Introduction
Problem statement and adjoint shape sensitivity analysis
IGA for transient heat conduction problems
Numerical error induced by time integration schemes
Accuracy
Stability
Oscillations
Numerical studies
Minimum boundary problem
Problem description
Stability and oscillations in the transient analysis
Location | Weight | | Location | Weight | |||
---|---|---|---|---|---|---|---|
(1, 1) | 0.0100 | 0.0000 | 1.00 | (4, 2) | 0.0091 | 0.0121 | 0.85 |
(2, 1) | 0.0100 | 0.0026 | 0.90 | (5, 2) | 0.0039 | 0.0150 | 0.90 |
(3, 1) | 0.0080 | 0.0061 | 0.85 | (6, 2) | 0.0000 | 0.0150 | 1.00 |
(4, 1) | 0.0061 | 0.0080 | 0.85 | (1, 3) | 0.0200 | 0.0000 | 1.00 |
(5, 1) | 0.0026 | 0.0100 | 0.90 | (2, 3) | 0.0200 | 0.0100 | 1.00 |
(6, 1) | 0.0000 | 0.0100 | 1.00 | (3, 3) | 0.0238 | 0.0213 | 1.00 |
(1, 2) | 0.0150 | 0.0000 | 1.00 | (4, 3) | 0.0213 | 0.0238 | 1.00 |
(2, 2) | 0.0150 | 0.0039 | 0.90 | (5, 3) | 0.0100 | 0.0200 | 1.00 |
(3, 2) | 0.0121 | 0.0091 | 0.85 | (6, 3) | 0.0000 | 0.0200 | 1.00 |
Referential sensitivity calculation
Tab.2 Sensitivity analysis using FD with s for different and the referential sensitivity of the minimum boundary problem |
Component | FD | Referential FD | |||
---|---|---|---|---|---|
1 | 135874.5067 | 135876.4566 | 135887.5961 | 135879.5198 | |
1 | 243335.5767 | 243333.5103 | 243350.4778 | 243339.8549 | |
2 | –37170.7865 | –37162.2082 | –37166.9958 | –37166.6635 | |
1 | 897323.9019 | 897326.7759 | 897308.0330 | 897319.5703 | |
2 | 306189.0384 | 306194.4444 | 306182.1844 | 306188.5557 | |
1 | 306189.1548 | 306194.2916 | 306176.7857 | 306186.7440 | |
2 | 897323.3635 | 897312.5441 | 897306.3377 | 897314.0818 | |
1 | –37165.4896 | –37167.1631 | –37163.8816 | –37165.5115 | |
2 | 243336.8791 | 243328.4244 | 243347.0800 | 243337.4611 | |
2 | 135881.9864 | 135871.1306 | 135886.7594 | 135879.9588 |
Numerical adjoint sensitivity analysis convergence with respect to the number of time-steps for different
Tab.3 Computational time of different time-step sizes for the minimum boundary problem |
Time-step size/s | Number of time-steps | Computational time/s |
---|---|---|
60.00 | 5 | 0.4633 |
30.00 | 10 | 0.5676 |
15.00 | 20 | 0.6906 |
10.00 | 30 | 0.7631 |
5.00 | 60 | 1.1828 |
3.00 | 100 | 1.7065 |
2.00 | 150 | 2.0199 |
1.00 | 300 | 3.7158 |
0.50 | 600 | 6.8297 |
0.30 | 1000 | 11.2075 |
0.10 | 3000 | 36.3381 |
0.01 | 30000 | 429.7075 |
Numerical error of adjoint sensitivity analysis with respect to at a given time interval discretization
Plunger shape design problem
Problem description
Location | Weight | | Location | Weight | |||
---|---|---|---|---|---|---|---|
(1, 1) | 0.00 | 100.00 | 1.00 | (5, 2) | 90.00 | 20.00 | 1.00 |
(2, 1) | 0.00 | 80.00 | 1.00 | (6, 2) | 145.00 | 20.00 | 1.00 |
(3, 1) | 0.00 | 30.00 | 1.00 | (7, 2) | 200.00 | 20.00 | 1.00 |
(4, 1) | 0.00 | 0.00 | 0.71 | (1, 3) | 30.00 | 100.00 | 1.00 |
(5, 1) | 30.00 | 0.00 | 1.00 | (2, 3) | 30.00 | 80.00 | 1.00 |
(6, 1) | 140.00 | 0.00 | 1.00 | (3, 3) | 30.00 | 65.00 | 1.00 |
(7, 1) | 200.00 | 0.00 | 1.00 | (4, 3) | 30.00 | 45.00 | 1.00 |
(1, 2) | 15.00 | 100.00 | 1.00 | (5, 3) | 70.00 | 45.00 | 1.00 |
(2, 2) | 15.00 | 80.00 | 1.00 | (6, 3) | 120.00 | 45.00 | 1.00 |
(3, 2) | 15.00 | 65.00 | 1.00 | (7, 3) | 200.00 | 45.00 | 1.00 |
(4, 2) | 15.00 | 20.00 | 1.00 |
Stability and oscillations in the transient analysis
Referential sensitivity analysis calculation
Tab.5 Sensitivity analysis using FD with s for different and the referential sensitivity of the plunger design problem |
Component | FD | Referential FD | |||
---|---|---|---|---|---|
1 | –6.2906×105 | –6.2891×105 | –6.2885×105 | –6.2894×105 | |
2 | 2.0000×10 | 3.0000×10 | 1.0000×10 | 2.0000×10 | |
1 | –3.2439×105 | –3.2434×105 | –3.2422×105 | –3.2432×105 | |
2 | 3.4358×105 | 3.4351×105 | 3.4350×105 | 3.4353×105 | |
1 | 2.0760×106 | 2.0759×106 | 2.0758×106 | 2.0759×106 | |
2 | 1.1210×106 | 1.1210×106 | 1.1210×106 | 1.1210×106 | |
1 | 8.6030×104 | 8.6030×104 | 8.6010×104 | 8.6020×104 | |
2 | –6.2714×105 | –6.2713×105 | –6.2714×105 | –6.2714×105 | |
1 | 2.0000×10 | 1.0000×10 | 1.0000×10 | 2.0000×10 | |
2 | 1.5853×106 | 1.5852×106 | 1.5849×106 | 1.5851×106 |
Numerical adjoint sensitivity analysis convergence with respect to the number of time-steps for different
Tab.6 Computational time of different time step-sizes for the plunger design case |
Time-step size/s | Number of time-steps | Computational time/s |
---|---|---|
25.00 | 20 | 11.5991 |
10.00 | 50 | 27.7114 |
5.00 | 100 | 54.6364 |
2.50 | 200 | 108.7865 |
1.00 | 500 | 278.4205 |
0.50 | 1000 | 565.3433 |
0.25 | 2000 | 1110.6680 |
0.10 | 5000 | 2794.8650 |
0.05 | 10000 | 5704.8800 |