skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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

Abstract

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. Here, we also design a single-constituent isotropic material with negative Poisson's ratio and a light, stiff material comprised of 2 constituent solidsmore » plus void space.« less

Authors:
ORCiD logo [1];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Univ. of Illinois at Urbana-Champaign, Urbana, IL (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1410062
Alternate Identifier(s):
OSTI ID: 1400835
Report Number(s):
LLNL-JRNL-701297
Journal ID: ISSN 0029-5981
Grant/Contract Number:  
AC52-07NA27344; 14-SI-005 and 17-SI-005
Resource Type:
Accepted Manuscript
Journal Name:
International Journal for Numerical Methods in Engineering
Additional Journal Information:
Journal Volume: 112; Journal Issue: 11; Journal ID: ISSN 0029-5981
Publisher:
Wiley
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; elasticity; finite element methods; topology design

Citation Formats

Watts, Seth, and Tortorelli, Daniel A. A geometric projection method for designing three-dimensional open lattices with inverse homogenization. United States: N. p., 2017. Web. doi:10.1002/nme.5569.
Watts, Seth, & Tortorelli, Daniel A. A geometric projection method for designing three-dimensional open lattices with inverse homogenization. United States. doi:10.1002/nme.5569.
Watts, Seth, and Tortorelli, Daniel A. Thu . "A geometric projection method for designing three-dimensional open lattices with inverse homogenization". United States. doi:10.1002/nme.5569. https://www.osti.gov/servlets/purl/1410062.
@article{osti_1410062,
title = {A geometric projection method for designing three-dimensional open lattices with inverse homogenization},
author = {Watts, Seth and Tortorelli, Daniel A.},
abstractNote = {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. Here, 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.},
doi = {10.1002/nme.5569},
journal = {International Journal for Numerical Methods in Engineering},
number = 11,
volume = 112,
place = {United States},
year = {2017},
month = {4}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 5 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

An n -material thresholding method for improving integerness of solutions in topology optimization : An
journal, June 2016

  • Watts, Seth; Tortorelli, Daniel A.
  • International Journal for Numerical Methods in Engineering, Vol. 108, Issue 12
  • DOI: 10.1002/nme.5265

Achieving minimum length scale in topology optimization using nodal design variables and projection functions
journal, August 2004

  • Guest, J. K.; Prévost, J. H.; Belytschko, T.
  • International Journal for Numerical Methods in Engineering, Vol. 61, Issue 2
  • DOI: 10.1002/nme.1064

Doing Topology Optimization Explicitly and Geometrically—A New Moving Morphable Components Based Framework
journal, May 2014

  • Guo, Xu; Zhang, Weisheng; Zhong, Wenliang
  • Journal of Applied Mechanics, Vol. 81, Issue 8
  • DOI: 10.1115/1.4027609

A geometry projection method for the topology optimization of plate structures
journal, May 2016

  • Zhang, Shanglong; Norato, Julián A.; Gain, Arun L.
  • Structural and Multidisciplinary Optimization, Vol. 54, Issue 5
  • DOI: 10.1007/s00158-016-1466-6

Generating optimal topologies in structural design using a homogenization method
journal, November 1988

  • Bendsøe, Martin Philip; Kikuchi, Noboru
  • Computer Methods in Applied Mechanics and Engineering, Vol. 71, Issue 2
  • DOI: 10.1016/0045-7825(88)90086-2

H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations
journal, January 1990

  • Tartar, Luc
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 115, Issue 3-4
  • DOI: 10.1017/S0308210500020606

Explicit feature control in structural topology optimization via level set method
journal, April 2014

  • Guo, Xu; Zhang, Weisheng; Zhong, Wenliang
  • Computer Methods in Applied Mechanics and Engineering, Vol. 272
  • DOI: 10.1016/j.cma.2014.01.010

Optimizing the layout of discrete objects in structures and materials: A projection-based topology optimization approach
journal, January 2015


Shape optimization by the homogenization method
journal, March 1997

  • Allaire, Grégoire; Bonnetier, Eric; Francfort, Gilles
  • Numerische Mathematik, Vol. 76, Issue 1
  • DOI: 10.1007/s002110050253

Design of materials with extreme thermal expansion using a three-phase topology optimization method
journal, June 1997


Optimal lattice-structured materials
journal, November 2016


Ultralight, ultrastiff mechanical metamaterials
journal, June 2014


A new three-dimensional topology optimization method based on moving morphable components (MMCs)
journal, December 2016


Topology optimization with multiple phase projection
journal, December 2009


Multidimensional Architectures for Functional Optical Devices
journal, February 2010

  • Arpin, Kevin A.; Mihi, Agustin; Johnson, Harley T.
  • Advanced Materials, Vol. 22, Issue 10
  • DOI: 10.1002/adma.200904096

On projection methods, convergence and robust formulations in topology optimization
journal, December 2010

  • Wang, Fengwen; Lazarov, Boyan Stefanov; Sigmund, Ole
  • Structural and Multidisciplinary Optimization, Vol. 43, Issue 6
  • DOI: 10.1007/s00158-010-0602-y

Armadillo: a template-based C++ library for linear algebra
journal, June 2016

  • Sanderson, Conrad; Curtin, Ryan
  • The Journal of Open Source Software, Vol. 1, Issue 2
  • DOI: 10.21105/joss.00026

An investigation concerning optimal design of solid elastic plates
journal, January 1981


The structural performance of near-optimized truss core panels
journal, July 2002


Material microstructure optimization for linear elastodynamic energy wave management
journal, February 2012

  • Le, Chau; Bruns, Tyler E.; Tortorelli, Daniel A.
  • Journal of the Mechanics and Physics of Solids, Vol. 60, Issue 2
  • DOI: 10.1016/j.jmps.2011.09.002

Inverse homogenization for evaluation of effective properties of a mixture
journal, July 2001


Effective properties of the octet-truss lattice material
journal, August 2001

  • Deshpande, V. S.; Fleck, N. A.; Ashby, M. F.
  • Journal of the Mechanics and Physics of Solids, Vol. 49, Issue 8
  • DOI: 10.1016/S0022-5096(01)00010-2

Filters in topology optimization
journal, January 2001

  • Bourdin, Blaise
  • International Journal for Numerical Methods in Engineering, Vol. 50, Issue 9
  • DOI: 10.1002/nme.116

An alternative interpolation scheme for minimum compliance topology optimization
journal, September 2001

  • Stolpe, M.; Svanberg, K.
  • Structural and Multidisciplinary Optimization, Vol. 22, Issue 2
  • DOI: 10.1007/s001580100129

Nonlinear structural design using multiscale topology optimization. Part I: Static formulation
journal, July 2013

  • Nakshatrala, P. B.; Tortorelli, D. A.; Nakshatrala, K. B.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 261-262
  • DOI: 10.1016/j.cma.2012.12.018

Shape optimization with topological changes and parametric control
journal, January 2007

  • Chen, Jiaqin; Shapiro, Vadim; Suresh, Krishnan
  • International Journal for Numerical Methods in Engineering, Vol. 71, Issue 3
  • DOI: 10.1002/nme.1943

A geometry projection method for continuum-based topology optimization with discrete elements
journal, August 2015

  • Norato, J. A.; Bell, B. K.; Tortorelli, D. A.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 293
  • DOI: 10.1016/j.cma.2015.05.005

Materials with prescribed constitutive parameters: An inverse homogenization problem
journal, September 1994


Self-assembly of nanoscale cuboctahedra by coordination chemistry
journal, April 1999

  • Olenyuk, Bogdan; Whiteford, Jeffery A.; Fechtenkötter, Andreas
  • Nature, Vol. 398, Issue 6730, p. 796-799
  • DOI: 10.1038/19740

“Color” level sets: a multi-phase method for structural topology optimization with multiple materials
journal, February 2004

  • Wang, Michael Yu; Wang, Xiaoming
  • Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 6-8
  • DOI: 10.1016/j.cma.2003.10.008

Design of manufacturable 3D extremal elastic microstructure
journal, February 2014


Morphology-based black and white filters for topology optimization
journal, January 2007


On domain symmetry and its use in homogenization
journal, June 2017

  • Barbarosie, Cristian; Tortorelli, Daniel A.; Watts, Seth
  • Computer Methods in Applied Mechanics and Engineering, Vol. 320
  • DOI: 10.1016/j.cma.2017.01.009

Topology optimization of non-linear elastic structures and compliant mechanisms
journal, March 2001

  • Bruns, Tyler E.; Tortorelli, Daniel A.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 190, Issue 26-27
  • DOI: 10.1016/S0045-7825(00)00278-4

A new approach to variable-topology shape design using a constraint on perimeter
journal, February 1996

  • Haber, R. B.; Jog, C. S.; Bends�e, M. P.
  • Structural Optimization, Vol. 11, Issue 1-2
  • DOI: 10.1007/BF01279647

Manufacturing tolerant topology optimization
journal, March 2009


A level set method for structural topology optimization
journal, January 2003

  • Wang, Michael Yu; Wang, Xiaoming; Guo, Dongming
  • Computer Methods in Applied Mechanics and Engineering, Vol. 192, Issue 1-2
  • DOI: 10.1016/S0045-7825(02)00559-5