High-Order Eulerian Simulations of Multimaterial Elastic–Plastic Flow

[+] Author and Article Information
Akshay Subramaniam

Department of Aeronautics & Astronautics,
Stanford University,
Stanford, CA 94305
e-mail: akshays@stanford.edu

Niranjan S. Ghaisas

Center for Turbulence Research,
Stanford University,
Stanford, CA 94305
e-mail: nghaisas@stanford.edu

Sanjiva K. Lele

Department of Aeronautics & Astronautics,
Stanford University,
Stanford, CA 94305
e-mail: lele@stanford.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 1, 2016; final manuscript received May 4, 2017; published online December 22, 2017. Assoc. Editor: Ben Thornber.

J. Fluids Eng 140(5), 050904 (Dec 22, 2017) (9 pages) Paper No: FE-16-1787; doi: 10.1115/1.4038399 History: Received December 01, 2016; Revised May 04, 2017

We develop a new high-order numerical method for continuum simulations of multimaterial phenomena in solids exhibiting elastic–plastic behavior using the diffuse interface numerical approximation. This numerical method extends an earlier single material high-order formulation that uses a tenth-order high-resolution compact finite difference scheme in conjunction with a localized artificial diffusivity (LAD) method for shock and contact discontinuity capturing. The LAD method is extended here to the multimaterial formulation and is shown to perform well for problems involving shock waves, material interfaces and interactions between the two. Accuracy of the proposed approach in terms of formal order (eighth-order) and numerical resolution is demonstrated using a suite of test problems containing smooth solutions. Finally, the Richtmyer–Meshkov (RM) instability between copper and aluminum is simulated in two-dimensional (2D) and a parametric study is performed to assess the effect of initial perturbation amplitude and yield stress.

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Fig. 1

Interface advection test problem. Spatial profiles at two time instants of (a) mixture density, (b) volume fraction in logarithmic scale of material 1 using NX = 400, dt = 2.5 × 10−4, and (c) eighth-order spatial accuracy in different species-specific quantities.

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Fig. 2

Interaction of a linear plane-strain wave with a material interface of density ratio 2. Density profiles focusing on (a) initial and reflected waves, and (b) transmitted wave. (c) Order of accuracy evaluated with respect to finely resolved (NX = 12,800) numerical solution for fixed interface thickness 0.01, and with respect to sharp-interface analytical solution for interface thicknesses 1/NX.

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Fig. 3

Perturbation amplitude normalized by its initial value plotted as a function of time for the elastic Cu–Al RM problem

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Fig. 4

Time period of the amplitude oscillation plotted as a function of κ−1∕2

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Fig. 5

Initial density contours in the Cu–Al RM problem with 128 points in the transverse direction

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Fig. 6

(a) Spike (upper set of lines) and bubble locations (lower set of lines) and (b) net integrated positive vorticity plotted for the three different grid resolutions. Number of points in the transverse direction is represented in the legend. (c) x − t diagram showing evolution of shock (solid lines) and interface (dashed line) locations with time.

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Fig. 7

Density and vorticity contours in the Cu–Al RM problem with 128 points in the transverse direction: (a) density profile at t = 1, (b) density profile at t = 3, (c) vorticity profile at t = 1, and (d) vorticity profile at t = 3

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Fig. 8

Density contours at t = 3 in the copper–aluminum RM problem with (a) 192 × 32, (b) 384 × 64, and (c) 768 × 128 grid points

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Fig. 9

Cu mass fraction contours at t = 4 using 384 × 64 grid points with initial interface amplitudes η0k of (a) 0.4, (b) 0.3, (c) 0.2, and (d) 0.1

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Fig. 10

Plot of (a) the mixing width and (b) the net integrated positive vorticity Γ+ as a function of time for different initial perturbation amplitudes. The legend indicates the value of η0k for each line.

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Fig. 11

Cu mass fraction contours at t = 4 using 384 × 64 grid points with yield stress amplification κY of (a)1, (b) 2, (c) 4, and (d) 16

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Fig. 12

Density and vorticity contours in the elastic Cu–Al RM problem with 128 points in the transverse direction: (a) density profile at t = 0.5, (b) density profile at t = 1, (c) vorticity profile at t = 0.5, and (d) vorticity profile at t = 1




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