of these suffer from either: being very small in the third dimension [107,122]; having severely restricted design freedom in the third dimension [164,168,172,173]; and using very coarse discretisations [14,23,46,56,132,150]. Truly three-dimensional problems inherently carry a large computational cost with them, and this is discussed in a number of papers, where high performance parallel computing [15,25,33,55,76,82,126,129,151,153,160], graphics processing units [119,173] and adaptive meshes [61] have been proposed as solutions.

Future papers treating three-dimensional problems should include a discussion of the computational cost involved. Since the research community should be moving towards more complicated flow problems including transient and turbulent flows, the concern of computational cost becomes even more dominant. Thus, even though simple problems may be treated in future papers, the computational cost and limitations of the method must be discussed in the context of tackling large scale three-dimensional problems.

#### *4.11. Simplified Models or Approximations*

Since all of the above problem areas all carry a large computational cost, it is beneficial for the community to work on simplified models or approximations to the complex physics.

#### 4.11.1. 2D Simplification of 3D

It is very common in the included papers to treat two-dimensional academic problems. However, some works directly approximate a three-dimensional plane problem, with a small thickness, using a two-dimensional simplification, either stated explicitly [43,64,67,68,106,112,116,134,140,142,143,149] or implicitly [50,67,87,102,107]. Out of these, only three papers [106,140,143] include the viscous resistance from the friction due to the out-of-plane viscous boundary layers. This is despite the fact that a simple expression is given in the original works on the subject [7,8]. If the out-of-plane dimension is large, e.g., [152,158], the friction will go to zero. However, for small thicknesses, the friction cannot be neglected [143].

For forced convection cooling of heat sinks, pseudo-3D models have been proposed [113,127,128,141,143] consisting of two layers in order to approximate the temperature of both the heat source and a cross-section of the heat sink. Furthermore, a cross-sectional model for forced convection has also been proposed for flow that is fully-developed in the out-of-plane direction [121].

Common to all of the above dimensional simplifications is that the design is assumed to be constant in the out-of-plane direction and physical fields are assumed to vary polynomially in the out-of-plane direction. However, as shown by Dilgen et al. [129], the error introduced by this assumption can be rather large and, therefore, designs must be validated.

#### 4.11.2. Simplified Flow Models

A number of the included papers use a simplified flow model compared to the situation that they wish to model. One examples is to use Stokes flow instead of turbulent flow, e.g., [116]; however, this is severely limited in capturing the correct physics. One suggestion to approximate the thin boundary layers for turbulent flow is by instead using a Darcy flow model [123] with an artificial permeability in the fluid region. However, inertia is still not captured. Another recent example uses laminar flow to approximate turbulent flow in a multifidelity approximation framework [148]. For natural convection, a potential-like model is derived by reducing the Navier–Stokes equations by assuming the buoyancy term to be dominant [157,160].

Using simplified models to treat complex problems is one way to reduce the computational cost, but it is extremely important to validate the design performance for the final design using the real model. Neglecting terms and phenomena leads to lower accuracy, but combining the simplified and full models in a sequential optimisation approach can reduce the cost significantly, while still retaining the accuracy of the full model for some steps of the optimisation. In engineering practise, a low-fidelity model is often used in the initial stages to make fast design progress and then a high-fidelity model is used to refine the design at the end [148]. However, Pollini et al. [160] recently proposed a sequential optimisation approach using the full model initially to point the gradient-based optimiser in the correct direction and then refine the design features using an approximate model.

#### *4.12. Numerical Verification*

For all papers treating the topology optimisation of fluid-based problems, numerical verification must be performed for the final design using an independent solver with a body-fitted mesh, sufficient mesh resolution and a fully descriptive physical model. This is a bare minimum for all future papers.

It is especially important for density- or Ersatz-material based approaches, where the boundary is not necessarily captured accurately on regular meshes. It is also important if the mesh used for optimisation is relatively coarse or where a simplified or reduced model has been used to ensure fast computations.
