Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm
Abstract
:1. Introduction
2. Problem Formulation
2.1. Mathematical Model for the Reheating Furnace
2.2. Inverse Heat Conduction Problem
3. The Fréchet Gradient Based on the FOTD Approach
3.1. The Adjoint PDEs Based on Weak Solutions
3.2. Fréchet Gradient of the Cost Functional
3.3. Lipschitz Continuous
4. The Improved Hybrid Conjugate Gradient Algorithm
Algorithm 1: IHCG algorithm based on the FOTD approach. | |
Begin | |
1: | Set parameters . Choose an initial point . |
2: | Solve the direct problem (2) to determine the temperature distribution , and estimate the cost function ; |
3: | Solve the adjoint problem (5) to determine the gradient of the cost function . If or , stop the algorithm. |
4: | Estimate the descent depth based on and Equation (23). |
5: | Determine a step length by the standard Wolfe line search method.
|
6: | Update , and go to Step 2. |
end |
4.1. Property of the Proposed Algorithm
4.2. Global Convergence
5. Simulation and Analysis
5.1. FOTD Approach vs. FDTO Approach
5.2. Comparisons between IHCG and Other CG Algorithms
5.3. Application in Reheating Furnace
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Proof of Lemma 3
References
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Simulation Time (s) | Iteration Steps | Final Cost Functional (J) | |
---|---|---|---|
FOTD | 300.92 | 713 | 0.6761 |
FDTO | 42035.76 | 317 | 0.3710 |
Algorithms | VFR | SCG | HS | LS | MCD | MPRP | NTTCG | IHCG |
---|---|---|---|---|---|---|---|---|
Iteration steps | 708 | 896 | 971 | 713 | 783 | 1044 | 791 | 334 |
Simulation time (s) | 306.72 | 409 | 456.44 | 300.92 | 334.64 | 474.74 | 331.32 | 140.38 |
Algorithms | ||||||
---|---|---|---|---|---|---|
S | S | S | ||||
VFR | 2.73 | 1.48 | 6.52 | 3.52 | 4.15 | 1.13 |
SCG | 3.18 | 1.72 | 5.74 | 3.10 | 3.89 | 8.05 |
HS | 3.16 | 1.70 | 5.90 | 3.19 | 3.88 | 8.07 |
LS | 3.10 | 1.67 | 5.73 | 3.09 | 3.89 | 8.07 |
MCD | 3.12 | 1.68 | 5.72 | 3.09 | 3.88 | 7.99 |
MPRP | 2.95 | 1.59 | 9.62 | 5.19 | 3.89 | 8.07 |
NTTCG | 7.15 | 3.86 | 6.51 | 3.52 | 1.48 | 3.23 |
IHCG | 2.94 | 1.58 | 5.71 | 3.08 | 3.85 | 7.98 |
Algorithms | Cost Function (J) | Iteration Steps | Simulation Time (s) |
---|---|---|---|
VFR | 1308.07 | 935 | 508.15 |
SCG | 30.31 | 948 | 495.18 |
HS | 13.53 | 1477 | 823.55 |
LS | 89.32 | 360 | 179.94 |
MCD | 28.56 | 980 | 525.86 |
MPRP | 31.6 | 808 | 408.31 |
NTTCG | 7258.97 | 60 | 29.72 |
IHCG | 7.58 | 781 | 407.42 |
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Yang, Z.; Luo, X.; Liu, P.; Qiao, J.; Liu, M. Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm. Mathematics 2022, 10, 4074. https://doi.org/10.3390/math10214074
Yang Z, Luo X, Liu P, Qiao J, Liu M. Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm. Mathematics. 2022; 10(21):4074. https://doi.org/10.3390/math10214074
Chicago/Turabian StyleYang, Zhi, Xiaochuan Luo, Pengbo Liu, Jinwei Qiao, and Ming Liu. 2022. "Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm" Mathematics 10, no. 21: 4074. https://doi.org/10.3390/math10214074
APA StyleYang, Z., Luo, X., Liu, P., Qiao, J., & Liu, M. (2022). Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm. Mathematics, 10(21), 4074. https://doi.org/10.3390/math10214074