尤拉梁正反算問題之創新無網格計算方法研究

Abstract

本論文發展出五套數值方法分別求解尤拉梁偏微分方程式的正反算問題，此正反算問題係屬於病態問題，乃是因為其解若存在，而其解卻不連續仰賴於量測數據點，而若因量測數據具有微量的干擾，其解必產生極大的誤差。雖有學者提出許多的改善此病態問題之方法，病態問題卻未能有效改善。於正算問題中，本文發展出兩套方法，分別為弱形式積分方程法以及配點法求解奇異攝動梁問題；在反算問題上，本文發展出三套方法，分別為穩定數值噪音數據微分器、非迭代之空間恢復力法以及邊界泛函法求解外力源恢復之梁問題。 奇異攝動梁問題指其材料本身之剛性遠小於外部張力之作用，因而衍伸出小參數的問題，但此問題不能夠直接以把小參數設為零而求近似解。若依常規攝動法把小參數設為零，將會導致方程降階，而不得該問題的近似解，故此為微小參數之變動所造成的病態問題。弱形式積分方程法先對四階常微分方程式採用弱形式方法進行積分，建構一個測試函數滿足梁的邊界，將常微分方程式的微分運算子利用分部積分法，將運算子作用於連續且可微之伴隨邊界測試函數上，並進一步推導出線性系統求解試函數的未知係數，此係數帶回即得梁位移之函數。指數多項式函數之配點法不同於弱形式積分方程法，此法採用指數多項式函數為解之基底，針對佈點代回四階常微分方程式而導出線性系統求解試函數的未知係數，此係數代回即得梁位移之函數。 梁外力恢復問題係指在已知位移而外力源未知之反算問題。此反算問題若於量測位移數據具有微量的干擾，再對其噪音數據微分，其外力源恢復的結果必產生極大的誤差。本文基於反算病態問題，提出穩定數值噪音數據微分器，此法運用了弱形式積分方程法之概念來解決數據微分噪音的反算問題，建構一個滿足邊界方程式的測試函數，並導入邊界型函數之概念，使外力試函數更具有彈性去描述外力源問題。非迭代之空間恢復力法，先採格林第二恆等式將域問題轉為邊界積分問題，透過選取伴隨特徵函數作為測試函數，在滿足控制方程式以及邊界條件下，此函數還避免了格林函數之奇異性；再者，在巧妙的選擇伴隨特徵函數與外力試函數，此法可求得係數的閉合解析解，在位移數據具有噪音干擾的情況下，有效恢復未知空間外力源問題。邊界泛函法則考量力與位移的功能轉換，並借助量測空間邊界外力作為附加條件，本法導出一系列的空間邊界函數，此空間邊界函數透過與隱格式時間函數的作用下，不僅滿足梁方程式的邊界條件，更能於時間上保守能。在所有邊界函數皆保守能的情況下，可建立一個包含零元素構成的線性空間。透過對線性系統求解空間邊界函數的未知係數，此係數帶回即得梁外力源之函數。 本文透過將空間上與時間上的數據點加入噪音，透過噪音的放大過程，可瞭解數據噪音對於正反算問題的敏感程度，並討論數據噪音在各種梁形式下，試函數以及測試函數展開項次的影響性，進而在描述梁受外力模擬時需仔細考量之因素。最後以解析解與方法論之數值模擬結果比較，驗證本方法論於正反算病態梁問題中，確實可有效且準確地求解空間與時間項的外力恢復源問題與提高計算精度，而對於帶有噪訊干擾的量測數據，進行深入的解析與探討該方法論對解的穩定性與計算誤差。

In this dissertation we investigate the ill-posed problems about the direct problem and the inverse problem in Euler-Bernoulli beam. The solution of direct problem is existent and unique; however, it does not rely on the continuously measurement data. On the other hand, the inverse problem, deducing a force distribution from the final measurement data, is highly sensitively dependent on the data. Despite that many researchers are dedicated to improve the solution of ill-posed problems, the recent results are still disappointing. In this dissertation, there are two methods adopted to deal with the direct problem of singular perturbed beam, and there are three methods adopted to deal with on the inverse problem of recovery force on the beam. The singularly perturbed boundary value problems (SPBVPs) in Euler-Bernoulli beam is dominated by tension instead of rigidity. It contains a small perturbing parameter in its highest order derivative term. The small parameter is not considered as zero. Otherwise, the order of solution would be reduced. The weak-form integral equation method which contains the exponential and polynomial trial solutions weaken the governed beam equation and subsequently translate the differential operator to the continuous test functions. The collocation method is associated with the exponentially and polynomially fitted basis functions. It is different from the weak-form integral equation method. Instead of translating the differential operator to the test function, this method is adopted to directly satisfy the fourth-order ordinary differential equation at the collocation points. The force recovery problem refers to the inverse problem of the measured displacement data with noise, where the external force is unknown. If the measurement displacement data has a small disturbance, the result of the recovery external force will lead a serious consequence on error. The purpose of this dissertation is finding an appropriate approach to the inverse problem. The stable numerical differentiator to the fourth-order derivative method has the same concept of weak form integral equation method. The test functions which satisfy the boundary conditions are established. In addition, the trial external force functions are described by boundary conditions and shape functions. With these two features, the force expansion trial functions are flexible to recover the external force. The simple noniterative method is adopted the adjoint eigenfunctions as the test functions. Impressively, the test functions automatically satisfy the beam equation and the boundary conditions in one. With orthogonality of the adjoint eigenfunctions, a closed-form solution of the expansion coefficients can be determined. Consequently, the noniterative method to recover the unknown force at final time displacement data is obtained. The boundary functional method (BFM) is considering the work translation between force and displacement. With additional measurement data on the spatial force at the initial and final time, a series of spatial boundary functions are derived. Impressively, the boundary functions satisfy the governing equation and boundary conditions in one. Moreover, with the aid of the implicit time functions, the energy of work can be conservative for any boundary functions. In the case where all boundary functions are conservative, a linear space consisting of zero elements can be created. By solving the unknown coefficients of the spatial boundary function, we can reconstruct the external force. The more noise amplified, the more complicated the direct problems and inverse problems. In this dissertation, the adaptation of expansion terms on test function and expansion should be reconsidered with different types of the beam while the measurement is polluted with random noise. By comparing numerical computation and exact solution, this dissertation has deeply discussed all the adoption, and demonstrate the applications under the random noise in direct and inverse problems.