Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques
Abstract
:1. Introduction
2. Mathematic Model
2.1. Direct Problem Description
2.2. Description of the Inverse Problem
2.3. Method of Solving the Inverse Problem
2.3.1. Sensitivity Coefficient Matrix
2.3.2. Stopping Criteria
2.3.3. Algorithm for Sequential Regularization Method
3. Model Verification
3.1. Validation for Solving Direct Problem and Sensitivity Coefficient Problem
3.2. Validation for Inverse Problem
3.2.1. Methodology of Validation
3.2.2. Validation
4. Results and Discussion
4.1. Effect of Number of Future Time Steps
4.2. Effect of Regularization Parameter
4.3. Effect of Spatial Regularization Method (d) and Discrete Grids (nx × ny)
4.4. Effect of Time Step Size (dt)
5. Conclusions
- Increasing the number of future time steps results in a more pronounced phase shift between the peak of the predicted heat flux and the exact heat flux, where the predicted peak of heat flux rises earlier than the exact one and then decreases before it;
- Zeroth- and first-order spatial regularization provides higher accuracy than second-order spatial regularization, and zeroth-order regularization has comparable accuracy to first-order regularization;
- Reconstructing a sharply changing heat flux is more challenging than reconstructing a smoothly changing heat flux. First-order spatial regularization provides better accuracy for reconstructing heat flux with sharp spatial variations than both zeroth- and second-order spatial regularization;
- Using a coarser grid reduces the CPU time required for inverse analysis. The impact of the order of spatial regularization on CPU time is not significant;
- Decreasing the time step size initially increases the accuracy of inverse analysis, but after a certain point, the accuracy starts decreasing, and the CPU time required increases. The time step size of 1/2fs is recommended, where fs is the sampling rate of temperature.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Nomenclature
c | heat capacity (J/kg·K) |
d | order of spatial regularization, i.e., zeroth, first, and second |
dt | time step size (dt = tj+1−tj) |
dT | a scalar about temperature (-) |
epred | the relative error |
f(t) | temperature boundary condition (-) |
fs | the sampling frequency of temperature (Hz) |
gexa | a pre-set heat flux |
H | height (-) |
hd | dth-order derivative operator |
hd,i | dth-order derivative operator with dimension ni |
element in the m-th row and n-th column of a matrix sensitivity matrix at time ti | |
Ji | sensitivity matrix at time ti (-) |
L | width (m) |
M | the total number of measurements |
nj | number of heat flux components at the boundary Γj, j = 1,2,3 |
nx × ny | number of discrete grids |
N | N = n1 + n2 + n3 |
n | the outer normal of boundary |
the n-th component of heat flux qj at the time tj (-) | |
heat flux of Γi at time tj (-) | |
qj | vector of heat flux at time tj (-) |
q* | an assumed heat flux (-) |
r | the number of future time steps (-) |
Rc(qj), Rd(qj), | spatial regularization term (-) |
s | objective function (-) |
t | time (-) |
tj | the j-th time step |
Tini | initial temperature (-) |
the m-th component of Tj (-) | |
Tj | vector of estimated temperature at time tj (-) |
W | width (-) |
x, y | Cartesian spatial coordinates (-) |
tangential direction of boundary | |
Yexa | the true temperature |
Ymax | maximum measured temperature (K) |
the m-th component of Yj (-) | |
Y | vector of the measured temperature (-) |
GREEK SYMBOLS | |
α, α’, α* | regularization parameters |
Γ1, Γ2, Γ3, Γ4 | boundary of the calculated domain Ω |
δ(.) | Dirac delta function |
Δ | Laplace operator |
ε1, ε2 | tolerance |
λ | thermal conductivity (W/m·K) |
ρ | density (kg·m−3) |
Ω | calculated domain |
SUBSCRIPTS | |
ini | initial value |
j | time at tj |
m | number of thermocouples |
ref | reference value |
SUPERSCRIPTS | |
j | time at tj |
k | iteration number |
T | transpose |
* | the dimensional, initial, or optimal variable |
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Regularization Method | Integral Form Rc(qj) | Equivalent Discrete Form | Derivative Operator , i = 1, 2 3 |
---|---|---|---|
Zeroth order (d = 0) | |||
First order (d = 1) | . | ||
Second order (d = 2) |
Algorithm Sequential regularization method for the inverse problem | |
0: | Initialize the order of spatial regularization d, the number of future time steps r, the values for ε1 and ε2; set the time step j is 1 and the regularization parameter α is 100; input the measured temperature Y, and the number of discrete grids. |
1: | while α > 10−12 do |
2: | while j does not reach the end of the time step do |
3: | Take an initial guess for the heat flux q*, then qj = q*. |
4: | Solve the direct problem given by Equations (6)–(9) with the guess heat flux qj for in the time interval [tj, tj+r−1]. |
5: | Solve the sensitivity coefficient problem given by Equations (18)–(21) for the sensitivity coefficient matrix Jj during the time from 0 to tr. |
6: | if the stopping criteria given by Equations (22) and (23) are satisfied |
7: | qj could be regarded as the predicted heat flux at time tj. |
8: | else |
9: | Update qj using Equation (17) with and Jj, and return to Step 4. |
10: | end if |
11: | Replace j by j + l. |
12: | endwhile |
13: | Replace α by 0.2α, and set j is 1. |
14: | endwhile |
Parameter | Variations |
---|---|
Case of heat flux to be predicted | Case 1 (Equation (25)), Case 2 (Equation (26)), and Case 3 (Equation (27)) |
Number of future time steps (r) | 1, 2, 4, 6, 8, 10 |
Regularization parameter (α) | 10−9, 10−8, 10−7, 10−6, 10−5, 10−4, 5×10−4, 10−3, 5×10−3, 10−2, 5×10−2, 1×10−1 |
Order of spatial regularization (d) | Zeroth, first, and second |
Number of discrete grids (nx × ny) | 9 × 22 (msh1), 17 × 43 (msh2), and 25 × 64 (msh3) |
Time step size (dt) | 0.1 (1/fs), 0.05 (1/2fs) and 0.02 (1/5fs) seconds |
The sampling frequency of temperature fs is 10 Hz |
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Zhang, H.; Zou, J.; Xiao, P. Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques. Metals 2023, 13, 1685. https://doi.org/10.3390/met13101685
Zhang H, Zou J, Xiao P. Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques. Metals. 2023; 13(10):1685. https://doi.org/10.3390/met13101685
Chicago/Turabian StyleZhang, Haihui, Jiawei Zou, and Pengcheng Xiao. 2023. "Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques" Metals 13, no. 10: 1685. https://doi.org/10.3390/met13101685