Abstract
高分子流體力學是高分子加工過程中有必要加以研究探討的課題。隨著電腦軟硬體技術以及數值方法的發展，吾人得以採用數值模擬的方式求解高分子流體力學問題，協助吾人進行深入的研究。其中尤以黏彈性流體流動問題的求解更具有挑戰性，隨其流動彈性效應的增強，數值方法出現不穩定甚至發散的現象。本論文的目的即為發展具有準確及有效率的數值方法，以求解黏彈性流體的流動問題。本論文內容大約涵蓋四個部份：(1)黏彈性流體數值模擬的現況與回顧(2)頻譜元素法的理論與施行(3)運用頻譜元素法於對流-擴散問題以及牛頓流體流動問題(4)運用頻譜元素法於黏彈性流體流動問題。本研究首先發展了可變階數型頻譜元素法。頻譜元素法將ｐ型頻譜方法的準確性及快速收斂性與ｈ型有限元素法的幾何彈性加以有機結合，因此對於變量梯度明顯以及具內層及邊界層的流場具有極高解析能力。本文將傳統的協調型算法加以改善，使其具有可變階數能力，如此於流場計算中可採用不同階數元素，可以提高局部解析能力並節省不必要的計算量，因此大幅提升了頻譜元素法的使用效率。本論文將此一非協調型算法運用於許多計算流體力學問題，均獲致相當的成功。本研究同時引入高階迎風加權函數，以穩定在對流效應╱彈性效應明顯的情形時可能引發的計算不穩定性。最小平方加權餘值法，協調型加權餘值法(SUPG)以及非協調型加權餘值法(SU)均加以引進並測試性能。在黏彈性流體求解方面，除原始的MIX1型式法外，尚採用了EEME列式法於頻譜元素法的計算架構中。由數值實驗發現，EEME列式法可得到較高的彈性收斂上限，配合SU/SUPG 迎風加權，當可進一步提高此一算法之計算穩定性。Variable-order spectral element method is first developed inthis study. Spectral element method combines the accuracy andfast convergence rate of p-type spectral methods with thegeometrical flexibility of h-type finite element method. It'spowerful to the resolution of complicated flow field wheresharp solution gradient、inner of boundary layer of solutionmay appear. In this work, a variable-order version is developedto promote the flexibility and effcicency of the spectralelement method. Depending on resolution requirement, variable-order elements could be selected for different region, thus anaccerate solution can be obtained without expending too manycomputational resource. The performance of this method isexamined by many flow problems of Newtonian fluids. Improvedflexibility and efficiency of this method make it a highlypromising candidate for solving practical fluid dynamicproblems. A p-type streamline-upwinding weighted residualmethod is also introduced to stabilize the numericalinstability appeared in the convection /elastic effectdominanting situations. Least-squares weighted residualformulation, consisitent (SUPG) and non-consistent (SU)streamline-upwind formulations are introduced to examine theirperformance. A comparison is also made of the conventional MIX1formulation with that of the EEME formulation of the upper-convected Maxwell (UCM) flud flow problem in the context ofspectral element method. Numerical experiments indicate thatEEME/SU and EEME/SUPG-spectral element methods are numericallystable and convergent for the test problem. The upper limit ofnumerical convergence should be further promoted if a properstreamline-upwinding and sufficiently high spectral orders areadopted in the calculation.