Robust topology optimization of vibrating structures considering random diffuse regions via a phase-field method

Abstract This paper investigates a robust topology optimization method for structural dynamic problems by considering random diffuse-region widths between different material phases using a phase-field model. Herein, the spatial distribution of the widths of diffuse regions in a multi-material structure is first represented by a random field and then discretized into uncorrelated stochastic variables using the expansion optimal linear estimation method; stochastic response analysis is then conducted with polynomial chaos expansion. Furthermore, a robust topology optimization formulation of structural dynamic problems is proposed on the basis of the phase-field method, where the design domain is represented with the phase-field function and the explicit phase-field curve is updated by solving the Allen–Cahn equation. A weighted summation of the mean value and standard deviation of the structural dynamic performance is taken as the objective function of the robust optimization problem, where three types of the dynamic performance functions are considered, including the structural dynamic compliance, the fundamental frequency or frequency gap, and the transient displacement under impact loads. The stochastic structural dynamic responses and the corresponding sensitivities are evaluated by polynomial chaos expansion based on finite element analysis at each sampling point. Numerical examples show that the proposed method generates meaningful optimal topologies for structural dynamic robust optimization problems with the framework of the phase-field method. Additionally, some influence factors that affect the optimal solutions are discussed.

[1]  Min Song,et al.  Robustness analysis of responses of vibration control structures with uncertain parameters using interval algorithm , 2007 .

[2]  O. Sigmund,et al.  Topology optimization approaches , 2013, Structural and Multidisciplinary Optimization.

[3]  Jakob S. Jensen,et al.  On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases , 2006 .

[4]  Wei Chen,et al.  Concurrent topology optimization of multiscale structures with multiple porous materials under random field loading uncertainty , 2017, Structural and Multidisciplinary Optimization.

[5]  Mehdi Jalalpour,et al.  Robust topology optimization of frame structures under geometric or material properties uncertainties , 2017 .

[6]  Hu Liu,et al.  A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations , 2015 .

[7]  N. Kikuchi,et al.  Solutions to shape and topology eigenvalue optimization problems using a homogenization method , 1992 .

[8]  A. Doostan,et al.  Least squares polynomial chaos expansion: A review of sampling strategies , 2017, 1706.07564.

[9]  Wei Chen,et al.  A new level-set based approach to shape and topology optimization under geometric uncertainty , 2010 .

[10]  Carl-Johan Thore,et al.  A general framework for robust topology optimization under load-uncertainty including stress constraints , 2017 .

[11]  O. Sigmund,et al.  Robust topology optimization accounting for spatially varying manufacturing errors , 2011 .

[12]  G. Cheng,et al.  On topology optimization of damping layer in shell structures under harmonic excitations , 2012 .

[13]  Jie Gao,et al.  Robust topology optimization for structures under interval uncertainty , 2016, Adv. Eng. Softw..

[14]  Zhan Kang,et al.  Dynamic topology optimization of piezoelectric structures with active control for reducing transient response , 2014 .

[15]  Ole Sigmund,et al.  Systematic design of phononic band–gap materials and structures by topology optimization , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[16]  Z. Kang,et al.  Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model , 2009 .

[17]  Takayuki Yamada,et al.  A topology optimization method based on the level set method incorporating a fictitious interface energy , 2010 .

[18]  W. Carter,et al.  A continuum model of grain boundaries , 2000 .

[19]  S. Nishiwaki,et al.  Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques , 2006 .

[20]  Shiwei Zhou,et al.  Topology optimization for 3D microstructures of viscoelastic composite materials , 2021 .

[21]  Daniel A. Tortorelli,et al.  Topology optimization under uncertainty via non-intrusive polynomial chaos expansion , 2017 .

[22]  Jakob Søndergaard Jensen,et al.  Topology optimization of dynamics problems with Padé approximants , 2007 .

[23]  Arun L. Gain,et al.  Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation , 2012 .

[24]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[25]  Martin Burger,et al.  Phase-Field Relaxation of Topology Optimization with Local Stress Constraints , 2006, SIAM J. Control. Optim..

[26]  N. Olhoff,et al.  Reliability-based topology optimization , 2004 .

[27]  Zhan Kang,et al.  Robust shape and topology optimization considering geometric uncertainties with stochastic level set perturbation , 2017 .

[28]  A. Kiureghian,et al.  OPTIMAL DISCRETIZATION OF RANDOM FIELDS , 1993 .

[29]  Shinji Nishiwaki,et al.  Shape and topology optimization based on the phase field method and sensitivity analysis , 2010, J. Comput. Phys..

[30]  Hae Chang Gea,et al.  Robust topology optimization under multiple independent unknown-but-bounded loads , 2018 .

[31]  Glaucio H. Paulino,et al.  Multi-material continuum topology optimization with arbitrary volume and mass constraints , 2018, Computer Methods in Applied Mechanics and Engineering.

[32]  G. Yoon.  Structural topology optimization for frequency response problem using model reduction schemes , 2010 .

[33]  James K. Guest,et al.  Achieving minimum length scale in topology optimization using nodal design variables and projection functions , 2004 .

[34]  D. Tortorelli,et al.  Gradient based design optimization under uncertainty via stochastic expansion methods , 2016 .

[35]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[36]  Ole Sigmund,et al.  Manufacturing tolerant topology optimization , 2009 .

[37]  N. Kikuchi,et al.  Topological design for vibrating structures , 1995 .

[38]  G. Allaire,et al.  A level-set method for vibration and multiple loads structural optimization , 2005 .

[39]  Z. Kang,et al.  Topology optimization of magnetorheological fluid layers in sandwich plates for semi-active vibration control , 2015 .

[40]  W. Carter,et al.  Extending Phase Field Models of Solidification to Polycrystalline Materials , 2003 .

[41]  Sung-Kie Youn,et al.  Multi-material topology optimization of viscoelastically damped structures under time-dependent loading , 2017 .

[42]  L. Xia,et al.  Topology optimization for maximizing the fracture resistance of quasi-brittle composites , 2018 .

[43]  N. Olhoff,et al.  Topological design of freely vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps , 2007 .

[44]  Akihiro Takezawa,et al.  Phase field method to optimize dielectric devices for electromagnetic wave propagation , 2014, J. Comput. Phys..

[45]  A. Chambolle,et al.  Design-dependent loads in topology optimization , 2003 .

[46]  R. Kobayashi.  Modeling and numerical simulations of dendritic crystal growth , 1993 .

[47]  Bo Ping Wang,et al.  Topology optimization of plate structures subject to initial excitations for minimum dynamic performance index , 2016 .

[48]  Niels Olhoff,et al.  Minimization of sound radiation from vibrating bi-material structures using topology optimization , 2007 .

[49]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[50]  M. Wang,et al.  Level Set Based Structural Topology Optimization for Minimizing Frequency Response , 2011 .

[51]  Shengfeng Zhu,et al.  A multi-mesh finite element method for phase-field based photonic band structure optimization , 2018, J. Comput. Phys..

[52]  Stephen J. Ludwick,et al.  Natural frequency optimization of 3D printed variable-density honeycomb structure via a homogenization-based approach , 2017 .

[53]  Shengli Xu,et al.  Volume preserving nonlinear density filter based on heaviside functions , 2010 .

[54]  I. Doltsinis,et al.  ROBUST DESIGN OF STRUCTURES USING OPTIMIZATION METHODS , 2004 .

[55]  Harald Garcke,et al.  A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints , 2017, Advances in Computational Mathematics.

[56]  D. Xiu.  Numerical Methods for Stochastic Computations: A Spectral Method Approach , 2010 .

[57]  James K. Guest,et al.  Topology optimization of continuum structures under uncertainty – A Polynomial Chaos approach , 2012 .

[58]  Héctor D. Ceniceros,et al.  Computation of multiphase systems with phase field models , 2002 .

[59]  Tong Gao,et al.  A mass constraint formulation for structural topology optimization with multiphase materials , 2011 .

[60]  Z. Luo,et al.  Level-set topology optimization for mechanical metamaterials under hybrid uncertainties , 2017 .

[61]  Dong-Hoon Choi,et al.  Development of a novel phase-field method for local stress-based shape and topology optimization , 2014 .

[62]  Shiwei Zhou,et al.  Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition , 2006 .

[63]  Sang-Hoon Lee,et al.  Level set based robust shape and topology optimization under random field uncertainties , 2010 .