Efficient Infinite Elements based on Jacobi Polynomials

In this contribution an optimized version of the so–called mapped wave envelope elements, also known as Astley–Leis elements, is presented and its practical usability is assessed. The elements are based on Jacobi polynomials in the direction of radiation, which leads to a low conditioning of the resulting system matrices and to a superior performance in conjunction with iterative solvers. This is shown for practically relevant simulations in the frequency as well as in the time domain.

[1]  O. C. Zienkiewicz,et al.  Diffraction and refraction of surface waves using finite and infinite elements , 1977 .

[2]  R. Astley.  Infinite elements for wave problems: a review of current formulations and an assessment of accuracy , 2000 .

[3]  O. von Estorff,et al.  Effectiveness and robustness of improved infinite elements for exterior acoustics , 2006 .

[4]  Isaac Harari,et al.  A survey of finite element methods for time-harmonic acoustics , 2006 .

[5]  Benjamin S. Kirk,et al.  Library for Parallel Adaptive Mesh Refinement / Coarsening Simulations , 2006 .

[6]  E. Turkel,et al.  Absorbing PML boundary layers for wave-like equations , 1998 .

[7]  J. P. Moitinho de Almeida,et al.  Upper bounds of the error in local quantities using equilibrated and compatible finite element solutions for linear elastic problems , 2006 .

[8]  R. J. Astley,et al.  Conditioning of infinite element schemes for wave problems , 2001 .

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

[10]  K. Gerdes,et al.  A REVIEW OF INFINITE ELEMENT METHODS FOR EXTERIOR HELMHOLTZ PROBLEMS , 2000 .

[11]  M. Guddati,et al.  Continued fraction absorbing boundary conditions for convex polygonal domains , 2006 .

[12]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[13]  Isaac Harari,et al.  Studies of FE/PML for exterior problems of time-harmonic elastic waves , 2004 .

[14]  David S. Burnett,et al.  Prolate and oblate spheroidal acoustic infinite elements , 1998 .

[15]  J. C. Marques,et al.  Infinite elements in quasi-static materially nonlinear problems , 1984 .

[16]  Daniel Dreyer.  Efficient Infinite Elements for Exterior Acoustics , 2004 .

[17]  Udo Nackenhorst,et al.  A finite element approach for the simulation of tire rolling noise , 2008 .

[18]  David S. Burnett,et al.  An ellipsoidal acoustic infinite element , 1998 .

[19]  Ivo Babuška,et al.  A comparison of approximate boundary conditions and infinite element methods for exterior Helmholtz problems , 1998 .

[20]  D. Givoli.  High-order local non-reflecting boundary conditions: a review☆ , 2004 .

[21]  L. Trefethen,et al.  Numerical linear algebra , 1997 .

[22]  R. Freund,et al.  QMR: a quasi-minimal residual method for non-Hermitian linear systems , 1991 .

[23]  O. von Estorff,et al.  Improved conditioning of infinite elements for exterior acoustics , 2003 .

[24]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[25]  Richard Barrett,et al.  Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods , 1994, Other Titles in Applied Mathematics.

[26]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[27]  Leszek Demkowicz,et al.  Solution of 3D-Laplace and Helmholtz equations in exterior domains using hp-infinite elements , 1996 .

[28]  R. J. Astley.  TRANSIENT WAVE ENVELOPE ELEMENTS FOR WAVE PROBLEMS , 1996 .

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

[30]  J. Coyette,et al.  Treatment of frequency-dependent admittance boundary conditions in transient acoustic finite/infinite-element models. , 2001, The Journal of the Acoustical Society of America.

[31]  L. Thompson.  A review of finite-element methods for time-harmonic acoustics , 2006 .

[32]  R. J. Astley,et al.  The performance of spheroidal infinite elements , 2001 .

[33]  Saikat Dey,et al.  Acoustic infinite elements for non-separable geometries , 2002 .

[34]  R. J. Astley,et al.  NUMERICAL STUDIES OF CONJUGATED INFINITE ELEMENTS FOR ACOUSTICAL RADIATION , 2000 .

[35]  Frank Ihlenburg,et al.  ON FUNDAMENTAL ASPECTS OF EXTERIOR APPROXIMATIONS WITH INFINITE ELEMENTS , 2000 .

[36]  Mardochée Magolu monga Made,et al.  Incomplete factorization-based preconditionings for solving the Helmholtz equation , 2001 .

[37]  Marcus J. Grote,et al.  On nonreflecting boundary conditions , 1995 .

[38]  S. Petersen,et al.  Assessment of finite and spectral element shape functions for efficient iterative simulations of interior acoustics , 2006 .

[39]  Roland W. Freund,et al.  A Transpose-Free Quasi-Minimal Residual Algorithm for Non-Hermitian Linear Systems , 1993, SIAM J. Sci. Comput..

[40]  Karl Meerbergen,et al.  Time integration for spherical acoustic finite–infinite element models , 2005 .

[41]  Thomas E. Giddings,et al.  A finite element model for acoustic scattering from objects near a fluid-fluid interface , 2006 .

[42]  M. Benzi.  Preconditioning techniques for large linear systems: a survey , 2002 .

[43]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[44]  K. Gerdes.  The conjugated vs. the unconjugated infinite element method for the Helmholtz equation in exterior domains , 1998 .

[45]  David S. Burnett,et al.  A three‐dimensional acoustic infinite element based on a prolate spheroidal multipole expansion , 1994 .

[46]  J. A. Hamilton,et al.  The stability of infinite element schemes for transient wave problems , 2006 .

[47]  Jean-Pierre Berenger,et al.  A perfectly matched layer for the absorption of electromagnetic waves , 1994 .

[48]  R. J. Astley,et al.  Three-dimensional wave-envelope elements of variable order for acoustic radiation and scattering. Part I. Formulation in the frequency domain , 1998 .

[49]  Leszek Demkowicz,et al.  Convergence of the infinite element methods for the Helmholtz equation in separable domains , 1998 .

[50]  J. Keller,et al.  Exact non-reflecting boundary conditions , 1989 .

[51]  R. Leis,et al.  Initial Boundary Value Problems in Mathematical Physics , 1986 .