Robust topology optimization accounting for misplacement of material

The use of topology optimization for structural design often leads to slender structures. Slender structures are sensitive to geometric imperfections such as the misplacement or misalignment of material. The present paper therefore proposes a robust approach to topology optimization taking into account this type of geometric imperfections. A density filter based approach is followed, and translations of material are obtained by adding a small perturbation to the center of the filter kernel. The spatial variation of the geometric imperfections is modeled by means of a vector valued random field. The random field is conditioned in order to incorporate supports in the design where no misplacement of material occurs. In the robust optimization problem, the objective function is defined as a weighted sum of the mean value and the standard deviation of the performance of the structure under uncertainty. A sampling method is used to estimate these statistics during the optimization process. The proposed method is successfully applied to three example problems: the minimum compliance design of a slender column-like structure and a cantilever beam and a compliant mechanism design. An extensive Monte Carlo simulation is used to show that the obtained topologies are more robust with respect to geometric imperfections.

[1]  A. N. Kolmogorov,et al.  Foundations of the theory of probability , 1960 .

[2]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[3]  E. Gaylord,et al.  Design of Steel Structures , 1972 .

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

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

[6]  R. Ghanem,et al.  Stochastic Finite Elements: A Spectral Approach , 1990 .

[7]  M. Zhou,et al.  Generalized shape optimization without homogenization , 1992 .

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

[9]  O. Ditlevsen.  Dimension Reduction and Discretization in Stochastic Problems by Regression Method , 1996 .

[10]  Arkadi Nemirovski,et al.  Robust Truss Topology Design via Semidefinite Programming , 1997, SIAM J. Optim..

[11]  Ole Sigmund,et al.  On the Design of Compliant Mechanisms Using Topology Optimization , 1997 .

[12]  J. Petersson,et al.  Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima , 1998 .

[13]  Dimitris Diamantidis,et al.  The Joint Committee on Structural Safety (JCSS) Probabilistic Model Code for New and Existing Structures , 1999 .

[14]  B. Bourdin.  Filters in topology optimization , 2001 .

[15]  T. E. Bruns,et al.  Topology optimization of non-linear elastic structures and compliant mechanisms , 2001 .

[16]  JCSS PROBABILISTIC MODEL CODE PART 3: RESISTANCE MODELS , 2001 .

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

[18]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[19]  D. Hartmann,et al.  Optimization of slender structures considering geometrical imperfections , 2006 .

[20]  O. Sigmund.  Morphology-based black and white filters for topology optimization , 2007 .

[21]  Bernhard Sendhoff,et al.  Robust Optimization - A Comprehensive Survey , 2007 .

[23]  James K. Guest,et al.  Structural optimization under uncertain loads and nodal locations , 2008 .

[24]  Bruno Sudret,et al.  Global sensitivity analysis using polynomial chaos expansions , 2008, Reliab. Eng. Syst. Saf..

[25]  Nozomu Kogiso,et al.  Robust Topology Optimization for Compliant Mechanisms Considering Uncertainty of Applied Loads , 2008 .

[26]  Chanakya Arya,et al.  Eurocode 3: Design of steel structures , 2018, Design of Structural Elements.

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

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

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

[30]  James K. Guest,et al.  Optimal design of trusses with geometric imperfections: Accounting for global instability , 2011 .

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

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

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

[34]  James K. Guest,et al.  Eliminating beta-continuation from Heaviside projection and density filter algorithms , 2011 .

[35]  Jakob S. Jensen,et al.  Robust topology optimization of photonic crystal waveguides with tailored dispersion properties , 2011 .

[36]  Ole Sigmund,et al.  On projection methods, convergence and robust formulations in topology optimization , 2011, Structural and Multidisciplinary Optimization.

[37]  M. Bendsøe,et al.  Topology Optimization: "Theory, Methods, And Applications" , 2011 .

[38]  James K. Guest,et al.  Robust topology optimization of structures with uncertainties in stiffness - Application to truss structures , 2011 .

[39]  Mattias Schevenels,et al.  Robust design of large-displacement compliant mechanisms , 2011 .

[40]  P. Papalambros,et al.  A heuristic sequencing procedure for sequential solution of decomposed optimal design problems , 2012 .

[41]  Mattias Schevenels,et al.  Topology optimization with geometric uncertainties by perturbation techniques , 2012 .

[42]  Daniel A. Tortorelli,et al.  Minmax topology optimization , 2012 .

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