Simultaneous material, shape and topology optimization

Fernandez, Felipe; Barker, Andrew T.; Kudo, Jun; Lewicki, James P.; Swartz, Kenneth; Tortorelli, Daniel A.; Watts, Seth; White, Daniel A.; Wong, Jonathan

doi:10.1016/j.cma.2020.113321

Simultaneous material, shape and topology optimization

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

DOI:https://doi.org/10.1016/j.cma.2020.113321· OSTI ID:1735801

^[1]; ^[1]; Kudo, Jun ^[1]; Lewicki, James P. ^[1]; Swartz, Kenneth ^[2]; Tortorelli, Daniel A. ^[2]; ^[1]; White, Daniel A. ^[1]; Wong, Jonathan ^[1]

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

Using three design fields we develop an optimization environment that can simultaneously optimize material, shape and topology. We use the implicit representation of the boundaries with level-set functions that define the shape and topology. Differentiable R-functions allow us to combine these shapes and topology descriptions with Boolean operations. Additionally, we incorporate design dependent-stiffness materials with another design field. Notably, this framework accommodates design dependent loads, has the ability to introduce holes, and ensures the satisfaction of optimality criteria. It builds upon the fictitious domain, ersatz material, material interpolation and level-set methods. Additionally, it also borrows from parameterized density-based topology optimization methods. Since analytical sensitivities can be computed, we use efficient nonlinear programming algorithms to update the design instead of the Hamilton–Jacobi’s scheme of level-set methods. We illustrate the features of our framework by designing a cantilever beam with octet truss microlattice, a dam with design-dependent loads, and a composite clevis plate.

View Accepted Manuscript (DOE)

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

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1735801

Alternate ID(s):: OSTI ID: 1809463

Report Number(s):: LLNL-JRNL--750458; 935328

Journal Information:: Computer Methods in Applied Mechanics and Engineering, Journal Name: Computer Methods in Applied Mechanics and Engineering Journal Issue: no. 1 Vol. 371; ISSN 0045-7825

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (85)

Slope constrained topology optimization Petersson, Joakim; Sigmund, Ole International Journal for Numerical Methods in Engineering, Vol. 41, Issue 8 https://doi.org/10.1002/(SICI)1097-0207(19980430)41:8<1417::AID-NME344>3.0.CO;2-N	journal	April 1998
Optimal design and relaxation of variational problems, II Kohn, Robert V.; Strang, Gilbert Communications on Pure and Applied Mathematics, Vol. 39, Issue 2 https://doi.org/10.1002/cpa.3160390202	journal	March 1986
A geometry projection method for shape optimization Norato, J.; Haber, R.; Tortorelli, D. International Journal for Numerical Methods in Engineering, Vol. 60, Issue 14 https://doi.org/10.1002/nme.1044	journal	July 2004
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
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
Radial basis functions and level set method for structural topology optimization Wang, Shengyin; Wang, Michael Yu International Journal for Numerical Methods in Engineering, Vol. 65, Issue 12 https://doi.org/10.1002/nme.1536	journal	January 2006
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 level set-based parameterization method for structural shape and topology optimization Luo, Zhen; Wang, Michael Yu; Wang, Shengyin International Journal for Numerical Methods in Engineering, Vol. 76, Issue 1 https://doi.org/10.1002/nme.2092	journal	October 2008
A level set based model for damage growth: The thick level set approach Moës, N.; Stolz, C.; Bernard, P. -E. International Journal for Numerical Methods in Engineering, Vol. 86, Issue 3 https://doi.org/10.1002/nme.3069	journal	December 2010
Filters in topology optimization based on Helmholtz-type differential equations Lazarov, B. S.; Sigmund, O. International Journal for Numerical Methods in Engineering, Vol. 86, Issue 6 https://doi.org/10.1002/nme.3072	journal	December 2010
An embedded mesh method using piecewise constant multipliers with stabilization: mathematical and numerical aspects: AN EMBEDDED MESH METHOD: MATHEMATICAL AND NUMERICAL ASPECTS Puso, M. A.; Kokko, E.; Settgast, R. International Journal for Numerical Methods in Engineering, Vol. 104, Issue 7 https://doi.org/10.1002/nme.4796	journal	October 2014
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
Parametric shape and topology optimization: A new level set approach based on cardinal basis functions Jiang, Long; Chen, Shikui; Jiao, Xiangmin International Journal for Numerical Methods in Engineering, Vol. 114, Issue 1 https://doi.org/10.1002/nme.5733	journal	December 2017
Topology optimization with implicit functions and regularization Belytschko, T.; Xiao, S. P.; Parimi, C. International Journal for Numerical Methods in Engineering, Vol. 57, Issue 8 https://doi.org/10.1002/nme.824	journal	January 2003
Structural Boundary Design via Level Set and Immersed Interface Methods Sethian, J. A.; Wiegmann, Andreas Journal of Computational Physics, Vol. 163, Issue 2 https://doi.org/10.1006/jcph.2000.6581	journal	September 2000
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
Optimal shape design as a material distribution problem Bendsøe, M. P. Structural Optimization, Vol. 1, Issue 4 https://doi.org/10.1007/BF01650949	journal	December 1989
Bubble method for topology and shape optimization of structures Eschenauer, H. A.; Kobelev, V. V.; Schumacher, A. Structural Optimization, Vol. 8, Issue 1 https://doi.org/10.1007/BF01742933	journal	August 1994
On thickness and orientational design with orthotropic materials Pedersen, P. Structural Optimization, Vol. 3, Issue 2 https://doi.org/10.1007/BF01743275	journal	June 1991
Adaptive topology optimization Maute, K.; Ramm, E. Structural Optimization, Vol. 10, Issue 2 https://doi.org/10.1007/BF01743537	journal	October 1995
An optimal design problem with perimeter penalization Ambrosio, Luigi; Buttazzo, Giuseppe Calculus of Variations and Partial Differential Equations, Vol. 1, Issue 1 https://doi.org/10.1007/BF02163264	journal	March 1993
Topology optimization using a topology description function de Ruiter, M. J.; van Keulen, F. Structural and Multidisciplinary Optimization, Vol. 26, Issue 6 https://doi.org/10.1007/s00158-003-0375-7	journal	April 2004
Topological optimization of continuum structures with design-dependent surface loading ? Part I: new computational approach for 2D problems Du, J.; Olhoff, N. Structural and Multidisciplinary Optimization, Vol. 27, Issue 3 https://doi.org/10.1007/s00158-004-0379-y	journal	May 2004
Topological optimization of continuum structures with design-dependent surface loading ? Part II: algorithm and examples for 3D problems Du, J.; Olhoff, N. Structural and Multidisciplinary Optimization, Vol. 27, Issue 3 https://doi.org/10.1007/s00158-004-0380-5	journal	May 2004
Density-based topological design of structures subjected to water pressure using a parametric loading surface Fuchs, M. B.; Shemesh, N. N. Y. Structural and Multidisciplinary Optimization, Vol. 28, Issue 1 https://doi.org/10.1007/s00158-004-0406-z	journal	July 2004
Note on topology optimization of continuum structures including self-weight Bruyneel, M.; Duysinx, P. Structural and Multidisciplinary Optimization, Vol. 29, Issue 4 https://doi.org/10.1007/s00158-004-0484-y	journal	November 2004
Application of spectral level set methodology in topology optimization Gomes, Alexandra A.; Suleman, Afzal Structural and Multidisciplinary Optimization, Vol. 31, Issue 6 https://doi.org/10.1007/s00158-006-0005-2	journal	April 2006
Stress concentration minimization of 2D filets using X-FEM and level set description Van Miegroet, Laurent; Duysinx, Pierre Structural and Multidisciplinary Optimization, Vol. 33, Issue 4-5 https://doi.org/10.1007/s00158-006-0091-1	journal	January 2007
A topological derivative method for topology optimization Norato, Julian A.; Bendsøe, Martin P.; Haber, Robert B. Structural and Multidisciplinary Optimization, Vol. 33, Issue 4-5 https://doi.org/10.1007/s00158-007-0094-6	journal	February 2007
A critical review of established methods of structural topology optimization Rozvany, G. I. N. Structural and Multidisciplinary Optimization, Vol. 37, Issue 3 https://doi.org/10.1007/s00158-007-0217-0	journal	February 2008
Levelset based fluid topology optimization using the extended finite element method Kreissl, Sebastian; Maute, Kurt Structural and Multidisciplinary Optimization, Vol. 46, Issue 3 https://doi.org/10.1007/s00158-012-0782-8	journal	March 2012
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 mesh evolution algorithm based on the level set method for geometry and topology optimization Allaire, Grégoire; Dapogny, Charles; Frey, Pascal Structural and Multidisciplinary Optimization, Vol. 48, Issue 4 https://doi.org/10.1007/s00158-013-0929-2	journal	April 2013
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
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
Topology optimization using an explicit interface representation Christiansen, Asger Nyman; Nobel-Jørgensen, Morten; Aage, Niels Structural and Multidisciplinary Optimization, Vol. 49, Issue 3 https://doi.org/10.1007/s00158-013-0983-9	journal	August 2013
Industrial application of topology optimization for combined conductive and convective heat transfer problems Zhou, Mingdong; Alexandersen, Joe; Sigmund, Ole Structural and Multidisciplinary Optimization, Vol. 54, Issue 4 https://doi.org/10.1007/s00158-016-1433-2	journal	April 2016
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
Structural complexity control in topology optimization via moving morphable component (MMC) approach Zhang, Weisheng; Zhou, Jianhua; Zhu, Yichao Structural and Multidisciplinary Optimization, Vol. 56, Issue 3 https://doi.org/10.1007/s00158-017-1736-y	journal	June 2017
Stress-based topology optimization using spatial gradient stabilized XFEM Sharma, Ashesh; Maute, Kurt Structural and Multidisciplinary Optimization, Vol. 57, Issue 1 https://doi.org/10.1007/s00158-017-1833-y	journal	October 2017
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
Topology optimization for design-dependent hydrostatic pressure loading via the level-set method Picelli, Renato; Neofytou, A.; Kim, H. Alicia Structural and Multidisciplinary Optimization, Vol. 60, Issue 4 https://doi.org/10.1007/s00158-019-02339-y	journal	July 2019
Topology optimization of continuum structures subjected to pressure loading Hammer, V. B.; Olhoff, N. Structural and Multidisciplinary Optimization, Vol. 19, Issue 2 https://doi.org/10.1007/s001580050088	journal	April 2000
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
Topology optimization of 3D shell structures with porous infill Clausen, Anders; Andreassen, Erik; Sigmund, Ole Acta Mechanica Sinica, Vol. 33, Issue 4 https://doi.org/10.1007/s10409-017-0679-2	journal	June 2017
Experimental Validation of an Additively Manufactured Stiffness-Optimized Short-Fiber Reinforced Composite Clevis Joint Saito, Y.; Fernandez, F.; Tortorelli, D. A. Experimental Mechanics, Vol. 59, Issue 6 https://doi.org/10.1007/s11340-019-00514-2	journal	April 2019
Regularized formulation for optimal design of axisymmetric plates Keno-Tung, Cheng; Olhoff, Niels International Journal of Solids and Structures, Vol. 18, Issue 2 https://doi.org/10.1016/0020-7683(82)90023-3	journal	January 1982
Optimal design of elastic plates with a constraint on the slope of the thickness function Niordson, Frithiof International Journal of Solids and Structures, Vol. 19, Issue 2 https://doi.org/10.1016/0020-7683(83)90005-7	journal	January 1983
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
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
On CAD-integrated structural topology and design optimization Olhoff, Niels; Bendsøe, Martin P.; Rasmussen, John Computer Methods in Applied Mechanics and Engineering, Vol. 89, Issue 1-3 https://doi.org/10.1016/0045-7825(91)90044-7	journal	August 1991
Multiphase composites with extremal bulk modulus Gibiansky, Leonid V.; Sigmund, Ole Journal of the Mechanics and Physics of Solids, Vol. 48, Issue 3 https://doi.org/10.1016/S0022-5096(99)00043-5	journal	March 2000
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 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
Topological sensitivity analysis Novotny, A. A.; Feijóo, R. A.; Taroco, E. Computer Methods in Applied Mechanics and Engineering, Vol. 192, Issue 7-8 https://doi.org/10.1016/S0045-7825(02)00599-6	journal	February 2003
The shape and topological optimizations connection Céa, Jean; Garreau, Stéphane; Guillaume, Philippe Computer Methods in Applied Mechanics and Engineering, Vol. 188, Issue 4 https://doi.org/10.1016/S0045-7825(99)00357-6	journal	August 2000
Topology optimization with design-dependent loads Chen, Bing-Chung; Kikuchi, Noboru Finite Elements in Analysis and Design, Vol. 37, Issue 1 https://doi.org/10.1016/S0168-874X(00)00021-4	journal	January 2001
Topology optimization using a mixed formulation: An alternative way to solve pressure load problems Sigmund, O.; Clausen, P. M. Computer Methods in Applied Mechanics and Engineering, Vol. 196, Issue 13-16 https://doi.org/10.1016/j.cma.2006.09.021	journal	March 2007
An embedded mesh method in a multiple material ALE Puso, Michael A.; Sanders, Jessica; Settgast, Randy Computer Methods in Applied Mechanics and Engineering, Vol. 245-246 https://doi.org/10.1016/j.cma.2012.07.014	journal	October 2012
Topology optimization in B-spline space Qian, Xiaoping Computer Methods in Applied Mechanics and Engineering, Vol. 265 https://doi.org/10.1016/j.cma.2013.06.001	journal	October 2013
Shape optimization with a level set based mesh evolution method Allaire, G.; Dapogny, C.; Frey, P. Computer Methods in Applied Mechanics and Engineering, Vol. 282 https://doi.org/10.1016/j.cma.2014.08.028	journal	December 2014
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
Feature-driven topology optimization method with signed distance function Zhou, Ying; Zhang, Weihong; Zhu, Jihong Computer Methods in Applied Mechanics and Engineering, Vol. 310 https://doi.org/10.1016/j.cma.2016.06.027	journal	October 2016
Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons Guo, Xu; Zhang, Weisheng; Zhang, Jian Computer Methods in Applied Mechanics and Engineering, Vol. 310 https://doi.org/10.1016/j.cma.2016.07.018	journal	October 2016
Parametric structural shape & topology optimization with a variational distance-regularized level set method Jiang, Long; Chen, Shikui Computer Methods in Applied Mechanics and Engineering, Vol. 321 https://doi.org/10.1016/j.cma.2017.03.044	journal	July 2017
Stress-based topology optimization with discrete geometric components Zhang, Shanglong; Gain, Arun L.; Norato, Julián A. Computer Methods in Applied Mechanics and Engineering, Vol. 325 https://doi.org/10.1016/j.cma.2017.06.025	journal	October 2017
A comprehensive study of feature definitions with solids and voids for topology optimization Zhang, Weihong; Zhou, Ying; Zhu, Jihong Computer Methods in Applied Mechanics and Engineering, Vol. 325 https://doi.org/10.1016/j.cma.2017.07.004	journal	October 2017
Multiscale topology optimization using neural network surrogate models White, Daniel A.; Arrighi, William J.; Kudo, Jun Computer Methods in Applied Mechanics and Engineering, Vol. 346 https://doi.org/10.1016/j.cma.2018.09.007	journal	April 2019
Optimal design of fiber reinforced composite structures and their direct ink write fabrication Fernandez, Felipe; Compel, W. Scott; Lewicki, James P. Computer Methods in Applied Mechanics and Engineering, Vol. 353 https://doi.org/10.1016/j.cma.2019.05.010	journal	August 2019
Topology optimization of multiple deformable bodies in contact with large deformations Fernandez, Felipe; Puso, Michael A.; Solberg, Jerome Computer Methods in Applied Mechanics and Engineering, Vol. 371 https://doi.org/10.1016/j.cma.2020.113288	journal	November 2020
Procedural function-based modelling of volumetric microstructures Pasko, Alexander; Fryazinov, Oleg; Vilbrandt, Turlif Graphical Models, Vol. 73, Issue 5 https://doi.org/10.1016/j.gmod.2011.03.001	journal	September 2011
Topology optimization of convection-dominated, steady-state heat transfer problems Bruns, T. E. International Journal of Heat and Mass Transfer, Vol. 50, Issue 15-16 https://doi.org/10.1016/j.ijheatmasstransfer.2007.01.039	journal	July 2007
Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection Iga, A.; Nishiwaki, S.; Izui, K. International Journal of Heat and Mass Transfer, Vol. 52, Issue 11-12 https://doi.org/10.1016/j.ijheatmasstransfer.2008.12.013	journal	May 2009
Structural optimization using sensitivity analysis and a level-set method Allaire, Grégoire; Jouve, François; Toader, Anca-Maria Journal of Computational Physics, Vol. 194, Issue 1 https://doi.org/10.1016/j.jcp.2003.09.032	journal	February 2004
A review of optimization techniques used in the design of fibre composite structures for civil engineering applications Awad, Ziad K.; Aravinthan, Thiru; Zhuge, Yan Materials & Design, Vol. 33 https://doi.org/10.1016/j.matdes.2011.04.061	journal	January 2012
Semi-analytic geometry with R-functions Shapiro, Vadim Acta Numerica, Vol. 16 https://doi.org/10.1017/S096249290631001X	journal	April 2007
3D-Printing of Meso-structurally Ordered Carbon Fiber/Polymer Composites with Unprecedented Orthotropic Physical Properties Lewicki, James P.; Rodriguez, Jennifer N.; Zhu, Cheng Scientific Reports, Vol. 7, Issue 1 https://doi.org/10.1038/srep43401	journal	March 2017
Design-dependent loads in topology optimization Bourdin, Blaise; Chambolle, Antonin ESAIM: Control, Optimisation and Calculus of Variations, Vol. 9 https://doi.org/10.1051/cocv:2002070	journal	January 2003
Fictitious domain methods using cut elements: III. A stabilized Nitsche method for Stokes’ problem Burman, Erik; Hansbo, Peter ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 48, Issue 3 https://doi.org/10.1051/m2an/2013123	journal	April 2014
An Integrated Environment for Structural Configuration Design Papalambros, Panos Y.; Chirehdast, Mehran Journal of Engineering Design, Vol. 1, Issue 1 https://doi.org/10.1080/09544829008901645	journal	January 1990
Fixed grid finite elements in elasticity problems García‐Ruíz, M. J.; Steven, G. P. Engineering Computations, Vol. 16, Issue 2 https://doi.org/10.1108/02644409910257430	journal	March 1999
Topology optimization of continuum structures: A review* Eschenauer, Hans A.; Olhoff, Niels Applied Mechanics Reviews, Vol. 54, Issue 4 https://doi.org/10.1115/1.1388075	journal	July 2001
Lagrangian Description Based Topology Optimization—A Revival of Shape Optimization Zhang, Weisheng; Zhang, Jian; Guo, Xu Journal of Applied Mechanics, Vol. 83, Issue 4 https://doi.org/10.1115/1.4032432	journal	January 2016
On the Topological Derivative in Shape Optimization Sokolowski, J.; Zochowski, A. SIAM Journal on Control and Optimization, Vol. 37, Issue 4 https://doi.org/10.1137/S0363012997323230	journal	January 1999

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