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Shape Sensitivity Analysis for a Microchannel Cooling System

Sebastian Blauth, Christian Leithäuser, René Pinnau

Abstract

We analyze the theoretical framework of a shape optimization problem for a microchannel cooling system. To this end, a cost functional based on the tracking of absorbed energy by the cooler as well as some desired flow on a subdomain of the cooling system is introduced. The flow and temperature of the coolant are modeled by a Stokes system coupled to a convection diffusion equation. We prove the well-posedness of this model on a domain transformed by the speed method. Further, we rigorously prove that the cost functional of our optimization problem is shape differentiable and calculate its shape derivative by means of a recent material derivative free adjoint approach.

Shape Sensitivity Analysis for a Microchannel Cooling System

Abstract

We analyze the theoretical framework of a shape optimization problem for a microchannel cooling system. To this end, a cost functional based on the tracking of absorbed energy by the cooler as well as some desired flow on a subdomain of the cooling system is introduced. The flow and temperature of the coolant are modeled by a Stokes system coupled to a convection diffusion equation. We prove the well-posedness of this model on a domain transformed by the speed method. Further, we rigorously prove that the cost functional of our optimization problem is shape differentiable and calculate its shape derivative by means of a recent material derivative free adjoint approach.

Paper Structure

This paper contains 15 sections, 12 theorems, 84 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Basic Notations
  4. Mathematical Model
  5. The Optimization Problem
  6. Shape Calculus
  7. Basic Results
  8. A Material Derivative Free Adjoint Approach
  9. Analysis of the State System
  10. Reformulation as an Abstract Problem
  11. Well-Posedness of the Abstract Problem
  12. Shape Differentiability of the Problem
  13. Relation to the Abstract Differentiability Result of Section \ref{['sec:sturm']}
  14. Verification of the Conditions of Theorem \ref{['thm:sturm']}
  15. Conclusion and Outlook

Key Result

Lemma 3.2

Let $\tau > 0$ be sufficiently small, $\mathcal{V} \in C^k_0(\mathbb{R}^d;\mathbb{R}^d)$ with $k\geq 1$, and $\Phi_t$ be its associated flow.

Figures (2)

  • Figure 1: Two-dimensional slice of the domain $\Omega$, as used in blauth, with inlet (green, top left), wall boundary (gray), and outlet (red, bottom right). The subdomain $\Omega^\text{sub}$ (orange) corresponds to the region of the microchannels.
  • Figure 2: Optimized geometry of the cooling system from blauth showing the temperature of the coolant.

Theorems & Definitions (29)

  • Definition 3.1
  • Remark
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Remark
  • Lemma 4.2
  • ...and 19 more