直拉式单晶炉PDE降维模型的生长界面温度控制

doi:10.12034/j.issn.1009-606X.216331

过程工程学报 ›› 2017, Vol. 17 ›› Issue (3): 571-577.DOI: 10.12034/j.issn.1009-606X.216331

直拉式单晶炉PDE降维模型的生长界面温度控制

丛其然¹，闫鹏^2*

1. 北京航空航天大学自动化科学与电气工程学院，北京 100191；2. 山东大学机械科学与工程学院，山东济南 250061

收稿日期:2016-10-24 修回日期:2016-12-06 出版日期:2017-06-20 发布日期:2017-06-14
通讯作者: 闫鹏 15811442058@163.com
基金资助:
面向量子功能器件制备的直写式真空蒸发系统

Interface Temperature Control of Czochralski Crystal Furnace Based on Novel Reduced Order PDE Model

Qiran CONG¹， Peng YAN^2*

1. School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China;
2. School of Mechanical Engineering, Shandong University, Jinan, Shandong 250061, China

Received:2016-10-24 Revised:2016-12-06 Online:2017-06-20 Published:2017-06-14

摘要/Abstract

摘要：

基于抛物线型偏微分方程(PDE)研究了直拉式单晶炉中生长界面的温度控制，将Galerkin法应用于PDE的处理，求解中使用空间映射对较复杂形状的空间区域进行简化，并用Karhunen-Loève (KL)分解法求取经验特征方程；针对非齐次边界条件，设计了一种高效的模型分解方法，基于优化模型引入最优控制方法实现了温度控制，验证了模型的有效性.

关键词: 晶体生长, KL-Galerkin法, PDE模型分解, 最优控制

Abstract:

In order to deal with the interface temperature control problem of Czochralski crystal furnace, the associated parabolic partial differential equation (PDE) model is proposed, where the Galerkin method is used for the parabolic PDE order reduction. Furthermore Karhunen-Loève (KL) decomposition method is applied to generate the optimal empirical eigenfunctions, where the irregular domain is mapped to a fixed rectangular domain to facilitate the decomposition process. A method is proposed that decomposes the PDE into a homogeneous part and inhomogeneous parts to improve the computation efficiency. Optimal control method with the proposed model is employed where numerical simulations are provided to verified the effectiveness.

Key words: Crystal remelt, KL-Galerkin method, PDE model decomposition, optimal control

丛其然闫鹏. 直拉式单晶炉PDE降维模型的生长界面温度控制[J]. 过程工程学报, 2017, 17(3): 571-577.

Qiran CONG Peng YAN. Interface Temperature Control of Czochralski Crystal Furnace Based on Novel Reduced Order PDE Model[J]. Chin. J. Process Eng., 2017, 17(3): 571-577.

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直拉式单晶炉PDE降维模型的生长界面温度控制

Interface Temperature Control of Czochralski Crystal Furnace Based on Novel Reduced Order PDE Model

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