Document Zbl 1440.74455

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Methods Appl. Mech. Eng.</a> <a href="/?q=in%3A448300" title="Articles in this Issue">334, 56-78 (2018)</a>. </div> <div class="abstract">Summary: This work aims at developing a topology optimization method for the design of self-supporting structures that can be directly realized by additive manufacturing (AM). In the proposed method, polygon-featured holes are introduced as basic design primitives whose movements, deformations and intersections allow to control the structural topology. In addition, overhang constraints are introduced directly into the geometry description, so that the inclined angles along the overhang boundaries are not less than the critical overhang angle (COA). Meanwhile, unprintable V-shaped areas caused by polygon intersections are gradually eliminated through polygon modifications and re-optimization. Numerical examples are tested to demonstrate the effectiveness of the proposed method.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7186831">Cited in <strong>20</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A74S05" title="MSC2020">74S05</a> </td> <td class="space"> Finite element methods applied to problems in solid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A74P15" title="MSC2020">74P15</a> </td> <td class="space"> Topological methods for optimization problems in solid mechanics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Atopology+optimization">topology optimization</a>; <a href="/?q=ut%3Apolygon+feature">polygon feature</a>; <a href="/?q=ut%3AV-shaped+area">V-shaped area</a>; <a href="/?q=ut%3Aadditive+manufacturing">additive manufacturing</a>; <a href="/?q=ut%3Aself-supporting">self-supporting</a></div> <div class="software"> <h3>Software:</h3><a href="/software/11037">BOSS-Quattro</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1440.74455" data-ciurl="/ci/07186831" data-biburl="/bibtex/07186831.bib" data-amsurl="/amsrefs/07186831.bib" data-xmlurl="/xml/07186831.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07186831.pdf" title="Zbl 1440.74455 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.cma.2018.01.037" aria-label="DOI for “Topology optimization of self-supporting structures with polygon features for additive manufacturing”" title="10.1016/j.cma.2018.01.037">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Li, Q.; Chen, W.; Liu, S.; Tong, L., Structural topology optimization considering connectivity constraint, Struct. Multidiscip. Optim., 54, 971-984 (2016)</td> </tr><tr> <td>[2]</td> <td class="space">O. Diegel, S. Singamneni, S. Reay, A. Withell, Tools for sustainable product design: additive manufacturing, 2010.; O. Diegel, S. Singamneni, S. Reay, A. Withell, Tools for sustainable product design: additive manufacturing, 2010.</td> </tr><tr> <td>[3]</td> <td class="space">Gao, W.; Zhang, Y.; Ramanujan, D.; Ramani, K.; Chen, Y.; Williams, C. B.; Wang, C. C.; Shin, Y. C.; Zhang, S.; Zavattieri, P. D., The status, challenges, and future of additive manufacturing in engineering, Comput. Aided Des., 69, 65-89 (2015)</td> </tr><tr> <td>[4]</td> <td class="space">Qattawi, A.; Ablat, M. A., Design consideration for additive manufacturing: fused deposition modelling, Open J. Appl. Sci., 7, 291 (2017)</td> </tr><tr> <td>[5]</td> <td class="space">Calignano, F., Design optimization of supports for overhanging structures in aluminum and titanium alloys by selective laser melting, Mater. Des., 64, 203-213 (2014)</td> </tr><tr> <td>[6]</td> <td class="space">Morgan, H.; Cherry, J.; Jonnalagadda, S.; Ewing, D.; Sienz, J., Part orientation optimisation for the additive layer manufacture of metal components, Int. J. Adv. Manuf. Technol., 86, 1679-1687 (2016)</td> </tr><tr> <td>[7]</td> <td class="space">Strano, G.; Hao, L.; Everson, R.; Evans, K., A new approach to the design and optimisation of support structures in additive manufacturing, Int. J. Adv. Manuf. Technol., 1-8 (2013)</td> </tr><tr> <td>[8]</td> <td class="space">Das, P.; Chandran, R.; Samant, R.; Anand, S., Optimum part build orientation in additive manufacturing for minimizing part errors and support structures, Proc. Manuf., 1, 343-354 (2015)</td> </tr><tr> <td>[9]</td> <td class="space">Hussein, A.; Hao, L.; Yan, C.; Everson, R.; Young, P., Advanced lattice support structures for metal additive manufacturing, J. Mater Process. Technol., 213, 1019-1026 (2013)</td> </tr><tr> <td>[10]</td> <td class="space">Vanek, J.; Galicia, J. A.G.; Benes, B., Clever support: Efficient support structure generation for digital fabrication, (Computer graphics forum (2014), Wiley Online Library), 117-125</td> </tr><tr> <td>[11]</td> <td class="space">Dumas, J.; Hergel, J.; Lefebvre, S., Bridging the gap: automated steady scaffoldings for 3D printing, ACM Trans. Graph., 33, 98 (2014)</td> </tr><tr> <td>[12]</td> <td class="space">Barnett, E.; Gosselin, C., Weak support material techniques for alternative additive manufacturing materials, Additive Manuf., 8, 95-104 (2015)</td> </tr><tr> <td>[13]</td> <td class="space">Hu, K.; Jin, S.; Wang, C. C., Support slimming for single material based additive manufacturing, Comput. Aided Des., 65, 1-10 (2015)</td> </tr><tr> <td>[14]</td> <td class="space">Leary, M.; Merli, L.; Torti, F.; Mazur, M.; Brandt, M., Optimal topology for additive manufacture: a method for enabling additive manufacture of support-free optimal structures, Mater. Des., 63, 678-690 (2014)</td> </tr><tr> <td>[15]</td> <td class="space">D. Brackett, I. Ashcroft, R. Hague, Topology optimization for additive manufacturing, in: Proceedings of the Solid Freeform Fabrication Symposium, Austin, TX, S, 2011, pp. 348-362.; D. Brackett, I. Ashcroft, R. Hague, Topology optimization for additive manufacturing, in: Proceedings of the Solid Freeform Fabrication Symposium, Austin, TX, S, 2011, pp. 348-362.</td> </tr><tr> <td>[16]</td> <td class="space">Clausen, A., Topology Optimization for Additive Manufacturing (2016), Technical University of Denmark</td> </tr><tr> <td>[17]</td> <td class="space">A.T. Gaynor, J.K. Guest, Topology optimization for additive manufacturing: considering maximum overhang constraint, in: 15th AIAA/ISSMO multidisciplinary analysis and optimization conference, 2014, pp. 16-20.; A.T. Gaynor, J.K. Guest, Topology optimization for additive manufacturing: considering maximum overhang constraint, in: 15th AIAA/ISSMO multidisciplinary analysis and optimization conference, 2014, pp. 16-20.</td> </tr><tr> <td>[18]</td> <td class="space">Gaynor, A. T.; Guest, J. K., Topology optimization considering overhang constraints: Eliminating sacrificial support material in additive manufacturing through design, Struct. Multidiscip. Optim., 54, 1157-1172 (2016)</td> </tr><tr> <td>[19]</td> <td class="space">Guo, X.; Zhou, J.; Zhang, W.; Du, Z.; Liu, C.; Liu, Y., Self-supporting structure design in additive manufacturing through explicit topology optimization, Comput. Methods Appl. Mech. Engrg. (2017) · <a href="/1439.74273" class="nowrap">Zbl 1439.74273</a></td> </tr><tr> <td>[20]</td> <td class="space">Langelaar, M., Topology optimization of 3D self-supporting structures for additive manufacturing, Additive Manuf., 12, 60-70 (2016)</td> </tr><tr> <td>[21]</td> <td class="space">Langelaar, M., An additive manufacturing filter for topology optimization of print-ready designs, Struct. Multidiscip. Optim., 55, 871-883 (2017)</td> </tr><tr> <td>[22]</td> <td class="space">Mirzendehdel, A. M.; Suresh, K., Support structure constrained topology optimization for additive manufacturing, Comput. Aided Des., 81, 1-13 (2016)</td> </tr><tr> <td>[23]</td> <td class="space">Zegard, T.; Paulino, G. H., Bridging topology optimization and additive manufacturing, Struct. Multidiscip. Optim., 53, 175-192 (2016)</td> </tr><tr> <td>[24]</td> <td class="space">Clausen, A.; Aage, N.; Sigmund, O., Topology optimization with flexible void area, Struct. Multidiscip. Optim., 50, 927-943 (2014)</td> </tr><tr> <td>[25]</td> <td class="space">Guo, X.; Zhang, W.; Zhong, W., Doing topology optimization explicitly and geometrically—a new moving morphable components based framework, J. Appl. Mech., 81, 081009 (2014)</td> </tr><tr> <td>[26]</td> <td class="space">W. Yang, W. Zhang, X. Guo, Explicit structural topology optimization via moving morphable voids (MMV) approach, in: 2016 Asian Congress of Structural and Multidisciplinary Optimization, Nagasaki, Japan, 2016, p. 98.; W. Yang, W. Zhang, X. Guo, Explicit structural topology optimization via moving morphable voids (MMV) approach, in: 2016 Asian Congress of Structural and Multidisciplinary Optimization, Nagasaki, Japan, 2016, p. 98.</td> </tr><tr> <td>[27]</td> <td class="space">Zhang, W.; Zhao, L.; Gao, T., CBS-based topology optimization including design-dependent body loads, Comput. Methods Appl. Mech. Engrg., 322, 1-22 (2017) · <a href="/1439.74308" class="nowrap">Zbl 1439.74308</a></td> </tr><tr> <td>[28]</td> <td class="space">Zhang, W.; Zhao, L.; Gao, T.; Cai, S., Topology optimization with closed B-splines and Boolean operations, Comput. Methods Appl. Mech. Engrg., 315, 652-670 (2017) · <a href="/1439.74309" class="nowrap">Zbl 1439.74309</a></td> </tr><tr> <td>[29]</td> <td class="space">Zhou, Y.; Zhang, W.; Zhu, J.; Xu, Z., Feature-driven topology optimization method with signed distance function, Comput. Methods Appl. Mech. Engrg., 310, 1-32 (2016) · <a href="/1439.74315" class="nowrap">Zbl 1439.74315</a></td> </tr><tr> <td>[30]</td> <td class="space">Kreisselmeier, G.; Steinhauser, R., Systematic control design by optimizing a vector performance index, IFAC Proc. Vol., 12, 113-117 (1979) · <a href="/0443.49028" class="nowrap">Zbl 0443.49028</a></td> </tr><tr> <td>[31]</td> <td class="space">Kranz, J.; Herzog, D.; Emmelmann, C., Design guidelines for laser additive manufacturing of lightweight structures in TiAl6V4, J. Laser Appl., 27, S14001 (2015)</td> </tr><tr> <td>[32]</td> <td class="space">Mertens, R.; Clijsters, S.; Kempen, K.; Kruth, J.-P., Optimization of scan strategies in selective laser melting of aluminum parts with downfacing areas, J. Manuf. Sci. Eng., 136, 061012 (2014)</td> </tr><tr> <td>[33]</td> <td class="space">Osher, S.; Fedkiw, R.; Piechor, K., Level set methods and dynamic implicit surfaces, Appl. Mech. Rev., 57, B15 (2004)</td> </tr><tr> <td>[34]</td> <td class="space">Rvachev, V., Theory of R-functions and Some Applications, Vol. 552 (1982), Naukova Dumka · <a href="/0521.65084" class="nowrap">Zbl 0521.65084</a></td> </tr><tr> <td>[35]</td> <td class="space">Shapiro, V., Theory of R-functions and Applications: A Primer (1991), Cornell University</td> </tr><tr> <td>[36]</td> <td class="space">Ricci, A., A constructive geometry for computer graphics, Comput. J., 16, 157-160 (1973) · <a href="/0251.68053" class="nowrap">Zbl 0251.68053</a></td> </tr><tr> <td>[37]</td> <td class="space">Cai, S.; Zhang, W.; Zhu, J.; Gao, T., Stress constrained shape and topology optimization with fixed mesh: a B-spline finite cell method combined with level set function, Comput. Methods Appl. Mech. Engrg., 278, 361-387 (2014) · <a href="/1423.74741" class="nowrap">Zbl 1423.74741</a></td> </tr><tr> <td>[38]</td> <td class="space">Parvizian, J.; Düster, A.; Rank, E., Finite cell method, Comput. Mech., 41, 121-133 (2007) · <a href="/1162.74506" class="nowrap">Zbl 1162.74506</a></td> </tr><tr> <td>[39]</td> <td class="space">Parvizian, J.; Düster, A.; Rank, E., Topology optimization using the finite cell method, Optim. Eng., 13, 57-78 (2012) · <a href="/1293.74357" class="nowrap">Zbl 1293.74357</a></td> </tr><tr> <td>[40]</td> <td class="space">Zhang, W.; Zhao, L.; Cai, S., Shape optimization of Dirichlet boundaries based on weighted B-spline finite cell method and level-set function, Comput. Methods Appl. Mech. Engrg., 294, 359-383 (2015) · <a href="/1423.74730" class="nowrap">Zbl 1423.74730</a></td> </tr><tr> <td>[41]</td> <td class="space">K. Svanberg, A globally convergent version of MMA without linesearch, in: Proceedings of the First World Congress of Structural and Multidisciplinary Optimization, Goslar, Germany, 1995, pp. 9-16.; K. Svanberg, A globally convergent version of MMA without linesearch, in: Proceedings of the First World Congress of Structural and Multidisciplinary Optimization, Goslar, Germany, 1995, pp. 9-16.</td> </tr><tr> <td>[42]</td> <td class="space">Radovcic, Y.; Remouchamps, A., BOSS QUATTRO: an open system for parametric design, Struct. Multidiscip. Optim., 23, 140-152 (2002)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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