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; Tortorelli, Daniel A.; White, Daniel A.

doi:10.1007/s00158-019-02297-5

Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design

Research Paper
Published: 25 May 2019

Volume 60, pages 1887–1920, (2019)
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Structural and Multidisciplinary Optimization Aims and scope Submit manuscript

Seth Watts ORCID: orcid.org/0000-0001-9086-9700¹,
William Arrighi²,
Jun Kudo¹,
Daniel A. Tortorelli^3,4 &
…
Daniel A. White¹

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A Correction to this article was published on 07 January 2020

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Abstract

Elastic meta-materials are those whose unique properties come from their micro-architecture, rather than, e.g., from their chemistry. The introduction of such architecture, which is increasingly able to be fabricated due to advances in additive manufacturing, expands the design domain and enables improved design, from the most complex multi-physics design problems to the simple compliance design problem that is our focus. Unfortunately, concurrent design of both the micro-scale and the macroscale is computationally very expensive when the former can vary spatially, particularly in three dimensions. Instead, we provide simple, accurate surrogate models of the homogenized linear elastic response of the isotruss, the octet truss, and the ORC truss based on high-fidelity continuum finite element analyses. These surrogate models are relatively accurate over the full range of relative densities, in contrast to analytical models in the literature, which we show lose accuracy as relative density increases. The surrogate models are also simple to implement, which we demonstrate by modifying Sigmund’s 99-line code to solve a three-dimensional, multiscale compliance design problem with spatially varying relative density. We use this code to generate examples in both two and three dimensions that illustrate the advantage of elastic meta-materials over structures with a single length scale, i.e., those without micro-architectures.

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Change history

07 January 2019
The original version of this paper unfortunately contains three errors in the topology optimization code that was used to generate the examples. Line numbers refer to the code as it appears in Appendix D of the original paper.
07 January 2020
The original version of this paper unfortunately contains three errors in the topology optimization code that was used to generate the examples. Line numbers refer to the code as it appears in Appendix D of the original paper.

References

Aage N, Andreassen E, Lazarov BS, Sigmund O (2017) Giga-voxel computational morphogenesis for structural design. Nature 550(7674):84
Google Scholar
Allaire G (2002) Shape optimization by the homogenization method. Springer, New York
MATH Google Scholar
Allaire G, Jouve F, Toader AM (2004) Structural optimization using sensitivity analysis and a level-set method. J Comput Phys 164(1):363–393
MathSciNet MATH Google Scholar
Alzahrani M, Choi SK, Rosen DW (2015) Design of truss-like cellular structures using relative density mapping method. Mater Des 85:349–360
Google Scholar
Bendsøe M, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224
MathSciNet MATH Google Scholar
Bendsøe M, Sigmund O (2003) Topology optimization: theory, methods, and applications. Springer, Berlin
MATH Google Scholar
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Structural Optimization 1 (4):193–202
Google Scholar
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9-10):635– 654
MATH Google Scholar
Berger JB, Wadley HNG, McMeeking RM (2017) Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness. Nature 543:233–537
Google Scholar
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158
MathSciNet MATH Google Scholar
Bower AF (2009) Applied mechanics of solids, 1st edn. CRC Press, Boca Raton
Google Scholar
Bruns T, Tortorelli D (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26):3443–3459
MATH Google Scholar
Chang PS, Rosen DW, Chang PS, Rosen DW (2011) An improved size, matching, and scaling method for the design of deterministic mesoscale truss structures. In: ASME 2011 International design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers, pp 697–707
Coelho PG, Fernandes PR, Guedes JM, Rodrigues HC (2008) A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidiscip Optim 35(2):107–115
Google Scholar
Cowin S, Mehrabadi M (1995) Anisotropic symmetries of linear elasticity. Appl Mech Rev 48(5):247–285
MATH Google Scholar
Deshpande VS, Fleck NA, Ashby MF (2001) Effective properties of the octet-truss lattice material. J Mech Phys Solids 49(8):1747–1769
MATH Google Scholar
Feppon F, Michailidis G, Sidebottom MA, Allaire G, Krick BA, Vermaak N (2017) Introducing a level-set based shape and topology optimization method for the wear of composite materials with geometric constraints. Struct Multidiscip Optim 55(2):547–568
MathSciNet Google Scholar
Francfort GA, Murat F (1986) Homogenization and optimal bounds in linear elasticity. Arch Ration Mech Anal 94(4):307–334
MathSciNet MATH Google Scholar
Fuller RB (1961) Synergetic building construction. U.S. Patent No. 2,986, 241
Gaynor AT, Guest JK, Gaynor AT, Guest JK (2016) Topology optimization considering overhang constraints: eliminating sacrificial support material in additive manufacturing through design. Struct Multidiscip Optim 54(5):1157–1172
MathSciNet Google Scholar
Graf GC, Chu J, Engelbrecht S, Rosen DW (2009) Synthesis methods for lightweight lattice structures. In: ASME 2009 International design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers, pp 579–589
Hashin Z, Shtrikman S (1963) A variational approach to the theory of the elastic behaviour of multiphase materials. J Mech Phys Solids 11(2):127–140
MathSciNet MATH Google Scholar
Horn TJ, Harrysson OLA (2012) Overview of current additive manufacturing technologies and selected applications. Sci Prog 95(3):255–282
Google Scholar
Li H, Luo Z, Zhang N, Gao L, Brown T (2016) Integrated design of cellular composites using a level-set topology optimization method. Comput Methods Appl Mech Eng 309:453–475
MathSciNet MATH Google Scholar
Liu J, Cheng L, To AC (2017) Arbitrary void feature control in level set topology optimization. Comput Methods Appl Mech Eng 324:595–618
MathSciNet MATH Google Scholar
Messner MC (2016) Optimal lattice-structured materials. J Mech Phys Solids 96:162–183
MathSciNet Google Scholar
Messner MC, Barham MI, Kumar M, Barton NR (2015) Wave propagation in equivalent continuums representing truss lattice materials. Int J Solids Struct 73-74:55–66
Google Scholar
MFEM (2018) Modular finite element methods library, http://mfem.org
Mirzendehdel AM, Suresh K (2016) Support structure constrained topology optimization for additive manufacturing. Comput Aided Des 81:1–13
Google Scholar
Nakshatrala P, Tortorelli D, Nakshatrala K (2013) Nonlinear structural design using multiscale topology optimization. Part I: static formulation. Comput Methods Appl Mech Eng 261-262:167– 176
MathSciNet MATH Google Scholar
Rupp CJ, Evgrafov A, Maute K, Dunn ML (2009) Design of piezoelectric energy harvesting systems: a topology optimization approach based on multilayer plates and shells. J Intell Mater Syst Struct 20(16):1923–1939
Google Scholar
Sigmund O (1997) On the design of compliant mechanisms using topology optimization. J Struct Mech 25 (4):493–524
Google Scholar
Sigmund O (2001) A 99 line topology optimization code written in matlab. Struct Multidiscip Optim 21 (2):120–127
Google Scholar
Sigmund O (2011) On the usefulness of non-gradient approaches in topology optimization. Struct Multidiscip Optim 43(5):589–596
MathSciNet MATH Google Scholar
Sigmund O, Petersson J (1998) Numerical instabilities in toplogy optimization: a survey on procedures dealing with checkerboards, mesh-dependencies, and local minima. Struct Multidiscip Optim 16(1):68–75
Google Scholar
Sigmund O, Torquato S (1997) Design of materials with extreme thermal expansion using three-phase topology optimization method. J Mech Phys Solids 45(6):1037–1067
MathSciNet Google Scholar
Sivapuram R, Dunning PD, Kim HA (2016) Simultaneous material and structural optimization by multiscale topology optimization. Struct Multidiscip Optim 54(5):1267–1281
MathSciNet Google Scholar
Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim 22(2):116–124
Google Scholar
Tancogne-Dejean T, Diamantopoulou M, Gorji MB, Bonatti C, Mohr D (2018) 3d platelattices: an emerging class of lowdensity metamaterial exhibiting optimal isotropic stiffness. Adv Mater 30(45):1803334
Google Scholar
Wang M, Wang X, Guo D (2003) A level set method for structural topology optimization. Comput Methods Appl Mech Eng 192(1):227–246
MathSciNet MATH Google Scholar
Wang MY, Wang X (2004) “color” level sets: a multi-phase method for structural topology optimization with multiple materials. Comput Methods Appl Mech Eng 193(6–8):469–496
MathSciNet MATH Google Scholar
Watts S, Tortorelli DA (2016) An n-material thresholding method for improving integerness of solutions in topology optimization. Int J Numer Methods Eng 108(12):1498–1524
MathSciNet Google Scholar
Watts S, Tortorelli DA (2017) A geometric projection method for designing three-dimensional open lattices with inverse homogenization. Int J Numer Methods Eng 112(11):1564– 1588
MathSciNet Google Scholar
White DA, Arrighi WJ, Kudo J, Watts SE (2019) Multiscale topology optimization using neural network surrogate models. Comput Meth Appl Mech Eng 346:1118–1135
MathSciNet MATH Google Scholar
Xia L, Breitkopf P (2014) Concurrent topology optimization design of material and structure within nonlinear multiscale analysis framework. Comput Methods Appl Mech Eng 278:524–542
MathSciNet MATH Google Scholar
Zener C (1948) Elasticity and anelasticity of metals. University of Chicago Press, Chicago
MATH Google Scholar
Zheng X, Lee H, Weisgraber TH, Shusteff M, DeOtte J, Duoss EB, Kuntz JD, Biener MM, Ge Q, Jackson JA et al (2014) Ultralight, ultrastiff mechanical metamaterials. Science 344(6190):1373–1377
Google Scholar

Download references

Acknowledgments

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Funding

This work received funding from LDRD number 17-SI-005. LLNL-JRNL-758077.

Author information

Authors and Affiliations

Computational Engineering Division, Lawrence Livermore National Laboratory, L-227, PO Box 808, Livermore, CA, 94551, USA
Seth Watts, Jun Kudo & Daniel A. White
Applications, Simulations, and Quality Division, Lawrence Livermore National Laboratory, Livermore, CA, USA
William Arrighi
Center for Design and Optimization, Lawrence Livermore National Laboratory, Livermore, CA, USA
Daniel A. Tortorelli
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Daniel A. Tortorelli

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Seth Watts
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William Arrighi
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Jun Kudo
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Daniel A. Tortorelli
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Daniel A. White
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Correspondence to Seth Watts.

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Appendices

Appendix 1. Tabulated data for isotruss

Constituent	Relative	Rod	Relative Young’s	Poisson’s	Relative shear	Zener
Poisson’s ratio	density	diameter	modulus	ratio	modulus	ratio
ν _S	ρ	d	E^h/E_S	ν ^h	G^h/G_S	A ^h
0.2	0.005	0.02369	0.001	0.251	0.001	0.997
	0.010	0.03380	0.002	0.251	0.002	0.995
	0.050	0.07781	0.010	0.250	0.009	0.989
	0.100	0.11261	0.022	0.247	0.021	0.982
	0.200	0.16518	0.052	0.239	0.048	0.967
	0.300	0.20899	0.091	0.228	0.085	0.954
	0.400	0.24911	0.141	0.217	0.132	0.945
	0.500	0.28774	0.207	0.206	0.194	0.942
	0.600	0.32640	0.291	0.195	0.276	0.944
	0.700	0.36660	0.400	0.188	0.385	0.954
	0.800	0.41060	0.544	0.185	0.534	0.968
	1.000	0.58000	1.000	0.200	1.000	1.000
0.3	0.005	0.02369	0.001	0.253	0.001	0.997
	0.010	0.03380	0.002	0.253	0.002	0.994
	0.050	0.07781	0.010	0.256	0.010	0.989
	0.100	0.11261	0.022	0.256	0.022	0.982
	0.200	0.16518	0.052	0.254	0.052	0.969
	0.300	0.20899	0.091	0.250	0.091	0.958
	0.400	0.24911	0.142	0.245	0.141	0.952
	0.500	0.28774	0.207	0.241	0.206	0.951
	0.600	0.32640	0.291	0.240	0.292	0.956
	0.700	0.36660	0.400	0.242	0.405	0.967
	0.800	0.41060	0.545	0.251	0.555	0.980
	1.000	0.58000	1.000	0.300	1.000	1.000
0.4	0.005	0.02369	0.001	0.254	0.001	0.997
	0.010	0.03380	0.002	0.256	0.002	0.994
	0.050	0.07781	0.010	0.262	0.011	0.989
	0.100	0.11261	0.022	0.265	0.024	0.983
	0.200	0.16518	0.052	0.269	0.056	0.972
	0.300	0.20899	0.091	0.271	0.097	0.963
	0.400	0.24911	0.142	0.273	0.150	0.959
	0.500	0.28774	0.208	0.277	0.219	0.961
	0.600	0.32640	0.292	0.284	0.309	0.968
	0.700	0.36660	0.402	0.297	0.426	0.981
	0.800	0.41060	0.547	0.318	0.577	0.992
	1.000	0.58000	1.000	0.400	1.000	1.000

Appendix 2. Tabulated data for octet truss

Constituent	Relative	Rod	Relative Young’s	Poisson’s	Relative shear	Zener
Poisson’s ratio	density	diameter	modulus	ratio	modulus	ratio
ν _S	ρ	d	E^h/E_S	ν ^h	G^h/G_S	A ^h
0.2	0.005	0.01960	0.001	0.333	0.001	1.981
	0.010	0.02786	0.001	0.333	0.002	1.972
	0.050	0.06430	0.007	0.330	0.012	1.915
	0.100	0.09308	0.015	0.325	0.026	1.851
	0.200	0.13673	0.038	0.311	0.060	1.731
	0.300	0.17331	0.070	0.296	0.105	1.616
	0.400	0.20709	0.115	0.278	0.163	1.506
	0.500	0.23978	0.177	0.258	0.237	1.403
	0.600	0.27292	0.264	0.239	0.334	1.306
	0.700	0.30805	0.385	0.222	0.458	1.212
	0.800	0.34804	0.556	0.209	0.616	1.115
	1.000	0.70000	1.000	0.200	1.000	1.000
0.3	0.005	0.01960	0.001	0.335	0.001	1.983
	0.010	0.02786	0.001	0.335	0.002	1.975
	0.050	0.06430	0.007	0.335	0.013	1.920
	0.100	0.09308	0.015	0.332	0.028	1.859
	0.200	0.13673	0.038	0.324	0.065	1.740
	0.300	0.17331	0.070	0.313	0.112	1.626
	0.400	0.20709	0.115	0.302	0.173	1.515
	0.500	0.23978	0.177	0.290	0.252	1.412
	0.600	0.27292	0.263	0.280	0.351	1.315
	0.700	0.30805	0.384	0.275	0.478	1.221
	0.800	0.34804	0.555	0.277	0.633	1.121
	1.000	0.70000	1.000	0.300	1.000	1.000
0.4	0.005	0.01960	0.001	0.336	0.001	1.986
	0.010	0.02786	0.001	0.337	0.002	1.978
	0.050	0.06430	0.007	0.339	0.014	1.927
	0.100	0.09308	0.015	0.339	0.030	1.868
	0.200	0.13673	0.038	0.336	0.069	1.752
	0.300	0.17331	0.070	0.331	0.120	1.638
	0.400	0.20709	0.115	0.325	0.185	1.527
	0.500	0.23978	0.177	0.321	0.267	1.423
	0.600	0.27292	0.263	0.321	0.370	1.326
	0.700	0.30805	0.385	0.327	0.499	1.230
	0.800	0.34804	0.556	0.344	0.652	1.126
	1.000	0.70000	1.000	0.400	1.000	1.000

Appendix 3. Tabulated data for ORC truss

Constituent	Relative	Rod	Relative Young’s	Poisson’s	Relative shear	Zener
Poisson’s ratio	density	diameter	modulus	ratio	modulus	ratio
ν _S	ρ	d	E^h/E_S	ν ^h	G^h/G_S	A ^h
0.2	0.005	0.02773	0.001	0.332	0.001	1.963
	0.010	0.03964	0.001	0.331	0.002	1.944
	0.050	0.09158	0.007	0.323	0.012	1.819
	0.100	0.13314	0.017	0.311	0.025	1.680
	0.200	0.19717	0.044	0.284	0.059	1.426
	0.300	0.25229	0.087	0.252	0.101	1.211
	0.400	0.30466	0.152	0.219	0.155	1.041
	0.500	0.35831	0.243	0.191	0.225	0.917
	0.600	0.41804	0.359	0.177	0.317	0.864
	0.700	0.48908	0.489	0.179	0.439	0.882
	0.800	0.57554	0.640	0.187	0.601	0.929
	1.000	1.00000	1.000	0.200	1.000	1.000
0.3	0.005	0.02773	0.001	0.333	0.001	1.958
	0.010	0.03964	0.001	0.333	0.002	1.936
	0.050	0.09158	0.007	0.327	0.013	1.801
	0.100	0.13314	0.017	0.317	0.027	1.653
	0.200	0.19717	0.045	0.294	0.063	1.390
	0.300	0.25229	0.089	0.268	0.107	1.172
	0.400	0.30466	0.155	0.244	0.163	1.005
	0.500	0.35831	0.249	0.226	0.234	0.888
	0.600	0.41804	0.367	0.224	0.327	0.840
	0.700	0.48908	0.498	0.238	0.450	0.862
	0.800	0.57554	0.647	0.260	0.612	0.916
	1.000	1.00000	1.000	0.300	1.000	1.000
0.4	0.005	0.02773	0.001	0.335	0.001	1.953
	0.010	0.03964	0.001	0.334	0.002	1.929
	0.050	0.09158	0.007	0.331	0.013	1.783
	0.100	0.13314	0.017	0.323	0.029	1.626
	0.200	0.19717	0.046	0.305	0.066	1.353
	0.300	0.25229	0.091	0.285	0.113	1.133
	0.400	0.30466	0.159	0.269	0.170	0.969
	0.500	0.35831	0.256	0.262	0.243	0.858
	0.600	0.41804	0.376	0.273	0.338	0.816
	0.700	0.48908	0.509	0.299	0.462	0.842
	0.800	0.57554	0.657	0.334	0.623	0.903
	1.000	1.00000	1.000	0.400	1.000	1.000

Appendix 4. MATLAB code for multiscale topology optimization

1.1 MIT License

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % A 99 LINE TOPOLOGY OPTIMIZATION CODE BY OLE SIGMUND, OCTOBER 1999 % MODIFIED FOR 3D MULTISCALE DESIGN VIA SURROGATE MODEL, LLNL, JULY 2018 % % This work was produced under the auspices of the U.S. Department of Energy by % Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. % % This work was prepared as an account of work sponsored by an agency of the % United States Government. Neither the United States Government nor Lawrence % Livermore National Security, LLC, nor any of their employees makes any warranty, % expressed or implied, or assumes any legal liability or responsibility for the % accuracy, completeness, or usefulness of any information, apparatus, product, or % process disclosed, or represents that its use would not infringe privately owned % rights. Reference herein to any specific commercial product, process, or service % by trade name, trademark, manufacturer, or otherwise does not necessarily % constitute or imply its endorsement, recommendation, or favoring by the United % States Government or Lawrence Livermore National Security, LLC. The views and % opinions of authors expressed herein do not necessarily state or reflect those % of the United States Government or Lawrence Livermore National Security, LLC, % and shall not be used for advertising or product endorsement purposes. % % LLNL-CODE-757968 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function top(nelx, nely, nelz, volfrac, rmin, truss, Es, vs, minVF, maxVF, maxit) % INITIALIZE x(1:nelx, 1:nely, 1:nelz) = volfrac; Gs = Es / (2*(1+vs)); loop = 0; change = 1.0; nnx = nelx+1; nny = nely+1; nnz = nelz+1; colormap(gray); caxis([0.0, 1.0]); % START ITERATION while change > 0.01 && loop < maxit loop = loop + 1; xold = x; % FE-ANALYSIS [U] = FE(nelx, nely, nelz, nnx, nny, nnz, x, truss, Es, vs, Gs); % OBJECTIVE FUNCTION AND SENSITIVITY ANALYSIS c = 0.0; dc = zeros(nelx, nely, nelz); for elz = 1:nelz; for ely = 1:nely for elx = 1:nelx [KE] = get_KE(truss, x, Es, vs, Gs, elx, ely, elz, 0); [DKE] = get_KE(truss, x, Es, vs, Gs, elx, ely, elz, 1); [dofs] = get_elem_dofs(nnx, nny, nnz, elx, ely, elz); Ue = U(dofs,1); c = c + Ue'*KE*Ue; dc(elx,ely,elz) = -Ue'*DKE*Ue; end; end; end % FILTERING OF SENSITIVITIES [dc] = check(nelx, nely, nelz, rmin, x, dc); % DESIGN UPDATE BY THE OPTIMALITY CRITERIA METHOD [x] = OC(nelx, nely, nelz, x, volfrac, dc, minVF, maxVF); % PRINT RESULTS change = max(max(max(abs(x-xold)))); disp([’ It.: ’ sprintf(’%4i’,loop) ’ Obj.: ’ sprintf(’%10.4f’,c) ... ’ Vol.: ’ sprintf(’%6.3f’,sum(sum(sum(x)))/(nelx*nely*nelz)) ... ’ ch.: ’ sprintf(’%6.3f’,change )]) % PLOT DENSITIES viz3d(nelx, nely, nelz, x, volfrac, nelx==1); % SAVE PARAMETER VALUES (ELEMENT DENSITIES AND ROD DIAMETERS) xOut = reshape(x,[],1); save('-ascii','elVolFrac.txt', 'xOut'); dOut = reshape(get_d(truss,x),[],1); save('-ascii','elRodDiam.txt','dOut'); end %%%%%%%%% OPTIMALITY CRITERIA UPDATE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [xnew] = OC(nelx, nely, nelz, x, volfrac, dc, minVF, maxVF) l1 = 0; l2 = 100000; move = 0.2; while (l2-l1 > 1e-4) lmid = 0.5*(l2 + l1); xnew = max(minVF, max(x-move, min(maxVF, min(x+move,x.*sqrt(-dc./lmid))))); if sum(sum(sum(xnew))) - volfrac*nelx*nely*nelz > 0; l1 = lmid; else

l2 = lmid; end end %%%%%%%%% MESH-INDEPENDENCY FILTER %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [dcn] = check(nelx, nely, nelz, rmin, x, dc) dcn=zeros(size(dc)); for elz = 1:nelz; for ely = 1:nely; for elx = 1:nelx sum = 0.0; for k = max(elz-round(rmin),1):min(elz+round(rmin),nelz) for j = max(ely-round(rmin),1):min(ely+round(rmin),nely) for i = max(elx-round(rmin),1):min(elx+round(rmin),nelx) fac = rmin - sqrt((elx-i)^2+(ely-j)^2+(elz-k)^2); sum = sum + max(0,fac); dcn(elx,ely,elz) = dcn(elx,ely,elz) + max(0,fac)*x(i,j,k)*dc(i,j,k); end end end dcn(elx,ely,elz) = dcn(elx,ely,elz) / (x(elx,ely,elz)*sum); end; end; end %%%%%%%%% FE-ANALYSIS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [U] = FE(nelx, nely, nelz, nnx, nny, nnz, x, truss, Es, vs, Gs) K = sparse(3*nnx*nny*nnz, 3*nnx*nny*nnz); F = sparse(3*nnx*nny*nnz,1); U = sparse(3*nnx*nny*nnz,1); for elz = 1:nelz; for ely = 1:nely; for elx = 1:nelx [KE] = get_KE(truss, x, Es, vs, Gs, elx, ely, elz, 0); [dofs] = get_elem_dofs(nnx, nny, nnz, elx, ely, elz); K(dofs,dofs) = K(dofs,dofs) + KE; end; end; end % DEFINE LOADS AND SUPPORTS (HALF MBB-BEAM) coords = zeros(nnx*nny*nnz,3); n = 0; for k = 1:nnz; for j = 1:nny; for i = 1:nnx n = n+1; coords(n,1) = i-1; coords(n,2) = j-1; coords(n,3) = k-1; end; end; end midplane_nodes = find(coords(:,2)==0); loaded_nodes = intersect(find(coords(:,3)==nelz), find(coords(:,2)==0)); fixed_nodes = intersect(find(coords(:,3)==0), find(coords(:,2)==nely)); fixeddofs = zeros(size(midplane_nodes,1) + 2*size(fixed_nodes,1),1); for i = loaded_nodes'; F(3*(i-1)+3) = -1.0/nnx; end n = 1; for i = midplane_nodes'; for j=[2]; fixeddofs(n,1) = 3*(i-1)+j; n =n+1; end; end for i = fixed_nodes'; for j=[1,3]; fixeddofs(n,1) = 3*(i-1)+j; n =n+1; end; end alldofs = [1:3*nnx*nny*nnz]; freedofs = setdiff(alldofs,fixeddofs); % SOLVING U(freedofs,:) = K(freedofs,freedofs) \ F(freedofs,1); U(fixeddofs,:) = 0; %%%%%%%%% ELEMENT AND NODE NUMBERING IN 3D MESH %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [num] = get_num(nx, ny, nz, i, j, k) num = (nx*ny)*(k-1) + nx*(j-1) + i;

%%%%%%%%% GLOBAL DOFS FOR A GIVEN ELEMENT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [dofs] = get_elem_dofs(nnx, nny, nnz, elx, ely, elz) n = get_num(nnx, nny, nnz, elx, ely, elz); N = [n; n+1; n+nnx+1; n+nnx; n+nnx*nny; n+nnx*nny+1; n+nnx*nny+nnx+1; n+nnx*nny+nnx]; dofs = zeros(24,1); for j = 1:8; for i = 1:3; dofs(3*(j-1)+i) = 3*(N(j)-1)+i; end; end; %%%%%%%%% INTEGRATE ELASTICITY TENSOR CE TO GET KE %%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [KE] = get_KE(truss, x, Es, vs, Gs, i, j, k, deriv) KE = zeros(24,24); CE = get_CE(truss, x, Es, vs, Gs, i, j, k, deriv); for l = 1:8 r = (sqrt(3)/3) * (-1 + 2*any([2,3,6,7]==l)); rp = (1+r); rm = (1-r); s = (sqrt(3)/3) * (-1 + 2*any([3,4,7,8]==l)); sp = (1+s); sm = (1-s); t = (sqrt(3)/3) * (-1 + 2*any([5,6,7,8]==l)); tp = (1+t); tm = (1-t); DN = [-sm*tm, -rm*tm, -rm*sm; sm*tm, -rp*tm, -rp*sm; sp*tm, rp*tm, -rp*sp; -sp*tm, rm*tm, -rm*sp; -sm*tp, -rm*tp, rm*sm; sm*tp, -rp*tp, rp*sm; sp*tp, rp*tp, rp*sp; -sp*tp, rm*tp, rm*sp] / 8; B = DN * 2*eye(3); G = kron(B', eye(3)); KE = KE + G' * CE * G / 4; end %%%%%%%%% DEFINE ELASTICITY TENSOR FOR DIFFERENT TRUSSES %%%%%%%%%%%%%%%%%%%%%%% function [CE] = get_CE(truss, x, Es, vs, Gs, i, j, k, D) p = x(i, j, k); if strcmpi(truss, 'iso'); TM = @(p,Es,vs,Gs,D) iso_moduli(p,Es,vs,Gs,D); elseif strcmpi(truss, 'octet'); TM = @(p,Es,vs,Gs,D) octet_moduli(p,Es,vs,Gs,D); elseif strcmpi(truss, 'orc'); TM = @(p,Es,vs,Gs,D) orc_moduli(p,Es,vs,Gs,D); elseif strcmpi(truss, 'bound'); TM = @(p,Es,vs,Gs,D) bound_moduli(p,Es,vs,Gs,D); else; TM = @(p,Es,vs,Gs,D) simp_moduli(p,Es,vs,Gs,D); end [E, v, G] = TM(p,Es,vs,Gs,0); if D; [DE, Dv, DG] = TM(p,Es,vs,Gs,1); end if D == 0 C1111 = E * (1.0 - v) / (1.0 - v - 2*v^2); C1122 = (E * v) / (1.0 - v - 2*v^2); C1212 = G; else % return the derivatives instead C1111 = ((DE*(1-v)-E*Dv)*(1-v-2*v^2)-E*(1-v)*(-Dv-4*v*Dv)) / (1-v-2*v^2)^2; C1122 = ((DE*v+E*Dv)*(1-v-2*v^2)-E*(1-v)*(-Dv-4*v*Dv)) / (1-v-2*v^2)^2; C1212 = DG; end CE = [C1111 0 0 0 C1122 0 0 0 C1122; 0 C1212 0 C1212 0 0 0 0 0 ; 0 0 C1212 0 0 0 C1212 0 0 ; 0 C1212 0 C1212 0 0 0 0 0 ; C1122 0 0 0 C1111 0 0 0 C1122; 0 0 0 0 0 C1212 0 C1212 0 ; 0 0 C1212 0 0 0 C1212 0 0 ; 0 0 0 0 0 C1212 0 C1212 0 ; C1122 0 0 0 C1122 0 0 0 C1111];

%%%%%%%%% TRUSS-SPECIFIC MECHANICS MODELS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [E,v,G] = iso_moduli(p, Es, vs, Gs, deriv) E = Es * (2.05292e-01 - 3.30265e-02*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (8.12145e-02 + 2.72431e-01*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (6.49737e-01 - 2.42374e-01*vs) * (p^(3-deriv)) * (1+2*deriv); v = (2.47760e-01 + 1.69804e-02*vs) * (1-deriv) + ... (-1.59293e-01 + 7.38598e-01*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (-1.86279e-01 - 4.83229e-01*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (9.77457e-02 + 7.26595e-01*vs) * (p^(3-deriv)) * (1+2*deriv); G = Gs * (1.63200e-01 + 1.27910e-01*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (6.00810e-03 + 4.13331e-01*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (7.22847e-01 - 3.56032e-01*vs) * (p^(3-deriv)) * (1+2*deriv); function [E,v,G] = octet_moduli(p, Es, vs, Gs, deriv) E = Es * (1.36265e-01 - 1.22204e-02*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (8.57991e-02 + 6.63677e-02*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (7.39887e-01 - 6.26129e-02*vs) * (p^(3-deriv)) * (1+2*deriv); v = (3.29529e-01 + 1.86038e-02*vs) * (1-deriv) + ... (-1.42155e-01 + 4.57806e-01*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (-3.29837e-01 + 5.59823e-02*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (1.41233e-01 + 4.72695e-01*vs) * (p^(3-deriv)) * (1+2*deriv); G = Gs * (2.17676e-01 + 7.22515e-02*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (-7.63847e-02 + 1.31601e+00*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (9.11800e-01 - 1.55261e+00*vs) * (p^(3-deriv)) * (1+2*deriv); function [E,v,G] = orc_moduli(p, Es, vs, Gs, deriv) E = Es * (1.34332e-01 - 7.06384e-02*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (2.59957e-01 + 8.51515e-01*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (6.53902e-01 - 7.29803e-01*vs) * (p^(3-deriv)) * (1+2*deriv); v = (3.38525e-01 + 7.04361e-03*vs) * (1-deriv) + ... (-4.25721e-01 + 4.14882e-01*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (-7.68215e-02 + 5.58948e-01*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (1.64073e-01 + 3.98374e-02*vs) * (p^(3-deriv)) * (1+2*deriv); G = Gs * (1.96762e-01 + 1.66705e-01*vs) * (p^(1-deriv)) * (1+0*deriv) + ... (1.30938e-01 + 1.72565e-01*vs) * (p^(2-deriv)) * (1+1*deriv) + ... (6.45455e-01 - 2.87424e-01*vs) * (p^(3-deriv)) * (1+2*deriv); function [E,v,G] = bound_moduli(p, Es, vs, Gs, deriv) Ks = 1.0 / (3*(1-2*vs)); K = Ks + (1-p) / (-1.0/Ks + p/(Ks + (4.0*Gs)/3.0) ); G = Gs + (1-p) / (-1.0/Gs + (2.0*p*(Ks+2.0*Gs)) / (5.0*Gs*(Ks+(4.0*Gs)/3.0)) ); E = 9*K*G/(3*K+G); v = (3*K-2*G) / (2*(3*K+G)); if deriv DK = (p - 1)/(((4*Gs)/3 + Ks)*(p/((4*Gs)/3 + Ks) - 1/Ks)^2) - ... 1/(p/((4*Gs)/3 + Ks) - 1/Ks); DG = 1/(1/Gs - (2*p*(2*Gs + Ks))/(5*Gs*((4*Gs)/3 + Ks))) + ... (2*(2*Gs + Ks)*(p - 1))/(5*Gs*((4*Gs)/3 + Ks)*(1/Gs - ... (2*p*(2*Gs + Ks))/(5*Gs*((4*Gs)/3 + Ks)))^2); DE = (9*(3*K+G)*(DK*G+K*DG) - 9*K*G*(3*DK+DG) ) / (3*K+G)^2; Dv = (2*(3*K+G)*(3*DK-2*DG) - 2*(3*K-2*G)*(3*DK+DG) ) / (2*(3*K+G))^2; G = DG; E = DE; v = Dv; end

function [E,v,G] = simp_moduli(p, Es, vs, Gs, deriv) E = Es * p^(3-deriv) * (1+2*deriv); v = vs * (1-deriv); G = Gs * p^(3-deriv) * (1+2*deriv); %%%%%%%%% TRUSS-SPECIFIC ROD DIAMETERS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function [d] = get_d(truss, p) if strcmpi(truss, 'iso') d = 2.04920e-02 + 1.05076e+00*p - 1.59468e+00*(p.^2) + 1.09799e+00*(p.^3); elseif strcmpi(truss, 'octet') d = 1.64505e-02 + 9.23773e-01*p - 1.61345e+00*(p.^2) + 1.23729e+00*(p.^3); elseif strcmpi(truss, 'orc') d = 2.32950e-02 + 1.31602e+00*p - 2.28842e+00*(p.^2) + 1.90225e+00*(p.^3); else d = -1*ones(size(p)); end %%%%%%%%% 3D VISUALIZATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% function viz3d(nelx, nely, nelz, x, volfrac, is2D) y = zeros(nelx+2, nely+2, nelz+2); y(2:nelx+1, 2:nely+1, 2:nelz+1) = x; if is2D; T=0; A=90; E=0; else; T=volfrac; A=142.5; E=30; end; nf = nelx*nely*(nelz+1) + nelx*(nely+1)*nelz + (nelx+1)*nely*nelz; n = 0; X = zeros(4,nf); Y = zeros(4,nf); Z = zeros(4,nf); C = zeros(1,nf); for k = 1:nelz+1; for j = 1:nely+1; for i = 1:nelx+1; I = i-1; J = j-1; K = k-1; L = i+1; M = j+1; N = k+1; cz = max(y(L,M,k:N)); cy = max(y(L,j:M,N)); cx = max(y(i:L,M,N)); dz = min(y(L,M,k:N)); dy = min(y(L,j:M,N)); dx = min(y(i:L,M,N)); if cz > T && dz < T+is2D; n = n+1; C(1,n) = 1-cz; X(:,n) = [I,i,i,I]'; Y(:,n) = [J,J,j,j]'; Z(:,n) = [K,K,K,K]'; end if cy > T && dy < T+is2D; n = n+1; C(1,n) = 1-cy; X(:,n) = [I,i,i,I]'; Y(:,n) = [J,J,J,J]'; Z(:,n) = [K,K,k,k]'; end if cx > T && dx < T+is2D; n = n+1; C(1,n) = 1-cx; X(:,n) = [I,I,I,I]'; Y(:,n) = [J,j,j,J]'; Z(:,n) = [K,K,k,k]'; end end; end; end patch(X(:,1:n), Y(:,1:n), Z(:,1:n), C(1,1:n), 'EdgeColor', 'none'); view(A,E); axis equal; axis tight; axis off; pause(1e-3);

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Watts, S., Arrighi, W., Kudo, J. et al. Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design. Struct Multidisc Optim 60, 1887–1920 (2019). https://doi.org/10.1007/s00158-019-02297-5

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Received: 13 September 2018
Revised: 12 February 2019
Accepted: 30 April 2019
Published: 25 May 2019
Issue Date: November 2019
DOI: https://doi.org/10.1007/s00158-019-02297-5

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Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design

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Topology optimization of multi-scale structures: a review

A Novel Equation of Motion to Predict Elastoplastic Deformation of 1-D Stochastic Bars

Stress-constrained topology optimization using the constrained natural element method

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07 January 2019

07 January 2020

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Appendices

Appendix 1. Tabulated data for isotruss

Appendix 2. Tabulated data for octet truss

Appendix 3. Tabulated data for ORC truss

Appendix 4. MATLAB code for multiscale topology optimization

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Simple, accurate surrogate models of the elastic response of three-dimensional open truss micro-architectures with applications to multiscale topology design

Abstract

Access this article

Similar content being viewed by others

Topology optimization of multi-scale structures: a review

A Novel Equation of Motion to Predict Elastoplastic Deformation of 1-D Stochastic Bars

Stress-constrained topology optimization using the constrained natural element method

Change history

07 January 2019

07 January 2020

References

Acknowledgments

Funding

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Authors and Affiliations

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Additional information

Publisher’s note

Appendices

Appendix 1. Tabulated data for isotruss

Appendix 2. Tabulated data for octet truss

Appendix 3. Tabulated data for ORC truss

Appendix 4. MATLAB code for multiscale topology optimization

1.1 MIT License

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