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Analytical solution of the diffusion model and the inverse problem
PMiKNoM seminars - class 9
Introduction
The methods for establishing the diffusion coefficient in concretes:Fick's second Law - analytical solution
Assumptions
- Semi-infinite medium
- Concentration at the right boundary `c_R=0`
Analytical solution
` c_k^\text{model} \equiv c^\text{model} (x_k,t) = c_L (1-erf (\frac{x_k}{2 \sqrt{D \cdot t}} )) `Comparison of the solutions
- Comparison between analytical solution and the numerical solution created during class 5.
The inverse problem - formulation
The goal function is given as:
` Gf(D,c_L) = \sum_{k=1}^r (c_k^{exp} - c_k^\text{model})^2 `
After inserting the analytical solution:
` Gf(D,c_L) = \sum_{k=1}^r ( c_k^{exp} - c_L (1-erf (\frac{x_k}{2 \sqrt{D \cdot t}} )))^2 `
The inverse problem - exercises using MS Excel Solver
Data
$$
\begin{array}{lcccccc}
\text{Distance from the left boundary: }& 6mm & 12mm & 18mm & 24mm & 30mm & 36mm \\
\text{Measured concentration: } & 0,58\text{%} & 0,32 \text{%} & 0,23 \text{%} & 0,08 \text{%} & 0,06 \text{%} & 0,04 \text{%}
\end{array}
$$
Measurement time: `t = 10 months = 26562000s`
Parameters to be optimised
`D \in [10^{-16};10^{-9}] \text{ and } c_L \in [0;2\text{%}] `
Different approaches for parametrs search
- linear
- logarithmic