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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.

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 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 \(c=log(C_m)\).