The History of the Topology Optimisation

The History of the Topology Optimisation

In the 16^th and 17^th century, in his book Discorsi e Dimonstrazioni Matematich, Galileo Galilei introduced the first concepts about optimal forms of structural elements. He started to investigate a brittle fracture process, where the bodies forms were designed considering local strengths (Figure 1). 

Figure 1 - Galileo Galilei optimal forms studies. (Galilei, 1638)

 Gottfried Wilhelm Leibniz´s (1646-1716) works introduced the basis of analytic procedure, and Leonard Euler´s (1707-1783) works, with the theory of extremes, could provide the basis for the calculus of variations development. Following Euler´s work contributions, Lagrange (1736-1813) and Hamilton (1805-1865) contributed in completing the variational calculus, which becomes the basis of several optimisations problems, once the theory of topology optimisation combines mechanics, variational calculus and mathematical programming (Johnsen, 2013).

Maxwell in 1870 (Maxwell, 1870), focused on civil engineering problems, proposed to design bridges with less material as possible using elasticity theory to find the ideal material distribution through principal stress field. In directions of principal stress. Since there is only normal stress, without shear, the optimal structure could be made of frame elements aligned with these stress directions. In 1904 Michell (Michell, 1904) continued Maxwell´ studies, to create optimal structures (Figure 2). In that time, Michell´ structures were considered very difficult to manufacture, and these become only a reference for academic studies. Currently, these structures can be used as an analytical benchmark for bi-dimensional topology optimisation problems when volume tends to zero (Rozvany, 1998).

 

Figure 2 - (a), (b) Michell frame bridge structures. (c) Michell optimal crank structure. (Michell, 1904)

  

Some 70 years later, Rozvany and his research group, published papers extending Michell´s theory to beam systems (Rozvany, 1972a, 1972b). Based on these papers, Prager and Rozvany formulated the first general theory of topology optimisation, termed “optimal layout theory” (Prager & Rozvany, 1977).

In 1988, Bendsøe and Kikuchi proposed the homogenization method (Bendsøe & Kikuchi, 1988) that is considered a landmark for TO. Since this paper, this field attracted wide industrial and academic interest due to its massive potential in engineering applications and its intrinsic mathematical challenges. Several developments were made and many different mathematical methods and practical approaches have been observed:

Density (Bendsøe, 1989; Mlejnek, 1992; Zhou & Rozvany, 1991) later renamed the base method to Solid Isotropic Microstructure with Penalization (Rozvany, Zhou, & Birker, 1992) and followed by Solid Isotropic Material with Penalization (SIMP) (Bendsøe & Sigmund, 2003). This method gained popularity and has received extensive research. Today SIMP is the standard approach method of the most of commercial TO software. One of the reasons for the success of this approach is the possibility of integration of manufacturing restrictions (Fiebig & Axmann, 2013).

The Soft Kill Option (SKO) method (Baumgartner, Harzheim, & Mattheck, 1992), in turn, is inspirited on biological growth rules of trees and bones, wherein highly stressed regions the material is addited and in lower stressed regions material is removed.

Evolutionary approaches are another prominent example of structural optimisation methods (Xie & Steven, 1993). The Evolutionary Structural Optimization (ESO) is focused on removing unnecessary material from too conservatively designed parts. For ESO, it is only possible to remove material. Querin introduced the Additive Evolutionary Structural Optimization method (AESO) (Querin, Steven, & Xie, 2000). AESO adds material to areas in order to improve the structure. The combination of ESO and AESO leads to the Bidirectional Evolutionary Structural Optimization (BESO) method. The main idea behind ESO, AESO and BESO is to remove lowly stressed elements and adding material to higher stressed regions (Fiebig & Axmann, 2013).

            Other approaches are topological derivative (Sokołowski & Zochowski, 1999), Level Set (Allaire, Jouve, & Toader, 2002, 2004; Wang, Wang, & Guo, 2003), Phase Field (Bourdin & Chambolle, 2003). Hybrid approaches have appeared, such as Level Set Method (LSM) can that uses Shape Derivatives for design updates or Iso-Geometric Analysis (IGA) (Qian, 2013). Also, Filtered Density Fields used in recent projection techniques (Sigmund & Maute, 2013). In special, the IGA is a recent emerging technology. This method uses B-Splines and Non-Uniform B-Splines (NURBS) to describe the geometry, more common in CAD approaches, allowing to eliminate the gap between CAD and FEM to define the geometry (CAD and FEM describe the same geometry differently). With IGA, the CAD geometry is precisely and efficiently represented (different from FEM mesh, that is an approximation, most based on Lagrange polynomial) (Figure 3), allowing to simplify the analysis with a better approximation of properties and accuracy of solution, integration of design in CAD system (is possible to use the same CAD models for structural analysis and optimization) and a faster refinement process, avoiding the long time demanded meshing the geometry at FEM. The combining of TO and IGA has the potential to generate an algorithm with faster convergence rate in comparison with the other hybrid approaches (Roodsarabi, Khatibinia, & Sarafrazi, 2016).     

Figure 3 - Representation of differences between FEM and IGA. In IGA, the geometry is more smooth and accurate. FONT: Terrific.

In recent years, with progressive advances and maturing of methods and mathematical approaches, and the accessibility of computer processing power, the TO has been increasingly introduced in the industry. Industries, including automotive, aerospace, heavy industry, energy, etc. Can now take advantage of the successful development and promotion of topology optimisation packages from FEA commercial software providers, within CAD/CAE frameworks such as Optistruct-Hyperworks (Altair), Ansys, Simula-Abaqus (Dassault), Simula-Tosca (Dassault), Fusion 360 (Autodesk), Genesis (VR&D), Inspire (Solidthinking), MSC Nastran (MSCsoftware), Generate-NX (Frustum-Siemens), TopShape, VirtualPyxis, ToOptix-Blender (open source), TopOpt (open source), Z88Arion (freeware) and several others (Topology Optimization Guide, n.d.), or as its incorporation in several other CAD software SolidEdge (Siemens), Inventor (Autodesk), Nastran (Autodesk), SolidWorks (Dassault), Catia (Dassault) and several others.

Bibliography:

Allaire, G., Jouve, F., & Toader, A. M. (2002). A level-set method for shape optimization. C R Math, 334(12):1125–1130. https://doi.org/10.1016 /S1631- 073X(02)02412-3

Allaire, G., Jouve, F., & Toader, A. M. (2004). Structural optimization using sensitivity analysis and a level-set method. J Comput Phys, 194(1):363–393.

Baumgartner, A., Harzheim, H., & Mattheck, C. (1992). SKO (soft kill option): the biological way to find an optimum structure topology. Int. Journey Fatigue.

Bendsøe, M. P. (1989). Optimal shape design as a material distribution problem. Struct Optim, 1:193–202.

Bendsøe, M. P., & Kikuchi, N. (1988). Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71, 197–224.

Bendsøe, M. P., & Sigmund, O. (2003). Topology optimization: theory, methods, and applications. Engineering (Vol. 2nd Editio). https://doi.org/10.1063/1.3278595

Bourdin, B., & Chambolle, A. (2003). Design-dependent loads in topology optimization. ESAIM Control Optim Calc Var, 9:19–48.

Fiebig, S., & Axmann, J. K. (2013). Using a binary material model for stress constraints and nonlinearities up to crash in topology optimization. 10th World Congress on Structural and Multidisciplinary Optimization, 2013.

Galilei, G. (1638). Discorsi e Dimonstrazioni Matematich, Intorno à Due Nuove Scienze Attenenti alla Mecanica & i Movimenti Locali. (ELSEVIER, Ed.). Leiden, Netherlands.

Johnsen, S. (2013). Structural Topology Optimization, (June). https://doi.org/10.1533/ijcr.2004.0288

Maxwell, J. C. (1870). On reciprocal figures, frames and diagrams of forces. Transactions of the Royal Society of Edinburgh, Vol. XXVI, 1–40.

Michell, A. G. M. (1904). The limits of economy of material in frame structures. Philisophical Magazine, 8(Series 6):589–597.

Mlejnek, H. P. (1992). Some aspects of the genesis of structures. Struct Optim, 5:64–69.

Prager, W., & Rozvany, G. I. N. (1977). Optimization of the structural geometry. In In: Bednarek AR, Cesari L (eds) Dynamical systems (p. pp 265–293). Gainsville, Florida.

Qian, X. (2013). Topology optimization in B-spline space. Computer Methods in Applied Mechanics and Engineering, 265 (2013) 15–35.

Querin, Q. M., Steven, G. P., & Xie, Y. M. (2000). Evolutionary structural optimisation using an additive algorithm. Finite Elements in Analysis and Design, 34: 291-308.

Roodsarabi, M., Khatibinia, M., & Sarafrazi, S. R. (2016). Isogeometric Topology Optimization of Structures Using Level Set Method Incorporating, 6(3), 405–422.

Rozvany, G. I. N. (1972a). Grillages of maximum strength and maximum stiffness. Int J Mech Sci, 14:1217–1222.

Rozvany, G. I. N. (1972b). Optimal load transmission by flexure. Methods Appl Mech Eng, 1:253–263.

Rozvany, G. I. N. (1998). Exact analytical solutions for some popular benchmark problems in topology optimization. Structural and Multidisciplinary Optimization.

Rozvany, G. I. N., Zhou, M., & Birker, T. (1992). Generalized shape optimization without homogenization. Structural and Multidisciplinary Optimization, 4, 250–254.

Sigmund, O., & Maute, K. (2013). Topology optimization approaches: A comparative review. Structural and Multidisciplinary Optimization, 48(6), 1031–1055. https://doi.org/10.1007/s00158-013-0978-6

Sokołowski, J., & Zochowski, A. (1999). On the topological derivative in shape optimization. SIAM J Control Opt, 37:1251–1272.

Topology Optimization Guide. (n.d.). Software list. Retrieved March 11, 2018, from http://www.topology-opt.com/software-list/

Wang, M., Wang, X., & Guo, D. (2003). level set method for structural topology optimization. Comput Methods Appl Mech Eng, 192(1– 2):227–246.

Xie, Y. M., & Steven, G. P. (1993). A simple evolutionary procedure for structural optimization. Comput Struct, 49:885–896.

Zhou, M., & Rozvany, G. I. N. (1991). The COC algorithm, part II: topological, geometry and generalized shape optimization. Methods Appl Mech Eng, 89(1–3):309–336.