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Research Papers

Conservation Properties of the Trapezoidal Rule in Linear Time Domain Analysis of Acoustics and Structures

[+] Author and Article Information
C. S. Jog

Facility for Research in Technical
Acoustics (FRITA),
Department of Mechanical Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mail: jogc@mecheng.iisc.ernet.in

Arup Nandy

Facility for Research in Technical
Acoustics (FRITA),
Department of Mechanical Engineering,
Indian Institute of Science,
Bangalore 560012, India

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.

J. Vib. Acoust 137(2), 021010 (Apr 01, 2015) (17 pages) Paper No: VIB-14-1143; doi: 10.1115/1.4029075 History: Received April 20, 2014; Revised November 07, 2014; Online December 18, 2014

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.

Copyright © 2015 by ASME
Topics: Acoustics , Stress , Pressure
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References

Newmark, N. M., 1959, “A Method of Computation for Structural Dynamics,” ASCE J. Eng. Mech. Div., 85(3), pp. 67–94.
Simo, J. C., and Tarnow, N., 1992, “The Discrete Energy–Momentum Method. Conserving Algorithms for Nonlinear Elastodynamics,” Z. Angew. Math. Phys., 43(5), pp. 757–792. [CrossRef]
Laursen, T. A., and Chawla, V., 1997, “Design of Energy Conserving Algorithms for Frictionless Dynamic Contact Problems,” Int. J. Numer. Methods Eng., 40(5), pp. 863–886. [CrossRef]
West, M., Kane, C., Marsden, J. E., and Ortiz, M., 2000, “Variational Integrators, the Newmark Scheme, and Dissipative Systems,” International Conference on Differential Equations, Berlin, Germany, Aug. 1–7, 1999, B.Fiedler, K., Groger, J., and Sprekels, eds., World Scientific, Singapore, pp. 1009–1011.
Krenk, S., 2006, “Energy Conservation in Newmark Based Time Integration Algorithms,” Comput. Methods Appl. Mech. Eng., 195(44–47), pp. 6110–6124. [CrossRef]
Pian, T. H. H., and Sumihara, K., 1984, “Rational Approach for Assumed Stress Finite Elements,” Int. J. Numer. Methods Eng., 20(9), pp. 1685–1695. [CrossRef]
Pian, T. H. H., and Tong, P., 1986, “Relations Between Incompatible Displacement Model and Hybrid Stress Model,” Int. J. Numer. Methods Eng., 22(1), pp. 173–181. [CrossRef]
Jog, C. S., 2005, “A 27-Node Hybrid Brick and a 21-Node Hybrid Wedge Element for Structural Analysis,” Finite Elem. Anal. Des., 41(11–12), pp. 1209–1232. [CrossRef]
Jog, C. S., 2010, “Improved Hybrid Elements for Structural Analysis,” J. Mech. Mater. Struct., 5(3), pp. 507–528. [CrossRef]
Hughes, T. J. R., 2009, The Finite Element Method, Dover Publications, New York.
Hilber, H. M., Hughes, T. J. R., and Taylor, R. L., 1977, “Improved Numerical Dissipation for Time Integration Algorithms in Structural Dynamics,” Earthquake Eng. Struct. Dyn., 5(3), pp. 283–292. [CrossRef]
Krenk, S., 2006, “State-Space Time Integration With Energy Control and Fourth-Order Accuracy for Linear Dynamic Systems,” Int. J. Numer. Methods Eng., 65(5), pp. 595–619. [CrossRef]
Krenk, S., 2008, “Extended State-Space Time Integration With High-Frequency Energy Dissipation,” Int. J. Numer. Methods Eng., 73(12), pp. 1767–1787. [CrossRef]
Shishvan, S. S., Noorzad, A., and Ansari, A., 2009, “A Time Integration Algorithm for Linear Transient Analysis Based on the Reproducing Kernel Method,” Comput. Methods Appl. Mech. Eng., 198(41–44), pp. 3361–3377. [CrossRef]
Manoj, K. G., and Bhattacharyya, S. K., 1999, “Transient Acoustic Radiation From Impulsively Accelerated Bodies by the Finite Element Method,” J. Acous. Soc. Am., 107(3), pp. 1179–1188. [CrossRef]
Pinsky, P. M., and Abboud, N. N., 1991, “Finite Element Solution of the Transient Exterior Structural Acoustics Problem Based on the Use of Radially Asymptotic Boundary Operators,” Comput. Methods Appl. Mech. Eng., 85(3), pp. 311–348. [CrossRef]
Yue, B., and Gudati, M. N., 2005, “Dispersion-Reducing Finite Elements for Transient Acoustics,” J. Acoust. Soc. Am., 118(4), pp. 2132–2141. [CrossRef]
Ainsworth, M., and Wajid, H. A., 2010, “Optimally Blended Spectral-Finite Element Scheme for Wave Propagation and Nonstandard Reduced Integration,” SIAM J. Numer. Anal., 48(1), pp. 346–371. [CrossRef]
Everstine, G. C., 1997, “Finite Element Formulations of Structural Acoustics Problems,” Comput. Struct., 60(3), pp. 307–321. [CrossRef]
Jog, C. S., 2013, “An Outward-Wave-Favouring Finite Element Based Strategy for Exterior Acoustical Problems,” Int. J. Acoust. Vib., 18(1), pp. 27–38.
Wu, S. F., 1993, “Transient Sound Radiation From Impulsively Accelerated Bodies,” J. Acoust. Soc. Am., 94(1), pp. 542–553. [CrossRef]
Soedel, W., 2005, Vibrations of Shells and Plates, Marcel Dekker, New York.
Akkas, N., Akay, H. U., and Yilmaz, C., 1978, “Applicability of General-Purpose Finite Element Programs in Solid–Fluid Interaction Problems,” Comput. Struct., 10(5), pp. 773–783. [CrossRef]

Figures

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Fig. 1

Pressure variation along the length of the duct at different time instants: (a) Sinusoidal loading and (b) impulsive loading

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Fig. 2

Conservation of Eac in the impulsive surface acceleration case in the straight duct problem; Eac and A(t) are normalized with respect to their respective maximum values of 20 and 1432

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Fig. 3

Pressure variation along radius at different time instants in the pulsating sphere (exterior acoustic) example: (a) Sinusoidal surface acceleration and (b) exponentially decaying surface acceleration

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Fig. 4

Variation of Eac in the exponential loading case in the pulsating sphere (exterior acoustic) problem; the acceleration and Eac are normalized with respect to their respective maximum values of 384.32 and 17.03 × 106

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Fig. 5

Pressure variation along radius for different time instants in the pulsating sphere (interior acoustic) problem: (a) Sinusoidal loading and (b) impulsive loading

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Fig. 6

Variation of Eac in the pulsating sphere (interior acoustic) problem; the energy has been normalized using the maximum value of 5.005 × 106

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Fig. 7

Oscillating sphere

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Fig. 8

Pressure variation along radius for different times in the oscillating sphere (exterior acoustic) problem. (a) θ = 135 deg and (b) θ = 180 deg.

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Fig. 9

Variation of Eac in the oscillating sphere (exterior acoustic) problem; A(t) and Eac are normalized against their maximum values of 384.32 and 51,210, respectively

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Fig. 10

Pressure variation along radius for different times in the oscillating sphere (interior acoustic) problem. (a) θ = 135 deg and (b) θ = 180 deg.

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Fig. 11

Center deflection for a ring pressure load of 2 sin 500t in the clamped circular plate problem: (a) Initial transient response and (b) periodic steady-state response

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Fig. 12

Force and energy variations for a ring pressure load of 2e−200t in the clamped circular plate example: (a) Center deflection and (b) conservation of energy

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Fig. 13

Skew plate problem: (a) Geometry and (b) center point deflection

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Fig. 14

Variation of force and linear and angular momenta as a function of time; “Con” and “Hyb” denote conventional and hybrid formulations, respectively

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Fig. 15

Variation of energy with time in the plate subjected to exponentially decaying loads example

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Fig. 16

Pressure variation with time at different distances from the shell; solid lines and black dots represent the conventional and hybrid results, respectively

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Fig. 17

Domain used for meshing in the coupled cylindrical shell problem

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Fig. 18

Variation of pressure at (r, z) − (0.5, 0.5) with time in the coupled cylindrical shell problem

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