Topology optimization using material-field series expansion and Kriging-based algorithm: An effective non-gradient method

Abstract Topology optimization is now a very effective and important tool for designing the layouts of various structural and multidisciplinary problems, but most existing methods require information about the sensitivity of the performance function with respect to an enormous number of design variables. This paper presents an efficient non-gradient approach to the topology optimization of structures when no information is available about design sensitivity. Based on the material-field series expansion (MFSE), the problem of topology optimization is constructed as a constrained minimization model with the series expansion coefficients as the design variables, thereby involving a considerable reduction of design variables. The Kriging-based optimization algorithm incorporating two infill criteria is used to solve the optimization problem. A special strategy of (i) using a self-adjusting design domain and (ii) remodeling the surrogate function is proposed to improve the searching efficiency of the Kriging-based algorithm. Several examples are given in the form of linear, nonlinear, and fluid topology optimization problems to demonstrate the effectiveness and applicability of the proposed Kriging-based MFSE method.

[1]  Zhan Kang,et al.  Wrinkle-free design of thin membrane structures using stress-based topology optimization , 2017 .

[2]  Y. Xie,et al.  A simple evolutionary procedure for structural optimization , 1993 .

[3]  Zhan Kang,et al.  Robust topology optimization of vibrating structures considering random diffuse regions via a phase-field method , 2019, Computer Methods in Applied Mechanics and Engineering.

[4]  M. Bendsøe,et al.  Generating optimal topologies in structural design using a homogenization method , 1988 .

[5]  Donald R. Jones,et al.  Efficient Global Optimization of Expensive Black-Box Functions , 1998, J. Glob. Optim..

[6]  Z. Kang,et al.  Structural topology optimization based on non-local Shepard interpolation of density field , 2011 .

[7]  K. Maute,et al.  Conceptual design of aeroelastic structures by topology optimization , 2004 .

[8]  M. Wang,et al.  Radial basis functions and level set method for structural topology optimization , 2006 .

[9]  Jakob S. Jensen,et al.  Maximizing band gaps in plate structures , 2006 .

[10]  Z. Kang,et al.  Non-probabilistic uncertainty quantification and response analysis of structures with a bounded field model , 2019, Computer Methods in Applied Mechanics and Engineering.

[11]  Xiaoming Wang,et al.  A level set method for structural topology optimization , 2003 .

[12]  Z. Kang,et al.  Topology optimization of geometrically nonlinear structures based on an additive hyperelasticity technique , 2015 .

[13]  Andy J. Keane,et al.  Engineering Design via Surrogate Modelling - A Practical Guide , 2008 .

[14]  Chun-Yin Wu,et al.  Topology optimization of structures using modified binary differential evolution , 2010 .

[15]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[16]  J. Petersson,et al.  Topology optimization of fluids in Stokes flow , 2003 .

[17]  O. Sigmund,et al.  Topology optimization of channel flow problems , 2005 .

[18]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[19]  Xu Guo,et al.  Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework , 2014 .

[20]  Jamie Paik,et al.  Stretchable Materials for Robust Soft Actuators towards Assistive Wearable Devices , 2016, Scientific Reports.

[21]  Ramana V. Grandhi,et al.  A survey of structural and multidisciplinary continuum topology optimization: post 2000 , 2014 .

[22]  James T. Allison,et al.  Topology optimization for heat conduction using generative design algorithms , 2017 .

[23]  Z. Kang,et al.  Multi-material topology optimization considering interface behavior via XFEM and level set method , 2016 .

[24]  Yangjun Luo,et al.  Buckling assessment of thin-walled plates with uncertain geometrical imperfections based on non-probabilistic field model , 2019 .

[25]  R. M. Natal Jorge,et al.  A Comparative Study of Several Material Models for Prediction of Hyperelastic Properties: Application to Silicone‐Rubber and Soft Tissues , 2006 .

[26]  M. Zhou,et al.  The COC algorithm, Part II: Topological, geometrical and generalized shape optimization , 1991 .

[27]  V. R. Joseph,et al.  ORTHOGONAL-MAXIMIN LATIN HYPERCUBE DESIGNS , 2008 .

[28]  Shigeru Obayashi,et al.  Topology optimization of fluid problems using genetic algorithm assisted by the Kriging model , 2017 .

[29]  Olivier Bruls,et al.  Topology and generalized shape optimization: Why stress constraints are so important? , 2008 .

[30]  Shiwei Zhou,et al.  Designing broad phononic band gaps for in-plane modes , 2018 .

[31]  Z. Kang,et al.  An adaptive refinement approach for topology optimization based on separated density field description , 2013 .

[32]  Zhonghua Han,et al.  Efficient aerodynamic shape optimization of transonic wings using a parallel infilling strategy and surrogate models , 2016, Structural and Multidisciplinary Optimization.

[33]  Zhan Kang,et al.  Robust topology optimization of phononic crystals with random field uncertainty , 2018 .

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

[35]  Mariusz Bujny,et al.  Kriging-assisted topology optimization of crash structures , 2019, Computer Methods in Applied Mechanics and Engineering.

[36]  Jianbin Du,et al.  A generalized DCT compression based density method for topology optimization of 2D and 3D continua , 2018, Computer Methods in Applied Mechanics and Engineering.

[37]  Zhan Kang,et al.  Achieving directional propagation of elastic waves via topology optimization , 2018, Ultrasonics.

[38]  James K. Guest,et al.  Optimizing multifunctional materials: Design of microstructures for maximized stiffness and fluid permeability , 2006 .

[39]  Anders Clausen,et al.  Efficient topology optimization in MATLAB using 88 lines of code , 2011 .

[40]  D. Krige.  A statistical approach to some basic mine valuation problems on the Witwatersrand, by D.G. Krige, published in the Journal, December 1951 : introduction by the author , 1951 .

[41]  K. Tai,et al.  Structural topology design optimization using Genetic Algorithms with a bit-array representation , 2005 .

[42]  Krister Svanberg,et al.  A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations , 2002, SIAM J. Optim..

[43]  Zhan Kang,et al.  Velocity field level‐set method for topological shape optimization using freely distributed design variables , 2019, International Journal for Numerical Methods in Engineering.

[44]  M. Bendsøe.  Optimal shape design as a material distribution problem , 1989 .

[45]  Ole Sigmund,et al.  On the usefulness of non-gradient approaches in topology optimization , 2011 .

[46]  Anirban Chaudhuri,et al.  Parallel surrogate-assisted global optimization with expensive functions – a survey , 2016 .

[47]  Hongliang Ren,et al.  Topology Optimized Design, Fabrication, and Characterization of a Soft Cable-Driven Gripper , 2018, IEEE Robotics and Automation Letters.

[48]  Z. Kang,et al.  A level set method for structural shape and topology optimization using radial basis functions , 2009 .

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

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

[51]  Yangjun Luo,et al.  A material-field series-expansion method for topology optimization of continuum structures , 2019 .

[52]  O. Sigmund,et al.  Stiffness design of geometrically nonlinear structures using topology optimization , 2000 .