Research Papers

Optimizing Topology and Gradient Orthotropic Material Properties Under Multiple Loads

[+] Author and Article Information
Anthony Garland

Department of General Engineering,
Clemson University,
Clemson, SC 29634
e-mail: apg@clemson.edu

Georges Fadel

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634
e-mail: fgeorge@clemson.edu

Manuscript received September 19, 2017; final manuscript received September 26, 2018; published online February 6, 2019. Assoc. Editor: Rahul Rai.

J. Comput. Inf. Sci. Eng 19(2), 021007 (Feb 04, 2019) (12 pages) Paper No: JCISE-17-1188; doi: 10.1115/1.4041744 History: Received September 19, 2017; Revised September 26, 2018

The goal of this research is to optimize an object's macroscopic topology and localized gradient material properties (GMPs) subject to multiple loading conditions simultaneously. The gradient material of each macroscopic cell is modeled as an orthotropic material where the elastic moduli in two local orthogonal directions we call x and y can change. Furthermore, the direction of the local coordinate system can be rotated to align with the loading conditions on each cell. This orthotropic material is similar to a fiber-reinforced material where the number of fibers in the local x and y-directions can change for each cell, and the directions can as well be rotated. Repeating cellular unit cells, which form a mesostructure, can also achieve these customized orthotropic material properties. Homogenization theory allows calculating the macroscopic averaged bulk properties of these cellular materials. By combining topology optimization with gradient material optimization and fiber orientation optimization, the proposed algorithm significantly decreases the objective, which is to minimize the strain energy of the object subject to multiple loading conditions. Additive manufacturing (AM) techniques enable the fabrication of these designs by selectively placing reinforcing fibers or by printing different mesostructures in each region of the design. This work shows a comparison of simple topology optimization, topology optimization with isotropic gradient materials, and topology optimization with orthotropic gradient materials. Finally, a trade-off experiment shows how different optimization parameters, which affect the range of gradient materials used in the design, have an impact on the final objective value of the design. The algorithm presented in this paper offers new insight into how to best take advantage of new AM capabilities to print objects with gradient customizable material properties.

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Fig. 1

Bridge macrostructure with varying mesostructure

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Fig. 2

Increasing the range of material properties allowed in the design decreases the objective function

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Fig. 3

Evaluation of the objective function value of an element shown as the solid black line. The vertical dotted line shows the actual minimum. The shaded region shows a local minimum region trap to avoid.

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Fig. 4

Equivalent states of the design variables are shown

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Fig. 5

Optimization flow chart

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Fig. 6

The top left image shows the discretization of the domain. In the remaining images, the six arrows represent six different loading cases where the arrow shows the location of the load. The bottom left image shows four loading conditions where the load is applied on the top of the structure and distributed over one quarter of the columns. The left images show two sheer cases where the load is distributed over the entire top node row and the load direction is to the left or right. The arrows do not represent point loads, but instead show the region where a distributed load is applied. The bottom of the design domain is fixed in the X and Y directions.

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Fig. 7

Output of the algorithm for the bridge design

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Fig. 8

“Vol Target” is ρtarget, “Actual Vol” is the current average ρ in the domain, “E avg” is 1/2Etarget∑e=1N(Exx+Eyyρ/2), “E target” is 1/2EtargetEtarget, and “Elast Obj” is objective J. The x-axis is optimizer calls, so three points represent one complete loop from Fig. 5. The objective is normalized by dividing by the greatest objective value.

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Fig. 9

Comparison of topology optimization techniques. Left: topology, Exx, Eyy, and rotation; center: topology and isotropic gradient materials; right: topology only.

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Fig. 10

Output design of the load on the left and clamped on the right. The output is represented using Eqs. (32) and (33).

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Fig. 11

Optimization metrics plot for the design in Fig. 10. “Vol Target” is ρtarget, “Actual Vol” is the current average ρ in the domain, “E avg” is 1/2Etarget∑e=1N(Exx+Eyyρ/2), “E target” is 1/2EtargetEtarget, and “Elast Obj” is objective J. The x-axis is optimizer calls, so three points represent one complete loop from Fig. 5. The objective is normalized by dividing by the greatest objective value.

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Fig. 12

The bar plot shows the relationship between the objective and ρtarget. Three of the final designs are also shown.



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