Boundary infinite elements for the Helmholtz equation in exterior domains

A novel approach to the development of infinite element formulations for exterior problems of time-harmonic acoustics is presented. This approach is based on a functional which provides a general framework for domain-based computation of exterior problems. Special cases include non-reflecting boundary conditions (such as the DtN method). A prominent feature of this formulation is the lack of integration over the unbounded domain, simplifying the task of discretization. The original formulation is generalized to account for derivative discontinuities across infinite element boundaries, typical of standard infinite element approximations. Continuity between finite elements and infinite elements is enforced weakly, precluding compatibility requirements. Various infinite element approximations for two-dimensional configurations with circular interfaces are presented. Implementation requirements are relatively simple. Numerical results demonstrate the good performance of this scheme. © 1998 John Wiley & Sons, Ltd.

[1]  Paul E. Barbone,et al.  FINITE ELEMENT FORMULATIONS FOR EXTERIOR PROBLEMS : APPLICATION TO HYBRID METHODS, NON-REFLECTING BOUNDARY CONDITIONS, AND INFINITE ELEMENTS , 1997 .

[2]  Robert E. Jones.  A generalization of the direct-stiffness method of structural analysis , 1964 .

[3]  Thomas J. R. Hughes,et al.  Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains , 1992 .

[4]  Richard H. Gallagher,et al.  A critical assessment of the Simplified Hybrid Displacement Method , 1977 .

[5]  P. Pinsky,et al.  A galerkin least-squares finite element method for the two-dimensional Helmholtz equation , 1995 .

[6]  Ivo Babuška,et al.  A Generalized Finite Element Method for solving the Helmholtz equation in two dimensions with minimal pollution , 1995 .

[7]  O. C. Zienkiewicz,et al.  Mapped infinite elements for exterior wave problems , 1985 .

[8]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[9]  P. Pinsky,et al.  Complex wave-number dispersion analysis of Galerkin and Galerkin least-squares methods for fluid-loaded plates , 1994 .

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

[11]  H. S. Chen,et al.  OSCILLATIONS AND WAVE FORCES IN A MAN-MADE HARBOR IN THE OPEN SEA , 1976 .

[12]  I. Babuska.  The finite element method with Lagrangian multipliers , 1973 .

[13]  Dan Givoli,et al.  A finite element method for large domains , 1989 .

[14]  Charbel Farhat,et al.  Using a reduced number of Lagrange multipliers for assembling parallel incomplete field finite element approximations , 1992 .

[15]  Rolf Stenberg,et al.  On some techniques for approximating boundary conditions in the finite element method , 1995 .

[16]  D. Givoli.  Numerical Methods for Problems in Infinite Domains , 1992 .

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

[18]  A mixed finite element method for boundary flux computation , 1986 .

[19]  Thomas J. R. Hughes,et al.  Recent developments in finite element methods for structural acoustics , 1996 .

[20]  R. Astley.  FE mode-matching schemes for the exterior Helmholtz problem and their relationship to the FE-DtN approach , 1996 .

[21]  Peter M. Pinsky,et al.  Finite element dispersion analysis for the three‐dimensional second‐order scalar wave equation , 1992 .

[22]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[23]  S. W. Key,et al.  A specialization of Jones' generalization of the direct-stiffness method of structural analysis , 1971 .

[24]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[25]  Helio J. C. Barbosa,et al.  The finite element method with Lagrange multiplier on the boundary: circumventing the Babuscka-Brezzi condition , 1991 .

[26]  Mohammad A. Aminpour,et al.  A coupled analysis method for structures with independently modelled finite element subdomains , 1995 .

[27]  H. Mang,et al.  Admissible and Inadmissible Simplifications of Variational Methods in Finite Element Analysis , 1980 .

[28]  Calvin H. Wilcox,et al.  Scattering Theory for the d'Alembert Equation in Exterior Domains , 1975 .

[29]  Mohamed Masmoudi,et al.  Numerical solution for exterior problems , 1987 .

[30]  Isaac Harari,et al.  Finite element analysis of stiffened plates , 1985 .

[31]  T. Hughes,et al.  Studies of domain‐based formulations for computing exterior problems of acoustics , 1994 .

[32]  R. Courant.  Variational methods for the solution of problems of equilibrium and vibrations , 1943 .

[33]  T. Hughes.  Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[34]  Thomas J. R. Hughes,et al.  Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains , 1992 .

[35]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

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