Modelling of short wave diffraction problems using approximating systems of plane waves

This paper describes a finite element model for the solution of Helmholtz problems at higher frequencies that offers the possibility of computing many wavelengths in a single finite element. The approach is based on partition of unity isoparametric elements. At each finite element node the potential is expanded in a discrete series of planar waves, each propagating at a specified angle. These angles can be uniformly distributed or may be carefully chosen. They can also be the same for all nodes of the studied mesh or may vary from one node to another. The implemented approach is used to solve a few practical problems such as the diffraction of plane waves by cylinders and spheres. The wave number is increased and the mesh remains unchanged until a single finite element contains many wavelengths in each spatial direction and therefore the dimension of the whole problem is greatly reduced. Issues related to the integration and the conditioning are also discussed.

[1]  Armel de La Bourdonnaye,et al.  High frequency approximation of integral equations modeling scattering phenomena , 1994 .

[2]  I. Babuska,et al.  The design and analysis of the Generalized Finite Element Method , 2000 .

[3]  P. Ortiz,et al.  An improved partition of unity finite element model for diffraction problems , 2001 .

[4]  C. Farhat,et al.  The Discontinuous Enrichment Method , 2000 .

[5]  Masayuki Kikuchi,et al.  Inversion of complex body waves , 1982 .

[6]  R. J. Astley,et al.  Wave envelope and infinite elements for acoustical radiation , 1983 .

[7]  R C MacCamy,et al.  Wave forces on piles: a diffraction theory , 1954 .

[8]  A. N. Williams,et al.  Wave Forces on an Elliptic Cylinder , 1985 .

[9]  Omar Laghrouche,et al.  Short wave modelling using special finite elements , 2000 .

[10]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[11]  A. De La Bourdonnaye.  UNE METHODE DE DISCRETISATION MICROLOCALE ET SON APPLICATION A UN PROBLEMEDE DIFFRACTION , 1994 .

[12]  Wim Desmet,et al.  A novel prediction technique for coupled structural-acoustic radiation problems , 2000 .

[13]  Bernard Peseux,et al.  A numerical integration scheme for special finite elements for the Helmholtz equation , 2003 .

[14]  Jan Mandel,et al.  The Finite Ray Element Method for the Helmholtz Equation of Scattering: First Numerical Experiments , 1997 .

[15]  Chiang C. Mei,et al.  Wave Forces on a Stationary Platform of Elliptical Shape , 1972 .

[16]  Edmund Chadwick,et al.  Wave envelope examples for progressive waves , 1995 .

[17]  P. Bettess,et al.  A new mapped infinite wave element for general wave diffraction problems and its validation on the ellipse diffraction problem , 1998 .

[18]  P. Bettess,et al.  The effectiveness of dampers for the analysis of exterior scalar wave diffraction by cylinders and ellipsoids , 1984 .

[19]  Charbel Farhat,et al.  A discontinuous finite element method for the helmholtz equation , 2000 .

[20]  Ismael Herrera,et al.  Connectivity as an alternative to boundary integral equations: Construction of bases. , 1978, Proceedings of the National Academy of Sciences of the United States of America.

[21]  I. Babuska,et al.  Finite Element Solution of the Helmholtz Equation with High Wave Number Part II: The h - p Version of the FEM , 1997 .

[22]  C. Mei,et al.  Scattering and Radiation of Gravity Waves by an Elliptical Cylinder. , 1971 .

[23]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[24]  R. J. Astley,et al.  Mapped Wave Envelope Elements for Acoustical Radiation and Scattering , 1994 .

[25]  P. Bettess,et al.  Analysis of the conditioning number of the plane wave approximation for the helmholtz equation , 2000 .

[26]  Omar Laghrouche,et al.  Solving short wave problems using special finite elements - Towards an adaptive approach , 2000 .

[27]  A. Sommerfeld.  Mathematische Theorie der Diffraction , 1896 .

[28]  P. Bettess,et al.  Diffraction of short waves modelled using new mapped wave envelope finite and infinite elements , 1999 .

[29]  Thomas Henry Havelock,et al.  The pressure of water waves upon a fixed obstacle , 1940, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[30]  F. Ihlenburg.  Finite Element Analysis of Acoustic Scattering , 1998 .

[31]  W. Desmet.  A wave based prediction technique for coupled vibro-acoustic analysis , 1998 .

[32]  Edmund Chadwick,et al.  Modelling of progressive short waves using wave envelopes , 1997 .

[33]  P. Morse,et al.  Methods of theoretical physics , 1955 .

[34]  R. J. Astley,et al.  Finite element formulations for acoustical radiation , 1983 .