基於多尺度技術的穩態熱分析域分解方法

Abstract

本論文針對現代超大規模積體電路（VLSI）設計中所面臨的熱模擬挑戰提出了一套解決方案。隨著積體電路在結構複雜度與功率密度上的不斷提升，傳統熱分析方法因記憶體需求過大與運算時間過長而變得效率低下。為克服此一瓶頸，本研究引入基於域分解與多尺度技術的新穎框架，將整體熱問題拆分為數個可獨立計算的子域分別計算，降低峰值記憶體用量。模擬結果顯示，與傳統有限元素分析相比，所提出之求解器在記憶體使用量上可減少 7.79 倍，同時在誤差極小的前提下維持高準確度。此外，運算效能的顯著提升，亦使得包含數百萬元素之IC封裝模擬成為可能。 本論文另一項核心貢獻在於構建了一個具高度彈性的元方法（meta-method）框架，容許以不同方法替換核心有限元素求解器，例如整合商用電子設計自動化（EDA）工具或基於機器學習的偏微分方程求解器，以進一步提升整體效能。本論文的多樣化策略不僅加速了迭代求解過程的收斂，亦為未來整合更多物理現象（如電行為與機械應力）於多物理場模擬中奠定了堅實基礎。

This thesis presents a solution to the thermal simulation challenges inherent in modern VLSI design. As integrated circuits grow in complexity and power density, traditional thermal analysis methods become inefficient due to excessive memory requirements and prolonged runtimes. To overcome these limitations, this work introduces a novel framework based on domain decomposition and multiscale techniques that partitions the global thermal problem into manageable subdomains. Simulation results demonstrate that the proposed solver achieves a 7.79× reduction in memory usage compared to conventional finite element analysis, while maintaining high accuracy with minimal error. In addition, significant improvements in runtime performance enable efficient simulation of IC packages comprising millions of elements. A key contribution of this thesis is the development of a flexible, meta-method framework that supports the replacement of the core finite element solver with alternative approaches, such as commercial EDA tools or machine learning–based PDE solvers, to further enhance performance. This versatility not only accelerates the convergence of the iterative solution process but also opens the door to integrating additional physical phenomena, such as electrical behavior and mechanical stress, in future multiphysics simulations.