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Research Papers

Topology Synthesis of Extrusion-Based Nonlinear Transient Designs

[+] Author and Article Information
Neal M. Patel1

Design Automation Laboratory, Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556npatel@nd.edu

Charles L. Penninger

Design Automation Laboratory, Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556cpenning@nd.edu

John E. Renaud1

Design Automation Laboratory, Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556jrenaud@nd.edu

1

Corresponding author.

J. Mech. Des 131(6), 061003 (Apr 28, 2009) (11 pages) doi:10.1115/1.3116255 History: Received March 30, 2008; Revised February 08, 2009; Published April 28, 2009

Many practical structural designs require that the structure is easily manufactured. Design concepts synthesized using conventional topology optimization methods are typically not easily manufactured, in that multiple finishing processes are required to construct the component. A manufacturing technique that requires only minimal effort is extrusion. Extrusion is a manufacturing process used to create objects of a fixed cross-sectional profile. The result of using this process is lower costs for the manufacture of the final product. In this paper, a hybrid cellular automaton algorithm is developed to synthesize constant cross section structures that are subjected to nonlinear transient loading. The novelty of the proposed method is the ability to generate constant cross section topologies for plastic-dynamic problems since the issue of complex gradients can be avoided. This methodology is applied to extrusions with a curved sweep along the direction of extrusion as well. Three-dimensional examples are presented to demonstrate the efficiency of the proposed methodology in synthesizing these structures. Both static and dynamic loading cases are studied.

Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 1

An illustration of the proposed method for grouping material elements. A three-dimensional structure is discretized in the plane that is normal to the extrusion axis, ξ and an extrusion row of elements span the length of the structure along this axis, as shown in (a). The IED of the elements in each extrusion row is summed along the extrusion direction, as shown in (b).

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Figure 2

Illustration of material elements in the density approach for a discretized continuum structure

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Figure 3

An illustration of a piecewise linear approximation of the stress-strain curve for a given material. A bilinear model is shown in (a). The multilinear model, shown in (b), better approximates the strain-hardening behavior of the material.

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Figure 4

The von Neumann CA neighborhoods: (a) 2D (N̂=4) and (b) 3D (N̂=6)

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Figure 5

Illustration of the HCA method for structural synthesis. Note that P is a general parameter that represents material properties, i.e., E for linear-static problems and E, σY, Eh, and ρ for plastic-dynamic problems.

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Figure 6

Example problem where a cantilevered structure is subjected to a static load. The beam is discretized into 80×18×18 cube elements.

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Figure 7

The topology generated for the single static applied load shown in Fig. 6. The raw topology is shown in (a), and an interpretation is shown in (b). The topology was synthesized after 33 iterations. A split view of the structure is shown in (c).

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Figure 8

The topology generated for the single static applied load when enforcing a constant cross section along the x-axis. The cross section is displayed in (b), and an interpretation of the cross section is shown in (c). The topology was synthesized after 30 iterations.

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Figure 9

The summed IED of each extrusion row for the problem in Fig. 6

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Figure 10

The topology synthesis histories for the constant cross section and the unconstrained cases in the static problem in Fig. 6. The total IED at each iteration is plotted.

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Figure 11

Design domain for the beam problem with (a) one and (b) three load cases. The beam is discretized into 10×10×10 mm3 cube elements. In both problems, the pole impacts the design domain with an initial velocity (v0) of 20 m/s.

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Figure 12

The final topology synthesized using the conventional topology synthesis method for a single dynamic loading event. The structure was synthesized after 43 iterations.

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Figure 13

The final topology generated for the single dynamic load case when enforcing a constant cross section along the x-axis. A three-dimensional view is shown in (a). The cross section and interpretation are shown in (b) and (c). The topology was synthesized after 44 iterations.

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Figure 14

The deformation of the structures generated using the (a) nonextrusion method and the (b) extrusion method

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Figure 15

The topology synthesis histories for the constant cross section and the unconstrained cases of the dynamic problem in Fig. 1. The total IED at each iteration is plotted.

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Figure 16

The final topology generated for the multiple dynamic load cases when enforcing constant cross section along the x-axis. The center load case displaces 47.3 mm and the two outside load cases displace 42.1 mm. A three-dimensional view is shown in (a). The cross section and interpretation are shown in (b) and (c). The final structure was synthesized after 83 iterations.

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Figure 17

The three-dimensional design domain for the curved bumper illustrated in (a). The two-dimensional material domain is shown in (b). At the time of impact, v0=5 m/s.

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Figure 18

The final topology generated for the bumper problem using the nonextrusion method. A three-dimensional view cut at the midspan of the structure is shown in (a) and (b). The deformation after t=0.015 s is shown in (c). The structure was synthesized after 45 iterations.

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Figure 19

The final topology generated using the extrusion-based HCA method for the bumper problem. A three-dimensional view cut at the midspan of the structure is shown in (a). The deformation after t=0.02 s is shown in (b). The cross section and interpretation are shown in (c) and (d). The structure was synthesized after 55 iterations.

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Figure 20

Plots of the plastic strain εp for topologies generated using the (a) nonextrusion and (b) extrusion methods. The maximum εp in (a) 1.51 and (b) 0.621.

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