Chapter 7 Conclusions and Extensions
7.2 Suggestions for future work
Though significant progress has been made towards the integration of topology optimization in the structural design industry, as described in this thesis, there are still several open topics that could provide further benefits for the modeling and optimization of high-rise buildings. The following problems are suggested as extensions of the work presented here.
Incorporation of shell finite elements
For realistic three-dimensional building structures, the framework presented here would ben- efit from including other types of elements to more accurately model the structural behavior, especially the combination of shell elements with three-dimensional beam elements to model large structural systems. Shell elements are more advantageous than the brick (B8) elements included here, as they have explicit terms (shape functions) to capture the moment and shear components necessary for the incorporation of the problem into current design codes, while B8 elements only account for nodal forces.
Other objective functions and loading scenarios
Though Chapter 4 discusses topology optimization for compliance, buckling and multi- objective problems, there is still a necessity to examine other objective functions. One of particular relevance to the high-rise design problem is the allowable stress level, especially if the stress limits of the material in tension and compression are not equal, as in the case of concrete or cable structures. Some analytical solutions for this type of problem are given for classical Michell truss structures in [150], but have not yet been incorporated into the topology optimization framework for high-rise buildings. Furthermore, using current AISC design codes [9], the stress level is also valuable in selecting an appropriate steel shape for the structural design. Other relevant objectives, including architectural criteria such as shading,
solar exposure, value of views, etc. might also be added for a more unified and global design; however, the quantification of these criteria and computation of their derivatives are not straightforward.
Throughout this thesis, for the studies conducted using the proposed framework for op- timization of compliance, buckling and multi-objective problems, the applied gravity and lateral (wind) loads were approximated as uniformly distributed loads. The examples might be further enhanced by applying non-uniform wind loads based on physical data from wind tunnel models or exploring other loading scenarios, such as applying several different loadings simultaneously or using the load cases designated by ASCE 7 [87].
Nonlinearities in topology optimization
Linear assumptions in topology optimization are not always valid or practical for structures subjected to realistic mechanical conditions [91]. For example, the buckling objective func- tion studied in Chapter 4 included only a linearized component; this work would benefit from accounting for the nonlinear behavior associated with large deflections present in high-rise buildings. Additionally, wind loads and gravity loads in this thesis were based on realistic constants. In the case where wind loads are modeled as a pressure along the exterior of a building, however, the resulting forces depend on the shape and material density distribution of the structure, which is unknown prior to the optimization. The gravity loads due to the self-weight of the structure are also dependent on the material distribution. This work would further benefit by incorporating the nonlinearities due to these loads, often referred to as design-dependent loads, into the high-rise optimization problem.
Integration with commercial software
Finally, a continuation of this work is to explore a more interactive framework between the structural engineering presented here and the architectural parameters both technical (need for stairs, elevators, mechanical openings, etc.) and aesthetic (design value of the structure). More specifically this work would benefit the structural engineering industry by extending its capabilities to integrate with architectural software, especially exporting the geometry of the results to AutoCADR or others. Additionally, a user friendly interface with input
parameters could be created to incorporate the structural engineering, architectural aspects and topology optimization problem for use in today’s design companies.
Appendix A
Graphic Statics Reciprocal Diagrams
The following table presents a summary of the reciprocity relationships between the form and force diagrams described in Chapter 6 for several optimal structures.
Form Diagram Force Diagram Dual Truss
I.1 A B C 1 2 F AB F BC F CA a 2 c b 1 a 2 c b 1 F A F B F C II.2 A B C 1 2 3 F AB F BC F CA a c b 1 2 3 A B C 1 2 III.3 A B C 1 2 3 4 5 7 8 9 10 11 F AB F BC F CA 6 a c b 1 2 3 4 5 6 7 8 9 10 11 a c b 1 2 3 4 5 6 7 8 9 10 11 F A F B F C
1I. Classical discrete Michell solution for the centrally-loaded beam [119, 80, 114] 2II. Variation on the classical discrete Michell centrally-loaded beam [80]
Form Diagram Force Diagram Dual Truss IV.4 A B C 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 F AB F BC F CA a c b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 a c b 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 F A F B F C V.5 A B C 1 2 3 4 5 6 7 8 9 10 11 12 F AB F BC F CA b c a 1 2 3 4 5 6 7 8 9 10 11 12 b c a 1 2 3 4 5 6 7 8 9 10 11 12 F A FB F C VI.6 F AB F BC F CA A B C 2 1 3 2 1 3 c a b 2 1 3 a b c F A F B F C VII.7 F AB B B C F CD F BC 9 10 D E F G F DE F EF F FG F GA a b c d e f g 10 9 FA FB a b e f g 10 9 c d FC FD FE FF FG
4IV. Self-reciprocal cantilever with two points of support [43, 114] 5V. Optimal shear bracing solution for high-rise problem [80, 175] 6VI. Two dual minimal weight trusses given in [137]
Bibliography
[1] Altair Engineering Web Page, 2013.
[2] W Achtziger, M P Bendsoe, A Ben-Tal, and J Zowe. Equivalent displacement based formulations for maximum strength truss topology design. IMPACT of Computing in Science and Engineering, 4(4):315–345, December 1992.
[3] N Adams, K Frampton, and T van Leeuwen. SOM Journal 7. Cantz, H, 2012.
[4] S Allahdadian and B Boroomand. Design and Retrofitting of Structures under Tran- sient Dynamic Loads by a Topology Optimization Scheme. In Proc 3rd Int Conf Seism Retrofit, number October, pages 1–9, 2010.
[5] S R M Almeida, G H Paulino, and E C N Silva. A simple and effective inverse projection scheme for void distribution control in topology optimization. Struct Multidisc Optim, 39(4):359–371, 2009.
[6] S R M Almeida, G H Paulino, and E C N Silva. Layout and material gradation in topology optimization of functionally graded structures: a global-local approach. Struct Multidisc Optim, 42(6):855–868, 2010.
[7] L Ambrosio and G Buttazzo. An optimal design problem with perimeter penalization. Calc Var Partial Differ Equ, 1(1):55–69, 1993.
[8] AMD. AMD Core Math Library (ACML), 2011.
[9] Inc American Institute of Steel Construction. AISC Steel Construction Manual. 13th edn. edition, 2005.
[10] O Amir, O Sigmund, B S Lazarov, and M Schevenels. Efficient reanalysis techniques for robust topology optimization. Comput Meth Appl Mech Eng, 245-246:217–231, 2012. [11] O Amir, M Stolpe, and O Sigmund. Efficient use of iterative solvers in nested topology
optimization. Struct Multidisc Optim, 42(1):55–72, 2009.
[12] J S Arora. Introduction to Optimum Design. Elsevier Inc., Oxford, 2012.
[13] W F Baker. Energy-Based Design of Lateral Systems. Struct Eng Int, 2:99–102, 1992. [14] W F Baker. Structural Innovation: Combining Classic Theories with New Technologies.
[15] W F Baker, L L Beghini, A Mazurek, J Carrion, and A Beghini. Maxwell’s Reciprocal Diagrams and Discrete Michell Frames. Struct Multidisc Optim, 2013.
[16] S Balay, J Brown, K Buschelman, V Eijkhout, W D Gropp, D Kaushik, M Knepley, L C McInnes, B F Smith, and H Zhang. PETSc Users Manual. Technical Report ANL-95/11 - Revision 3.2. Technical report, Argonne National Laboratory, 1997. [17] S Balay, K Buschelman, W D Gropp, D Kaushik, L C McInnes, and B F Smith. PETSc
Home Page, 2011.
[18] S Balay, W D Gropp, and L C McInnes. Efficient Management of Parallelism in Object-Oriented Numerical Software Libraries Programming. In E Arge, AM Bruaset, and HP Langtangen, editors, Mod Softw Tools Sci Comput, pages 163–202. Birkhauser Press, 1997.
[19] W Bangerth, C Burstedde, T Heister, and M Kronbichler. Algorithms and data struc- tures for massively parallel generic adaptive finite element codes. ACM Trans Math Softw, 38(2):1–28, 2011.
[20] Barcelona-Attractions.net. Sagrada Familia by Antonio Gaudi Barcelona, 2011. [21] M W Beall and M S Shephard. A general topology-based mesh data structure. Int J
Numer Meth Eng, 40(9):1573–1596, 1997.
[22] A Beghini, L L Beghini, J Carrion, A Mazurek, and W F Baker. Rankine’s Theorem for the Design of Cable Structures. Struct Multidisc Optim: under revision, 2013. [23] L L Beghini, J Carrion, A Beghini, A Mazurek, and W F Baker. Structural Optimiza-
tion Using Graphic Statics. Struct Multidisc Optim: Submitted, 2013.
[24] A Ben-Tal and M P Bendsoe. A new method for optimal truss topology design. SIAM Journal on Optimization, 3(2):322–358, 1993.
[25] A Ben-Tal and A Nemirovski. Robust optimization - methodology and applications. Math Program, 92(3):453, 2002.
[26] M P Bendsoe. Optimal shape design as a material distribution problem. Struct Optim, 1(4):193–202, 1989.
[27] M P Bendsoe and N Kikuchi. Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Engng, 71(2):197–224, 1988. [28] M P Bendsoe and O Sigmund. Material interpolation schemes in topology optimization.
Arch Appl Mech, 69(9-10):635–654, 1999.
[29] M P Bendsoe and O Sigmund. Topology Optimization: Theory, Methods and Applica- tions. Springer, 2002.
[30] P Block. Thrust Network Analysis: Exploring Three-dimensional equilibrium. PhD thesis, Massachusetts Institute of Technology, 2009.
[31] P Block, M DeJong, and J Ochsendorf. As Hangs the Flexible Line: Equilibrium of Masonry Arches. Nexus Network Journal, 8(2):13–24, 2006.
[32] P Block and J Ochsendorf. Thrust Network Analysis: A new methodology for three- dimensional equilibrium. J Int Assoc Shell Spatial Struct, 48(3):1–8, 2007.
[33] P Block and J Ochsendorf. Lower-bound analysis of masonry vaults. In Conf Struct Anal Hist Constr Preserv Saf Signif, pages 593–600, 2008.
[34] T Borrvall and J Petersson. Large-scale topology optimization in 3D using parallel computing. Comput Meth Appl Mech Eng, 190(46-47):6201–6229, 2001.
[35] T Borrvall and J Petersson. Topology optimization using regularized intermediate density control. Comput Meth Appl Mech Eng, 190(37-38):4911–4928, 2001.
[36] B Bourdin. Filters in topology optimization. Int J Numer Meth Eng, 50(9):2143–2158, 2001.
[37] R H Bow. Economics of construction in relation to framed structures. 1873.
[38] C Burstedde and L C Wilcox. p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM J Sci Comput, 33(3):1103–1133, 2011.
[39] J Cagan, K Shimada, and S Yin. A survey of computational approaches to three- dimensional layout problems. Comput Aided Design, 34(8):597–611, 2002.
[40] R C Carbonari, E C N Silva, and G H Paulino. Multi-actuated functionally graded piezoelectric micro-tools design: A multiphysics topology optimization approach. Int J Numer Meth Eng, 77(3):301–336, 2009.
[41] W Celes, G H Paulino, and R Espinha. A compact adjacency-based topological data structure for finite element mesh representation. Int J Numer Meth Eng, 64(11):1529– 1556, 2005.
[42] W Celes, G H Paulino, and R Espinha. Efficient Handling of Implicit Entities in Reduced Mesh Representations. J Comput Inf Sci Eng, 5(4):348–359, 2005.
[43] A S L Chan. The design of Michell optimum structures. Minist Aviat Aeronaut Res Counc Rep, (3303), 1960.
[44] P W Christensen and A Klarbring. An introduction to structural optimization. Springer, 2008.
[45] J Comba, N Max, J S B Mitchell, and P L Williams. Fast Polyhedral Cell Sorting for Interactive Rendering of Unstructured Grids. Eurographics, 18(3), 1999.
[46] R D Cook, D S Malkus, M E Plesha, and R J Witt. Concepts and Applications of Finite Element Analysis, 4th Edition. Wiley, 2001.
[47] H L Cox. The design of structures of least weight. Pergamon, 1965.
[48] L Cremona. Le figure reciproche nella grafica statica. Tipografia Giuseppe Bernardoni, Milano, 1872.
[49] L Cremona. Graphical statics: two treatises on the graphical calculus and reciprocal figures in graphical statics. Clarendon Press, 1890.
[50] K Culmann. Die graphische Statik. 1864.
[51] I Das and J E Dennis. A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct Optim, 14(1):63–69, 1997.
[52] T A Davis. A column pre-ordering strategy for the unsymmetric-pattern multifrontal method. ACM Trans Math Softw, 30(2):165–195, 2004.
[53] T A Davis. Unsymmetric-Pattern Multifrontal Method. ACM Trans Math Softw, 30(2):196–199, 2004.
[54] T A Davis and I S Duff. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J Matrix Anal Appl, 18(1):140–158, 1997.
[55] T A Davis and I S Duff. A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans Math Softw, 25(1):1–20, 1999.
[56] E De Sturler, G H Paulino, and S Wang. Topology optimization with adaptive mesh refinement. In Proc 6th Int Conf Comput Shell Spatial Struct, number May, pages 1–4, 2008.
[57] A R Diaz and N Kikuchi. Solutions to shape and topology eigenvalue optimization problems using a homogenization method. Int J Numer Meth Eng, 35(7):1487–1502, 1992.
[58] W Dorn, R Gomory, and M Greenberg. Automatic design of optimal structures. J. de Mechanique, 3:25–52, 1964.
[59] Y Dubois-Pèlerin and T Zimmermann. Object-oriented finite element programming: III. An efficient implementation in C++. Comput Meth Appl Mech Eng, 108(1-2):165– 183, 1993.
[60] Y Dubois-Pèlerin, T Zimmermann, and P Bomme. Object-oriented finite element programming: II. A prototype program in smalltalk. Comput Meth Appl Mech Eng, 98(3):361–397, 1992.
[61] R Espinha, W Celes, N Rodriguez, and G H Paulino. ParTopS: compact topological framework for parallel fragmentation simulations. Eng Comput, 25(4):345–365, 2009. [62] D Eyheramendy. Object-oriented finite elements II. A symbolic environment for auto-
matic programming. Comput Meth Appl Mech Eng, 132(3-4):277–304, 1996.
[63] D Eyheramendy. Object-oriented finite elements. IV. Symbolic derivations and auto- matic programming of nonlinear formulations. Comput Meth Appl Mech Eng, 190(22- 23):2729–2751, 2001.
[64] D Eyheramendy and T Zimmermann. Object-oriented finite elements III. Theory and application of automatic programming. Comput Meth Appl Mech Eng, 7825(97), 1998. [65] J S R A Filho and P R B Devloo. Object oriented programming in scientific compu-
[66] P Fleron. The minimum weight of trusses. Bygningsstatiske Meddelelser, 35:81–96, 1964.
[67] J Folgado and H C Rodrigues. Structural optimization with a non-smooth buckling load criterion. Control Cyber, 27(2):235–253, 1998.
[68] M Fournie, N Renon, Y Renard, and D Ruiz. CFD parallel simulation using Getfem++ and Mumps. Euro-Par 2010, pages 77–88, 2010.
[69] H Fredricson. Topology optimization of frame structures - joint penalty and material selection. Struct Multidisc Optim, 30(3):193–200, 2005.
[70] R V Garimella. Mesh data structure selection for mesh generation and FEA applica- tions. Int J Numer Meth Eng, 55(4):451–478, 2002.
[71] K Gerfen. 2009 R+D Awards: Oasis Generator. Archit Mag, pages 54–55, 2009. [72] A R Gersborg and C S Andreasen. An explicit parameterization for casting constraints
in gradient driven topology optimization. Struct Multidisc Optim, 44:875–881, 2011. [73] P Geyer. Component-oriented decomposition for multidisciplinary design optimization
in building design. Adv Eng Inform, 23(1):12–31, 2009.
[74] J K Guest. Imposing maximum length scale in topology optimization. Struct Multidisc Optim, 37(5):463–473, 2009.
[75] J K Guest, J H Prévost, and T Belytschko. Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Meth Eng, 61(2):238–254, 2004.
[76] R B Haber, C S Jog, and M P Bendsoe. A new approach to variable-topology shape design using a constraint on perimeter. Struct Optim, 11(1):1–12, 1996.
[77] R T Haftka and Z Gürdal. Elements of Structural Optimization (Solid Mechanics and Its Applications). Springer, 1991.
[78] S Hansen and G N Vanderplaats. An approximation method for configuration opti- mization of trusses. AIAA J, 28(1):161–168, 1988.
[79] J Haslinger and R A E Mäkinen. Introduction to shape optimization: theory, approx- imation, and computation, 2003.
[80] W S Hemp. Optimum Structures. Clarendon Press, Oxford, 1973.
[81] R Hill. The Mathematical Theory of Plasticity. Oxford University Press Inc., New York, 1950.
[82] B F Hobbs. A comparison of weighting methods in power plant siting. Decis Sci, 11(4):725–737, 1980.
[83] C Huang, X Han, C Wang, J Ji, and W Li. Parametric analysis and simplified cal- culating method for diagonal grid structural system. Jianzhu Jiegou Xuebao/J Build Struct, 31(1):70–77, 2010.
[84] X Huang and M Xie. Evolutionary Topology Optimization of Continuum Structures: Methods and Applications. Wiley, 2010.
[85] X Huang and Y M Xie. Optimal design of periodic structures using evolutionary topology optimization. Struct Multidisc Optim, 36(6):597–606, 2007.
[86] X Huang and Y M Xie. Topology optimization of nonlinear structures under displace- ment loading. Eng Struct, 30(7):2057–2068, 2008.
[87] Structural Engineering Institute. ASCE/SEI 7-10 Minimum Design Loads for Build- ings and Other Structures. American Society of Civil Engineers, 2011, 2010.
[88] K Ishii and S Aomura. Topology optimization for the extruded three dimensional structure with constant cross section. JSME Int J, 47(2):198–206, 2004.
[89] J Jonsmann, O Sigmund, and S Bouwstra. Compliant thermal microactuators. Sens Actuators A: Phys, 76(1-3):463–469, 1999.
[90] A Kaveh and M Shahrouzi. Graph Theoretical Topology Control in Structural Opti- mization of Frames with Bracing Systems. Sci Iran, 16(2):173–187, 2009.
[91] R Kemmler, A Lipka, and E Ramm. Large deformations and stability in topology optimization. Struct Multidisc Optim, 30(6):459–476, 2005.
[92] Y S Khan. Engineering Architecture: The Vision of Fazlur R. Khan. W. W. Norton & Company, New York, 2004.
[93] R V Kohn and G Strang. Optimal-design and relaxation of variational problems I. Commun Pure Appl Math, 39(1):113–137, 1986.
[94] R V Kohn and G Strang. Optimal-design and relaxation of variational problems II. Commun Pure Appl Math, 39(2):139–182, 1986.
[95] R V Kohn and G Strang. Optimal-design and relaxation of variational problems III. Commun Pure Appl Math, 39(3):353–377, 1986.
[96] I Kosaka and C C Swan. A symmetry reduction method for continuum structural topology optimization. Comput Struct, 70(1):47–61, 1999.
[97] L Krog, A Tucker, M Kemp, and R Boyd. Topology optimization of aircraft wing box ribs. In 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, pages 2004–4481, 2004.
[98] T Kunakote and S Bureerat. Multi-objective topology optimization using evolutionary algorithms. Eng Opt, 43(5):541–557, 2011.
[99] N Lagaros, L Psarras, M Papadrakakis, and G Panagiotou. Optimum design of steel structures with web openings. Eng Struct, 30(9):2528–2537, 2008.
[100] O S Lawlor, S Chakravorty, T L Wilmarth, N Choudhury, I Dooley, G Zheng, and L V Kalé. Parfum: A parallel framework for unstructured meshes for scalable dynamic physics applications. Eng Comput, 22(3):215–235, 2006.
[101] W J LeMessurier. A Practical Method of Second Order Analysis: Part I Pin-Jointed Systems. Eng J, AISC, 13(4):89–96, 1976.
[102] W J LeMessurier. A Practical Method of Second Order Analysis: Part II Rigid Frames. Eng J, AISC, 14(2):49–67, 1977.
[103] S E Leon, G H Paulino, A Pereira, I F M Menezes, and E N Lages. A Unified Library of Nonlinear Solution Schemes. Appl Mech Rev, 64(4), 2011.
[104] M Levy. La Statique Graphique Et Ses Applications Aux Constructions. Gauthier- Villars, Paris, 1888.
[105] Q Q Liang. Effects of continuum design domains on optimal bracing systems for mul- tistory steel building frameworks. In Proc 5th Australas Congr Appl Mech, volume 2, pages 794–799. Engineers Australia, 2007.
[106] Q Q Liang, Y M Xie, and G P Steven. Optimal topology design of bracing systems for multi-story steel frames. ASCE J Struct Eng, 126(7):823–829, 2000.
[107] J Lubliner. Plasticity theory. Macmillan Publishing Company, New York, 1990. [108] J Mackerle. Object-oriented programming in FEM and BEM: a bibliography (1990 -
2003). Adv Eng Softw, 35(6):325–336, 2004.
[109] R T Marler and J S Arora. Survey of multi-objective optimization methods for engi- neering. Struct Multidisc Optim, 26(6):369–395, 2004.
[110] LF Martha and E Parente Jr. An object-oriented framework for finite element pro- gramming. In Proceed Fifth World Congr Comput Mech, IACM, Vienna, Austria, 2002.
[111] K Martini. Harmony Search Method for Multimodal Size, Shape, and Topology Opti- mization of Structural Frameworks. J Struct Eng, 137(11):1332–1339, 2011.
[112] J C Maxwell. On Reciprocal Figures and Diagrams of Forces. Phil Mag, 26:250–261, 1864.
[113] J C Maxwell. On Reciprocal Figures, Frames, and Diagrams of Forces... Edinb Roy Soc Proc, 7:160–208, 1870.
[114] A Mazurek, W F Baker, and C Tort. Geometrical aspects of optimum truss like structures. Struct Multidisc Optim, 43(2):231–242, 2011.
[115] F McKenna. Object-Oriented Finite Element Programming: Frameworks for Analysis, Algorithms and Parallel Computing. PhD thesis, University of California, Berkeley, 1997.
[116] F McKenna. OpenSees: A Framework for Earthquake Engineering Simulation. Comput Sci Eng, 13(4):58–66, 2011.
[117] F McKenna and G L Fenves. An object-oriented software design for parallel structural analysis. In ASCE Adv Tech Struct Eng, pages 1–8. Structural Engineering Institute, ASCE, 2000.
[118] F McKenna, M H Scott, and G L Fenves. Nonlinear finite-element analysis software architecture using object composition. J Comput Civil Eng, 24(February):95, 2010. [119] A G M Michell. The Limits of Economy of Material in Frame-Structures. Phil Mag,
8(47):589–597, 1904.
[120] A R Mijar, C C Swan, J S Arora, and I Kosaka. Continuum topology optimization for concept design of frame bracing systems. ASCE J Struct Eng, 124(5):541–550, 1998. [121] S Min and N Kikuchi. Optimal reinforcement design of structures under the buckling
load using the homogenization design method. Struct Eng Mech, 5(5):565–576, 1997. [122] M M Neves, H C Rodrigues, and J M Guedes. Generalized topology design of structures
with a buckling load criterion. Struct Multidisc Optim, 10(2):71–78, 1995.
[123] M M Neves, O Sigmund, and M P Bendsoe. Topology optimization of periodic mi- crostructures with a penalization of highly localized buckling modes. Int J Numer Meth Eng, 54(6):809–834, 2002.
[124] T H Nguyen, G H Paulino, J Song, and C H Le. A computational paradigm for multiresolution topology optimization (MTOP). Struct Multidisc Optim, 41(4):525– 539, 2009.
[125] T H Nguyen, G H Paulino, J Song, and C H Le. Improving multiresolution topology optimization via multiple discretizations. Int J Numer Meth Eng, 92:507–530, 2012. [126] J H Nie, D A Hopkins, Y T Chen, and H T Hsieh. Development of an object-oriented
finite element program with adaptive mesh refinement for multi-physics applications. Adv Eng Softw, 41(4):569–579, 2010.
[127] B Niu, J Yan, and G D Cheng. Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Struct Multidisc Optim, 39(2):115–132, 2008.
[128] J M Oberndorfer, W Achtziger, and H Hörnlein. Two approaches for truss topology optimization: a comparison for practical use. Struct Multidisc Optim, 11(3):137–144, 1996.
[129] N Olhoff and J Du. On Topological Design Optimization of Vibrating Structures, 2012. [130] N Olhoff and S H Rasmussen. On bimodal optimum loads of clamped columns. Int J
Solids Struct, 13:605–614, 1977.
[131] V Pareto. Manuale di Economia Politica Societa Editrice Libraria. Milan, 1906. [132] G H Paulino, W Celes, R Espinha, and Z Zhang. A general topology-based framework
for adaptive insertion of cohesive elements in finite element meshes. Eng Comput, 24(1):59–78, 2007.
[133] G H Paulino, I F M Menezes, J B Cavalcante Neto, and LF Martha. A methodology for adaptive finite element analysis: towards an integrated computational environment. Comput Mech, 23(5):361–388, 1999.
[134] G H Paulino, K Park, W Celes, and R Espinha. Adaptive dynamic cohesive fracture simulation using nodal perturbation and edge-swap operators. Int J Numer Meth Eng, 84(11):1303–1343, 2010.
[135] N L Pedersen. Maximization of eigenvalues using topology optimization. Struct Mul- tidisc Optim, 20(1):2–11, 2000.
[136] J Petersson and O Sigmund. Slope constrained topology optimization. Int J Numer Meth Eng, 41(8):1417–1434, 1998.
[137] W Prager. Optimization of structural design. J Optim Theory Appl, 6(I), 1970. [138] W Prager. Optimal layout of cantilever trusses. J Optim Theory Appl, 23(1):111–117,
1977.
[139] W Prager. Nearly optimal design of trusses. Computers and Structures, 8:451–454, 1978.
[140] W Prager. Optimal layout of trusses with finite number of joints. J Mech Phys Solids, 26(4):241–250, 1978.
[141] H Rahami, A Kaveh, and Y Gholipour. Sizing, geometry and topology optimization of trusses via force method and genetic algorithm. Eng Struct, 30(9):2360–2369, 2008. [142] S F Rahmatalla. Continuum topology design of sparse structures and compliant mech-
anisms. PhD thesis, 2004.
[143] S F Rahmatalla and C C Swan. Form Finding of Sparse Structures with Continuum Topology Optimization. ASCE J Struct Eng, 129(12):1707, 2003.
[144] W J M Rankine. XVII. Principle of the equilibrium of polyhedral frames. Phil Mag, Series 4,:92, 1864.
[145] J F Remacle, B K Karamete, and M S Shephard. An algorithm oriented mesh database. Int J Numer Meth Eng, 58(2):349–374, 2003.
[146] J F Remacle, O Klaas, J E Flaherty, and M S Shephard. Parallel Algorithm Oriented Mesh Database. Eng Comput, 18(3):274–284, 2002.
[147] Y Renard and J Pommier. GetFEM++ Homepage, 2010.
[148] G I N Rozvany. Optimal Design of Flexural Systems. Pergamon Press, Oxford, 1976. [149] G I N Rozvany. Structural design via optimality criteria. Kluwer Academic Publishers
Group, Dordrecht. Boston. London., 1989.
[150] G I N Rozvany. Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Multidisc Optim, 11:213–217, 1996.
[151] G I N Rozvany. Aims, scope, methods, history and unified terminology of computer- aided topology optimization in structural mechanics. Struct Multidisc Optim, pages 90–108, 2001.
[152] G I N Rozvany. A critical review of established methods of structural topology opti- mization. Struct Multidisc Optim, 37(3):217–237, 2009.
[153] G I N Rozvany, Bendsoe M P, and U Kirsch. Layout optimization of structures. Appl Mech Rev, 48(2):41–119, 1995.
[154] G I N Rozvany, M Zhou, and T Birker. Generalized shape optimization without