Optimal Cooling Control of a Sphere

在物體的降溫過程中，邊界溫度下降的快慢會影響物體內部溫度梯度的大小。對於同一個物體而言，邊界溫度降得快溫度梯度大，溫度降得慢則溫度梯度小。本文目的是將最佳化控制理論用在一個簡化的熱傳導系統上，以得到某種快速但不至於使物體內部溫度梯度過大的降溫過程。我們定義一個目標函數表示最佳化所針對的是溫度和溫度梯度，並以權重值調整最佳化中溫度以及溫度梯度效應的比重。由於溫度和溫度梯度必須符合熱傳導定律，因此我們引入Lagrange乘數並擴增目標函數。為了導出最佳化的必要條件並算出目標函數的極小值，我們對目標函數作變分，由此得到由Euler-Lagrange方程式和熱傳導方程式所組成的聯立PDE與邊界條件，稱為最佳化熱傳導系統。接著我們以牛頓迭代法修正並計算出最佳化熱傳導系統的正確解。從與其他形式的降溫過程比較後，證實我們可由最佳化控制理論所得到符合要求的最佳降溫過程。In a cooling process,the rate of surface temperature drop will affect the magnitude of temperature gradient in the body. The sooner the surface temperature is lowered,the larger the temperature gradient is,and vice-versa.The objective of this thesis is to apply Optimal Control theory to a simplified heat conduction system to obtain an optimal way of cooling in a sense of archiving a fast temperature drop while not causing large temperature gradient in the solid body.We first define an functional constructed by temperature and temperature gradient as an object to be optimized. Since the temperature and temperature gradient must conform to the law of heat conduction, a Lagrangeultiplier is introduced and the object function is augmented with the multiplier.In order to derive the necessary conditions for optimization, method of Variation is applied to the object function, and a system of PDEs(composed of an Euler-Lagrange equation and heat conduction equation) and relevant boundary conditions are obtained. The equations are then solved numerically for the required optimal temperature drop on the body surface.After comparing with other pre-specified processes,it is confirmed that the resulting cooling process is indeeded an optimal one as expected.中文摘要...................................................................................................................i文摘要.................................................................................................................i i錄..........................................................................................................................iii圖目錄.......................................................................................................................v目錄......................................................................................................................vi一章 序論.....................................................................................................1二章 基本理論..........................................................................................3 2-1 系統方程式......................................................................................3 2-2 無因次化...........................................................................................6 2-3 系統模型...........................................................................................7三章 最佳化控制..................................................................................9 3-1 目標函數...........................................................................................9 3-2 限制條件..............................................................................................10 3-3 變分法..............................................................................................11 3-4 最佳化系統模型...........................................................................15四章 數值方法........................................................................................17 4-1 值的討論........................................................................................17 4-2 Global收斂......................................................................................18 4-3 解法和流程.......................................................................................20 4-4 轉換式................................................................................................22五章 結果與討論...................................................................................26 5-1 不同權重的比較..............................................................................26 5-2 與其它行程的比較.........................................................................30六章 結論與未來研究方向..............................................................32-1 結論.....................................................................................................32-2 未來研究方向..................................................................................33考文獻................................................................................................................34目錄(2.1) 球體在 和 時的溫度................................................................4(2.2) 球座標..........................................................................................................5(2.3) 數學模型.....................................................................................................7(3.1) 數學模型(最佳化模型) ........................................................................15(5.1) 不同權重值之下的球體邊界溫度曲線...........................................27(5.2) 不同權重值之下的球體邊界溫度梯度曲線..................................27(5.3) 球內溫度分布隨時間變化曲線(權重值0.02) ..............................28(5.4) 球內溫度分布隨時間變化曲線(權重值0.1) .................................29(5.5) 球內溫度分布隨時間變化曲線(權重值0.5) .................................29(5.6) 最佳化和其它行程的邊界溫度曲線比較.......................................30(5.7) 最佳化和其它行程的邊界溫度梯度曲線比較.............................31目錄(5.1) 最佳化和其它行程的 值比較..........................................................3