# Curve Fitting Theory¶

The section shows you the theoretical details of each curve or function.

## Basic Curves¶

The group of Basic contains all commonly used curves.

### Straight line¶

The function of this curve is given by

$y(x)=a+bx$

where $$a$$ and $$b$$ are constants to fit, $$x$$ and $$y$$ are the test data pair. This function is also called 1st order polynomial.

### Natural logarithm¶

The function of this curve is given by

$y(x)=a+b \cdot ln(x)$

where $$a$$ and $$b$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

Note

Independent variable $$x$$ must be larger than zero.

### Exponential¶

The function of this curve is given by

$y(x)=ae^{bx}$

where $$a$$ and $$b$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

Note

Dependent variable $$y$$ must be larger than zero.

### Power¶

The function of this curve is given by

$y(x)=ax^{b}$

where $$a$$ and $$b$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

Note

Variables $$x$$ and $$y$$ must be larger than zero.

### Gaussian¶

The function of this curve is given by

$y(x)=a \exp{(-\dfrac{(x-b)^2}{2c^2})}$

where $$a$$, $$b$$, and $$c$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

Note

Dependent variables $$y$$ must be larger than zero.

## Polynomial Curves¶

The group of Polynomial contains polynomial curves. The first-order polynomial is located in the Basic group as Straight Line.

### 2nd Order Polynomial¶

The function of this curve is given by

$y(x)=a+bx+cx^2$

where $$a$$, $$b$$, and $$c$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### 3rd Order Polynomial¶

The function of this curve is given by

$y(x)=a+bx+cx^2+dx^3$

where $$a$$, $$b$$, $$c$$, and $$d$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### 4th Order Polynomial¶

The function of this curve is given by

$y(x)=a+bx+cx^2+dx^3+ex^4$

where $$a$$, $$b$$, $$c$$, $$d$$, and $$e$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### 5th Order Polynomial¶

The function of this curve is given by

$y(x)=a+bx+cx^2+dx^3+ex^4+ex^5$

where $$a$$, $$b$$, $$c$$, $$d$$, $$e$$, and $$f$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

## Schulz-Flory functions¶

Schulz Flory distribution function to describe relative ratios of polymers after a polymerization process. The function of this curve is given by

$y(x) = \sum_{i=1}^{n} ln(10) \dfrac{a_i}{b_i^2} \exp{(4.6x-\dfrac{\exp{(2.3x)}}{b_i})}$

where $$a_i$$ and $$b_i$$ are constants to fit, $$x$$ and $$y$$ are the test data pair. The parameter must satisfy the condition: $$0<a_i<1$$.

## Nonlinear Curves¶

The group of Nonlinear curves contains nonlinear curves that do not belong to the polynomial.

### Symmetrical Sigmoidal¶

The function of this curve is given by

$y(x)=d + \dfrac{a-d}{1+(\dfrac{x}{c})^b}$

where $$a$$, $$b$$, $$c$$, and $$d$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### Asymmetrical Sigmoidal¶

The function of this curve is given by

$y(x)=d + \dfrac{a-d}{ (1+(\dfrac{x}{c})^b)^m }$

where $$a$$, $$b$$, $$c$$, $$d$$, and $$m$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### Rectangular Hyperbola¶

The function of this curve is given by

$y(x)=\dfrac{V_{max}x}{ K_m + x}$

where $$V_{max}$$ and $$K_m$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### Basic Exponential¶

The function of this curve is given by

$y(x)=a + be^{-cx}$

where $$a$$, $$b$$, and $$c$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### Half-Life Exponential¶

The function of this curve is given by

$y(x)=a + \dfrac{b}{2^{(\dfrac{x}{c})}}$

where $$a$$, $$b$$, and $$c$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### Proportional Rate Growth or Decrease¶

The function of this curve is given by

$y(x)=Y_0 + \dfrac{V_0}{K}(1-e^{-Kx})$

where $$Y_0$$, $$V_0$$, and $$K$$ are constants to fit, $$x$$ and $$y$$ are the test data pair.

### Log-Normal Particle Size Distribution¶

The function of this curve is given by

$\dfrac{dy(x)}{d\ln{x}}=\dfrac{C_t}{\sigma_g\sqrt{2}\pi} \exp{(-\dfrac{(\ln{x}-\ln{D_m})^2}{2\ln{\sigma_g}^2})}$

where $$D_m$$, $$\sigma_g$$, and $$C_t$$ are constants to fit, x and y are test data pair. In the computation, the Left-Hand-Side term ($$dy(x)/d\ln{x}$$) is calculated using finite difference scheme.

Note

Independent variables $$x$$ must be larger than zero. The number of input x-y pairs must be large than 3.

## Hyperelastic Material Model Curves¶

The group of hyperelastic material models contains the commonly used hyperelastic models in the finite element analysis. The test data pair is engineering strain and stress.

### Arruda-Boyce¶

The form of the strain-energy potential for Arruda-Boyce model is

$\begin{array}{ccl} W & = & \mu[\dfrac{1}{2}(\bar{I}_{1}-3)+\dfrac{1}{20\lambda_{m}^{2}}(\bar{I_{1}^{2}}-9)+\dfrac{11}{1050\lambda_{m}^{4}}(\bar{I_{1}^{3}}-27)\\ & + & \dfrac{19}{7000\lambda_{m}^{6}}(\bar{I_{1}^{4}}-81) + \dfrac{519}{673750\lambda_{m}^{8}}(\bar{I_{1}^{5}}-243)] \end{array}$

where $$\mu$$ is the initial shear modulus of the material, $$\lambda_{m}$$ is limiting network stretch.

### Gent¶

The form of the strain-energy potential for the Gent model is:

$W=-\frac{\mu J_{m}}{2}\mathrm{ln}\left(1-\frac{\bar{I}_{1}-3}{J_{m}}\right)$

where $$\mu$$ is the initial shear modulus of the material, $$J_m$$ is limiting value of $$\bar{I}_1-3$$.

### Mooney-Rivlin 2 3 5 and 9 Parameters¶

This model includes two-, three-, five-, and nine-terma Mooney-Rivlin models. The form of the strain-energy potential for a two-parameter Mooney-Rivlin model is

$W=C_{10}\left(\bar{I}_{1}-3\right)+C_{01}\left(\bar{I}_{2}-3\right)$

where $$C_{10}$$, $$C_{01}$$, and $$D_{1}$$ are the material constants.

The form of strain-energy potential for a three-parameter Mooney-Rivlin model is

$W=C_{10}\left(\bar{I}_{1}-3\right)+C_{01}\left(\bar{I}_{2}-3\right)+C_{11}\left(\bar{I}_{1}-3\right)\left(\bar{I}_{2}-3\right)$

where $$C_{10}$$, $$C_{01}$$, and $$C_{11}$$ are material constants.

The form of strain-energy potential for a five-parameter Mooney-Rivlin model is

$\begin{array}{ccl} W & = & C_{10}\left(\bar{I}_{1}-3\right)+C_{01}\left(\bar{I}_{2}-3\right)+C_{20}\left(\bar{I}_{1}-3\right)^{2}\\ & + & C_{11}\left(\bar{I}_{1}-3\right)\left(\bar{I}_{2}-3\right)+C_{02}\left(\bar{I}_{2}-3\right)^{2} \end{array}$

where $$C_{10}$$, $$C_{01}$$, $$C_{20}$$, $$C_{11}$$, and $$C_{02}$$ are material constants.

The form of strain-energy potential for a nine-parameter Mooney-Rivlin model is

$\begin{array}{ccl} W & = & C_{10}\left(\bar{I}_{1}-3\right)+C_{01}\left(\bar{I}_{2}-3\right)+C_{20}\left(\bar{I}_{1}-3\right)^{2}\\ & + & C_{11}\left(\bar{I}_{1}-3\right)\left(\bar{I}_{2}-3\right)+C_{02}\left(\bar{I}_{2}-3\right)^{2}+C_{30}\left(\bar{I}_{1}-3\right)^{3}\\ & + & C_{21}\left(\bar{I}_{1}-3\right)^{2}\left(\bar{I}_{2}-3\right)+C_{12}\left(\bar{I}_{1}-3\right)\left(\bar{I}_{2}-3\right)^{2}+C_{03}\left(\bar{I}_{2}-3\right)^{3} \end{array}$

where $$C_{10}$$, $$C_{01}$$, $$C_{20}$$, $$C_{11}$$, $$C_{02}$$, $$C_{30}$$, $$C_{21}$$, $$C_{12}$$, and $$C_{03}$$ are material constants.

### Neo-Hookean¶

The Neo-Hookean model is a well-known hyperelastic model with an expanded linear rule (Hooke rule) having isotropy so that it can respond to finite deformation problems. The elastic potential is as follows.

$W=\frac{\mu}{2}(\bar{I}_{1}-3)$

where $$\mu$$ is initial shear modulus of materials.

### Ogden¶

The strain-energy potential of the Ogden compressible foam model is based on the principal stretches of left Cauchy strain tensor, which has the form:

$W=\sum_{i=1}^{N}\frac{\mu_{i}}{\alpha_{i}}\left(\bar{\lambda}_{1}^{\alpha_{i}}+\bar{\lambda}_{2}^{\alpha_{i}}+\bar{\lambda}_{3}^{\alpha_{i}}-3\right)+\sum_{k=1}^{N}\frac{1}{D_{k}}\left(J-1\right)^{2k}$

where N determines the order of the polynomial, $$\mu_i$$, $$\alpha_i$$ are material constants. The reduced principal strench is defined by:

$\bar{\lambda}_{p}=J^{-\frac{1}{3}}\lambda_p,\; J=(\lambda_{1}\lambda_{2}\lambda_{3})^{\frac{1}{2}}$

When parameters N=1, $$\alpha_1$$=2, the Ogden model is converted to the neo-Hookean model. When parameters N=2, $$\alpha_1$$=2 and $$\alpha_2$$=-2, the Ogden model is converted to the 2-parameter Mooney-Rivlin model.

### Polynomial¶

The polynomial form of strain-energy potential is:

$W=\sum_{i+j=1}^{N}c_{ij}\left(\bar{I}_{1}-3\right)^{i}\left(\bar{I_{2}}-3\right)^{j}$

where $$N$$ determines the order of the polynomial, $$c_{ij}$$ are material constants.

The Polynomial model is converted to following models with specific parameters:

Parameters of Polynomial model Equivalent model
N=1, $$C_{01}$$=0 neo-Hookean
N=1 2-parameter Mooney-Rivlin
N=2 5-parameter Mooney-Rivlin
N=3 9-parameter Mooney-Rivlin

### Yeoh¶

The Yeoh model is also called the reduced polynomial form. The strain-energy potential is

$W=\sum_{i=1}^{N}c_{i0}\left(\bar{I}_{1}-3\right)^{i}$

where N denotes the order of the polynomial, $$C_{i0}$$ are material constants. When N=1, Yeoh becomes neo-Hookean model.

## Electromagnetic Model Curves¶

This group includes the commonly used fitting curves in the electromagnetic analysis.

### Electrical Steel¶

The iron-core loss without DC flux bias is expressed as the following:

$p_v=P_h+P_c+P_e=K_{f} f (B_m)^2 + K_c (fB_m)^2 + K_e(fB_m)^{1.5}$

where

• $$B_m$$ is the amplitude of the AC flux component,
• $$f$$ is the frequency,
• $$K_h$$ is the hysteresis core loss coefficient,
• $$K_c$$ is the eddy-current core loss coefficient, and
• $$K_e$$ is the excess core loss coefficient,

### Power Ferrite¶

The iron-core loss is expressed as the Steinmetz approximation

$p_v=C_m f^x B_m^y$

where $$p_v$$ is the average power density, $$f$$ is the excitation frequency, and $$B_m$$ is the peak flux density, is commonly used to characterize core loss data for sinusoidal excitation, but can also be applied to square-wave data.

To linearize the equation for curve fitting, we used base-10 logarithms. The equation above can be rewritten to

$log(p_v)=c + x\cdot log(f) + y \cdot(B_m)$

where $$c=log(C_m)$$.