A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale Algebraic Equations of a 3-D Field Problem
Abstract
:Featured Application
Abstract
1. Introduction
2. Proposed Order Reduction Methodology
2.1. Three-Dimensional Transient Temperature Field Problems
2.2. Moment Matching Method for 3-D Transient Field Problems
2.3. Multipoint Expansion
2.4. Adaptive Scheme
Algorithm 1. Main loop of the proposed scheme for 3-D transient field problems |
Input: The range of the broadband frequency (,), coefficient matrices of NTM model (,,), global error tolerance tolg and local error tolerance toll. |
1. 2. [] = S-POINT (, toll) 3. 4. while do [] = S-POINT (, toll) 5. end while 6. ,, Output: The coefficient matrices of a reduced system(,,) |
Algorithm 2. S-POINT: Block Arnoldi-based moment matching at a single frequency point |
Input: The current expansion point , local error tolerance toll and coefficient matrices of the high-fidelity model (,,) |
1. 2. 3. while do = EST-ERROR (,,A,B,P) 4. end while Output: |
Algorithm 3. EST-ERROR: Error estimator |
Input: ,, ,, local error: the single frequency point , global error, the range of the broadband frequency (,) |
1. ,, 2. Local error: Get from Equation (1) under , , Get from Equation (3) under , , 3. Global error: Get from Equation (1) under , , Get from Equation (3) under , , 4. Output: error |
3. Numerical Application
4. Conclusions
- (1)
- The model reduced from the proposed methodology could achieve a higher accuracy compared to one reduced from conventional block Arnoldi in a wideband frequency. For point 1, the maximum error of the proposed model was 0.3603, whereas that of the conventional reduced model can reach 3.2275. For point 2, the maximum error of the proposed approach is 0.0986, whereas that of the conventional reduced model can reach 0.9803. For point 3, the maximum error of the proposed model is 0.1459, whereas that of the conventional reduced model can reach 6.3752.
- (2)
- The ROM obtained using the proposed scheme can accurately replicate the transient temperature fields of the NTM high-fidelity model through a wide frequency range, while the computational time of the reduced-order model is less than 13% of the original one.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Analysis Point | NTM Model | Proposed | ||
---|---|---|---|---|
max (K) | min (K) | max (K) | min (K) | |
Point 1 | 40.9190 | −37.1739 | 40.9114 | −37.1928 |
Point 2 | 12.3376 | −9.8100 | 12.3112 | −9.8353 |
Point 3 | 19.9404 | −17.9279 | 19.9244 | −17.9785 |
Analysis Point | Block Arnoldi (Order = 30) | Proposed (Order = 28) | ||||
---|---|---|---|---|---|---|
max (K) | min (K) | Average (K) | max (K) | min (K) | Average (K) | |
Point 1 | 3.2275 | 0 | 0.1869 | 0.3603 | 0 | 0.0689 |
Point 2 | 0.9803 | 0 | 0.0543 | 0.0986 | 0 | 0.0246 |
Point 3 | 6.3752 | 0 | 0.1995 | 0.1459 | 0 | 0.0224 |
Analysis Point | Proposed (Order = 28) | Block Arnoldi (Order = 20) | Block Arnoldi (Order = 30) |
---|---|---|---|
Point 1 | 0.0066 | 0.0479 | 0.0246 |
Point 2 | 0.0065 | 0.0298 | 0.0190 |
Point 3 | 0.0088 | 0.2659 | 0.1627 |
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Wang, N.; Chen, J.; Wang, H.; Yang, S. A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale Algebraic Equations of a 3-D Field Problem. Appl. Sci. 2021, 11, 9435. https://doi.org/10.3390/app11209435
Wang N, Chen J, Wang H, Yang S. A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale Algebraic Equations of a 3-D Field Problem. Applied Sciences. 2021; 11(20):9435. https://doi.org/10.3390/app11209435
Chicago/Turabian StyleWang, Ning, Jiajia Chen, Huifang Wang, and Shiyou Yang. 2021. "A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale Algebraic Equations of a 3-D Field Problem" Applied Sciences 11, no. 20: 9435. https://doi.org/10.3390/app11209435
APA StyleWang, N., Chen, J., Wang, H., & Yang, S. (2021). A Block Arnoldi Algorithm Based Reduced-Order Model Applied to Large-Scale Algebraic Equations of a 3-D Field Problem. Applied Sciences, 11(20), 9435. https://doi.org/10.3390/app11209435