A Three-Dimensional Acoustic Infinite Element Based On A Prolate Spheroidal Multipole Expansion

Journal of the Acoustical Society of America

Volumn 96, Issue 5, 1994, Pages 2798-2816

  Burnett, David S ^a  

^a LUCENT TECHNOLOGIES   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACOUSTICS; ARTICLE; FOURIER TRANSFORMATION; MATHEMATICAL ANALYSIS; PHYSICS; PRIORITY JOURNAL; STRUCTURE ANALYSIS;

EID: 0028099393     PISSN: 00014966     EISSN: NA     Source Type: Journal    
DOI: 10.1121/1.411286     Document Type: Article

References (52)

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31. In this spherical mesh, the mapped infinite elements (i) satisfy the convergence theorem in Sec. II A, and (ii) give the same field representation as the new spheroidal elements described in this paper. Therefore, Ref. 11, in effect, compares these new spheroidal elements with the exponential decay elements.
32. Frequencies up to ka=5 have been analyzed, but only on a structure for which no independent verification was available. Results seemed reasonable, e.g., normal convergence behavior and presence of expected physical phenomena such as shadow zones and specular-like reflection.
33. All runs were done on a dedicated machine, i.e., nothing running in the background.
34. The data in Fig. 4 for meshes 1, 2, and 4 were previously published in the proceedings of the 7th World Congress on Finite Element Methods in Monaco in November 1993 [FEM: Today and the Future, edited by J. Robinson, Robinson and Assoc, Great Bidlake Manor, Bridestowe, Oke-hampton, Devon EX20 4NT, England]. The proceedings reported the actual run times, i.e., 4X the octant times and 2X the quadrant time, which were therefore one-half the maximum possible times shown in Fig. 4. The proceedings also, incorrectly, reported the number of DOF in meshes that would have covered the entire shell in the absence of any planes of symmetry, whereas Fig. 4 reports the actual number of DOF on the octant or quadrant meshes. Neither of these differences in the reported data affect the ratio of BEM to IEM times, which is the sole purpose of the comparison.
35. It would also apply to the mapped element in the special case when it lies outside the minimum circumscribing sphere and its sides conform to radial lines of a single spherical coordinate system.
36. F. V. Atkinson, “On Sommerfeld's radiation condition, “ Philos. Mag. 40, 645-651 (1949).
37. R. B. Barrar and A. F. Kay, “A series development of a solution of the wave equation in powers of 1/r, “ Technical Research Group, New York City, unpublished (cited in Ref. 38).
38. C. H. Wilcox, “A generalization of theorems of Rellich and Atkinson, “ Proc. Am. Math. Soc. 7, 271-276 (1956). Wilcox only states Eqs. (2) and (3) without proof, which he attributes to Atkinson36 and Barrar and Kay.37 Reference 39 tightens Atkinson's region of convergence from [formula omitted].
39. C. H. Wilcox, “An expansion theorem for electromagnetic fields, “Comm. Pure Appl. Math. 9, 115-134 (1956).
40. The coordinate systems also have open coordinate surfaces, e.g., hyperbo-loids, planes, or cones, as described in Sec. III C 1, but only the closed surfaces, which can circumscribe the body, are relevant to choosing an appropriate coordinate system.
41. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), p. 511.
42. R. L. Holford (AT&T Bell Laboratories, Whippany, NJ), “A multipole expansion for the acoustic field exterior to a prolate or oblate spheroid, “ in preparation.
43. D. S. Burnett, “A 3-D acoustic infinite element based on an oblate spheroidal multipole expansion, “ in preparation.
44. B. A. Szabo and I. Babuska, Finite Element Analysis (Wiley, New York, 1991).
45. D. S. Burnett, Finite Element Analysis: From Concepts to Applications (Addison-Wesley, Reading, MA, 1987).
46. Ibid., p. 578.
47. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions (U.S. NBS, Washington, DC, 1965), AMS55, Chap. 5.
48. Because of these properties, it does not appear that wave envelope elements,16-18 which are noted for being much cheaper to generate than other infinite elements, offer any advantage in computational cost over this infinite element. (In addition, they have the disadvantage of nonsym-metric matrices.)
49. A non-finite-element mathematical model was developed by G. V. Bor-giotti (George Washington University, Washington, DC, structural theory), W. Wasylkiwskyj (George Washington University, Washington, DC, acoustic theory, viz., BEM), and P. S. Gural (S.A.I.C., Arlington, VA, algorithmic integration). See G. V. Borgiotti, “Effect of the compliance on the scattering of an elastic object immersed in a fluid: A general formulation, “ J. Acoust. Soc. Am. 87, 1055-1061 (1990); also, G. V. Borgiotti and E. M. Rosen, “The state vector approach to the wave analysis of the forced vibration of a cylindrical shell. Part II: Finite cylinders in vacuum, “ J. Acoust. Soc. Am. 93, 864-874 (1993).
50. The model in note 49 was enhanced by G. Korzeniewski (S.A.I.C., Arlington, VA) to include elastic hemispherical end caps. [Theory from C. Prasad, “On the vibration of spherical shells, “ J. Acoust. Soc. Am. 36, 489-494 (1964).]
51. Joseph J. Shirron (Naval Research Laboratory, Washington, DC), private communication. Mr. Shirron has been doing independent research on the prolate spheroidal infinite element since attending a seminar on the subject given by the author at the University of Maryland in April 1993.
52. D. S. Burnett, “3-D structural acoustic modeling using a new high accuracy acoustic infinite element, and a cost comparison with the BEM, “ in preparation.