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

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.

Key words： topology optimization; solid isotropic material with penalization; transient response; aggregation function; second-order Arnoldi reduction

Acknowledgements

The authors acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 11902232).

References

1 Bendsøe M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization. Computer Methods in Applied Mechanics and Engineering, 1988, 71(2): 197–224

2 Deaton J D, Grandhi R V. A survey of structural and multidisciplinary continuum topology optimization: post 2000. Structural and Multidisciplinary Optimization, 2014, 49(1): 1–38

3 Sigmund O, Maute K. Topology optimization approaches. Structural and Multidisciplinary Optimization, 2013, 48(6): 1031–1055

4 Chen Z, Long K, Wen P, . Fatigue-resistance topology optimization of continuum structure by penalizing the cumulative fatigue damage. Advances in Engineering Software, 2020, 150: 102924

5 Zargham S, Ward T A, Ramli R, . Topology optimization: A review for structural designs under vibration problems. Structural and Multidisciplinary Optimization, 2016, 53(6): 1157–1177

6 Díaaz A R, Kikuchi N. Solution to shape and topology eigenvalue optimization problems using a homogenization method. International Journal for Numerical Methods in Engineering, 1992, 35(7): 1487–1502

7 Pedersen N L. Maximization of eigenvalues using topology optimization. Structural and Multidisciplinary Optimization, 2000, 20(1): 2–11

8 Du J, Olhoff N. Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Structural and Multidisciplinary Optimization, 2007, 34(2): 91–110

9 Li Q, Wu Q, Liu J, . Topology optimization of vibrating structures with frequency band constraints. Structural and Multidisciplinary Optimization, 2021, 63(3): 1203–1218

10 Niu B, Yan J, Cheng G. Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Structural and Multidisciplinary Optimization, 2009, 39(2): 115–132

11 Long K, Han D, Gu X. Concurrent topology optimization of composite macrostructure and microstructure constructed by constituent phases of distinct Poisson’s ratios for maximum frequency. Computational Materials Science, 2017, 129: 194–201

12 Ma Z D, Kikuchi N, Cheng H C. Topological design for vibrating structures. Computer Methods in Applied Mechanics and Engineering, 1995, 121(1‒4): 259–280

13 Jog C S. Topology design of structures subject to periodic loading. Journal of Sound and Vibration, 2002, 253(3): 687–709

14 Olhoff N, Du J. Generalized incremental frequency method for topological design of continuum structures for minimum dynamic compliance subject to forced vibration at a prescribed low or high value of the excitation frequency. Structural and Multidisciplinary Optimization, 2016, 54(5): 1113–1141

15 Niu B, He X, Shan Y, . On objective functions of minimizing the vibration response of continuum structures subject to external harmonic excitation. Structural and Multidisciplinary Optimization, 2018, 57(6): 2291–2307

16 Yoon G H. Structural topology optimization for frequency response problem using model reduction schemes. Computer Methods in Applied Mechanics and Engineering, 2010, 199(25‒28): 1744–1763

17 Liu H, Zhang W, Gao T. A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations. Structural and Multidisciplinary Optimization, 2015, 51(6): 1321–1333

18 Zhu J, He F, Liu T, . Structural topology optimization under harmonic base acceleration excitations. Structural and Multidisciplinary Optimization, 2018, 57(3): 1061–1078

19 Long K, Wang X, Liu H. Stress-constrained topology optimization of continuum structures subjected to harmonic force excitation using sequential quadratic programming. Structural and Multidisciplinary Optimization, 2019, 59(5): 1747–1759

20 Niu B, Olhoff N, Lund E, . Discrete material optimization of vibrating laminated composite plates for minimum sound radiation. International Journal of Solids and Structures, 2010, 47(16): 2097–2114

21 Du J, Olhoff N. Minimization of sound radiation from vibrating bi-material structures using topology optimization. Structural and Multidisciplinary Optimization, 2007, 33(4‒5): 305–321

22 Du J, Olhoff N. Topological design of vibrating structures with respect to optimum sound pressure characteristics in a surrounding acoustic medium. Structural and Multidisciplinary Optimization, 2010, 42(1): 43–54

23 Dilgen C B, Dilgen S B, Aage N, . Topology optimization of acoustic mechanical interaction problems: A comparative review. Structural and Multidisciplinary Optimization, 2019, 60(2): 779–801

24 Kang B S, Park G J, Arora J S. A review of optimization of structures subjected to transient loads. Structural and Multidisciplinary Optimization, 2006, 31(2): 81–95

25 Min S, Kikuchi N, Park Y, . Optimal topology design of structures under dynamic loads. Structural and Multidisciplinary Optimization, 1999, 17(2‒3): 208–218

26 Turteltaub S. Optimal non-homogeneous composites for dynamic loading. Structural and Multidisciplinary Optimization, 2005, 30(2): 101–112

27 Zhao J P, Wang C J. Topology optimization for minimizing the maximum dynamic response in the time domain using aggregation functional method. Computers & Structures, 2017, 190: 41–60

28 Zhao J P, Wang C J. Dynamic response topology optimization in the time domain using model reduction method. Structural and Multidisciplinary Optimization, 2016, 53(1): 101–114

29 Zhao J P, Yoon H, Youn B D. Concurrent topology optimization with uniform microstructure for minimizing dynamic response in the time domain. Computers & Structures, 2019, 222: 98–117

30 Zhao L, Xu B, Han Y, . Continuum structural topological optimization with dynamic stress response constraints. Advances in Engineering Software, 2020, 148: 102834

31 Zhao L, Xu B, Han Y, . Structural topological optimization with dynamic fatigue constraints subject to dynamic random loads. Engineering Structures, 2020, 205(15): 110089

32 Jang H, Lee H A, Lee J Y, . Dynamic response topology optimization in the time domain using equivalent static loads. AIAA Journal, 2012, 50(1): 226–234

33 Kang B S, Choi W S, Park G J. Structural optimization under equivalent static loads transformed from dynamic loads based on displacement. Computers & Structures, 2001, 79 (2): 145–154

34 Choi W S, Park G J. Structural optimization using equivalent static loads at all time intervals. Computer Methods in Applied Mechanics and Engineering, 2002, 191(19‒20): 2105–2122

35 Kim E, Kim H, Baek S, . Effective structural optimization based on equivalent static loads combined with system reduction method. Structural and Multidisciplinary Optimization, 2014, 50(5): 775–786

36 Xu B, Huang X, Xie Y M. Two-scale dynamic optimal design of composite structures in the time domain using equivalent static loads. Composite Structures, 2016, 142: 335–345

37 Stolpe M. On the equivalent static loads approach for dynamic response structural optimization. Structural and Multidisciplinary Optimization, 2014, 50(6): 921–926

38 Stolpe M, Verbart A, Rojas-Labanda S. The equivalent static loads method for structural optimization does not in general generate optimal designs. Structural and Multidisciplinary Optimization, 2018, 58(1): 139–154

39 Lee H A, Park G J. Nonlinear dynamic response topology optimization using equivalent static loads method. Computer Methods in Applied Mechanics and Engineering, 2015, 283: 956–970

40 Bai Y C, Zhou H S, Lei F, . An improved numerically-stable equivalent static loads (ESLs) algorithm based on energy-scaling ratio for stiffness topology optimization under crash loads. Structural and Multidisciplinary Optimization, 2019, 59(1): 117–130

41 Bai Z, Su Y. Dimension reduction of large-scale second-order dynamical systems via a second-order Arnoldi method. SIAM Journal on Scientific Computing, 2005, 26(5): 1692–1709

42 Wang X, Tang X B, Mao L Z. A modified second-order Arnoldi method for solving the quadratic eigenvalue problems. Computers & Mathematics with Applications (Oxford, England), 2017, 73(2): 327–338

43 Zhou P, Peng Y, Du J. Topology optimization of bi-material structures with frequency-domain objectives using time-domain simulation and sensitivity analysis. Structural and Multidisciplinary Optimization, 2021, 63(2): 575–593

44 Kennedy G J, Hicken J E. Improved constraint-aggregation methods. Computer Methods in Applied Mechanics and Engineering, 2015, 289: 332–354

45 Wang F, Lazarov B S, Sigmund O. On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization, 2011, 43(6): 767–784

46 da Silva G A, Beck A T, Sigmund O. Stress-constrained topology optimization considering uniform manufacturing uncertainties. Computer Methods in Applied Mechanics and Engineering, 2019, 344: 512–537

47 da Silva G A, Beck A T, Sigmund O. Topology optimization of compliant mechanisms considering stress constraints, manufacturing uncertainty and geometric nonlinearity. Computer Methods in Applied Mechanics and Engineering, 2020, 365: 112972

48 Svanberg K. The method of moving asymptotes—A new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24(2): 359–373

49 Da D, Xia L, Li G, . Evolutionary topology optimization of continuum structures with smooth boundary representation. Structural and Multidisciplinary Optimization, 2018, 57(6): 2143–2159

50 Xiao M, Lu D, Breitkopf P, . On-the-fly model reduction for large-scale structural topology optimization using principal components analysis. Structural and Multidisciplinary Optimization, 2020, 62(1): 209–230

51 Xiao M, Lu D, Breitkopf P, . Multi-grid reduction-order topology optimization. Structural and Multidisciplinary Optimization, 2020, 61(6): 1–23