A geometric projection method for designing three-dimensional open lattices with inverse homogenization

Watts, Seth; Tortorelli, Daniel A.

doi:10.1002/nme.5569

Title: A geometric projection method for designing three-dimensional open lattices with inverse homogenization

Journal Article · Thu Apr 13 00:00:00 EDT 2017 · International Journal for Numerical Methods in Engineering

DOI:https://doi.org/10.1002/nme.5569· OSTI ID:1410062

^[1]; Tortorelli, Daniel A. ^[2]

Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Univ. of Illinois at Urbana-Champaign, Urbana, IL (United States)

Summary Topology optimization is a methodology for assigning material or void to each point in a design domain in a way that extremizes some objective function, such as the compliance of a structure under given loads, subject to various imposed constraints, such as an upper bound on the mass of the structure. Geometry projection is a means to parameterize the topology optimization problem, by describing the design in a way that is independent of the mesh used for analysis of the design's performance; it results in many fewer design parameters, necessarily resolves the ill‐posed nature of the topology optimization problem, and provides sharp descriptions of the material interfaces. We extend previous geometric projection work to 3 dimensions and design unit cells for lattice materials using inverse homogenization. We perform a sensitivity analysis of the geometric projection and show it has smooth derivatives, making it suitable for use with gradient‐based optimization algorithms. The technique is demonstrated by designing unit cells comprised of a single constituent material plus void space to obtain light, stiff materials with cubic and isotropic material symmetry. We also design a single‐constituent isotropic material with negative Poisson's ratio and a light, stiff material comprised of 2 constituent solids plus void space.

View Accepted Manuscript (DOE)

View Accepted Manuscript (Publisher)

Cite

Export

Save

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

Sponsoring Organization:: USDOE

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1410062

Alternate ID(s):: OSTI ID: 1400835

Report Number(s):: LLNL-JRNL-701297

Journal Information:: International Journal for Numerical Methods in Engineering, Vol. 112, Issue 11; ISSN 0029-5981

Publisher:: WileyCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 50 works

Citation information provided by
Web of Science

References (39)

An n -material thresholding method for improving integerness of solutions in topology optimization : An Watts, Seth; Tortorelli, Daniel A. International Journal for Numerical Methods in Engineering, Vol. 108, Issue 12 https://doi.org/10.1002/nme.5265	journal	June 2016
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
Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework Guo, Xu; Zhang, Weisheng; Zhong, Wenliang Journal of Applied Mechanics, Vol. 81, Issue 8 https://doi.org/10.1115/1.4027609	journal	May 2014
A geometry projection method for the topology optimization of plate structures Zhang, Shanglong; Norato, Julián A.; Gain, Arun L. Structural and Multidisciplinary Optimization, Vol. 54, Issue 5 https://doi.org/10.1007/s00158-016-1466-6	journal	May 2016
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
H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations Tartar, Luc Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 115, Issue 3-4 https://doi.org/10.1017/S0308210500020606	journal	January 1990
Explicit feature control in structural topology optimization via level set method Guo, Xu; Zhang, Weisheng; Zhong, Wenliang Computer Methods in Applied Mechanics and Engineering, Vol. 272 https://doi.org/10.1016/j.cma.2014.01.010	journal	April 2014
Optimizing the layout of discrete objects in structures and materials: A projection-based topology optimization approach Guest, James K. Computer Methods in Applied Mechanics and Engineering, Vol. 283 https://doi.org/10.1016/j.cma.2014.09.006	journal	January 2015
Shape optimization by the homogenization method Allaire, Grégoire; Bonnetier, Eric; Francfort, Gilles Numerische Mathematik, Vol. 76, Issue 1 https://doi.org/10.1007/s002110050253	journal	March 1997
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
Optimal lattice-structured materials Messner, Mark C. Journal of the Mechanics and Physics of Solids, Vol. 96 https://doi.org/10.1016/j.jmps.2016.07.010	journal	November 2016
Ultralight, ultrastiff mechanical metamaterials Zheng, X.; Lee, H.; Weisgraber, T. H. Science, Vol. 344, Issue 6190 https://doi.org/10.1126/science.1252291	journal	June 2014
A new three-dimensional topology optimization method based on moving morphable components (MMCs) Zhang, Weisheng; Li, Dong; Yuan, Jie Computational Mechanics, Vol. 59, Issue 4 https://doi.org/10.1007/s00466-016-1365-0	journal	December 2016
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
Multidimensional Architectures for Functional Optical Devices Arpin, Kevin A.; Mihi, Agustin; Johnson, Harley T. Advanced Materials, Vol. 22, Issue 10 https://doi.org/10.1002/adma.200904096	journal	February 2010
On projection methods, convergence and robust formulations in topology optimization Wang, Fengwen; Lazarov, Boyan Stefanov; Sigmund, Ole Structural and Multidisciplinary Optimization, Vol. 43, Issue 6 https://doi.org/10.1007/s00158-010-0602-y	journal	December 2010
Armadillo: a template-based C++ library for linear algebra Sanderson, Conrad; Curtin, Ryan The Journal of Open Source Software, Vol. 1, Issue 2 https://doi.org/10.21105/joss.00026	journal	June 2016
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
The structural performance of near-optimized truss core panels Chiras, S.; Mumm, D. R.; Evans, A. G. International Journal of Solids and Structures, Vol. 39, Issue 15 https://doi.org/10.1016/S0020-7683(02)00241-X	journal	July 2002
Material microstructure optimization for linear elastodynamic energy wave management Le, Chau; Bruns, Tyler E.; Tortorelli, Daniel A. Journal of the Mechanics and Physics of Solids, Vol. 60, Issue 2 https://doi.org/10.1016/j.jmps.2011.09.002	journal	February 2012
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
Effective properties of the octet-truss lattice material Deshpande, V. S.; Fleck, N. A.; Ashby, M. F. Journal of the Mechanics and Physics of Solids, Vol. 49, Issue 8 https://doi.org/10.1016/S0022-5096(01)00010-2	journal	August 2001
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
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
Shape optimization with topological changes and parametric control Chen, Jiaqin; Shapiro, Vadim; Suresh, Krishnan International Journal for Numerical Methods in Engineering, Vol. 71, Issue 3 https://doi.org/10.1002/nme.1943	journal	January 2007
A geometry projection method for continuum-based topology optimization with discrete elements Norato, J. A.; Bell, B. K.; Tortorelli, D. A. Computer Methods in Applied Mechanics and Engineering, Vol. 293 https://doi.org/10.1016/j.cma.2015.05.005	journal	August 2015
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
Self-assembly of nanoscale cuboctahedra by coordination chemistry Olenyuk, Bogdan; Whiteford, Jeffery A.; Fechtenkötter, Andreas Nature, Vol. 398, Issue 6730, p. 796-799 https://doi.org/10.1038/19740	journal	April 1999
“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
Design of manufacturable 3D extremal elastic microstructure Andreassen, Erik; Lazarov, Boyan S.; Sigmund, Ole Mechanics of Materials, Vol. 69, Issue 1 https://doi.org/10.1016/j.mechmat.2013.09.018	journal	February 2014
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
On domain symmetry and its use in homogenization Barbarosie, Cristian; Tortorelli, Daniel A.; Watts, Seth Computer Methods in Applied Mechanics and Engineering, Vol. 320 https://doi.org/10.1016/j.cma.2017.01.009	journal	June 2017
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
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
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

Cited By (6)

A Heaviside function‐based density representation algorithm for truss‐like buckling‐induced mechanism design Deng, Hao International Journal for Numerical Methods in Engineering, Vol. 119, Issue 11 https://doi.org/10.1002/nme.6084	journal	June 2019
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
Design for additive manufacturing: cellular structures in early-stage aerospace design Opgenoord, Max M. J.; Willcox, Karen E. Structural and Multidisciplinary Optimization, Vol. 60, Issue 2 https://doi.org/10.1007/s00158-019-02305-8	journal	June 2019
Generalized Geometry Projection: A Unified Approach for Geometric Feature Based Topology Optimization Coniglio, Simone; Morlier, Joseph; Gogu, Christian Archives of Computational Methods in Engineering, Vol. 27, Issue 5 https://doi.org/10.1007/s11831-019-09362-8	journal	October 2019
Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials Kazemi, Hesaneh; Vaziri, Ashkan; Norato, Julián A. Journal of Mechanical Design, Vol. 140, Issue 11 https://doi.org/10.1115/1.4040624	journal	September 2018
Stiff isotropic lattices beyond the Maxwell criterion Chen, Wen; Watts, Seth; Jackson, Julie A. Science Advances, Vol. 5, Issue 9 https://doi.org/10.1126/sciadv.aaw1937	journal	September 2019

Similar Records

Elastic response of hollow truss lattice micro-architectures

Journal Article · Tue Dec 01 00:00:00 EST 2020 · International Journal of Solids and Structures · OSTI ID:1410062

Watts, Seth

Nonlinear homogenization for topology optimization

Journal Article · Mon Jun 01 00:00:00 EDT 2020 · Mechanics of Materials · OSTI ID:1410062

Wallin, Mathias; Tortorelli, Daniel A.

On domain symmetry and its use in homogenization

Journal Article · Wed Mar 08 00:00:00 EST 2017 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:1410062

Barbarosie, Cristian A.; Tortorelli, Daniel A.; Watts, Seth E.

Related Subjects

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

Title: A geometric projection method for designing three-dimensional open lattices with inverse homogenization

Citation Formats

References (39)

Cited By (6)

Similar Records

Related Subjects