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# Journal Club Theme of January 2015: Topology Optimization for Materials Design

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Processing technologies are rapidly advancing and manufacturers now have the ability to control material architecture, or topology, at unprecedented length scales. This opens up the design space and provides exciting opportunities for tailoring material properties through design of the material’s topology. But as seen many times in history with advancements in materials and processing technologies, the natural default is to rely on familiar shapes and structure topologies. A new manufacturing process perhaps lets us create an existing shape at lower cost; increases in base material strength allows us to thin member sizes and reduce material usage in an existing design. But what is missed in defaulting to the familiar is that new processes may enable realization of previously unachievable topologies, or that thinning members to leverage higher base material strengths may ultimately lead to issues of stability that can be addressed more efficiently with new topologies. Technological advancements may thus not be fully leveraged unless we are willing to re-think design - similar evidence can be found in the evolution of structural forms [1].

Topology optimization offers an alternative to ad hoc, intuition, and experience-driven design approaches, providing a systematic, rigorous framework for exploring the design space. The goal, put simply, is to identify whether or not material exists at each point in space within the design domain. For periodic materials, the design domain is the characteristic unit cell and one uses some form of upscaling to estimate the effective properties of the bulk material, which are the properties to be optimized and/or constrained. This upscaling is performed numerically, typically with finite elements, and the goal of determining material concentration at “every point in space” then becomes determining material concentration in every finite element. The connectivity of the elements containing material then ends up defining the topology. In this sense, one can think of topology optimization as the process of building the material unit cell from scratch out of white and black Legos, where the white bricks indicate voids and the black bricks indicate solid material. Computationally, we can change the color of any brick, from white to black or black to white, evolving topology as we do so. These “Legos” may also come in an array of colors (as in design of composite unit cells containing multiple materials) and need not be prismatic, but can be any shape or combination of shapes and sizes (as in unstructured meshes).

The capability to alter the connectivity is what differentiates topology optimization from shape and sizing optimization. Consider, for example, materials that can be characterized as micro-truss lattices, which have garnering significant interest [2-5]. For a given truss topology (pyramidal, tetrahedral, octet), one can optimize the size of the truss struts using sizing optimization, optimize the angle of struts with geometric optimization, or even add strut tapering with shape optimization. The goal of topology optimization, on all the other hand, is to consider all of these, in addition to the actual connectivity of the truss system. This of course adds computational expense, but also significant design freedom.

To fully leverage this freedom, the design problem is posed and solved as a formal optimization problem. The objective function is some scalar metric of material performance (effective modulus, effective thermal conduction, mass, etc.) and the constraints can include bounds on such metrics. Additionally, the governing mechanics are embedded in this optimization formulation - both the mechanics governing the unit cell behavior and the upscaling of these mechanics to estimate the material’s effective properties.

The resulting optimization problem is discrete - what material is located in each finite element - and one can imagine if we are allowing topological evolution and fine scale features we need a high dimension design space (lots and lots of design variables). Traditionally, the number of design variables scales with the number of finite elements (although this need not always be the case), and so hundreds of thousands of design variables is quite typical. Random search algorithms tend to fail in such high dimensions, giving strong preference for gradient-based optimizers. The Method of Moving Asymptotes, for example, is the most widely-used in the field [6]. This requires that the discrete condition on the design variables be relaxed (we must allow mixtures of materials in each element (grey Lego bricks)), we must approximate the constitutive behavior of such mixtures (interpolate the mechanics), and further must mathematically make discrete black-white solutions ‘more desirable’ than mixtures. The SIMP method, essentially a power-law approach, is the most popular approach for achieving this [7,8]. Gradient-based optimizers also require gradients (of course!), or a sensitivity analysis, which are typically estimated using adjoint methods or direct differentiation (see e.g., [9,10]). To add further challenge, optimization has the wonderful trait of revealing deficiencies in your problem formulation or underlying numerical instabilities [11], although we now have a good handle on many of these.

So, to summarize, we discretize the domain serving as the material's unit cell and construct the design variables to indicate each element’s composition. We perform an upscaling of the resulting unit cell design to estimate effective material properties, and perform a sensitivity analysis to indicate how each design variable will influence these effective properties. The output of these analyses are used by the gradient-based optimizer to change the design and evolve the topology. Figure 1 illustrates the general framework and illustrates that the design may evolve from an uneducated, ‘blurry’ guess to a clear representation of topology (note we’ve traded in greyscale Legos for color).

**Examples in Literature**

The pioneering work of Ole Sigmund at Technical University of Denmark was the first to apply the topology optimization strategy to materials design. Sigmund discretized the design domain with truss and frame elements and optimized the connectivity of these elements (essentially building the unit cell out of K’nex, rather than Legos). He successfully designed materials with tailored elastic properties, including material topologies with negative Poisson’s ratio [12-14]. These ideas were later extended to continuum representations of material topology, where significantly more design freedom is introduced. This included porous materials with negative Poisson’s ratio ([12], and later fabricated and tested at -0.8 in [15]) and optimized elastic moduli [16]. It should be mentioned that elastic properties have also been optimized for composite materials [17] and functionally graded materials [18].

Although its roots are in solid mechanics, topology optimization has undergone rapid expansion to consider other fields over the past decade. Material design examples include three-phase materials with negative thermal expansion [19], optimizing electric and thermal conduction of composites [20], fluid permeability of porous materials [21], and various combinations as may be needed in multifunctional materials, such as elastic moduli and thermal conduction [22,23] and elastic moduli and fluid permeability [24]. Various combinations of the latter have been manufactured and proposed for bone scaffolds [25-27]. A few additional examples include the design of electromagnetic metamaterials with negative permeability [28,29] and phononic materials [30,31]; a recent review of materials design with topology optimization can be found in [32].

It is important to also note these topology optimization problems are nonconvex, making it impossible to guarantee identification of a global minimum. There are, however, many examples of solutions being shown (computationally) to achieve theoretical bounds, such Hashin-Shtikman bounds for elastic moduli (e.g., [12,16,33]) and Gibiansky-Torquato stiffness-conductivity bounds in [22], giving confidence in the optimality of solutions. Interestingly, this design tool can also be used to influence the development of theoretical or estimated bounds. In [19], Sigmund and Torquato tell an interesting story that topology optimization was unable to identify solutions near the Schapery-Rosen-Hashin bounds on effective thermal strain coefficient for three-phase materials, ultimately motivating further study and development of new, tighter bounds [34]. Topology optimization was also recently used to probe the relationship between elastic stiffness and permeability in an attempt to estimate the upper bounds of this cross-property space [33].

**Manufacturability**

The primary disadvantage of the design freedom inherent in topology optimization is that solutions may be complex structures that are quite challenging, or even impossible, to manufacture. This has been one of the major obstacles in bringing topology optimization into practice, and has garnered significant attention by topology optimization researchers. In the last ~10 years, various algorithms have been developed for controlling the minimum and maximum length scale of designed features in materials [35-39] so as to match manufacturing resolution capabilities, for handling size and shape restrictions in inclusion-based materials [40,41], and for materials manufactured by 3D weaving processes [42]. Algorithms have also been developed for other manufacturing processes (machining, casting, etc), but thus far have not been extended to 3D periodic material domains. It must be noted that while 3D printing has greatly opened up the space of what is realizable, manufacturing constraints will be omnipresent and require treatment within the optimization formulation if solutions are to be optimized for as-built conditions.

**Future Challenges **

As the fundamental need for topology optimization is computationally efficient upscaling of unit cell mechanics, it is not surprising that the above-mentioned material design works assume deterministic conditions and domains governed by linear continuum mechanics, enabling implementation of (for example) elastic homogenization [43,44]. However, topology optimization methods and algorithms for ‘structure’ design (where design is conducted at the same scale that performance is defined) are rapidly advancing, including algorithms for nonlinear mechanics [45-48] and designing robust structures in the presence of material and geometric uncertainties [49-55]. Some of these ideas have been implemented in the material domains recently, such as design for negative Poisson’s ratio under geometric uncertainties in the form of thinning or thickening of unit cell features [56] and optimizing energy absorption metrics in cellular materials [57], and it is fully expected that this trend will continue as computational methods for upscaling under these conditions advance. Driving design at even smaller scales, beyond continuum scale ‘architecture’, such as in [58], is another challenge that surely will be drawing interest in the future, particularly as our understanding of and control over processing continues to grow.

In summary, topology optimization is a systematic tool for guiding distribution of base materials within a design domain, such as the unit cell of a periodic material. Its power comes from the fact that (1) design decisions are driven by mathematical programming, rather than ad hoc rules, and (2) that design evolves in a ‘free-form’ manner, where structural connectivity (topology) is free to change during the design evolution. These properties enable exploration into new regions of the design space and, potentially, enable the discovery of new unit cell topologies offering previously unrealizable combinations of effective material properties.

For those that want to learn more, in addition to the above referenced works, you might consider a recent broad survey of applications and methods [59], an in-depth discussion of the properties of topology optimization methods and algorithms [60], and some basic MATLAB code [61] that is excellent to play with.

**References**

1. Billington D.P. (1983). The Tower and The Bridge. Princeton University Press, Princeton, NJ.

2. Fleck N.A., Deshpande V.S., and Ashby M.F. (2011). Micro-architectured materials: past, present and future. Proceedings of the Royal Society A 466: 2495–2516.

3. Wadley H.N.G. (2006). Multifunctional periodic cellular metals. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 364: 31-68.

4. Schaedler T.A., Jacobsen A.J., Torrents A., Sorensen A.E., Lian J., Greer J.R., Valdevit L., and Carter W. B. (2011). Ultralight metallic microlattices. Science 334: 962–965.

5. Torrents A., Schaedler T.A., Jacobsen A.J., Carter W.B., and Valdevit L. (2012). Characterization of nickel-based microlattice materials with structural hierarchy from the nanometer to the millimeter scale. Acta Materialia 60: 3511–3523.

6. Svanberg K. (1987). The method of moving asymptotes - a new method for structural optimization. International Journal for Numerical Methods in Engineering 24: 359-373.

7. Bendsøe M.P. (1989) Optimal shape design as a material distribution problem. Structural Optimization 1: 193– 202.

8. Zhou M., Rozvany G.I.N. (1991) The COC algorithm, part II: Topological, geometry and generalized shape optimization. Computer Methods in Applied Mechanics and Engineering 89: 309–336.

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10. Jameson A. (2001). A perspective on computational algorithms for aerodynamic analysis and design. Progress in Aerospace Sciences 37: 197-243.

11. Sigmund O., and Petersson J. (1998). Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Structural Optimization 16: 68–75.

12. Sigmund O. (1994). Design of material structures using topology optimization. Ph.D. Thesis, Department of Solid Mechanics, Technical University of Denmark.

13. Sigmund O. (1994). Materials with prescribed constitutive parameters: an inverse homogenization problem. International Journal of Solids and Structures 31: 2313–2329.

14. Sigmund O. (1995). Tailoring materials with prescribed elastic properties. Mechanics of Materials 20: 351-368.

15. Larsen U. D., Sigmund O., and Bouwstra S. (1997). Design and fabrication of compliant mechanisms and material structures with negative Poissons ratio. Journal of MicroElectroMechanical Systems 6: 99-106.

16. Sigmund O. (2000). A new class of extremal composites. Journal of the Mechanics and Physics of Solids 48: 397–428.

17. Gibiansky L.V., and Sigmund O. (2000). Multiphase composites with extremal bulk modulus. Journal of the Mechanics and Physics of Solids 48: 461–498.

18. Paulino G.H., Silva E.C.N., and Le C.H. (2009). Optimal design of periodic functionally graded composites with prescribed properties. Structural and Multidisciplinary Optimization 38: 469-489.

19. Sigmund O., and Torquato S. (1997). Design of materials with extreme thermal expansion using a three-phase topology optimization method. Journal of the Mechanics and Physics of Solids 45(6): 1037–1067.

20. Torquato S., Hyun S., and Donev A. (2002). Multifunctional composites: optimizing microstructures for simultaneous transport of heat and electricity. Physical Review Letters 89: 266601-1–266601-4.

21. Guest J.K., and Prévost J.H. (2007). Design of maximum permeability material structures. Computer Methods in Applied Mechanics and Engineering 196: 1006–1017.

22. Challis V.J., Roberts A.P., and Wilkins A.H. (2008). Design of three dimensional isotropic microstructures for maximized stiffness and conductivity. International Journal of Solids and Structures 45: 4130–4146.

23. de Kruijf N., Zhou S., Li Q., and Mai Y.-W. (2007). Topological design of structures and composite materials with multiobjectives. International Journal of Solids and Structures 44: 7092–7109.

24. Guest J.K., and Prévost J.H. (2006). Optimizing multifunctional materials: design of microstructures for maximized stiffness and fluid permeability. International Journal of Solids and Structures 43: 7028–7047.

25. Hollister S.J. (2009). Scaffold design and manufacturing: from concept to clinic. Advanced Engineering Materials 21: 3330–3342.

26. Challis V.J., Roberts A.P., Grotowski J.F., Zhang L.C., and Sercombe, T.B. (2010). Prototypes for bone implant scaffolds designed via topology optimization and manufactured by solid freeform fabrication. Advanced Engineering Materials 12: 1106–1110.

27. Chen Y., Zhou S., and Li Q. (2011). Microstructure design of biodegradable scaffold and its effect on tissue regeneration. Biomaterials 32: 5003–5014.

28. Diaz A. and Sigmund O. (2010). A topology optimization method for design of negative permeability metamaterials. Structural and Multidisciplinary Optimization 41: 163–177.

29. Zhou S., Li W., Chen Y., Sun G., and Li Q. (2011). Topology optimization for negative permeability metamaterials using level-set algorithm. Acta Materialia 59: 2624-2636,

30. Sigmund O. and Jensen J.S. (2003). Systematic design of phononic band–gap materials and structures by topology optimization. Philosophical Transactions of the Royal Society A 361: 1001–1019.

31. Rupp C.J., Evgrafov A., Maute K., and Dunn M.L. (2007). Design of Phononic Materials/Structures for Surface Wave Devices Using Topology Optimization. Structural and Multidisciplinary Optimization, 34: 111-122.

32. Cadman J., Zhou S., Chen Y., and Li Q. (2013). On design of multi-functional microstructural materials. Journal of Materials Science 48: 51–66.

33. Challis V.J., Guest J.K., Grotowski J.F., and Roberts A.P. (2012). Computationally generated cross-property bounds for stiffness and fluid permeability using topology optimization, International Journal of Solids and Structures 49: 3397-3408.

34. Gibiansky L.V. and Torquato S. (1997) Thermal expansion of isotropic multi-phase composites and polycrystals. Journal of the Mechanics and Physics of Solids 45: 1223-1252.

35. Poulsen T.A. (2003). A new scheme for imposing minimum length scale in topology optimization. International Journal of Numerical Methods in Engineering 57: 741–760.

36. Guest J.K., Prévost J.H., and Belytschko T. (2004). Achieving minimum length scale in topology optimization using nodal design variables and projection functions. International Journal for Numerical Methods in Engineering 61: 238-254.

37. Sigmund O (2007) Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization 33: 401–424

38. Guest J.K. (2009). Topology optimization with multiple phase projection. Computer Methods in Applied Mechanics and Engineering 199: 123-135.

39. Guest J.K. (2009). Imposing maximum length scale in topology optimization. Structural and Multidisciplinary Optimization. 37: 463-473.

40. Guest J.K. (2015). Optimizing Discrete Object Layouts in Structures and Materials: A Projection-Based Topology Optimization Approach. Computer Methods in Applied Mechanics and Engineering 283: 330-351.

41. Ha S. and Guest J.K. (2014). Optimizing inclusion shapes and patterns in periodic materials using Discrete Feature Projection. Structural and Multidisciplinary Optimization 50: 65-80.

42. Zhao L., Ha S.H., Sharp K.W., Geltmacher A.B., Fonda R.W., Kinsey A., Zhang Y., Ryan S., Erdeniz D., Dunand D.C., Hemker K.J., Guest J.K., and Weihs T.P. (2014). Permeability measurements and modeling of topology-optimized metallic 3D woven lattices. Acta Materialia 81: 326-336.

43. Bensoussan A., Lions J., and Papanicolaou G. (1978). Asymptotic Analysis for Periodic Structures. Elsevier, North-Holland, Amsterdam.

44. Sanchez-Palencia, E. (1980). Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics, vol. 127. Springer, Berlin.

45. Maute K., Schwarz S., and Ramm E. (1998) Adaptive topology optimization of elastoplastic structures. Structural Optimization 2: 81-91.

46. Buhl T., Pedersen C.B.W., and Sigmund O. (2000). Stiffness design of geometrically nonlinear structures using topology optimization. Structural and Multidisciplinary Optimization 19: 93-104.

47. Bruns T.E., and Tortorelli D.A. (2001). Topology optimization of non-linear elastic structures and compliant mechanisms, Computer Methods in Applied Mechanics and Engineering 190: 3443–3459.

48. Wang F., Lazarov B.S., Sigmund O., Jensen J.S. (2014). Interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems. Computer Methods in Applied Mechanics and Engineering 276: 453-472.

49. Guest J.K. and Igusa T. (2008). Structural optimization under uncertain loads and nodal locations. Computer Methods in Applied Mechanics and Engineering, 198(1):116-124.

50. Sigmund O (2009) Manufacturing tolerant topology optimization. Acta Mechanica Sinica 25: 227–239.

51. Chen S., Chen W., Lee S., (2010). Level set based robust shape and topology optimization under random field uncertainties, Structural and Multidisciplinary Optimization 41: 507–524.

52. Wang F., Jensen J.S., and Sigmund O. (2011). Robust topology optimization of photonic crystal waveguides with tailored dispersion properties. Journal of the Optical Society of America B: Optical Physics 28(3): 387–397.

53. M. Schevenels, B. S. Lazarov, and O. Sigmund (2011). Robust topology optimization accounting for spatially varying manufacturing errors. Computer Methods in Applied Mechanics and Engineering 200: 3613–3627.

54. Tootkaboni M., Asadpoure A., and Guest J.K. (2012). Topology Optimization of Continuum Structures under Uncertainty – A Polynomial Chaos Approach. Computer Methods in Applied Mechanics and Engineering 201-204(1): 263-275.

55. Jalalpour M., Guest J.K., and Igusa T. (2013). Reliability-based topology optimization of trusses with stochastic stiffness matrix. Journal of Structural Safety 43: 41-49.

56. Andreassen E., Lazarov B.S., Sigmund O. (2014). Design of manufacturable 3D extremal elastic microstructure. Mechanics of Materials 69: 1-10.

57. Lotfi R., Ha S., Carstensen J.V., and Guest J.K. (2014). Topology Optimization for Cellular Material Design. Proceedings of 2013 MRS Fall Meeting, Boston, MA, 1-6.

58. Evgrafov A., Maute K., Yang R.G., and Dunn M.L. (2009). Topology optimization for nano-scale heat transfer. International Journal for Numerical Methods in Engineering 77: 285-300.

59. Deaton J.D., Grandhi R.V. (2014). A survey of structural and multidisciplinary continuum topology optimization: post 2000. Structural and Multidisciplinary Optimization 49: 1-38.

60. Sigmund O. and Maute K. (2013). Topology optimization approaches: A comparative review. Structural and Multidisciplinary Optimization 48: 1031-1055.

61. Andreassen E., Clausen A., Schevenels M., Lazarov B.S., Sigmund O. (2011). Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization 43: 1-16.

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