Topology optimization of transient problem with maximum dynamic response constraint using SOAR scheme

doi:10.1007/s11465-021-0636-4

Front. Mech. Eng.

2021, Vol. 16

Issue (3) : 593-606 https://doi.org/10.1007/s11465-021-0636-4

RESEARCH ARTICLE

Topology optimization of transient problem with maximum dynamic response constraint using SOAR scheme

Kai LONG¹(

), Xiaoyu YANG¹, Nouman SAEED¹, Ruohan TIAN¹, Pin WEN², Xuan WANG³

¹. State Key Laboratory for Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China
². Hubei Key Laboratory of Theory and Application of Advanced Materials Mechanics, School of Science, Wuhan University of Technology, Wuhan 430070, China
³. Department of Engineering Mechanics, Hefei University of Technology, Hefei 230009, China

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Abstract

This paper proposes a novel method for the continuum topology optimization of transient vibration problem with maximum dynamic response constraint. An aggregated index in the form of an integral function is presented to cope with the maximum response constraint in the time domain. The density filter solid isotropic material with penalization method combined with threshold projection is developed. The sensitivities of the proposed index with respect to design variables are conducted. To reduce computational cost, the second-order Arnoldi reduction (SOAR) scheme is employed in transient analysis. Influences of aggregate parameter, duration of loading period, interval time, and number of basis vectors in the SOAR scheme on the final designs are discussed through typical examples while unambiguous configuration can be achieved. Through comparison with the corresponding static response from the final designs, the optimized results clearly demonstrate that the transient effects cannot be ignored in structural topology optimization.

Keywords topology optimization solid isotropic material with penalization transient response aggregation function second-order Arnoldi reduction

Corresponding Author(s): Kai LONG

Just Accepted Date: 21 June 2021 Online First Date: 03 August 2021 Issue Date: 24 September 2021

Cite this article:

Kai LONG,Xiaoyu YANG,Nouman SAEED, et al. Topology optimization of transient problem with maximum dynamic response constraint using SOAR scheme[J]. Front. Mech. Eng., 2021, 16(3): 593-606.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-021-0636-4
https://academic.hep.com.cn/fme/EN/Y2021/V16/I3/593

SOAR procedure
Solve $(K − ω 2 M) q ¯ 1 = F$
$q 1 = q ¯ 1 / ‖ q ¯ 1 ‖ 2$
$p 1 = 0$
for $i = 1, 2, ..., N$
Solve $(K − ω 2 M) r = (2 ω M) q i + M p i$ $s = q i$ for $j = 1, 2, ..., i$ $r = r − 〈 q i, r 〉 q j$ $s = s − 〈 q j, r 〉 q j$ endfor $q i + 1 = r / ‖ r ‖ 2$ $p i + 1 = s / ‖ r ‖ 2$
endfor The reduced order basis $P = {q 1, q 2, ..., q N}$

Fig.1 Half-sinusoidal force with various loading times.

Fig.2 Optimized topologies for t_L=0.05 s: (a) m=10, V/V₀=0.409, j(f)=0.146 m², and f_max=0.200 m²; (b) m=30, V/V₀=0.407, j(f)=0.178 m², and f_max=0.200 m²; (c) m=50, V/V₀=0.406, j(f)=0.185 m², and f_max=0.200 m².

Fig.3 Optimized topologies for t_L=0.03 s: (a) m=10, V/V₀=0.400, j(f)=0.146 m², and f_max=0.200 m²; (b) m=30, V/V₀=0.382, j(f)=0.180 m², and f_max=0.200 m²; (c) m=50, V/V₀=0.406, j(f)=0.185 m², and f_max=0.200 m².

Fig.4 Optimized topologies for t_L=0.01 s: (a) m=10, V/V₀=0.369, j(f)=0.138 m², and f_max=0.200 m²; (b) m=30, V/V₀=0.360, j(f)=0.174 m², and f_max=0.200 m²; (c) m=50, V/V₀=0.360, j(f)=0.183 m², and f_max=0.200 m².

Fig.5 Optimized topologies for t_L=0.01 s from (a) full analysis and (b) SOAR (N=10), (c) SOAR (N=30), and (d) SOAR (N=50) schemes.

Tab.1 Comparisons of the optimized results between full analysis and SOAR schemes with various numbers of basis vectors

Fig.6 Rectangular force.

Fig.7 Evolution of volume fraction with different upper bound dynamic responses and the corresponding structural topologies.

Fig.8 Evolution of volume fraction under static and dynamic requirements, with structural topologies inserted.

Fig.9 Time history curves of the maximum dynamic responses under different limits: (a) f_max≤0.300 m², (b) f_max≤0.500 m², (c) f_max≤0.700 m², and (d) f_max≤0.900 m².

Fig.10 Load with the interval time t.

Fig.11 Optimized topologies for varying

τ

: (a) t=0, V/V₀=0.332, f_max=0.400 m², and f_s=0.327 m²; (b) t=0.002 s, V/V₀=0.314, f_max=0.400 m², and f_s=0.347 m²; (c) t=0.004 s, V/V₀=0.263, f_max=0.400 m², and f_s=0.503 m²; (d) t=0.006 s, V/V₀=0.246, f_max=0.400 m², and f_s=0.641 m²; (e) t=0.008 s, V/V₀=0.246, f_max=0.400 m², and f_s=0.769 m².

Fig.12 Convergence history of objective and constraint function.

Fig.13 Final designs obtained from (a) full analysis and (b) SOAR (N=30), (c) SOAR (N=40), (d) SOAR (N=50), and (e) SOAR (N=60) schemes.

Tab.2 Comparison of the optimized results between full analysis and SOAR schemes with various numbers of basis vectors

$a 0$	Parameter defined in Eq. (27)
$A k$	Weight coefficient in integral operation
$B$	Width of the 3D structure
$C$	Global damping matrix
$c M$ , $c K$	Proportional damping coefficients
$c μ$	Adjustment coefficient
$C R$	Reduced damping matrix
$α$ , $δ$	Parameters in Newmark method
$β$ , $η$	Parameters of the threshold projection
$ϕ (f)$	A function that replace $f (t)$ to catch the maximum response
$Γ$	Unit vector, of which the concerned DOF is unity while all other components are zeros
$Δ t$	Time step
$Δ (e, i)$	Center-to-center distance
$E 0$	Material’s elastic modulus
$F$	Force vector
$f max$	Peak value of structural dynamic response in the time domain
$f ‾$	Corresponding upper limit
$f s$	Structure response from static analysis
$f (t)$	A structural behavior function used to measure structural dynamic response
$g (t)$	Time-varying value
$H$	Height of the structure
$H e i$	Weighting function
$k e$	Elemental stiffness matrix
$K^$	Effective stiffness matrix
$K$	Global stiffness matrix
$Κ R$	Reduced stiffness matrix
$L$	Length of the structure
$m e$	Elemental mass matrix
$M$	Global mass matrix
$M R$	Reduced mass matrix
$N e$	Number of total elements
$N ˜ e$	Neighborhood set of the eth elements
$N t$	Total step number for discretion of time zone
$P$	Projection matrix
$Q^$	Effective load matrix in Newmark method
$Q (t)$	Time-varying force
$r min$	Circular region of radius
$t k$	Time points in integral operation
$t L$	Loading time
$T$	Thickness of the structure
$u (t)$	Velocity vector
$u ˙ (t)$	Displacement vector
$u ¨ (t)$	Acceleration vector
$V (ρ)$	Structural volume
$z (t)$	Reduced solution given in differential Eq. (7)
$ρ 0$	Material’s density
$ρ e$	The eth elemental design variable
$ρ min$	Lower bound for $ρ$
$ρ ˜$	Filtered density
$ρ ‾$	Physical density from the threshold projection
$τ$	Time span between two loading sections
$ν$	Poisson’s ratio
$λ (t)$	An arbitrary vector
$μ$	An aggregation function parameter
$ω$	Circular frequency

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