應用逆向估測法於二維熱彈性問題之研究

The Study of Two Dimensional Thermoelasticity Problems by Inverse Estimation Method

二維熱傳導的溫度場藉由數值方法或實驗量測可以直接獲得。本研究採用逆向估測法，在變形量不易量測之條件下，量測模擬平板某處的溫度以計算其變形量或熱應變。本研究使用有限元素法，離散熱傳導方程式，在考量週遭干擾雜訊之統計下，建立卡爾曼濾波器中的狀態方程式與量測方程式。利用卡爾曼濾波器預測模擬平板中各節點的溫度，再來量測模擬平板的溫度更新量測點的預測溫度，同時濾波量測雜訊的共變異量。配合最小平方遞迴法從卡爾曼濾波器的狀態方程式中估測模擬平板所受的未知熱通量，便可以獲得所需的溫度場。並由最小勢能原理得到變形量與外力的關係。使模擬平板中各節點的溫度、變形量和元素的熱應變或熱應力能一併呈現在線上。結果顯示在模擬平板量測一點溫度，進而估算其熱應變或熱應力。其估測結果與正向解的差異甚小。 The temperature distribution can be solved directly by the numerical methods or by the experimental measurements for the two-dimensional heat conduction problems. For most of the times, one cannot measure the thermal deformation easily. we calculated the thermal deformation and the thermal strain by using measuring temperature at one point of a simulated flat plate We use finite element analysis to discretize the two-dimensional heat conduction equation, and to transfer heat conduction equation into the state equation and the measurement equation of the Kalman filter under considered surrounding noises. The Kalman filter predicts the temperature value of node in simulated plate. and innovates the predicted temperature with measuring the updated temperature in simulated plate It also filters out the measurement noises covariance The recursive least square algorithm were included to this method which inversely estimates the unknown heat flux by state equation of Kalman filter, so that we can get the exact temperature distribution For estimation of thermal deformation and thermal strain, the related matrices, such as stiffness and forces, of deformation and forces, were constructed by the principle of minimum potential energy. In this research, the two-dimensional simulated plate was used as the examples. The inverse estimate results are compared with the forward calculated solution, which shows very small value deviations.