Test

Fundamentally, design is the practice of defining a distribution of some resource(s) in a space to accomplish an objective. Many approaches have been developed to help enhance and streamline structural design over countless decades, but none do so with the directness of topology optimization (TO).

This approach to structural design was pioneered in the 1980’s by Bendsoe and Kikuchi [1] and considers the design problem in terms of:

This set of definitions is paired with an optimization algorithm to iteratively optimize the material distribution within the design domain. The resulting optimization problem can be expressed as

A common variation of this problem to minimize compliance by finding an optimal distribution of finite element densities, ${\rho }_e$, with a volume fraction constraint

In this problem formulation, an element stiffness of ${\rho }_e=1$ indicates that the element has the same stiffness as the material that it represents, while an element with ${\rho }_e=0$ has no stiffness and represent a region that is void. However, this problem formulation does not preclude the formation of regions with intermediate densities of $0<{\rho }_e<1$, which calls for a definition of a function for the constitutive properties as a function of ${\rho }_e$. For example, Figure 1 shows a topology optimization result for a case where

For compliance minimization, the sensitivity of the objective with respect to element density is

The element stiffness matrix, $K_e$, can be expressed as

where $K_e^0$ is the stiffness matrix for a totally solid element. Thus, the objective sensitivity becomes

The sensitivity of the constraint function is

where $v_e$ is finite element volume.

If intermediate density is allowed within the solution, then it is also necessary to define a physical interpretation of intermediate material density. One such interpretation is that intermediate density regions are regions populated by architected materials with volume fractions equal to ${\rho }_e$, and this interpretation is discussed in section 1.1. Alternatively, intermediate density can be assumed to be undesirable, and measures can be added to the problem formulation to avoid it in topology optimization solutions, as discussed in section 1.2.

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Figure 2 – Boundary conditions for the MBB beam problem with the design domain marked in gray

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An Architected Material Interpretation of Topology Optimization

Architected Materials Background

An “architected material” is a material that consists of small cellular “microstructures”. Due to the geometrical properties of these microstructures, such architected materials exhibit properties that are different from the properties of the material(s) from which they are made. Two examples of naturally occurring examples of architected materials that are bones and wood (Figure 3) [2]. Research into manmade architected microstructure materials began to gain traction in the latter half of the 20^th century [3]. These manmade architected materials can broadly be separated into two categories: random and structured (Figure 4), the latter of which can often be described in terms of “unit cells”. The impetus for exploring such materials is that they provide design engineers with access to material property combinations that are not attainable with naturally existing or conventional materials (Figure 5).

References

[1]  Bendsøe, M. P., and Kikuchi, N., 1988, “Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Comput. Methods Appl. Mech. Eng., 71(2), pp. 197–224.

[2]  Schaedler, T. A., and Carter, W. B., 2016, “Architected Cellular Materials,” Annu. Rev. Mater. Res., 46(1), pp. 187–210.

[3]  Fleck, N. A., Deshpande, V. S., and Ashby, M. F., 2010, “Micro-Architectured Materials: Past, Present and Future,” Proc. R. Soc. Math. Phys. Eng. Sci., 466(2121), pp. 2495–2516.

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