# Continuous Adjoint Topology Optimization of Duct Flow Configurations with Explicit Volume Constraint for Design Variable Update

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

^{®}version 1906 [17,18]. Acknowledging its limitations and restrictions, this solver has been selected for its accessible structure which allows for a transparent implementation of the presented method. The solver is developed from the simpleFoam (standard steady-state solver for incompressible isothermal turbulent flow of Newtonian fluids) by deploying the frozen turbulence assumption in the adjoint equations, adopting the duct flow configuration (the boundaries comprising the inlet, outlet, and wall surfaces $\mathsf{\Gamma}={\mathsf{\Gamma}}_{inlet}\cup {\mathsf{\Gamma}}_{outlet}\cup {\mathsf{\Gamma}}_{walls}$, according to the sketch in Figure 1a), and taking the total pressure loss between the inlet and outlet as the objective function to be minimized:

## 3. Explicit Volume Constraint Update

^{®}the step function is available as $pos\left(s\right)$, which evaluates to 1 if the scalar expression s yields a non-negative value, or 0 otherwise [17]. Finally, by suitably selecting the coefficients of the sigmoid function, it can be turned into the step function $pos\left(s\right)=pos(x-{x}_{L})$ to mark for blocking the part of the domain where the calculated sensitivity is greater than the limiting sensitivity value.

## 4. Topology Optimization Results

## 5. Conclusions and Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\alpha $ | design variable |

$\mathbf{v}$ | primal velocity vector |

p | primal kinematic pressure |

${\nu}_{eff}$ | kinematic turbulent viscosity |

$\mathbf{u}$ | adjoint velocity vector |

q | adjoint kinematic pressure |

${d}_{\alpha}$ | Darcy porosity coefficient |

J | objective function |

$\mathsf{\Gamma}$ | flow domain boundary |

$\mathsf{\Omega}$ | flow domain interior |

$sens$ | objective function sensitivity |

$vo{l}_{T}$ | target volume of blocking cells |

${n}_{loops}$ | number of search loops |

${n}_{cycles}$ | number of design cycles |

${C}_{0},{C}_{1},{C}_{2}$ | sigmoid function coefficients |

$\lambda $ | steepest gradient descent step |

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**Figure 1.**Topology optimization setup: available design space for a duct flow configuration (

**a**) and the topology optimization outcome (

**b**), with respective boundary conditions.

**Figure 2.**General principle of the design variable update based on steepest gradient descent method.

**Figure 3.**Sigmoid function with varying coefficients, ${C}_{0}$ for the inflection point shift (full, dash, dash–dot) and ${C}_{1}$ for modifying the slope (red, green, blue), and step function (black).

**Figure 5.**Flowchart of the topology optimization cycles using continuous adjoint method, with explicit design variable update.

**Figure 6.**The 3D flow distribution duct: geometry and boundary conditions (

**a**), streamlines and contour plots revealing flow features (

**b**).

**Figure 7.**Flow pattern (streamlines) and $\alpha $ distribution (left–back view, middle–front view) and sensitivity (right) for 3D flow distribution duct optimization with $vo{t}_{T}$ = 10% (

**a**), 20% (

**b**), 30% (

**c**).

**Figure 8.**Reduction of the objective function (relative to the non-optimized case) at different design optimization cycles.

**Figure 9.**Distribution of the Darcy porosity $\alpha {d}_{\alpha}$ (back view) with steepest descent method for varying step size $\lambda $: too small (

**a**), appropriate (

**b**), too big (

**c**).

**Figure 10.**Pin-plate cooler geometry (

**a**) and flow features (

**b**) with flow connector encircled white, and the topology optimization of flow connector ((

**c**) total view, (

**d**) zoom view).

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**MDPI and ACS Style**

Popovac, M.
Continuous Adjoint Topology Optimization of Duct Flow Configurations with Explicit Volume Constraint for Design Variable Update. *Energies* **2022**, *15*, 7349.
https://doi.org/10.3390/en15197349

**AMA Style**

Popovac M.
Continuous Adjoint Topology Optimization of Duct Flow Configurations with Explicit Volume Constraint for Design Variable Update. *Energies*. 2022; 15(19):7349.
https://doi.org/10.3390/en15197349

**Chicago/Turabian Style**

Popovac, Mirza.
2022. "Continuous Adjoint Topology Optimization of Duct Flow Configurations with Explicit Volume Constraint for Design Variable Update" *Energies* 15, no. 19: 7349.
https://doi.org/10.3390/en15197349