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Frontiers of Mechanical Engineering

Front. Mech. Eng.    2020, Vol. 15 Issue (2) : 279-293     https://doi.org/10.1007/s11465-019-0575-5
RESEARCH ARTICLE
An isogeometric numerical study of partially and fully implicit schemes for transient adjoint shape sensitivity analysis
Zhen-Pei WANG1,2, Zhifeng XIE3, Leong Hien POH1()
1. Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore
2. Institute of High Performance Computing (IHPC), Agency for Science, Technology and Research (A*STAR), Singapore 138632, Singapore
3. China Academy of Launch Vehicle Technology, Beijing Institute of Astronautical Systems Engineering, Beijing 100076, China
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Abstract

In structural design optimization involving transient responses, time integration scheme plays a crucial role in sensitivity analysis because it affects the accuracy and stability of transient analysis. In this work, the influence of time integration scheme is studied numerically for the adjoint shape sensitivity analysis of two benchmark transient heat conduction problems within the framework of isogeometric analysis. It is found that (i) the explicit approach (β = 0) and semi-implicit approach with β<0.5 impose a strict stability condition of the transient analysis; (ii) the implicit approach (β=1) and semi-implicit approach with β > 0.5 are generally preferred for their unconditional stability; and (iii) Crank–Nicolson type approach (β=0.5) may induce a large error for large time-step sizes due to the oscillatory solutions. The numerical results also show that the time-step size does not have to be chosen to satisfy the critical conditions for all of the eigen-frequencies. It is recommended to use β0.75 for unconditional stability, such that the oscillation condition is much less critical than the Crank–Nicolson scheme, and the accuracy is higher than a fully implicit approach.

Keywords isogeometric shape optimization      design-dependent boundary condition      transient heat conduction      implicit time integration      adjoint method     
Corresponding Authors: Leong Hien POH   
Online First Date: 23 March 2020    Issue Date: 25 May 2020
 Cite this article:   
Zhen-Pei WANG,Zhifeng XIE,Leong Hien POH. An isogeometric numerical study of partially and fully implicit schemes for transient adjoint shape sensitivity analysis[J]. Front. Mech. Eng., 2020, 15(2): 279-293.
 URL:  
http://journal.hep.com.cn/fme/EN/10.1007/s11465-019-0575-5
http://journal.hep.com.cn/fme/EN/Y2020/V15/I2/279
Fig.1  Schematics of initial design at s =0 (left) and updated design at s (right).
Fig.2  The initial plate design and the NURBS parameterization (values in m) [16,20]. Problem parameters are: θ0[x] =100°C, xΩs, θe =0°C, ρ =7800 kg/m3, c =420 J/(kg·°C), k =20 W/(m·°C) and h =50 W/(m2·°C).
Fig.3  Critical time-step size of the stability condition versus the time integration scheme coefficient β.
Fig.4  Critical time-step size of the oscillatory conditions for (a) all 288 eigenvalues and (b) the first 87 eigenvalues smaller than 20, with β =0.5 and 0.75, respectively.
Fig.5  Temperature oscillations at point A and C1 with different time-step sizes and β=0.5 for the first few iterative steps.
Fig.6  Temperature oscillations at point A and C1 with different time-step sizes and β=0.75 for the first few iterative steps.
I (i, j) Location Weight ?I (i, j) Location Weight
x1I x2I x1I x2I
(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
Tab.1  Initial locations of the design control points for the minimum boundary problem [16,20]
CI Component FD Referential FD
β =0.5 β =0.75 β =1
C1 1 135874.5067 135876.4566 135887.5961 135879.5198
C2 1 243335.5767 243333.5103 243350.4778 243339.8549
2 –37170.7865 –37162.2082 –37166.9958 –37166.6635
C3 1 897323.9019 897326.7759 897308.0330 897319.5703
2 306189.0384 306194.4444 306182.1844 306188.5557
C4 1 306189.1548 306194.2916 306176.7857 306186.7440
2 897323.3635 897312.5441 897306.3377 897314.0818
C5 1 –37165.4896 –37167.1631 –37163.8816 –37165.5115
2 243336.8791 243328.4244 243347.0800 243337.4611
C6 2 135881.9864 135871.1306 135886.7594 135879.9588
Tab.2  Sensitivity analysis using FD with Δt=0.01 s for different β and the referential sensitivity of the minimum boundary problem
Fig.7  The L2 norm of the relative difference of the adjoint sensitivity analysis versus number of time-steps 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
Tab.3  Computational time of different time-step sizes for the minimum boundary problem
Fig.8  The L2 norm of the relative difference of the adjoint sensitivity analysis versus β for the minimum boundary problem.
Fig.9  NURBS parameterization of the initial plunger model (values in mm) [16,20].
I (i,j) Location Weight ?I (i,j) Location Weight
x1I x2I x1I x2I
(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
Tab.4  Initial locations of the design control points for the plunger design problem [16,20]
Fig.10  Critical time-step size of the stability conditions versus the time integration scheme coefficient β.
Fig.11  Critical time-step size of the oscillatory conditions for (a) all 520 eigenvalues and (b) the first 445 eigenvalues smaller than 20, with β =0.5 and 0.75, respectively.
CI Component FD Referential FD
β =0.5 β =0.75 β =1
C1 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
C2 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
C3 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
C4 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
C5 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
Tab.5  Sensitivity analysis using FD with Δ t=0.05 s for different β and the referential sensitivity of the plunger design problem
Fig.12  The L2norm of the relative difference of the adjoint sensitivity analysis versus number of time-steps for the plunger design problem.
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
Tab.6  Computational time of different time step-sizes for the plunger design case
Fig.13  The L2 norm of the relative difference of the adjoint sensitivity analysis versus β for the plunger design problem.
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