利用 MS Excel 的 Solver 進行曲線擬合 (Curve Fitting Using Excel Solver)

當我們進行數據分析時，常常碰到「找最佳參數值」的狀況。以下介紹如何利用 Microsoft Excel 內建的 Solver (規劃求解)，快速完成黏度曲線擬合。

擬合公式為 Carreau-Yasuda 模型

(1)

此模型有五個參數，分別為 η_∞、η_0、λ、a、n，實驗數據如 Fig. 1 的黑色圓點。經使用 Excel Solver 求解後，可以得到 Fig. 1 的紅色曲線 (五個參數的最佳擬合值可見 Fig. 2 之結果)。對細節有興趣者，可參考文件 https://pages.mtu.edu/~fmorriso/cm4650/Using_Solver_in_Excel.pdf。

Figure 1

Figure 2

(註：在啟用規劃求解功能前，需至 Excel 的增益集新增該功能)

書籍 - 旋元佑英文字彙

字源大挪移一書由英文魔法師旋元佑所著的字彙書 (4 CDs)，約在 10 年前就絕版了，書中以字首、字根、字尾分門別類，有系統地逐一介紹，以幫助記憶 (學單字這樣背就對了!)。

好消息是，在今年年底，該書又重新出版，書名為旋元佑英文字彙。維持原來的風格以字首、字根分類，也拆成通用學術字彙、托福 TPO 字彙兩部分。另一個新增的特色為，針對例句中的搭配詞，作者使用底線標註，方便讀者記憶。此書已可於博客來訂購 (另有旋元佑文法書，也是再版)，書中的例句 MP3 免費下載網址 https://www.jwbooks.com.tw/DL/。

舊版的封面

舊版的內頁

2019 新版封面

書籍 - 朗文英文文法全集

由日本人寫的一本文法書，內容清楚且富含有趣的插圖說明。

Phan-Thien-Tanner (PTT) 黏彈模型預測之單軸拉伸黏度 (PTT Model Response in Elongation)

於單軸拉伸流場 (uniaxial elongational flow)，linear- 和 exponeitial-PTT 模型預測之拉伸黏度分別如 Figs. 1、2 所示。由計算結果發現，參數 ε 主要決定應變硬化的程度 (strain hardening)。

當 ε 越小，應變硬化程度越明顯；然而，ξ 對拉伸流場的預測影響不大 (虛線)。因此，ε 主要主導拉伸流的響應，需從拉伸黏度決定之，而 ξ 可從線性黏彈性和剪切黏度數據估計之 。
[註: 於剪切流場下，當 ε 越小，剪切致稀發生在越高的剪切速率]

Figure 1 Linear-PTT 模型的之預測 (ε = 0.01, 0.1, 1, 10 and ξ = 0 or 0.1)

Figure 2 Exponential-PTT 模型之預測 (ε = 0.01, 0.1, 1, 10 and ξ = 0 or 0.1)

由 Fig. 6-5 的比較可知，在高拉伸速率下，linear-PTT (1) 預測穩態拉伸黏度達到平台區；反觀，exponential-PTT (2) 則預測拉伸黏度有一最大值。因此，exponential-PTT 較適於描述實驗上觀察到的熔體單軸拉伸黏度數據。

Reference: RG Larson, Constitutive Equations for Polymer Melts and Solutions (Butterworths 1988).

Chapter 7 Defining Your Own Date Types

第 7 章基本 C++ 程式碼整理如下

Reference: I Horton, Ivor Horton's Beginning Visual C++ 2013 (Wiley 2014).


7.1 The Struct in C++
∎ Struct
// Struct definition
struct Rectangle
{
    int left;
    int top;
    int right;
    int bottom;
};


7.2 Types, Classes, Objects, and Instances
∎ Class
// Definition of a class
class CBox
{
    public:
        double m_length; // Data members
        double m_width;
        double m_height;
};

∎ Class
// Main.cpp
CBox box1;  // Declare object box1 of type CBox
CBox box2;

if (box1 > box2)  // ">" is an operator acting on the objects
    box1.fill();  // "fill()" is a member function acting on object 1
else
    box2.fill();


7.3 Understanding Classes
∎ Accessing the data members of a class
// Main.cpp
CBox box1;
CBox box2;
box1.m_length = box1.m_width = box1.m_height = 2.0;
box2 = box1;  // Member-wise copying from box1 to box2

∎ Adding a member function to CBox
// Member function
double volume()
{
    return m_length * m_width * m_height;
}


7.4 Class Constructors
∎ Adding a Constructor to the CBox class  
// Constructor definition
CBox(double lv, double wv, double hv)
{
    cout << "Constructor called." << endl;
    m_length = lv;  // Set values of data members
    m_width = wv;
    m_height = hv;
}

∎ Default constructor
// Default constructor definition
CBox()
{
    cout << "Default constructor called." << endl;
}

∎ Constructor initialization list
// Constructor definition
CBox(double lv = 1.0, double wv = 1.0, double hv = 1.0) :
             m_length {lv}, m_width{wv}, m_height{hv}
{
    cout << "Constructor called." << endl;
}
cf. CCandyBox(double lv, double wv, double hv, const char* str = "Candy") :
             : CBox {lv, wv, hv}  // In the inherited class CCandyBox

∎ Making a constructor explicit 
// Make a constructor more specific, instead of generalCBox box;
box = 99.0;
CBox(double side) : m_Length {side}, m_Width {side}, m_Height {side}  {}  // Will compile
explicit CBox(double side) : m_Length {side}, m_Width {side}, m_Height {side} {}  // Will not, unless CBox box {4.0} is used 


7.5 Private Members of a Class
∎ Private data members
// CBox.h
class CBox
{
    private:
        double m_length;
        double m_width;
        double m_height;
};

// Main.cpp
box1.m_length = 4.0; // Error message: cannot access private data member declared in class CBox

∎ Friend function of a class
// Friend function prototype (in CBox.h) 
friend double boxSurface(const CBox& aBox);

// A function that is not a member of the class, but with special privilege
double boxSurface(const CBox& aBox)
{
    return 2.0 * (aBox.m_length * aBox.m_width + aBox.m_width * aBox.m_height
               + aBox.m_height * aBox.m_length); 
}

∎ Copying information between instances
// Initialize box2 with box1 (done by the default copy constructor)
CBox box2{box1};

// .h
CBox (const CBox& initB);

// .cpp
CBox::CBox(const CBox& initB):
            m_length{initB.m_length}, m_width{initB.m_width}, m_height{initB.m_height}


7.6 The Pointer This
∎ Explicit use of this
// main.cpp
if (cigar.compare(match))
    cout << "cigar is larger than match" << endl;
else
    cout << "cigar is not larger than match" << endl;

// .cpp
bool compare(CBox& aBox)
{
    return this->volume() > aBox.volume();
}



7.7 Const Objects
∎ Constant member functions of a class
// Non-const object
CBox cigar{8.0, 5.0, 1.0};
bool CBox::compare(CBox& aBox)

// const object
const CBox cigar{8.0, 5.0, 1.0};
bool CBox::compare(const CBox& aBox) const

∎ Member function definitions outside the class


7.8 Array of Objects


7.9 Static Members of a Class

∎ Counting instances of a class type (ex7_12)
static int objectCount;  // Count of objects in existence
int CBox:: objectCount{};  // Initialize static data member of CBox class; no keyword "static"


7.10 Pointers and References to Objects


7.11 Summary