分段連續線性化降階演算法在非線性微熱傳效應之應用

Abstract

摘要

本篇論文討論片段線性模型降階法（trajectory piecewise linear model order reduction, TPWLMOR）在非線性熱傳模型的應用。首先我們發展出一個三次元非線性有限差分法（FDM）的熱傳計算程式，並考慮温度相依的材料性質及熱輻射效應的影響。之後，我們用片段線性模型降階法將FDM產生的數值模型降階成精簡模型。片段線性模型降階法是從片段線性簡化及Arnoldi降階演算法的觀念而來。Arnoldi演算法所產生的線性模型將用來找出線性展開的平衡點，根據這些平衡點所展開的線性模型將依權重而重新組合成為片段線性模型。最後再應用降階演算法，將片段線性模型縮降為片段線性精簡模型。 接下來應用片段線性模型降階法技巧來分析二個不同的微機電(MEMS)的熱致動器。穏態及暫態的熱傳特性，皆能由片段線性精簡模型所計算出的温度值準確並有效率的分析出來。此外，片段線性模型降階法在計算的效率，約為原來非線性模型的100倍。實驗量測結果，亦驗證片段線性精簡模型模擬的結果。本研究的成果顯示片段線性模型降階法可有效率的精確分析非線性的熱傳效應。

ABSTRACT

In this paper, we present a nonlinear heat-transfer macromodeling technique using the trajectory piecewise linear model order reduction (TPWLMOR) method. A 3D nonlinear heat-transfer model, which is capable of accounting for the temperature-dependent material properties as well as radiation effect, is implemented using the finite difference method (FDM). The numerical models generated by the FDM are reduced into compact models using the TPWLMOR technique, which is based on the concept of piecewise-linear approximation and an Arnoldi-based model order reduction (MOR) algorithm. Nonlinear macromodeling case studies of different MEMS thermal devices are demonstrated using the TPWLMOR technique. The calculated steady and transient characteristics of the thermal devices are discussed. In terms of computational cost, the TPWLMOR models are at least 2 orders of magnitude faster than the original nonlinear full-meshed models with negligible compromise in accuracy. The simulated results by the TPWLMOR models are also verified with the experimentally measured results. Keywords: model order reduction, piecewise linear, macromodel, system-level modeling, Arnoldi algorithm