A level set topology optimization method for the buckling of shell structures

Shell structures are some of the most widely used in engineering applications. Flat plates, stiffened panels, and wing ribs are each examples of components for which the design features may be dictated by the critical buckling load. Despite this practical significance, there exists only a handful of studies in the literature documenting applications of topology optimization which consider buckling performance. This is due to several issues innate to this domain, including mode switching, spurious behavior in void regions, and the presence of repeated eigenvalues. Herein, we propose a level set method capable of effectively optimizing structures despite these challenges in the context of linear buckling. We demonstrate the usefulness of such in the design of several common shell structures and explore the trade-off between stiffness and buckling load performance.

[1]  Rayleigh-Ritz optimal design of orthotropic plates for buckling , 1996 .

[2]  Ronald Fedkiw,et al.  Level set methods and dynamic implicit surfaces , 2002, Applied mathematical sciences.

[3]  G. I. N. Rozvany,et al.  Difficulties in truss topology optimization with stress, local buckling and system stability constraints , 1996 .

[4]  James A. Sethian,et al.  The Fast Construction of Extension Velocities in Level Set Methods , 1999 .

[5]  O. Sigmund,et al.  Revisiting topology optimization with buckling constraints , 2018, Structural and Multidisciplinary Optimization.

[6]  Bret Stanford,et al.  Aeroelastic Topology Optimization of Blade-Stiffened Panels , 2013 .

[7]  Archibald N. Sherbourne,et al.  MECHANICS OF SHAPE OPTIMIZATION IN PLATE BUCKLING , 1992 .

[8]  G. Kreisselmeier,et al.  SYSTEMATIC CONTROL DESIGN BY OPTIMIZING A VECTOR PERFORMANCE INDEX , 1979 .

[9]  Osama Bedair,et al.  Analysis and Limit State Design of Stiffened Plates and Shells: A World View , 2009 .

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

[11]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

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

[13]  M. Zhou,et al.  Difficulties in truss topology optimization with stress and local buckling constraints , 1996 .

[14]  Li-Juan Li,et al.  An Adaptive Continuation Method for Topology Optimization of Continuum Structures Considering Buckling Constraints , 2017 .

[15]  Grant P. Steven,et al.  An evolutionary method for optimization of plate buckling resistance , 1998 .

[16]  Peter D. Dunning,et al.  Optimal Topology of Aircraft Rib and Spar Structures Under Aeroelastic Loads , 2014 .

[17]  Julián A. Norato,et al.  Stress-based shape and topology optimization with the level set method , 2018 .

[18]  Alain Remouchamps,et al.  Discussion on some convergence problems in buckling optimisation , 2008 .

[19]  A. Peirce.  Computer Methods in Applied Mechanics and Engineering , 2010 .

[20]  Bret Stanford,et al.  Aeroelastic Wingbox Stiffener Topology Optimization , 2017 .

[21]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .

[22]  N. Olhoff,et al.  Multiple eigenvalues in structural optimization problems , 1994 .

[23]  K. Yamazaki,et al.  A new approach for the solution of singular optima in truss topology optimization with stress and local buckling constraints , 2001 .

[24]  Lars Krog,et al.  Application of Topology, Sizing and Shape Optimization Methods to Optimal Design of Aircraft Components , 2011 .

[25]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[26]  Bret Stanford,et al.  Structural Optimization of Platelike Aircraft Wings Under Flutter and Divergence Constraints , 2018, AIAA Journal.

[27]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[28]  Rakesh K. Kapania,et al.  Comparative Study on Optimal Stiffener Placement for Curvilinearly Stiffened Panels , 2011 .

[29]  K. Bathe,et al.  A four‐node plate bending element based on Mindlin/Reissner plate theory and a mixed interpolation , 1985 .

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

[31]  Ole Sigmund,et al.  Giga-voxel computational morphogenesis for structural design , 2017, Nature.

[32]  Y. Xie,et al.  An improved method for evolutionary structural optimisation against buckling , 2001 .

[33]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[34]  Niels Leergaard Pedersen,et al.  Buckling load optimization for 2D continuum models, with alternative formulation for buckling load estimation , 2018, Structural and Multidisciplinary Optimization.

[35]  W. J. Anderson,et al.  New mode tracking methods in aeroelastic analysis , 1995 .

[36]  Haitao Ma,et al.  Topology optimization of continuum structures under buckling constraints , 2015 .

[37]  Fred van Keulen,et al.  A unified aggregation and relaxation approach for stress-constrained topology optimization , 2017 .

[38]  Graeme J. Kennedy,et al.  Large-Scale Compliance-Minimization and Buckling Topology Optimization of the Undeformed Common Research Model Wing , 2016 .

[39]  Bret Stanford,et al.  Aerothermoelastic topology optimization with flutter and buckling metrics , 2013 .

[40]  Ole Sigmund,et al.  Buckling strength topology optimization of 2D periodic materials based on linearized bifurcation analysis , 2018, Computer Methods in Applied Mechanics and Engineering.

[41]  Jennifer A. Scott,et al.  Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver , 2016 .

[42]  S. Rahmatalla,et al.  CONTINUUM TOPOLOGY OPTIMIZATION OF BUCKLING-SENSITIVE STRUCTURES , 2002 .

[43]  Peter D. Dunning,et al.  Introducing the sequential linear programming level-set method for topology optimization , 2015 .