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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e-mail: yang.1130@osu.edu
e-mail: lowe.194@osu.edu
e-mail: yu.274@osu.edu
e-mail: bechtel.3@osu.edu
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October 2010
Research Papers
Numerical Solution by the CESE Method of a First-Order Hyperbolic Form of the Equations of Dynamic Nonlinear Elasticity
Lixiang Yang,
Lixiang Yang
Department of Mechanical Engineering,
e-mail: yang.1130@osu.edu
Ohio State University
, Columbus, OH 43210
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Robert L. Lowe,
Robert L. Lowe
Department of Mechanical Engineering,
e-mail: lowe.194@osu.edu
Ohio State University
, Columbus, OH 43210
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Sheng-Tao John Yu,
Sheng-Tao John Yu
Associate Professor
Department of Mechanical Engineering,
e-mail: yu.274@osu.edu
Ohio State University
, Columbus, OH 43210
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Stephen E. Bechtel
Stephen E. Bechtel
Professor
Department of Mechanical Engineering,
e-mail: bechtel.3@osu.edu
Ohio State University
, Columbus, OH 43210
Search for other works by this author on:
Lixiang Yang
Department of Mechanical Engineering,
Ohio State University
, Columbus, OH 43210e-mail: yang.1130@osu.edu
Robert L. Lowe
Department of Mechanical Engineering,
Ohio State University
, Columbus, OH 43210e-mail: lowe.194@osu.edu
Sheng-Tao John Yu
Associate Professor
Department of Mechanical Engineering,
Ohio State University
, Columbus, OH 43210e-mail: yu.274@osu.edu
Stephen E. Bechtel
Professor
Department of Mechanical Engineering,
Ohio State University
, Columbus, OH 43210e-mail: bechtel.3@osu.edu
J. Vib. Acoust. Oct 2010, 132(5): 051003 (12 pages)
Published Online: August 19, 2010
Article history
Received:
March 4, 2009
Revised:
January 14, 2010
Online:
August 19, 2010
Published:
August 19, 2010
Citation
Yang, L., Lowe, R. L., Yu, S. J., and Bechtel, S. E. (August 19, 2010). "Numerical Solution by the CESE Method of a First-Order Hyperbolic Form of the Equations of Dynamic Nonlinear Elasticity." ASME. J. Vib. Acoust. October 2010; 132(5): 051003. https://doi.org/10.1115/1.4001499
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