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題名 | 黏彈理論中Prony指數級數的近似問題=The Curve-Fitting Problem of Exponential Prony Series in Viscoelastic Applications: A New Algorithm |
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作者 | 龔皇光; | 書刊名 | 正修學報 |
卷期 | 12 1999.07[民88.07] |
頁次 | 頁117-129 |
分類號 | 440.2 |
關鍵詞 | 黏彈理論; 連續未限制最小化法; 對應理論; Prony 指數函數級數; Viscoelasticity; Optimization theory; Prony series; Correspondence principle; Creep relaxation; |
語文 | 中文(Chinese) |
中文摘要 | 高分子材料黏彈理論中,由不可逆熱動力學 (irreversible thermodynamics) 可推導出由指數 函數 (exponential function) 組成的Prony級數來描述材料潛變 (creep) 和應力張弛 (relaxation) 的機 械性質。但有趣的是如何將這些材料實際的黏彈機械性質以Prony級數來近似,文獻上幾乎找不到 相關論文部份來探討這個問題。究其原因,此問題若排除其所含物理意義,將變成僅如何將實驗資 料以多項指數函數近似 (curve fitting) 的單純數值分析問題。但綜觀所有數值分析書中,都僅侷限 於單項指數函數近似,對多項指數函數如Prony級數的近似,文獻中只找到一篇以最佳化理論中的 連續未限制最小化法 (Sequential Unconstrained Minimization Technique, SUMT) 來處理這個問題。但 伴隨而來的複雜電腦程式撰寫,解的收斂性與數值解析度問題,使得連續未限制最小化法未必能獲 得滿意程度的近似。且隨著所選取指數函數項目的增加,其解的解析度愈來愈差,迭代近似解愈來 愈難收斂,繼續利用此法來獲得較精確解,幾乎變得不可行。 本文中提出一套新方法理論,可使此Prony級數的近似問題變的相當簡單。此新方法主要是利 用黏彈理論中遲滯時間分佈函數 (retardation time distribution function) 和張弛時間分佈函數 (relaxation time distribution function) 的物理意義對Prony指數函數級數的瞭解運用,再配合一般最小 化原理求解即可。本法除可避開解的數值解析度與收斂與否問題外,更可獲得較連續未限制最小化 法更準確的解。 |
英文摘要 | To illustrate the real viscous characteristics of polymeric materials in viscoelastic range, combinations of many Voigh-Kelvin and Maxwell elements may be employed. In viscoelastic range, i.e., Boltzmans' superposition principle applied, the general constitutive equations may be represented by Prony or modified Prony series. The (modified) Prony series is especially essential for the use of correspondence principle, the well-known Laplace-Transform method, for studying the viscoelastic problems, while the time dependency is taken into account. To express the experimental data of the composite properties by proper Prony series, a new algorithm is developed instead of the sequential unconstrained minimization technique (SUMT), which is one of the commonly used optimization technique. The SUMT method, employed to minimize the discrepancies between experimental data and Prony series solution, need to appropriately select the proper initial values, mainly obtained by trial and error, and usually has difficulty to converge. Furthermore, with increase in number of terms in Prony series, the SUMT method to solve the problem efficiently becomes questionable. In this study, a new curve-fitting scheme is proposed to obtain the proper Prony series to describe experimental data. With the discovery of physical relation between experimental data and exponential function in Prony series, the coefficients in the exponential terms may be determined firstly. As a result, the number of terms and weighting coefficients in Prony series can be obtained subsequentially by conventional numerical methods. According to current new technique, the convergence problem and inefficiently mathematical complication in SUMT method can be completely resolved. The algorithm developed in this study not only simplify the fitting procedures mathematically but also increase the accuracy of solutions. In addition, this scheme provides a great tool in dealing with the transform problem of experimental data in correspondence principle. |
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