A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies

Maika Goto; Kazunori Kuwana; Yasuhide Uegata; Shigetoshi Yazaki

doi:10.3934/dcdss.2020233

Article Contents

2021, Volume 14, Issue 3: 881-891. Doi: 10.3934/dcdss.2020233

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A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies

1.
Faculty of Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa-shi, Yamagata 992-8510, Japan
2.
Graduate School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan
3.
School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan

^* Corresponding author. E-mail address : uegata@meiji.ac.jp (Yasuhide Uegata)
^* Corresponding author. E-mail address : uegata@meiji.ac.jp (Yasuhide Uegata)

Received: December 31, 2018

Revised: September 30, 2019

Early access: December 2019

Published: March 2021

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Abstract

We propose a simple and accurate procedure how to extract the values of model parameters in a flame/smoldering evolution equation from 2D movie images of real experiments. The procedure includes a novel method of image segmentation, which can detect an expanding smoldering front as a plane polygonal curve. The evolution equation is equivalent to the so-called Kuramoto-Sivashinsky (KS) equation in a certain scale. Our results suggest a valid range of parameters in the KS equation as well as the validity of the KS equation itself.

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Mathematics Subject Classification: Primary: 35R37, 68U10, 80A25; Secondary: 53C80.

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Figure 1. The photographs depict snapshots from an experimental movie of spreading flame/smoldering front along a sheet of paper placed near the floor at 200th, 400th, 1000th, 1600th, $ \cdots $, 4000th frames at the rate of 30 fps. Experiments were performed by the same method as [5]

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Figure 2. The left figure depicts numerical solutions to (1), with the normal velocity (2) in which the parameters are given by the right table and $ W $ is chosen for controlling the grid-point spacing to be uniform (see section 3). The solution curves evolve from inside to outside. The initial curve is a circle with the diameter $ R = R_\mathrm{ini} $ with 10% noise (see [4] in detail)

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Figure 3. (a) Jordan curve $ \Gamma $ (b) Jordan polygonal curve $ \mathcal{P} $

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Figure 4. The upper-left figure depicts selected segmentation curves at frames: 400, 1000, 1600, 2200, 2800, 3400, 4000, summarizing the front evolution in the other photographs from left to right, upper to lower. The blue curve in each photograph is a segmentation curve, and the background vague region is the same as that in FIGURE 1

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Figure 5. (Left) The total length of front $ \tilde{L} [\mathrm{mm}] $ vs. the actual time $ [\mathrm{second}] $. (Right) The enclosed area $ \tilde{A} [\mathrm{mm}^2] $ vs. the actual time $ [\mathrm{second}] $. Blue points indicate the actual values and red curves are the graphs of (23) and (24), respectively

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Figure 6. (Left) $ V^{(0)} $ vs. time, (Right) $ \alpha_\mathrm{eff} $ vs. time

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Table 1. Discretizations of length, normal/tangent vector, and normal velocity

$ r_i=\\|\boldsymbol{x}_i-\boldsymbol{x}_{i-1}\\| $	: The length of $ \mathcal{P}_i $
$ L= \sum\limits_{i=1}^Nr_i $	: The total length of $ \mathcal{P} $
$ \boldsymbol{t}_i=(\boldsymbol{x}_i-\boldsymbol{x}_{i-1})/r_i $	: The unit tangent vector on $ \mathcal{P}_i $
$ \boldsymbol{n}_i=-\boldsymbol{t}_i^\bot $	: The outward unit normal vector on $ \mathcal{P}_i $
$ v_i $	: A given representative normal velocity on $ \mathcal{P}_i $
$ \phi_i=\mathrm{sgn}(D_i)\arccos(\boldsymbol{t}_i\cdot\boldsymbol{t}_{i+1}) $	: The angle between the adjacent edges $ \mathcal{P}_{i} $ and $ \mathcal{P}_{i+1} $ where $ D_i=\det(\boldsymbol{t}_i, \boldsymbol{t}_{i+1}) $
$ \boldsymbol{T}_i=(\boldsymbol{t}_i+\boldsymbol{t}_{i+1})/(2\mathsf{cos}_i) $	: The unit tangent vector at $ \boldsymbol{x}_i $ where $ \mathsf{cos}_i=\cos(\phi_i/2)=\\|(\boldsymbol{t}_i+\boldsymbol{t}_{i+1})/2\\| $
$ \boldsymbol{N}_i=(\boldsymbol{n}_i+\boldsymbol{n}_{i+1})/(2\mathsf{cos}_i) $	: The outward unit normal vector at $ \boldsymbol{x}_i $
$ V_i=(v_i+v_{i+1})/(2\mathsf{cos}_i) $	: The normal velocity at $ \boldsymbol{x}_i $

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