A prediction-correction based iterative convolution-thresholding method for topology optimization of heat transfer problems

TL;DR

The paper addresses topology optimization for steady-state heat conduction by representing material distributions with a BV indicator $\\chi$ and minimizing a constrained energy $J^\\tau$ that couples complementary energy with a Gaussian-based perimeter regularization. It proposes a prediction-correction based iterative convolution-thresholding method (ICTM) that enforces monotone decay of the objective by adding a correction step to the standard ICTM; this mitigates oscillations observed under multi-physics constraints. The approach hinges on Gaussian kernel convolutions to approximate interface length and a convex-hull relaxation for efficient optimization, while maintaining a fixed volume fraction via a projection-like thresholding rule. Numerical experiments in 2D and 3D demonstrate robust convergence, parameter-robust feature resolution, and smooth decays of the objective, underscoring the method’s potential for broader multi-physics topology optimization tasks.

Abstract

In this paper, we propose an iterative convolution-thresholding method (ICTM) based on prediction-correction for solving the topology optimization problem in steady-state heat transfer equations. The problem is formulated as a constrained minimization problem of the complementary energy, incorporating a perimeter/surface-area regularization term, while satisfying a steady-state heat transfer equation. The decision variables of the optimization problem represent the domains of different materials and are represented by indicator functions. The perimeter/surface-area term of the domain is approximated using Gaussian kernel convolution with indicator functions. In each iteration, the indicator function is updated using a prediction-correction approach. The prediction step is based on the variation of the objective functional by imposing the constraints, while the correction step ensures the monotonically decreasing behavior of the objective functional. Numerical results demonstrate the efficiency and robustness of our proposed method, particularly when compared to classical approaches based on the ICTM.

Figures (25)

• Figure 1: $\Omega_1\in \Omega$ is the region of high conductive material with low heat-generation rate. $\Omega_2=\Omega\setminus\Omega_1\in\Omega$ is the region of low conductive material with high heat-generation rate, and it represents an area of the device that is filled with electrical components (cf. ikonen2018topology). See Section \ref{['sec:heat-transfer']}.
• Figure 2: The results uses ICTM on a $600\times 600$ grid. The parameters are set as $\tau = 1\times10^{-4}$, $\gamma = 30$ and $\xi=1\times10^{-5}$. Left: The snapshot of $\chi$ at the $99$-th iteration. Middle: The snapshot of $\chi$ at the $100$-th iteration. Right: The change of objective functional values during iterations. See Section \ref{['sec:failureICTM']}.
• Figure 3: The process of prediction-correction. Left: the $k$-th iteration. Middle: the prediction using ICTM ( i.e., sets $A$ and $B$). Right: the correction for the $(k+1)$-th iteration ( i.e., sets $A_1$ and $B_1$). See Section \ref{['sec:pcICTM']}.
• Figure 4: The initial distribution of $\chi_1.$ See Section \ref{['Area-to-point_ex']}.
• Figure 5: The optimal results with $\gamma = 50$, $\kappa_1=10,\ \kappa_2=1,\ q_1=1,\ q_2=100$, volume fraction $\beta = 0.2$ on a $600\times 600$ grid using ICTM. Left: Approximate optimal solution. Right: Objective functional curves. See Section \ref{['sec:comparison']}.

Theorems & Definitions (6)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3
• Remark 3.4