Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Model Hierarchy for the Shape Optimization of a Microchannel Cooling System

Sebastian Blauth, Christian Leithäuser, René Pinnau

TL;DR

This work addresses shape optimization of a microchannel cooling system by formulating a PDE-constrained problem on a geometry Ω and deriving reduced-cost functionals for three models: a full 3D physics model, a 3D porous-medium homogenized model, and two 2D dimension-reduced variants. Shape derivatives and adjoint systems are obtained via a material-derivative-free Lagrangian approach, enabling gradient-based optimization solved with a line-search algorithm. Numerical comparisons show that the reduced models closely approximate the full model while offering large reductions in mesh size, memory, and computation time, with the full 2D model delivering the best balance of accuracy and efficiency. The results demonstrate that efficient, accurate shape optimization of microchannel coolers is feasible, enabling rapid design iterations for applications in chemical reactors and electronics cooling.

Abstract

We model a microchannel cooling system and consider the optimization of its shape by means of shape calculus. A three-dimensional model covering all relevant physical effects and three reduced models are introduced. The latter are derived via a homogenization of the geometry in 3D and a transformation of the three-dimensional models to two dimensions. A shape optimization problem based on the tracking of heat absorption by the cooler and the uniform distribution of the flow through the microchannels is formulated and adapted to all models. We present the corresponding shape derivatives and adjoint systems, which we derived with a material derivative free adjoint approach. To demonstrate the feasibility of the reduced models, the optimization problems are solved numerically with a gradient descent method. A comparison of the results shows that the reduced models perform similarly to the original one while using significantly less computational resources.

Model Hierarchy for the Shape Optimization of a Microchannel Cooling System

TL;DR

This work addresses shape optimization of a microchannel cooling system by formulating a PDE-constrained problem on a geometry Ω and deriving reduced-cost functionals for three models: a full 3D physics model, a 3D porous-medium homogenized model, and two 2D dimension-reduced variants. Shape derivatives and adjoint systems are obtained via a material-derivative-free Lagrangian approach, enabling gradient-based optimization solved with a line-search algorithm. Numerical comparisons show that the reduced models closely approximate the full model while offering large reductions in mesh size, memory, and computation time, with the full 2D model delivering the best balance of accuracy and efficiency. The results demonstrate that efficient, accurate shape optimization of microchannel coolers is feasible, enabling rapid design iterations for applications in chemical reactors and electronics cooling.

Abstract

We model a microchannel cooling system and consider the optimization of its shape by means of shape calculus. A three-dimensional model covering all relevant physical effects and three reduced models are introduced. The latter are derived via a homogenization of the geometry in 3D and a transformation of the three-dimensional models to two dimensions. A shape optimization problem based on the tracking of heat absorption by the cooler and the uniform distribution of the flow through the microchannels is formulated and adapted to all models. We present the corresponding shape derivatives and adjoint systems, which we derived with a material derivative free adjoint approach. To demonstrate the feasibility of the reduced models, the optimization problems are solved numerically with a gradient descent method. A comparison of the results shows that the reduced models perform similarly to the original one while using significantly less computational resources.

Paper Structure

This paper contains 25 sections, 1 theorem, 78 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Description of the Geometry
  4. Mathematical Model
  5. The Optimization Problem
  6. Shape Calculus
  7. Formal Shape Optimization
  8. The Dimension Reduction Technique
  9. Description of the Model
  10. Application to the Shape Optimization Problem
  11. Porous Medium Model
  12. Description of the Geometry for the Darcy Model
  13. Description of the Model
  14. Application to the Shape Optimization Problem
  15. Dimension Reduction Applied to the Darcy Model
  16. ...and 10 more sections

Key Result

Theorem 2.2

Under the assumptions of sturm the shape derivative of eq:def_reduced_cost_function at $\Omega \subset D$ in direction $\mathcal{V}$ is given by where $(u, p, T) = U(\Omega)$ is the solution of the state system eq:weak_state and $(v, q, S) = P(\Omega)$ is the solution of the adjoint system eq:weak_adjoint.

Figures (6)

  • Figure 1: Two-dimensional domain of the cooling system $\tilde{\Omega}$.
  • Figure 2: Two-dimensional geometry for the Darcy model $\tilde{\Omega}^{\mathrm{por}}$, partitioned into the fluid part $\tilde{\Omega}^{\mathrm{por}}_\mathrm{f}$ (light gray) and porous medium part $\tilde{\Omega}^{\mathrm{por}}_\mathrm{d}$ (dark gray).
  • Figure 3: Comparison of the models' relative errors.
  • Figure 4: Results of the optimization for the full 3D model.
  • Figure 5: History of the optimization process.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Definition 2.1
  • Theorem 2.2
  • Remark