A High-Order Explicit Lie-Group Scheme on Solving the Three-Dimensional Nonlinear Backward Heat Conduction Problems in Time and Fictitious Time Domains

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Project title

Code/計畫編號

MOST107-2221-E019-029

Translated Name/計畫中文名

高階顯式李群法求解時間域及擬時間域三維非線性反向熱傳導方程

Project Coordinator/計畫主持人

Yung-Wei Chen

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Marine Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=12675257

Year

2018

Start date/計畫起

01-08-2018

Expected Completion/計畫迄

31-07-2019

Bugetid/研究經費

573千元

ResearchField/研究領域

機械工程

本計劃擬提出一完整高階顯式李群型式積分法求解三維時間反向非線性熱傳導問題。此一反算問題的困難在於反算問題的解若存在，會高度依賴給定的資料，特別是當量測資料有微小擾動時，將造成很大的震盪在於求解上。本計畫針對擬時間計算域及歐式空間計算域進行求解高維度及長時間數值驗證，實現理論中一步保群法之數值實驗，進一步突破國際間反問題求解程序必須利用疊代進行求得穩定解。本計畫第一年提出擬時間域『正-反算擬時間積分法』求解長時間三微熱傳反算問題，此方法透過擬時間變換與高階李群積分法建構一步演算法，稱前向擬時間積分法與後向擬時間積分法。首先，原本非線性與非齊次熱傳方程式透過擬時間變數變換為演化型式之熱傳方程式。因為傳統閩式空間下保群算法無法滿足光錐構造、李代數與李群特性限制條件，演算法之擬時間步長與黏性阻尼係數在計算過程將呈現非線性行為，且造成演算法之參數無法利用線性技巧將其線性化，導致解的收斂情況高度相依於黏性阻尼與擬時間步長而無法收斂。因此，本計畫將離散代數方程式之數值積分法在每一擬時間步長必須滿足上述三個限制條件。最後，本計畫提出具有物理架構之演算法進行求解，利用一極小擬時間步長促使李群積分法沿著微分流行一步收斂進而求得解。第一年計畫中，正-反算擬時間積分法分別針對「時間-空間相依」與「時間-空間獨立」的數值算例將被進行測試。此「時間-空間相依」問題直到目前尚未被完整解決及提出克服方法。本計畫第二年提出歐式空間下時間方向李群打靶法結合擬時間積分法求解長時間多維度反算熱傳導問題。對於此病態問題本計畫利用半離散差分法進行求解。透過離散技巧將原方程式透過高階歐式空間李群積分法往時間方向積分求解待定之初始值。李群打靶法根據李群元素G(tf)與廣義值原理G(r)進行建構. 根據G(tf)=G(r)，透過單參數r進行求解待求初始條件。然而李群打靶法當時間增加，仍然無法避免長時間計算與高維度噪音傳遞。其主要原因單參數G(r)進行求解時將發生三種特殊解情況：複數值、發散、及零解，將導致保群算法無法維持上述三種限制條件。本計畫重新推導滿足三種限制條件之二次方程的解，以維持李群積分法維持在光錐之微分流形上。李群打靶法結合正-反算擬時間積分法之數值精度、效率及噪音影響將被驗證與比較先前參考文獻。This project will develop the completely high-order explicit type Lie-group integral method for solving nonlinear three dimensional backward heat conduction problems (BHCPs). If the existence of a solution for an inverse heat transfer problem can be assured due to physical reasons, the requirement of uniqueness can only be formally proved for some special cases. Also, the solution of the inverse problem is highly dependent on the input data. Small perturbations in the input data, like random errors inherent to the measurements in the analysis, can cause large oscillations on the solution. In this project, the BHCPs in multi-dimensional and long time span problems are dealt with in time domain and fictitious time domains by the forward-backward fictitious time integral method (FTIM) and Lie group shooting method (LGSM). Numerical experiments will be conducted to verify the theory of one-step Lie group schemes, which can obtain efficient and stable solution, and do not need numerical iteration. In the first year, the FTIM will combine algebraic equations with a high-order Lie-group scheme to construct one-step algorithms, called the backward FTIM (BFTIM) and the forward FTIM (FFTIM). First, the original nonlinear and nonhomogeneous heat conduction equation can be transformed into a new heat conduction equation of an evolution type by introducing a fictitious time variable. Because the group preserving scheme (GPS) in Minkowski space cannot satisfy the constraints of the cone structure, Lie-group and Lie algebra such that these constraints for the fictitious time and viscosity-damping coefficient will result in nonlinear behaviors and cannot use linear technique to avoid their occurrence. Therefore, the numerical integration of the discretized algebraic equations must satisfy above three constrains at each fictitious time step. Finally, the present algorithms with the minimum fictitious time step along the manifold of the Lie-group scheme approach the true solution with one step. Several numerical examples, space-time dependent and -independent solution, will be tested by the BFTIM and the FFTIM.In the second year, this project will develop the time-direction LGSM combined with FTIM for solving multi-dimensional backward heat conduction problems　under long time span. The ill-posed problem is analysed by using the semi-discretization numerical schemes. The resulting ordinary differential equations in the discretized space are numerical integrated towards the time direction by the Lie-group shooting method to find the unknown initial conditions. The LGSM is based on a one-step Lie group element G(tf) and a generalized mid-point Lie-group element G(r). Then, according to G(tf)= G(r), we can search for the missing initial conditions through the weighting factor. However, the LGSM still fails when the computational time increases. The main reasons are that a real single-parameter G(tf) will lead to three conditions: zero, infinite and complex number, and the GPS cannot preserve above three constrains in time direction. Therefore, we will derive a solution of the quadratic equation for the three conditions and satisfy the three constrains of the GPS. The accuracy and efficiency of the present method will be validated by comparing the estimating results with previous literatures even under noisy measurement data.

Keyword(s)

熱傳導方程式
逆問題
病態問題
疊代法
Heat conduction equation
Inverse problem
Ill-posed problem
Iterative regularization method

DSpace CRIS

A High-Order Explicit Lie-Group Scheme on Solving the Three-Dimensional Nonlinear Backward Heat Conduction Problems in Time and Fictitious Time Domains

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