Abstract
Traditional building design is often done in a (pseudo-) sequential manner: the architect defines the form, the structural engineer defines the material and member dimensions, and the mechanical engineer defines the openings, clearances, and additional spaces that ensure proper operation of the building. The design process should ideally be linear, where each discipline receives a complete design from the previous. In reality, however, upstream revisions are usually substantive: significant work in the schematic design and design development phases are due to resolving upstream issues. That said, within the conceptual design and initial phase, the process is mostly linear. This work presents a set of tools that move towards an integrated design optimization, where the building’s form and structure are optimized together and not as separate stages in the design. This approach often results in a higher impact/gain in efficiency, safety, cost-savings, and ultimately results in innovative designs. This industrial application manuscript provides specific details on the implementation and experience gained from the development of various topology optimization tools for use in building engineering. These are all accompanied by examples of their use in applied building projects or more general structural engineering problems. Part of the success of this effort is attributed to the environment in which these tools are implemented, which is friendly to architects. In contrast, commercial tools for this purpose tend to cater to engineers instead.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
As structures become more extreme (taller, long cantilevers, long spans, etc), the architect is more open to contributions or comments from the structural engineer regarding the building geometry.
Skidmore, Owings and Merrill LLP (SOM) is an architecture+engineering firm with worldwide presence specializing in the design of super-tall buildings and large complexes, among others. It has a long-standing tradition of innovative and cutting edge designs, and has introduced significant changes to the practice of building engineering throughout its 80 years (at the time of this writing).
There if ongoing work to obtain analytical clean geometrical descriptions out of density-based topology optimization results. This is often referred to as obtaining a CAD file.
The marching cubes algorithm works with any hexahedra. Hence, a more appropriate name could be marching hexahedra.
The marching squares algorithm works with any quadrangle. Hence, a more appropriate name would be marching quadrangles.
Operations such as and, or, and subtract.
Refer to https://web.ornl.gov/sci/eere/amie/ and https://www.som.com/projects/amie for additional information on the project and all of the involved parties.
References
Achtziger W (2007) On simultaneous optimization of truss geometry and topology. Struct Multidiscip Optim 33(4-5):285–304. https://doi.org/10.1007/s00158-006-0092-0
Allaire G, Francfort G (1993) A numerical algorithm for topology and shape optimization. In: Bendsøe M, Mota Soares C (eds) Topology design of structures. Springer, Netherlands, pp 239–248
Allaire G, Kohn R (1993) Topology optimization and optimal shape design using homogenization. In: Bendsøe M, Mota Soares C (eds) Topology design of structures. Springer, Netherlands, pp 207–218
Altair Engineering, Inc. (2019a) HyperWorks 2017.2 (accessed January 11 2019). http://www.altairhyperworks.com/
Altair Engineering, Inc. (2019b) Inspire™ 2019 (accessed January 11, 2019). https://solidthinking.com/product/inspire/
Ambrosio L, Buttazzo G (1993) An optimal design problem with perimeter penalization. Calc Var Part Diff Equ 1(1):55–69. https://doi.org/10.1007/BF02163264
Arora J (2011) Introduction to optimum design, 3rd edn. Academic Press, Waltham
Baker WF, Beghini LL, Mazurek A, Carrion J, Beghini A (2013) Maxwell’s reciprocal diagrams and discrete Michell frames. Struct Multidiscip Optim 48(2):267–277. https://doi.org/10.1007/s00158-013-0910-0
Beghini LL, Beghini A, Katz N, Baker WF, Paulino GH (2014a) Connecting architecture and engineering through structural topology optimization. Eng Struct 59:716–726. https://doi.org/10.1016/j.engstruct.2013.10.032
Beghini LL, Carrion J, Beghini A, Mazurek A, Baker WF (2014b) Structural optimization using graphic statics. Struct Multidiscip Optim 49(3):351–366. https://doi.org/10.1007/s00158-013-1002-x
Ben-Tal A, Bendsøe M (1993) A new method for optimal truss topology design. SIAM J Optim 3(2):322–358
Bendsøe MP (1989) Optimal shape design as a material distribution problem. Struct Optim 1(4):193–202. https://doi.org/10.1021/jm1000584
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Engineering online library, 2nd edn. Springer, Berlin
Besserud K, Katz N, Beghini A (2013) Structural emergence: architectural and structural design collaboration at SOM. Arch Des 83(2):48–55
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158
Bow RH (1873) Economics of construction in relation to framed structures. ICE Publishing, London
Bruns TE, Tortorelli DA (2001) Topology optimization of non-linear elastic structures and compliant mechanisms. Comput Methods Appl Mech Eng 190(26–27):3443–3459. https://doi.org/10.1016/S0045-7825(00)00278-4
Christensen P, Klarbring A (2009) An introduction to structural optimization, 1st edn. Springer, Berlin
COMSOL Inc. (2019) COMSOL Multiphysics 5.4 (accessed January 11, 2019). https://www.comsol.com
Cremona L (1872) Le figure reciproche nella statica grafica. Tipografia Giuseppe Bernardoni, Milan, Italy
Dassault Systèmes (2019) Tosca – The Optimization Suite (accessed January 11, 2019). http://www.simulia.com/
Díaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Struct Optim 10(1):40–45
Doi A, Koide A (1991) An efficient method of triangulating equi-valued surfaces by using tetrahedral cells. IEICE Trans Inform Syst E74-D(1):214–224
Dombernowsky P, Sondergaard A (2009) Three-dimensional topology optimisation in architectural and structural design of concrete structures. In: Domingo A, Lazaro C (eds) International association for shell and spatial structures (IASS) symposium, Valencia, pp 1066–1077
Dorn WS, Gomory RE, Greenberg HJ (1964) Automatic design of optimal structures. J de Mecanique 3(1):25–52
Du Q, Emelianenko M, Ju L (2006) Convergence of the Lloyd algorithm for computing centroidal voronoi tessellations. SIAM J Numer Anal 44(1):102–119. https://doi.org/10.1137/040617364
Du Q, Faber V, Gunzburger M (1999) Centroidal voronoi tessellations: applications and algorithms. SIAM Rev 41(4):637–676. https://doi.org/10.1137/S0036144599352836
Eastman C, Teicholz P, Sacks R, Liston K (2011) BIM handbook: a guide to building information modeling for owners, managers, designers, engineers and contractors, 2nd edn. Wiley, Hoboken
Gehry Technologies Inc. (2019) Digital Project V1,R5 (accessed January 11, 2019). https://www.digitalproject3d.com/
Gilbert M, Tyas A (2003) Layout optimization of large-scale pin-jointed frames. Eng Comput 20(8):1044–1064. https://doi.org/10.1108/02644400310503017
GNU (2019) GLPK (GNU linear programming kit, version 4.65) (accessed January 11, 2019). http://www.gnu.org/software/glpk/
Golub GH, Van Loan CF (2013) Matrix computations, 4th edn. The Johns Hopkins University Press, Baltimore
Graczykowski C, Lewiński T (2005) The lightest plane structures of a bounded stress level transmitting a point load to a circular support. Control Cybern 34(1):227–253
Graczykowski C, Lewiński T (2006a) Michell cantilevers constructed within trapezoidal domains—part I: geometry of Hencky nets. Struct Multidiscip Optim 32(5):347–368. https://doi.org/10.1007/s00158-005-0599-9
Graczykowski C, Lewiński T (2006b) Michell cantilevers constructed within trapezoidal domains—part II: virtual displacement fields. Struct Multidiscip Optim 32(6):463–471. https://doi.org/10.1007/s00158-005-0600-7
Graczykowski C, Lewiński T (2006c) Michell cantilevers constructed within trapezoidal domains—part III: force fields. Struct Multidiscip Optim 33(1):1–19. https://doi.org/10.1007/s00158-005-0601-6
Graczykowski C, Lewiński T (2007) Michell cantilevers constructed within trapezoidal domains—part IV: complete exact solutions of selected optimal designs and their approximations by trusses of finite number of joints. Struct Multidiscip Optim 33(2):113–129. https://doi.org/10.1007/s00158-005-0602-5
Guest JK, Prévost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61(2):238–254. https://doi.org/10.1002/nme.1064
Haber RB, Jog CS, Bendsøe MP (1996) A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim 11(1–2):1–12
Hartz C, Mazurek A, Miki M, Zegard T, Mitchell T, Baker WF (2018) The application of 2D and 3D graphic statics in design. J Int Assoc Shell Spatial Struct 59(4):235–242
Heath MT (2002) Scientific computing: an introductory survey, 2nd edn. McGraw Hill, New York
Hemp WS (1973) Optimum structures, 1st edn. Oxford University Press, Oxford
Karmarkar N (1984) A new polynomial-time algorithm for linear programming. Combinatorica 4(4):373–395
Kintel M, Wolf C (2019) OpenSCAD – the programmers solid 3D CAD modeller (accessed January 11, 2019). http://www.openscad.org/
Kirsch F, Döllner J (2005) OpenCSG: a library for image-based CSG rendering. In: USENIX annual technical conference, FREENIX track
Lewiński T (2004) Michell structures formed on surfaces of revolution. Struct Multidiscip Optim 28(1):20–30. https://doi.org/10.1007/s00158-004-0419-7
Lewiński T, Rozvany GIN (2007) Exact analytical solutions for some popular benchmark problems in topology optimization—part II: three-sided polygonal supports. Struct Multidiscip Optim 33(4–5):337–349. https://doi.org/10.1007/s00158-007-0093-7
Lewiński T, Rozvany GIN (2008a) Analytical benchmarks for topological optimization—part IV: square-shaped line support. Struct Multidiscip Optim 36(2):143–158. https://doi.org/10.1007/s00158-007-0205-4
Lewiński T, Rozvany GIN (2008b) Exact analytical solutions for some popular benchmark problems in topology optimization—part III: L-shaped domains. Struct Multidiscip Optim 35(2):165–174. https://doi.org/10.1007/s00158-007-0157-8
Lewiński T, Zhou M, Rozvany GIN (1994a) Extended exact least-weight truss layouts—part II: unsymmetric cantilevers. Int J Mech Sci 36(5):399–419. https://doi.org/10.1007/978-3-319-95180-5
Lewiński T, Zhou M, Rozvany GIN (1994b) Extended exact solutions for least-weight truss layouts—part I: cantilever with a horizontal axis of symmetry. Int J Mech Sci 36(5):375–398
Lewiński T, Rozvany GIN, Sokół T, Bołbotowski K (2013) Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains revisited. Struct Multidiscip Optim 47 (6):937–942. https://doi.org/10.1007/s00158-012-0865-6
Lewiński T, Sokół T, Graczykowski C (2019) Michell structures. Springer, Cham
Lorensen WE, Cline HE (1987) Marching cubes: a high resolution 3D surface construction algorithm. In: Proceedings of the 14th annual conference on computer graphics and interactive techniques SIGGRAPH ’87. ACM, New York, pp 163–169, DOI https://doi.org/10.1145/37401.37422, (to appear in print)
Matsui K, Terada K (2004) Continuous approximation of material distribution for topology optimization. Int J Numer Methods Eng 59(14):1925–1944. https://doi.org/10.1002/nme.945
Maxwell JC (1864) On reciprocal figures and diagrams of forces. Philos Mag Series 4 27(182):250–261
Maxwell JC (1870) On reciprocal figures, frames, and diagrams of forces. Trans R Soc Edinburgh 26(1):1–40. https://doi.org/10.1017/S0080456800026351
Mazurek A (2012) Optimum distribution of material in structures with multiple optimization criteria. In: Structural engineers association of Illinois, Chicago
Nguyen TH, Paulino GH, Song J, Le CH (2009) A computational paradigm for multiresolution topology optimization (MTOP). Struct Multidiscip Optim 41(4):525–539. https://doi.org/10.1007/s00158-009-0443-8
Ohsaki M (2010) Optimization of finite dimensional structures, 1st edn. CRC Press, Boca Raton
Pereira A, Talischi C, Paulino GH, Menezes IF, Carvalho MS (2016) Fluid flow topology optimization in PolyTop: stability and computational implementation. Struct Multidiscip Optim 54(5):1345–1364. https://doi.org/10.1007/s00158-014-1182-z
Persson PO, Strang G (2004) A simple mesh generator in MATLAB. SIAM Rev 46(2):329–345
Petersson J (1999) A finite element analysis of optimal variable thickness sheets. SIAM J Numer Anal 36 (6):1759–1778. https://doi.org/10.1137/S0036142996313968
Prager W (1958) On a problem of optimal design. In: Olszak W (ed) IUTAM Symposium on non-homogeneity in elasticity and plasticity. Pergamon Press, Warsaw
Press WH, Teukolsky SA, Vetterling WT, Flannery BP (2007) Numerical recipes 3rd edition: the art of scientific computing, 3rd edn. Cambridge University Press, New York
Ramos AS, Paulino GH (1999) Convex topology optimization for hyperelastic trusses based on the ground-structure approach. Struct Multidiscip Optim 51(2):287–304. https://doi.org/10.1007/s00158-014-1147-2
Robert McNeel & Associates (2019a) Grasshopper v1.0.0007 (accessed January 11, 2019). https://www.grasshopper3d.com/
Robert McNeel & Associates (2019b) Rhinoceros 6 SR11 (accessed January 11, 2019). http://www.rhino3d.com/
Rocscience Inc. (2019) Dips 7.0 - graphical and statistical analysis of orientation data. Toronto, Canada (accessed January 11, 2019). https://www.rocscience.com/
Rozvany GIN (1998) Exact analytical solutions for some popular benchmark problems in topology optimization. Struct Optim 15(1):42–48
Rozvany G, Gollub W (1990) Michell layouts for various combinations of line supports—I. Int J Mech Sci 32(12):1021–1043
Rozvany G, Gollub W, Zhou M (1997) Exact Michell layouts for various combinations of line supports—part II. Struct Optim 14(2-3):138–149
Schöberl J (1997) NETGEN – an advancing front 2D/3D-mesh generator based on abstract rules. Comput Visual Sci 1(1):41–52. https://ngsolve.org/
Scorzelli G, Portuesi S, Milicchio F, Paoluzzi A (2019) PLaSM – Functional language for computing with geometry (accessed January 11, 2019). http://www.plasm.net/
Siek JG, Lee LQ, Lumsdaine A (2001) The boost graph library: user guide and reference manual. Addison-Wesley Professional, Boston. http://www.boost.org/libs/graph/
Sigmund O (1997) On the design of compliant mechanisms using topology optimization. Mech Struct Mach 25(4):493–524. https://doi.org/10.1080/08905459708945415
Sigmund O (2001) A 99 line topology optimization code written in Matlab. Struct Multidiscip Optim 21 (2):120–127
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidiscip Optim 33(4-5):401–424. https://doi.org/10.1007/s00158-006-0087-x
Sigmund O, Maute K (2013) Topology optimization approaches. Struct Multidiscip Optim 48(6):1031–1055. https://doi.org/10.1007/s00158-013-0978-6
Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct Optim 16(1):68–75
Smith CC, Gilbert M (2007) Application of discontinuity layout optimization to plane plasticity problems. Proc R Soc A: Math Phys Eng Sci 463(2086):2461–2484. https://doi.org/10.1098/rspa.2006.1788
Sokół T (2011) A 99 line code for discretized Michell truss optimization written in Mathematica. Struct Multidiscip Optim 43(2):181–190. https://doi.org/10.1007/s00158-010-0557-z
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):165–180. https://doi.org/10.1007/s00158-010-0563-1
Stromberg LL, Beghini A, Baker WF, Paulino GH (2012) Topology optimization for braced frames: combining continuum and beam/column elements. Eng Struct 37:106–124. https://doi.org/10.1016/j.engstruct.2011.12.034
Sukumar N, Malsch EA (2006) Recent advances in the construction of polygonal finite element interpolants. Arch Comput Methods Eng 13(1):129–163. https://doi.org/10.1007/BF02905933
Sukumar N, Tabarraei A (2004) Conforming polygonal finite elements. Int J Numer Methods Eng 61(12):2045–2066. https://doi.org/10.1002/nme.1141
Talischi C, Paulino GH, Pereira A, Menezes IFM (2012a) PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct Multidiscip Optim 45(3):309–328. https://doi.org/10.1007/s00158-011-0706-z
Talischi C, Paulino GH, Pereira A, Menezes IFM (2012b) PolyTop: a Matlab implementation of a general topology optimization framework using unstructured polygonal finite element meshes. Struct Multidiscip Optim 45 (3):329–357. https://doi.org/10.1007/s00158-011-0696-x
Topping BHV (1983) Shape optimization of skeletal structures: a review. J Struct Eng 109(8):1933–1951
Trudeau RJ (1994) Introduction to graph theory. Dover books on mathematics, 2nd edn. Dover Publications, New York
Tyas A, Gilbert M, Pritchard T (2006) Practical plastic layout optimization of trusses incorporating stability considerations. Comput Struct 84(3–4):115–126. https://doi.org/10.1016/j.compstruc.2005.09.032
US Army Corps of Engineers (2003) Slope stability. https://doi.org/10.1016/0148-9062(75)90139-4
Vanderplaats GN (2005) Numerical optimization techniques for engineering design, 4th edn. Vanderplaats R&D, Inc, Colorado Springs
Vanderplaats R&D, Inc. (2019) GENESIS 17.0 (accessed January 11, 2019). http://www.vrand.com/
Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidiscip Optim 43(6):767–784
Wenger R (2013) Isosurfaces: geometry, topology, and algorithms. CRC Press, Boca Raton
Wright MH (2004) The interior-point revolution in optimization: history, recent developments, and lasting consequences. Bull Am Math Soc 42(1):39–56. https://doi.org/10.1090/S0273-0979-04-01040-7
Xu S, Cai Y, Cheng G (2010) Volume preserving nonlinear density filter based on heaviside functions. Struct Multidiscip Optim 41(4):495–505. https://doi.org/10.1007/s00158-009-0452-7
Zalewski W, Allen E (1997) Shaping structures: statics. Wiley, New York, USA
Zegard T, Paulino GH (2014) GRAND – ground structure based topology optimization on arbitrary 2D domains using MATLAB. Struct Multidiscip Optim 50(5):861–882. https://doi.org/10.1007/s00158-014-1085-z
Zegard T, Paulino GH (2015) GRAND3 – ground structure based topology optimization on arbitrary 3D domains using MATLAB. Struct Multidiscip Optim 52(6):1161–1184
Zegard T, Paulino GH (2016) Bridging topology optimization and additive manufacturing. Struct Multidiscip Optim 53(1):175–192
Zhang XS, Paulino GH, Ramos AS Jr (2018) Multi-material topology optimization with multiple volume constraints: combining the ZPR update with a ground-structure algorithm to select a single material per overlapping set. Int J Numer Methods Eng 114(10):1053–1073. https://doi.org/10.1002/nme.5736
Zhou M, Rozvany GIN (1991) The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng 89(1–3):309–336. https://doi.org/10.1016/0045-7825(91)90046-9
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Responsible Editor: Ji-Hong Zhu
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Replication of results
The tools shown in the present manuscript were developed to satisfy the internal needs at Skidmore, Owings & Merril LLP and are thus proprietary. However, the methods used herein have been documented in previous publications, which are all referenced where appropriate. The replicability of the results, which applies to the method in question, is therefore left to their original contribution.
Rights and permissions
About this article
Cite this article
Zegard, T., Hartz, C., Mazurek, A. et al. Advancing building engineering through structural and topology optimization. Struct Multidisc Optim 62, 915–935 (2020). https://doi.org/10.1007/s00158-020-02506-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-020-02506-6