An n -material thresholding method for improving integerness of solutions in topology optimization

Watts, Seth; Tortorelli, Daniel A.

doi:10.1002/nme.5265

Title: An n -material thresholding method for improving integerness of solutions in topology optimization

Journal Article · Sun Apr 10 00:00:00 EDT 2016 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.5265· OSTI ID:1400087

^[1]; Tortorelli, Daniel A. ^[1]

Univ. of Illinois, Urbana-Champaign, IL (United States). Dept. of Mechanical Science & Engineering

It is common in solving topology optimization problems to replace an integer-valued characteristic function design field with the material volume fraction field, a real-valued approximation of the design field that permits "fictitious" mixtures of materials during intermediate iterations in the optimization process. This is reasonable so long as one can interpolate properties for such materials and so long as the final design is integer valued. For this purpose, we present a method for smoothly thresholding the volume fractions of an arbitrary number of material phases which specify the design. This method is trivial for two-material design problems, for example, the canonical topology design problem of specifying the presence or absence of a single material within a domain, but it becomes more complex when three or more materials are used, as often occurs in material design problems. We take advantage of the similarity in properties between the volume fractions and the barycentric coordinates on a simplex to derive a thresholding, method which is applicable to an arbitrary number of materials. As we show in a sensitivity analysis, this method has smooth derivatives, allowing it to be used in gradient-based optimization algorithms. Finally, we present results, which show synergistic effects when used with Solid Isotropic Material with Penalty and Rational Approximation of Material Properties material interpolation functions, popular methods of ensuring integerness of solutions.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE; Defense Advanced Research Projects Agency (DARPA)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1400087

Report Number(s):: LLNL-JRNL-683462; TRN: US1702980

Journal Information:: International Journal for Numerical Methods in Engineering, Vol. 108, Issue 12; ISSN 0029-5981

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 10 works

Citation information provided by
Web of Science

References (27)

Achieving minimum length scale in topology optimization using nodal design variables and projection functions Guest, J. K.; Prévost, J. H.; Belytschko, T. International Journal for Numerical Methods in Engineering, Vol. 61, Issue 2 https://doi.org/10.1002/nme.1064	journal	August 2004
A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate: BCP SCHEME FOR THE DISCRETE OPTIMAL ORIENTATION DESIGN OF THE LAMINATE Gao, Tong; Zhang, Weihong; Duysinx, Pierre International Journal for Numerical Methods in Engineering, Vol. 91, Issue 1 https://doi.org/10.1002/nme.4270	journal	May 2012
Topology optimization approaches: A comparative review Sigmund, Ole; Maute, Kurt Structural and Multidisciplinary Optimization, Vol. 48, Issue 6 https://doi.org/10.1007/s00158-013-0978-6	journal	August 2013
Generating optimal topologies in structural design using a homogenization method Bendsøe, Martin Philip; Kikuchi, Noboru Computer Methods in Applied Mechanics and Engineering, Vol. 71, Issue 2 https://doi.org/10.1016/0045-7825(88)90086-2	journal	November 1988
A survey of structural and multidisciplinary continuum topology optimization: post 2000 Deaton, Joshua D.; Grandhi, Ramana V. Structural and Multidisciplinary Optimization, Vol. 49, Issue 1 https://doi.org/10.1007/s00158-013-0956-z	journal	July 2013
Level-set methods for structural topology optimization: a review van Dijk, N. P.; Maute, K.; Langelaar, M. Structural and Multidisciplinary Optimization, Vol. 48, Issue 3 https://doi.org/10.1007/s00158-013-0912-y	journal	March 2013
A reevaluation of the SIMP method with filtering and an alternative formulation for solid–void topology optimization Bruns, T. E. Structural and Multidisciplinary Optimization, Vol. 30, Issue 6 https://doi.org/10.1007/s00158-005-0537-x	journal	August 2005
Design of materials with extreme thermal expansion using a three-phase topology optimization method Sigmund, O.; Torquato, S. Journal of the Mechanics and Physics of Solids, Vol. 45, Issue 6 https://doi.org/10.1016/S0022-5096(96)00114-7	journal	June 1997
Topology optimization with multiple phase projection Guest, James K. Computer Methods in Applied Mechanics and Engineering, Vol. 199, Issue 1-4 https://doi.org/10.1016/j.cma.2009.09.023	journal	December 2009
Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries Balay, Satish; Gropp, William D.; McInnes, Lois Curfman Modern Software Tools for Scientific Computing https://doi.org/10.1007/978-1-4612-1986-6_8	book	January 1997
An investigation concerning optimal design of solid elastic plates Cheng, Keng-Tung; Olhoff, Niels International Journal of Solids and Structures, Vol. 17, Issue 3 https://doi.org/10.1016/0020-7683(81)90065-2	journal	January 1981
A variational approach to the theory of the elastic behaviour of multiphase materials Hashin, Z.; Shtrikman, S. Journal of the Mechanics and Physics of Solids, Vol. 11, Issue 2 https://doi.org/10.1016/0022-5096(63)90060-7	journal	March 1963
Inverse homogenization for evaluation of effective properties of a mixture Cherkaev, Elena Inverse Problems, Vol. 17, Issue 4 https://doi.org/10.1088/0266-5611/17/4/341	journal	July 2001
A Simplex Method for Function Minimization Nelder, J. A.; Mead, R. The Computer Journal, Vol. 7, Issue 4 https://doi.org/10.1093/comjnl/7.4.308	journal	January 1965
Filters in topology optimization Bourdin, Blaise International Journal for Numerical Methods in Engineering, Vol. 50, Issue 9 https://doi.org/10.1002/nme.116	journal	January 2001
An alternative interpolation scheme for minimum compliance topology optimization Stolpe, M.; Svanberg, K. Structural and Multidisciplinary Optimization, Vol. 22, Issue 2 https://doi.org/10.1007/s001580100129	journal	September 2001
SFP—a new parameterization based on shape functions for optimal material selection: application to conventional composite plies Bruyneel, Michael Structural and Multidisciplinary Optimization, Vol. 43, Issue 1 https://doi.org/10.1007/s00158-010-0548-0	journal	August 2010
Nonlinear structural design using multiscale topology optimization. Part I: Static formulation Nakshatrala, P. B.; Tortorelli, D. A.; Nakshatrala, K. B. Computer Methods in Applied Mechanics and Engineering, Vol. 261-262 https://doi.org/10.1016/j.cma.2012.12.018	journal	July 2013
Materials with prescribed constitutive parameters: An inverse homogenization problem Sigmund, Ole International Journal of Solids and Structures, Vol. 31, Issue 17 https://doi.org/10.1016/0020-7683(94)90154-6	journal	September 1994
“Color” level sets: a multi-phase method for structural topology optimization with multiple materials Wang, Michael Yu; Wang, Xiaoming Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 6-8 https://doi.org/10.1016/j.cma.2003.10.008	journal	February 2004
Morphology-based black and white filters for topology optimization Sigmund, Ole Structural and Multidisciplinary Optimization, Vol. 33, Issue 4-5 https://doi.org/10.1007/s00158-006-0087-x	journal	January 2007
Topology optimization of non-linear elastic structures and compliant mechanisms Bruns, Tyler E.; Tortorelli, Daniel A. Computer Methods in Applied Mechanics and Engineering, Vol. 190, Issue 26-27 https://doi.org/10.1016/S0045-7825(00)00278-4	journal	March 2001
Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima Sigmund, O.; Petersson, J. Structural Optimization, Vol. 16, Issue 1 https://doi.org/10.1007/BF01214002	journal	August 1998
A new approach to variable-topology shape design using a constraint on perimeter Haber, R. B.; Jog, C. S.; Bends�e, M. P. Structural Optimization, Vol. 11, Issue 1-2 https://doi.org/10.1007/BF01279647	journal	February 1996
Manufacturing tolerant topology optimization Sigmund, Ole Acta Mechanica Sinica, Vol. 25, Issue 2 https://doi.org/10.1007/s10409-009-0240-z	journal	March 2009
A level set method for structural topology optimization Wang, Michael Yu; Wang, Xiaoming; Guo, Dongming Computer Methods in Applied Mechanics and Engineering, Vol. 192, Issue 1-2 https://doi.org/10.1016/S0045-7825(02)00559-5	journal	January 2003
On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming Wächter, Andreas; Biegler, Lorenz T. Mathematical Programming, Vol. 106, Issue 1 https://doi.org/10.1007/s10107-004-0559-y	journal	April 2005

Cited By (4)

A geometric projection method for designing three-dimensional open lattices with inverse homogenization: A geometric projection method Watts, Seth; Tortorelli, Daniel A. International Journal for Numerical Methods in Engineering, Vol. 112, Issue 11 https://doi.org/10.1002/nme.5569	journal	July 2017
Current and future trends in topology optimization for additive manufacturing Liu, Jikai; Gaynor, Andrew T.; Chen, Shikui Structural and Multidisciplinary Optimization, Vol. 57, Issue 6 https://doi.org/10.1007/s00158-018-1994-3	journal	May 2018
Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design Watts, Seth; Arrighi, William; Kudo, Jun Structural and Multidisciplinary Optimization, Vol. 60, Issue 5 https://doi.org/10.1007/s00158-019-02297-5	journal	May 2019
A geometric projection method for designing three-dimensional open lattices with inverse homogenization: A geometric projection method Watts, Seth; Tortorelli, Daniel A. International Journal for Numerical Methods in Engineering, Vol. 113, Issue 8 https://doi.org/10.1002/nme.5761	journal	January 2018

Similar Records

Toplogical optimization of structures using Fourier representations

Journal Article · Mon Apr 02 00:00:00 EDT 2018 · Structural and Multidisciplinary Optimization · OSTI ID:1400087

White, Daniel A.; Stowell, Mark L.; Tortorelli, Daniel A.

Novel Geometric Operations for Linear Programming

Technical Report · Sun Nov 01 00:00:00 EDT 2020 · OSTI ID:1400087

Ebeida, Mohamed; Abdelkader, Ahmed; Amenta, Nina; +4 more

Simultaneous material, shape and topology optimization

Journal Article · Tue Aug 18 00:00:00 EDT 2020 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:1400087

Fernandez, Felipe; Barker, Andrew T.; Kudo, Jun; +6 more

Related Subjects

42 ENGINEERING
97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE
topology design
elasticity
finite element methods
partition of unity

Title: An n -material thresholding method for improving integerness of solutions in topology optimization

Citation Formats

References (27)

Cited By (4)

Similar Records

Related Subjects