Abstract
Accurate prediction of stochastic responses of a structure caused by natural hazards or operations of non-structural components is crucial to achieve an effective design. In this regard, it is of great significance to incorporate the impact of uncertainty into topology optimization of structures under constraints on their stochastic responses. Despite recent technological advances, the theoretical framework remains inadequate to overcome computational challenges of incorporating stochastic responses to topology optimization. Thus, this paper presents a theoretical framework that integrates random vibration theories with topology optimization using a discrete representation of stochastic excitations. This paper also discusses the development of parameter sensitivity of dynamic responses in order to enable the use of efficient gradient-based optimization algorithms. The proposed topology optimization framework and sensitivity method enable efficient topology optimization of structures under stochastic excitations, which is successfully demonstrated by numerical examples of structures under stochastic ground motion excitations.
Similar content being viewed by others
Abbreviations
- f(t):
-
Stationary Gaussian input process
- v :
-
Vector of n uncorrelated standard normal random variables
- s(t):
-
Vector of deterministic basis functions
- h f (·):
-
Impulse-response function of a filter
- h s (·):
-
Impulse-response function of a system
- u(t):
-
Displacement time history
- a(t):
-
Vector of deterministic basis functions
- u 0 :
-
Threshold value
- E f :
-
Failure event
- P(E f ):
-
Failure probability
- P target f :
-
Target failure probability
- Φ[·]:
-
Cumulative distribution function of the standard normal distribution
- β:
-
Reliability index
- βtarget :
-
Target reliability index
- d :
-
Vector of deterministic design variables
- \( {\tilde{\rho}}_e\left(\mathbf{d}\right) \) :
-
Element density
- p :
-
Stiffness penalization parameter
- q :
-
Mass penalization parameter
- D(·):
-
Elastic tensor determined by material density function
- D 0 :
-
Elasticity tensor of the solid material
- E 0 :
-
Young’s modulus of the solid phase
- K e :
-
Element stiffness matrix
- M e :
-
Element mass matrix
- K :
-
Global stiffness matrix
- M :
-
Global mass matrix
- C :
-
Global damping matrix
- f :
-
Global load vector
- ü :
-
Global acceleration vector
- \( \dot{\mathbf{u}} \) :
-
Global velocity vector
- u :
-
Global displacement vector
- λ :
-
Adjoint variable vector
- Φ0 :
-
Power spectral density of the white noise process
- ω f :
-
Predominant frequency of a random process (a filter)
- ζ f :
-
Bandwidth of a random process (a filter)
- Δ i /L i :
-
Inter-story drift rations of the i-th floor evaluated at specified points
- L i :
-
i-th floor height
- Δ i :
-
i-th floor drift evaluated at a specified point
References
Allen M, Raulli M, Maute K, Frangopol DM (2004) Reliability-based analysis and design optimization of electrostatically actuated MEMS. Comp Struc 82(13–14):1007–1020
Almeida SRM, Paulino GH, Silva ECN (2010) Material gradation and layout in topology optimization of functionally graded structures: a global–local approach. Struct Multidiscip Optim 42(6):855–868
ASCE. (2010). Minimum design loads for buildings and other structures. ASCE 7–10, Reston, VA
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Multidiscip Optim 1(4):193–202
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71:197–224
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9):635–654
Bendsøe MP, Sigmund O (2003) Topology optimization – theory, methods and applications. Springer Verlag, New York
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158
Chan SP, Cox HL, Benfield WA (1962) Transient analysis of forced vibrations of complex structural mechanical systems. J R Aeronaut Sot 66:457–460
Chen X, Kareem A (2005) POD-based modeling, analysis, and simulation of dynamic wind load effects on structures. J Eng Mech 131(4):325–339
Chen L, Letchford CW (2005) Proper orthogonal decomposition of two vertical profiles of full-scale nonstationary downburst wind speeds[lzcl]. J Wind Eng Ind Aerodyn 93(3):187–216
Chen S, Chen W, Lee S (2010) Level set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41(4):507–524
Choi K, Kim N (2005) Structural sensitivity analysis and optimization 1. Springer-Verlag, New York
Clough R, Penzien J (1993) Dynamics of structures. McGraw Hill, New York
Deodatis G, Shinozuka M (1988) Autoregressive model for nonstationary stochastic processes. J Eng Mech 114(11):1995–2012
Der Kiureghian A (2000) The geometry of random vibrations and solutions by FORM and SORM. Probabilistic Eng Mech 15:81–90
Der Kiureghian A (2004) First- and second-order reliability methods. In: Nikolaidis E, Ghiocel DM, Singhal S (eds) Engineering Design Reliability Handbook. CRC Press, Boca Raton, FL, Chap. 14
Diaz AR, Kikuchi N (1992) Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int J Numer Methods Eng 35:1487–1502
Diaz AR, Sigmund O (1995) Checkerboard patterns in layout optimization. Struc Optim 10:40–45
Du J, Olhoff N (2007) Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps. Struct Multidiscip Optim 34:91–110
Fujimura K, Der Kiureghian A (2007) Tail-equivalent linearization method for nonlinear random vibration. Probabilistic Eng Mech 22(1):63–76
Gersch W, Yonemoto J (1977) Synthesis of multivariate random vibration systems: A two-stage least squares ARMA model approach. J Sound Vib 52(4):553–565
Grigoriu M (1993) On the spectral representation method in simulation. Probabilistic Eng Mech 8(2):75–90
Grigoriu M (2003) Algorithm for generating sampling of homogeneous Gaussian fields. J Eng Mech 129(1):43–49
Guest JK, Igusa T (2008) Structural optimization under uncertain loads and nodal locations. Comput Methods Appl Mech Eng 198(1):116–124
Guest JK, Prevost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254
Haftka RT, Gürdal Z (1992) Elements of structural optimization. 3rd edition, Springer
Haug E, Arora S (1978) Design sensitivity analysis of elastic mechanical systems. Comput Methods Appl Mech Eng 15(1):35–62
Haug J, Choi K, Komkov V (1986) Design sensitivity analysis of structural systems. Academic press, Orlando FL Allen M, Raulli M, Maute K, Frangopol, DM (2004) Reliability-based analysis and design optimization of electrostatically actuated MEMS. Computer & Structures, 82(13–14):1007–1020
Houbolt JC (1950) A recurrence matrix solution for the dynamic response of elastic aircraft. J Aeronaut Sci 17(9):540–550
Hulbert GM, Chung J (1996) Explicit time integration algorithms for structural dynamics with optimal numerical dissipation. Comput Methods Appl Mech Eng 137(2):175–188
Jalalpour M, Guest JK, Igusa T (2013) Reliability-based topology optimization of trusses with stochastic stiffness. Struct Saf 43:41–49
Jensen JS, Pedersen NL (2006) On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases. J Sound Vib 289(4–5):967–986
Jog CS, Haber RB (1996) Stability of finite element models for distributed-parameter optimization and topology design. Comput Methods Appl Mech Eng 130:203–226
Kang J, Kim C, Wang S (2004) Reliability-based topology optimization for electromagnetic systems. COMPEL: Int J Comput Math Elec Electron Eng 23(3):715–723
Kharmanda G, Olhoff N, Mohamed A, Lemaire M (2004) Reliability-based topology optimization. Struct Multidiscip Optim 26(5):295–307
Kim YY, Yoon GH (2000) Multi-resolution multi-scale topology optimization–a new paradigm. Int J Solids Struct 37:5529–5559
Kim C, Wang S, Rae K, Moon H, Choi KK (2006) Reliability-based topology optimization with uncertainties. J Mech Sci Technol 20(4):494–504
Kitagawa G (1996) Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J Comput Graph Stat 5(1):1–25
Kohn RV, Strang G (1986) Optimal design and relaxation of variational problems. Comm Pure Appl Math 39:113–137, 139–182, 353–377
Konakli K, Der Kiureghian A (2012) Simulation of spatially varying ground motions including incoherence, wave-passage and differential site-response effects. Earthq Eng Struct Dyn 41(3):495–513
Li CC, Der Kiureghian A (1993) Optimal discretization of random fields. J Eng Mech 119(6):1136–1154
Liu J, Chen R (1998) Sequential Monte Carlo methods for dynamic systems. J Am Stat Assoc 93(443):1032–1044
Lógó J, Ghaemi M, Rad MM (2009) Optimal topologies in case of probabilistic loading: the influence of load correlation. Mech Based Design Struc Mach 37(3):327–348
Luo Y, Kang Z, Luo Z, Li A (2009) Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct Multidiscip Optim 39(3):297–310
Lutes LD, Sarkani S (2003) Random vibrations: analysis of structural and mechanical systems. Elsevier Butterworth-Heinemann, Burlington MA
Ma ZD, Cheng HC, Kikuchi N (1994) Structural design for obtaining desired eigenfrequencies by using the topology and shape optimization method. Comput Syst Eng 5(1):77–89
Ma ZD, Kikuchi N, Cheng HC (1995) Topological design for vibrating structures. Comput Methods Appl Mech Eng 121(1–4):259–280
Maeda Y, Nishiwaki S, Izui K, Yoshimura M, Matsui K, Terada K (2006) Structural topology optimization of vibrating structures with specified eigenfrequencies and eigenmode shapes. Int J Numer Methods Eng 67:597–628
Maute K, Frangopol DM (2003) Reliability-based design of MEMS mechanisms by topology optimization. Comp Struct 81(8–11):813–824
Mignolet MP, Spanos PD (1987) Recursive simulation of stationary multivariate random processes-Part I. J Appl Mech 54(3):674–680
Min S, Kikuchi N, Park YC, Kim S, Chang S (1999) Optimal topology design of structures under dynamic loads. Struct Multidiscip Optim 17(2–3):208–218
NEHRP (2009) Recommended seismic provisions for new buildings and other structures (FEMA P-750). Federal Emergency Management Agency, Washington, D. C
Newmark NM (1959) A method of computation for structural dynamics. J Eng Mech 85:67–94
Nguyen TH, Song J, Paulino GH (2011) Single-loop system reliability-based topology optimization considering statistical dependence between limit-states. Struct Multidiscip Optim 44(5):593–611
Novak D, Stoyanoff S, Herda H (1995) Error assessment for wind histories generated by autoregressive method. Struct Saf 17(2):79–90
Pedersen NL (2000) Maximization of eigenvalues using topology optimization. Struct Multidiscip Optim 20:2–11
Poulsen A (2002) Topology optimization in wavelet space. Int J Numer Methods Eng 53:567–582
Rezaeian S, Der Kiureghian A (2008) A stochastic ground motion model with separable temporal and spectral nonstationarities. Earthq Eng Struc Dyn 37(13):1565–1584
Rezaeian S, Der Kiureghian A (2010) Simulation of synthetic ground motions for specified earthquake and site characteristics. Earthq Eng Struc Dyn 39(10):1155–1180
Rezaeian S, Der Kiureghian A (2012) Simulation of orthogonal horizontal ground motion components for specified earthquake and site characteristics. Earthq Eng Struc Dyn 41(2):335–353
Rozvany GIN (2008) Exact analytical solutions for benchmark problems in probabilistic topology optimization. In: EngOpt 2008–international conference on engineering optimization, Rio de Janeiro
Rozvany GIN, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Struct Multidiscip Optim 4(34):250–252
Rubio WM, Paulino GH, Silva ECN (2011) Tailoring vibration mode shapes using topology optimization and functionally graded material concepts. Smart Mater Struct 20(2):025009
Shinozuka M (1972) Monte Carlo solution of structural dynamics. Comp Struc 2(5–6):855–874
Shinozuka M, Deodatis G (1991) Simulation of the stochastic process by spectral representation. Appl Mech Rev 44(4):29–53
Shinozuka M, Deodatis G (1996) Simulation of multi-dimensional Gaussian stochastic fields by spectral representation. Appl Mech Rev 49(1):29–53
Shinozuka M, Jan CM (1972) Digital simulation of random processes and its applications. J Sound Vib 25(1):111–128
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4–5):401–424
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struc Optimization 16(1):68–75
Song J, Kang WH (2009) System reliability and sensitivity under statistical dependence by matrix-based system reliability method. Struct Saf 31(2):148–156
Spanos PD, Ghanem R (1989) Stochastic finite element expansion for random media. J Eng Mech 115(5):1035–1053
Spanos PD, Mignolet MP (1987) Recursive simulation of stationary multivariate random processes–Part II. J Appl Mech 54(3):681–687
Spanos PD, Mignolet MP (1990) Simulation of Stationary Random Processes: Two-Stage MA to ARMA Approach. J Eng Mech 116(3):620–641
Stromberg LL, Beghini A, Baker WF, Paulino GH (2011) Application of layout and topology optimization using pattern gradation for the conceptual design of buildings. Struct Multidiscip Optim 43(2):165–180
Svanberg K (1987) The method of moving asymptotes - a new method for structural optimization. Int J Numer Methods Eng 24(2):359–373
Zhang J, Ellingwood B (1994) Orthogonal series expansions of random fields in reliability analysis. J Eng Mech 120(12):2660–2677
Zienkiewicz OC (1977) A new look at the newmark, houbolt and other time stepping formulas. A weighted residual approach. Earthq Eng Struc Dyn 5(4):413–418
Acknowledgements
The authors gratefully acknowledge funding provided by the National Science Foundation (NSF) through project CMMI 1234243. We also acknowledge support from the Raymond Allen Jones Chair at the Georgia Institute of Technology. The second author also acknowledges the support from the Integrated Research Institute of Construction and Environmental Engineering at Seoul National University, and the National Research Foundation of Korea (NRF) Grant (No. 2015R1A5A7037372), funded by the Korean Government (MSIP). Any opinion, finding, conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors.
Author information
Authors and Affiliations
Corresponding author
Appendix: Derivation of (11)
Appendix: Derivation of (11)
To derive (11) with a uniform time step, the convolution integral in (9) is carried out for discrete time intervals. An entry of the vector a(t j ) can be written as follows:
where
Then, one can consider the following three cases:
-
Case 1:
i = j
-
Case 2:
i > j
-
Case 3:
i < j
Thus,
Therefore, the uniform step size (t n -t n-1 = ∆t, t n = t 0) in the convolution integral leads to the following expression:
Rights and permissions
About this article
Cite this article
Chun, J., Song, J. & Paulino, G.H. Structural topology optimization under constraints on instantaneous failure probability. Struct Multidisc Optim 53, 773–799 (2016). https://doi.org/10.1007/s00158-015-1296-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-015-1296-y