# A Review of Topology Optimisation for Fluid-Based Problems

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## Abstract

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## 1. Introduction

#### 1.1. Definitions for Inclusion

#### 1.1.1. Governing Equations

- Darcy, Forchheimer and Brinkman flow
- Stokes and Navier–Stokes flow
- Homogenised fluid equations
- Kinetic gas theory, Lattice Boltzmann and similar methods based on distributions
- Particle methods

- Species transport, e.g., microfluidic mixers,
- Reaction kinetics, e.g., ion transport in flow batteries,
- Temperature, e.g., heat exchangers,
- Structural mechanics, e.g., fluid–structure interaction.

#### 1.1.2. Literature Search

- fluid flow
- conjugate heat transfer
- convection
- fluid structure interaction
- microstructure
- homogenization

#### 1.1.3. Optimisation Methodology

#### 1.2. Layout of Paper

## 2. Literature Review

#### 2.1. Fluid Flow

#### 2.1.1. Steady Laminar Flow

#### 2.1.2. Unsteady Flow

#### 2.1.3. Turbulent Flow

#### 2.1.4. Non-Newtonian Fluids

#### 2.2. Species Transport

#### 2.3. Conjugate Heat Transfer

#### 2.3.1. Forced Convection

#### 2.3.2. Natural Convection

#### 2.4. Fluid–Structure Interaction

#### 2.5. Microstructure and Porous Media

#### 2.5.1. Material Microstructures

#### 2.5.2. Porous Media

## 3. Quantitative Analysis

#### 3.1. Total Publications

#### 3.2. Design Representations

#### 3.3. Discretisation Methods

#### 3.4. Problem Types

#### 3.5. Flow Types

#### 3.6. Three-Dimensional Problems

## 4. Recommendations

#### 4.1. Optimisation Methods

#### 4.2. Density-Based Approaches

#### 4.3. Level Set-Based Approaches

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

#### 4.5. Benchmarking

- accuracy of the geometric representation
- precision of solution and/or optimality
- algorithmic and/or computational efficiency
- parameter robustness and algorithmic stability

#### 4.6. Time-Dependent Problems

#### 4.7. Turbulent Flow

#### 4.8. Compressible Flow

#### 4.9. Fluid–Structure Interaction

#### 4.10. Three-Dimensional Problems

#### 4.11. Simplified Models or Approximations

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

#### 4.11.2. Simplified Flow Models

#### 4.12. Numerical Verification

#### 4.13. Experimental Validation

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Fluid nozzle illustrating the basic differences among design representations in topology optimisation: (

**a**) explicit boundary representation (body fitted mesh); (

**b**) density/ersatz material based representation; (

**c**) level set based X-FEM/cutFEM representation.

**Figure 2.**Illustration of a metallic block subjected to different heat transfer mechanism in the surrounding fluid. (

**a**) shows forced convection with a cold flow entering at the left-hand side; (

**b**,

**c**) show natural convection and pure diffusion, respectively, due to cold upper and side walls. Reproduced with permission from Alexandersen et al. [109].

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

**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.

**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.

**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.

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Alexandersen, J.; Andreasen, C.S.
A Review of Topology Optimisation for Fluid-Based Problems. *Fluids* **2020**, *5*, 29.
https://doi.org/10.3390/fluids5010029

**AMA Style**

Alexandersen J, Andreasen CS.
A Review of Topology Optimisation for Fluid-Based Problems. *Fluids*. 2020; 5(1):29.
https://doi.org/10.3390/fluids5010029

**Chicago/Turabian Style**

Alexandersen, Joe, and Casper Schousboe Andreasen.
2020. "A Review of Topology Optimisation for Fluid-Based Problems" *Fluids* 5, no. 1: 29.
https://doi.org/10.3390/fluids5010029