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Research Papers

Numerical Solution by the CESE Method of a First-Order Hyperbolic Form of the Equations of Dynamic Nonlinear Elasticity

[+] Author and Article Information
Lixiang Yang

Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210yang.1130@osu.edu

Robert L. Lowe

Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210lowe.194@osu.edu

Sheng-Tao John Yu1

Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210yu.274@osu.edu

Stephen E. Bechtel

Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210bechtel.3@osu.edu

1

Corresponding author.

J. Vib. Acoust 132(5), 051003 (Aug 19, 2010) (12 pages) doi:10.1115/1.4001499 History: Received March 04, 2009; Revised January 14, 2010; Published August 19, 2010; Online August 19, 2010

This paper reports the application of the space-time conservation element and solution element (CESE) method to the numerical solution of nonlinear waves in elastic solids. The governing equations consist of a pair of coupled first-order nonlinear hyperbolic partial differential equations, formulated in the Eulerian frame. We report their derivations and present conservative, nonconservative, and diagonal forms. The conservative form is solved numerically by the CESE method; the other forms are used to study the eigenstructure of the hyperbolic system (which reveals the underlying wave physics) and deduce the Riemann invariants. The proposed theoretical/numerical approach is demonstrated by directly solving two benchmark elastic wave problems: one involving linear propagating extensional waves, the other involving nonlinear resonant standing waves. For the extensional wave problem, the CESE method accurately captures the sharp propagating wavefront without excessive numerical diffusion or spurious oscillations, and predicts correct reflection characteristics at the boundaries. For the resonant vibrations problem, the CESE method captures the linear-to-nonlinear evolution of the resonant waves and the distribution of wave energy among multiple modes in the nonlinear regime.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematics of (a) the discretized space-time domain and (b) a representative SE and CE

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Figure 2

Numerical (x symbol) and theoretical (solid line) solutions of Burgers’ equation at (a) t=0.35, (b) t=0.875, (c) t=2.625, and (d) t=3.5 for Δt=0.0035 and Δx=0.04

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Figure 3

Numerical (x symbol) and theoretical (solid line) solutions of Burgers’ equation at (a) t=0.5, (b) t=0.9, (c) t=1.5, and (d) t=2.0 for Δt=0.001 and Δx=0.02

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Figure 4

Schematic illustrating our verification process for numerical solutions in the linear regime. The so-called nonlinear problem consists of the nonlinear governing equations along with appropriate initial and boundary conditions. The linearized problem consists of the linearized governing equations together with the same initial and boundary conditions.

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Figure 5

Numerical implementation of the boundary conditions in problem 4.1. (a) For the fixed boundary on the left end of the rod, ghost cells are added to the left of the computational domain. (b) The stress-free boundary-condition treatment on the right end of the rod.

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Figure 6

Analytical (solid line) and numerical (x symbol) stress and velocity profiles for problems 4.1.1 and 4.1.2, respectively, at t=0.171 ms ((a) and (d)), t=0.891 ms ((b) and (e)), and t=1.791 ms ((c) and (f)). The analytical solution is truncated at 2000 modes.

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Figure 7

Time histories of the analytical (solid line) and numerical (x symbol) solutions of stress and velocity at x=2 m for problems 4.1.1 and 4.1.2, respectively.

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Figure 8

Analytical (solid line) and numerical (x symbol) axial stress and axial velocity profiles for problems 4.2.1 and 4.2.2, respectively, at t=2.92 ms ((a) and (d)), t=5.68 ms ((b) and (e)), and t=10.05 ms ((c) and (f)). The analytical solution is truncated at 2500 modes.

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Figure 9

Analytical (solid line) and numerical (x symbol) axial stress and axial velocity profiles for problems 4.2.1 and 4.2.2, respectively, at t=239.20 ms ((a) and (d)), t=245.20 ms ((b) and (e)), and t=249.95 ms ((c) and (f)). The analytical solution is truncated at 2500 modes.

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Figure 10

Time histories of analytical (filled circles) and numerical (symbol x) stress and velocity at x=3.1 m for problems 4.2.1 and 4.2.2, respectively. Up to about t=50 ms, the resonant waves are in the linear regime. For t>50 ms, the effects of nonlinearity become significant, and the numerical and analytical solutions diverge.

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Figure 11

Power spectra generated from (a) the numerically obtained stress data for the nonlinear problem 4.2.2 and (b) the analytically obtained stress data for the linearized problem 4.2.1, both at t=249.95 ms, well into the nonlinear regime. Unlike the linear problem, where the wave energy is confined as a single mode, the wave energy is distributed among multiple superharmonics of the forcing frequency for the nonlinear problem.

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