An efficient Wave Based Method for 2D acoustic problems containing corner singularities

Computer Methods in Applied Mechanics and Engineering

Volumn 241-244, Issue , 2012, Pages 286-301

  Deckers, Elke ^a   Bergen, Bart ^a   Van Genechten, Bert ^a   Vandepitte, Dirk ^a   Desmet, Wim ^a  

^a UNIVERSITY OF LEUVEN   (Belgium)

Author keywords

2D acoustics; Corner singularities; Helmholtz equation; Trefftz; Wave Based Method

Indexed keywords

ACOUSTIC DOMAINS; ACOUSTIC PROBLEMS; COMPUTATIONAL EFFORT; CONVERGENCE RATES; CORNER SINGULARITY; DYNAMIC PRESSURE FIELDS; EXPANSION FUNCTION; FREQUENCY RANGES; NON-HOMOGENEOUS BOUNDARY CONDITIONS; PREDICTION TECHNIQUES; TREFFTZ; WAVE BASED METHOD;

HELMHOLTZ EQUATION;

ACOUSTIC FIELDS;

EID: 84864056852     PISSN: 00457825     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cma.2012.06.009     Document Type: Article

References (53)

1. Zienkiewicz O.C., Taylor R.L., Zhu J.Z. The Finite Element Method-Vol. 1: Its Basis & Fundamentals 2005, Butterworth-Heinemann. sixth ed.
2. Von Estorff O. Boundary Elements in Acoustics: Advance and Applications 2000, WIT press.
3. Li Z.C., Lu T.T. Singularities and treatment of elliptic boundary value problems. Math. Comput. Model. 2000, 31:97-145.
4. Babuška I., Oh H.S. The p-version of the finite element method for domains with corners and for infinite domains. Numer. Methods Partial Diff. Equat. 1990, 6:371-392.
5. Apel T., Sändig A.-M., Whiteman J.R. Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 1996, 19:63-85.
6. Apel T., Nicaise S. The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 1998, 21:219-549.
7. Maischak M., Mund P., Stephan E.P. Adaptive multilevel BEM for acoustic scattering. Comput. Methods Appl. Mech. Engrg. 1997, 150:351-367.
8. Jirousek J., Venkatesh A. A new fe approach for adaptive reliability assurance. Comput. Struct. 1990, 37:217-230.
9. Olson L.G., Georigiou G.C., Schultz W.W. An efficient finite element method for treating singularities in Laplace's equation. J. Comput. Phys. 1991, 96:391-410.
10. Wu X., Han H. A finite-element method for Laplace-and Helmholtz-type boundary value problems with singularities. SIAM J. Numer. Anal. 1997, 34:1037-1050.
11. Lucas T.R., Oh H.S. The method of auxiliary mapping for the finite element solutions of elliptic problems containing singularities. J. Comp. Phys. 1993, 108:327-342.
12. Givoli D., Rivkin L., Keller J.B. A finite element method for domains with corners. Int. J. Numer. Methods Engrg. 1990, 35:1329-1345.
13. Cai W., Lee H.C., Oh H.S. Coupling of spectral methods and the p-version of the finite element method for elliptic boundary value problems containing singularities. J. Comp. Phys. 1993, 108:314-326.
14. Chen J.T., Chen K.H. Dual integral formulation for determining the acoustic modes of a two-dimenstional cavity with a degenerate boundary. Engrg Anal. Bound. Elem. 1998, vol. 21:105-116.
15. Chen J.T., Liang M.T., Chen I.L., Chyuan S.W., Chen K.H. Dual boundary element analysis of wave scattering from singularities. Wave Motion 1999, 30:367-381.
16. Marin L., Lesnic D., Mantič V. Treatment of singularities in Helmholtz-type equations using the boundary element method. J. Sound Vibr. 2004, 278:39-62.
17. Harari I., Hughes T.J.R. A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics. Comput. Methods Appl. Mech. Engrg. 1992, 97:77-102.
18. Farhat C., Harari I., Franca L.P. The discontinuous enrichment method. Comput. Methods Appl. Mech. Engrg. 2001, 190:6455-6479.
19. Tezaur R., Zhang L., Farhat C. A discontinuous enrichment method for capturing evanescent waves in multiscale fluid and fluid/solid problems. Comput. Methods Appl. Mech. Engrg. 2008, 197:1680-1698.
20. E. Trefftz, Ein Gegenstück zum Ritzschen Verfahren, in: Proceedings of the Second International Congress on Applied Mechanics, Zurich, Switzerland, 1926, pp. 131-137.
21. Pluymers B., Van Hal B., Vandepitte D., Desmet W. Trefftz-based methods for time-harmonic acoustics. Arch. Comput. Methods Engrg. 2007, 14(4):343-381.
22. Riou H., Ladevéze P., Sourcis B. The multiscale VTCR approach applied to acoustic problems. J. Comput. Acoust. 2008, 16:487-505.
23. Huttunen T., Gamallo P., Astley R.J. Comparison of two wave element methods for the Helmholtz problem. Commun. Numer. Methods Engrg. 2009, 25:35-52.
24. Gamallo P., Astley R.J. A comparison of two Trefftz-type methods: the ultraweak variational formulation the least-squares method for solving shortwave 2-D Helmholtz problems. Int. J. Numer. Methods Engrg. 2007, 71:406-432.
25. Cessenat O., Depres B. Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J. Numer. Anal. 1998, 35:255-299.
26. Li Z.C. The Trefftz method for the Helmholtz equation with degeneracy. Appl. Numer. Math. 2008, 58:131-159.
27. Antunes P.R.S., Valtchev S.S. A meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks. J. Comput. Appl. Math. 2010, 234:1662-2646.
28. Marin L. Treatment of singularities in the method of fundamental solutions for two-dimensional Helmholtz-type equations. Appl. Math. Model. 2010, 34:1615-1633.
29. Stojek M. Least-squares Trefftz-type elements for the Helmholtz equation. Int. J. Numer. Methods Engrg. 1998, 41:831-849.
30. Barnett A.H., Betcke T. An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygnons. SIAM J. Sci. Comput. 2010, 32:1417-1441.
31. W. Desmet, A wave based prediction technique for coupled vibro-acoustic analysis, KULeuven, division PMA, PhD. thesis 98D12, 1998.
32. Van Genechten B., Vergote K., Vandepitte D., Desmet W. A multi-level wave based numerical modelling framework for the steady-state dynamic analysis of bounded Helmholtz problems with multiple inclusions. Comput. Methods Appl. Mech. Engrg. 2010, 199:1881-1905.
33. Van Genechten B., Bergen B., Vandepitte D., Desmet W. A Trefftz-based numerical modelling framework for Helmholtz problems with complex multiple scatterer configurations. J. Comput. Phys. 2010, 229(18):6623-6643.
34. Bergen B., Van Genechten B., Vandepitte D., Desmet W. An efficient Trefftz-based method for three-dimensional Helmholtz problems in unbounded domains. Comput. Model. Engrg. Sci. 2010, 61(2):155-175.
35. Bergen B., Pluymers B., Van Genechten B., Vandepitte D., Desmet W. A Trefftz based method for solving Helmholtz problems in semi-infinite domains. Engrg. Anal. Bound. Elem. 2012, 36:30-38.
36. Vanmaele C., Vandepitte D., Desmet W. An efficient wave based prediction technique for plate bending vibrations. Comput. Methods Appl. Mech. Engrg. 2007, 196:3178-3189.
37. Vanmaele C., Vandepitte D., Desmet W. An efficient wave based prediction technique for dynamic plate bending problems with corner stress singularities. Comput. Methods Appl. Mech. Engrg. 2009, 198:2227-2245.
38. Deckers E., Hörlin N.-E., Vandepitte D., Desmet W. A Wave Based Method for the efficient solution of the 2D poroelastic Biot equations. Comput. Methods Appl. Mech. Engrg. 2012, 201-204:245-262.
39. Deckers E., Van Genechten B., Vandepitte D., Desmet W. Efficient treatment of stress singularities in poroelastic wave based models using special purpose enrichment functions. Comput. Struct. 2011, 89:1117-1130.
40. Van Genechten B., Vandepitte D., Desmet W. A direct hybrid finite element-Wave based modelling technique for efficient coupled vibro-acoustic analysis. Comput. Methods Appl. Mech. Engrg. 2011, 200:742-764.
41. Vergote K., Van Genechten B., Vandepitte D., Desmet W. On the analysis of vibro-acoustic systems in the mid-frequency range using a hybrid deterministic-statistical approach. Comput. Struct. 2011, 89:868-877.
42. Pierce A. Acoustics: An Introduction to Its Physical Principles and Applications. McGraw-Hill Series in Mechanical Engineering 1981, McGraw-Hill.
43. Colton D., Kress R. Inverse Acoustic and Electromagnetic Scattering Theory 1998, Springer-Verlag, Berlin, Heidelbert, New York. second ed.
44. B. Pluymers, Wave Based Modelling Methods for Steady-state Vibro-acoustics, KULeuven, division PMA, Ph.D. Thesis 2006D04, 2006.
45. B. Bergen, Wave Based Modelling Techniques for Unbounded Acoustic Problems, KULeuven, Division PMA, Ph.D. Thesis 2011D07, 2011.
46. Herrera I. Boundary Methods: An Algebraic Theory 1984, Pitman Adv. Publ. Program, London.
47. Deckers E., Drofmans B., Van Genechten B., Bergen B., Vandepitte D., Desmet W. Spline-based boundaries: a first step towards generic geometric domain descriptions for effcient mid-frequency acoustic analysis using the Wave Based Method. J. Comput. Appl. Math. 2011, 235:2679-2693.
48. Millar R.F. Singularities of two-dimensional exterior solutions of the Helmholtz equation. Proc. Camb. Phil. Soc. 1971, 69:175-188.
49. Sinclair G.B. Stress singularities in classical elasticity-I: removal interpretation and analysis. Appl. Mech. Rev. 1993, 57:197-251.
50. Wolsson K. Linear dependence of a function set of m variables with vanishing generalized Wronskians. Linear Algebra Appl. 1989, 117:73-80.
51. Motz H. The treatment of singularities of partial differential equations by relaxation methods. Quart. Appl. Math. 1947, 4.
52. Moon P., Spencer D.E. Field Theory Handbook Including Coordinate Systems. Differential Equations And Their Solutions 1961, Prentice Hall, New Jersey.
53. Abramowitz M., Stegun A. Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables 1970, Dover publications, Inc., New York.