The trapezoidal rule, which is a special case of the Newmark family of algorithms, is one of the most widely used methods for transient hyperbolic problems. In this work, we show that this rule conserves linear and angular momenta and energy in the case of undamped linear elastodynamics problems, and an “energy-like measure” in the case of undamped acoustic problems. These conservation properties, thus, provide a rational basis for using this algorithm. In linear elastodynamics problems, variants of the trapezoidal rule that incorporate “high-frequency” dissipation are often used, since the higher frequencies, which are not approximated properly by the standard displacement-based approach, often result in unphysical behavior. Instead of modifying the trapezoidal algorithm, we propose using a hybrid finite element framework for constructing the stiffness matrix. Hybrid finite elements, which are based on a two-field variational formulation involving displacement and stresses, are known to approximate the eigenvalues much more accurately than the standard displacement-based approach, thereby either bypassing or reducing the need for high-frequency dissipation. We show this by means of several examples, where we compare the numerical solutions obtained using the displacement-based and hybrid approaches against analytical solutions.
Conservation Properties of the Trapezoidal Rule in Linear Time Domain Analysis of Acoustics and Structures
Department of Mechanical Engineering,
Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 20, 2014; final manuscript received November 7, 2014; published online December 18, 2014. Assoc. Editor: Liang-Wu Cai.
- Views Icon Views
- Share Icon Share
- Cite Icon Cite
- Search Site
Jog, C. S., and Nandy, A. (April 1, 2015). "Conservation Properties of the Trapezoidal Rule in Linear Time Domain Analysis of Acoustics and Structures." ASME. J. Vib. Acoust. April 2015; 137(2): 021010. https://doi.org/10.1115/1.4029075
Download citation file:
- Ris (Zotero)
- Reference Manager