# GRAND3 — Ground structure based topology optimization for arbitrary 3D domains using MATLAB

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## Abstract

Since its introduction, the ground structure method has been used in the derivation of closed–form analytical solutions for optimal structures, as well as providing information on the optimal load–paths. Despite its long history, the method has seen little use in three–dimensional problems or in problems with non–orthogonal domains, mainly due to computational implementation difficulties. This work presents a methodology for ground structure based topology optimization in arbitrary three–dimensional (3D) domains. The proposed approach is able to address concave domains and with the possibility of holes. In addition, an easy–to–use implementation of the proposed algorithm for the optimization of least–weight trusses is described in detail. The method is verified against three–dimensional closed–form solutions available in the literature. By means of examples, various features of the 3D ground structure approach are assessed, including the ability of the method to provide solutions with different levels of detail. The source code for a MATLAB implementation of the method, named GRAND3 — *GRound structure ANalysis and Design in 3D*, is available in the (electronic) *Supplementary Material* accompanying this publication.

### Keywords

Ground structure method Topology optimization of three–dimensional trusses Three-dimensional optimal structures Unstructured meshes Intersection tests## Notes

### Acknowledgments

The authors appreciate constructive comments and insightful suggestions from the anonymous reviewers. We are thankful to the support from the US National Science Foundation under grant CMMI #1335160. We also acknowledge the support from SOM (Skidmore, Owings and Merrill LLP) and from the Donald B. and Elizabeth M. Willett endowment at the University of Illinois at Urbana–Champaign. Any opinion, finding, conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors.

## Supplementary material

### References

- Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidiscip Optim 33(4–5):285–304CrossRefMathSciNetMATHGoogle Scholar
- Barber CB, Dobkin DP, Huhdanpaa H (1996) The quickhull algorithm for convex hulls. ACM Trans Math Softw 22(4):469–483. http://www.qhull.org/ CrossRefMathSciNetMATHGoogle Scholar
- Christensen P, Klarbring A (2009) An introduction to structural optimization, 1st edn. Springer, BerlinMATHGoogle Scholar
- Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. J Mech 3(1):25–52Google Scholar
- Ericson C (2004) Real-time collision detection, 1st edn. Morgan Kaufmann, San FranciscoGoogle Scholar
- Gerdes D (1994) Strukturoptimierung unter Anwendung der Optimalitätskriterien auf diskretisierte Tragwerke bei besonderer Berücksichtigung der Stabilität (in German). Phd thesis, Universitĺat EssenGoogle Scholar
- Gilbert M, Darwich W, Tyas A, Shepherd P (2005) Application of large-scale layout optimization techniques in structural engineering practice. In: 6th world congress of structural and multidisciplinary optimizartion, June:1–10Google Scholar
- Gilbert M, Tyas A (2003) Layout optimization of large-scale pin-jointed frames. Eng Comput 20(8):1044–1064CrossRefMATHGoogle Scholar
- Hemp WS (1973) Optimum structures, 1st edn. Oxford University Press, OxfordGoogle Scholar
- Hencky H (1923) Über einige statisch bestimmte Fälle des Gleichgewichts in plastischen Körpern. Z Angew Math Mech 747:241–251CrossRefGoogle Scholar
- Herceg M, Kvasnica M, Jones CN, Morari M (2013) Multi-parametric toolbox 3.0. In: Proceedings of the European control conference. http://control.ee.ethz.ch/~mpt, Zürich, pp 502–510
- Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395CrossRefMathSciNetMATHGoogle Scholar
- Kicher TP (1966) Optimum design-minimum weight versus fully stressed. ASCE Journal of Structural Division 92(ST 6):265–279Google Scholar
- Kirsch U (1993) Structural optimization: fundamentals and applications, 1st edn. Springer, BerlinCrossRefGoogle Scholar
- Lewiński T (2004) Michell structures formed on surfaces of revolution. Struct Multidiscip Optim 28(1):20–30CrossRefMathSciNetGoogle Scholar
- Lewiński T, Zhou M, Rozvany GIN (1994) Extended exact solutions for least-weight truss layouts–Part I: cantilever with a horizontal axis of symmetry. Int J Mech Sci 36(5):375–398CrossRefMATHGoogle Scholar
- Mazurek A, Baker WF, Tort C (2011) Geometrical aspects of optimum truss like structures. Struct Multidiscip Optim 43(2):231– 242CrossRefGoogle Scholar
- Michell AGM (1904) The limits of economy of material in frame-structures. Philosophical Magazine Series 6 8(47):589–597CrossRefMATHGoogle Scholar
- Ohsaki M (2010) Optimization of finite dimensional structures, 1st edn. CRC Press, Boca RatonCrossRefGoogle Scholar
- Rozvany GIN (1997) On the validity of Prager’s example of nonunique Michell structures. Structural Optimization 13(2–3):191– 194CrossRefGoogle Scholar
- Rycroft CH (2014) Voro++ v0.4.6: a three-dimensional Voronoi cell library in C++. http://math.lbl.gov/voro++/. Accessed 21 April 2014
- Smith ODS (1998) Generation of ground structures for 2D and 3D design domains. Eng Comput 15(4):462–500CrossRefMATHGoogle Scholar
- Sokół T (2011) A 99 line code for discretized Michell truss optimization written in mathematica. Struct Multidiscip Optim 43(2):181–190CrossRefMATHGoogle Scholar
- Stromberg LL, Beghini A, Baker WF, Paulino GH (2010) Application of layout and topology optimization using pattern gradation for the conceptual design of buildings. Struct Multidiscip Optim 43(2):165–180CrossRefGoogle Scholar
- Sved G (1954) The minimum weight of certain redundant structures. Austral J Appl Sci 5:1–9Google Scholar
- Tyas A, Gilbert M, Pritchard T (2006) Practical plastic layout optimization of trusses incorporating stability considerations. Comput Struct 84(3–4):115–126CrossRefGoogle Scholar
- Wright MH (2004) The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull Am Math Soc 42(1):39–56CrossRefGoogle Scholar
- Zegard T, Paulino GH (2014) GRAND – Ground structure based topology optimization on arbitrary 2D domains using MATLAB. Struct Multidiscip Optim 50(5):861–882CrossRefMathSciNetGoogle Scholar