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Research Papers

Triangular Cellular Automata for Computing Two-Dimensional Elastodynamic Response on Arbitrary Domains

[+] Author and Article Information
Ryan K. Hopman

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405

Michael J. Leamy1

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405michael.leamy@me.gatech.edu

1

Corresponding author.

J. Appl. Mech 78(2), 021020 (Dec 20, 2010) (10 pages) doi:10.1115/1.4002614 History: Received November 13, 2009; Revised September 21, 2010; Posted September 24, 2010; Published December 20, 2010; Online December 20, 2010

This study extends a recently developed cellular automata (CA) modeling approach (Leamy, 2008, “Application of Cellular Automata Modeling to Seismic Elastodynamics,” Int. J. Solids Struct., 45(17), pp. 4835–4849) to arbitrary two-dimensional geometries via the development of a rule set governing triangular automata (cells). As in the previous rectangular CA method, each cell represents a state machine, which updates in a stepped manner using a local “bottom-up” rule set and state input from neighboring cells. Notably, the approach avoids the need to develop and solve partial differential equations and the complexity therein. The elastodynamic responses of several general geometries and loading cases (interior, Neumann, and Dirichlet) are computed with the method and then compared with results generated using the earlier rectangular CA and finite element approaches. Favorable results are reported in all cases with numerical experiments indicating that the extended CA method avoids, importantly, spurious oscillations at the front of sharp wave fronts.

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Figures

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Figure 1

Nonuniform triangular mesh of a two-dimensional domain with an included hole. Geometry and mesh used in the studies of the Neumann and Dirichlet boundary conditions (see Figs.  1314).

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Figure 2

Illustration of geometrical differences between rectangular automata and triangular automata

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Figure 3

Angle θ relative to the global x-axis for each face determined by a line originating at the centroid and perpendicular to the face of interest. From this angle, the rotation transformation is determined for the normal and tangent directions.

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Figure 4

Identification of types I and II strain components (rectangular automata depicted)

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Figure 5

Illustration of the types I and II calculations for both rectangular and triangular automata

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Figure 6

Illustration of the parameters used to obtain Δn and Δs from centroid and face angle information

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Figure 7

Pseudocode depicting the fully object-oriented simulation approach used to update the automata states

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Figure 8

Method of applying a boundary cell. Any cell that fails a neighbor check gets a cell added to it with a parameter set reflecting that it is either displacement-imposed (Dirichlet) or traction-imposed (Neumann).

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Figure 9

Geometry of the uniform triangular mesh used in the elastic half-space study

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Figure 10

Comparison of displacement results computed for an elastic half-space using rectangular automata (top) and triangular automata (bottom): x-component (left) and y-component (right). The loading consists of a differentiated Gaussian pulse centered on the free surface.

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Figure 11

Triangular meshes used in the interior harmonic loading study: (a) coarse: 976 triangles, (b) fine: 3904 triangles, and (c) finer: 15,616 triangles

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Figure 12

Results at time 5.0×10−4 s for the y-component of displacement generated using CA (dark markers) and FE (surfaces) for the case of a harmonic interior load. Top to bottom: increasing mesh fineness.

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Figure 13

Results at time 5.0×10−4 s resulting from a Neumann boundary condition. Shown are the x-components of displacement: Two perspectives for the CA simulation (top row), FE (center row), and CA markers on top of FE surfaces (bottom row). The bottom row compares the CA results (markers) with the FE results (surfaces) at the same instant of time (left subfigure) and when the FE results plotted are 2.0×10−5 s earlier than the CA results (right subfigure).

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Figure 14

Results at time 1.3×10−4 s resulting from a Dirichlet boundary condition. Shown are the x-components of displacement: results displayed using isometric and x−z views for CA (top row) and FE (bottom row).

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