# 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 $β≈0.75$ 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. Corresponding Author(s): 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
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 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, $∀ x∈Ωs$, $θe =0$°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. Tab.1  Initial locations of the design control points for the minimum boundary problem [16,20] 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. 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]. 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. 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 $L2$norm of the relative difference of the adjoint sensitivity analysis versus number of time-steps for the plunger design problem. 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.