LEAST-SQUARES TREFFTZ-TYPE ELEMENTS FOR THE HELMHOLTZ EQUATION

Trefftz-type elements, or T-elements, are finite elements the internal field of which fulfills the governing differential equations of the problem a priori whereas the prescribed boundary conditions and the interelement continuity must be enforced by some suitable method. In this paper, the relevant matching is achieved by means of a least-squares procedure. The so-called ‘frameless’ or least-squares T-elements for Helmholtz's equation (related to the scattering of waves by offshore structures) in 2-D are developed and studied. The required accuracy of the solution can be obtained by increasing the number of either the subdomains or T-functions, which can be regarded as the h- or p-type approach, respectively. Convergence studies are performed with much attention to the use of special purpose elements for a doubly connected domain with a circular hole and for an angular corner subdomain. The most attractive features of the presented formulation are its simplicity and robustness. The matrix of the resulting linear system is always Hermitian and positive definite. © 1998 John Wiley & Sons, Ltd.

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

[2]  J. Jiroušek,et al.  Numerical assessment of a new T-element approach , 1995 .

[3]  J. Descloux,et al.  An accurate algorithm for computing the eigenvalues of a polygonal membrane , 1983 .

[4]  Theodore H. H. Pian,et al.  Basis of finite element methods for solid continua , 1969 .

[5]  J. Jirousek,et al.  Basis for development of large finite elements locally satisfying all field equations , 1978 .

[6]  J. A. Hendry.  Singular problems and the global element method , 1980 .

[7]  O. C. Zienkiewicz,et al.  Generalized finite element analysis with T-complete boundary solution functions , 1985 .

[8]  O. Zienkiewicz,et al.  The coupling of the finite element method and boundary solution procedures , 1977 .

[9]  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.

[10]  Theodore H. H. Pian,et al.  A hybrid‐element approach to crack problems in plane elasticity , 1973 .

[11]  J. Avilés,et al.  A boundary method for elastic wave diffraction: Application to scattering of SH waves by surface irregularities , 1982 .

[12]  J. Jirousek,et al.  A powerful finite element for plate bending , 1977 .

[13]  J. Jiroušek,et al.  Least‐squares T‐elments: Equivalent FE and BE forms of a substructure‐oriented boundary solution approach , 1994 .

[14]  J. Jirousek,et al.  T-elements: State of the art and future trends , 1996 .

[15]  P. Tong,et al.  Singular finite elements for the fracture analysis of V‐notched plate , 1980 .

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

[17]  C. Mei.  The applied dynamics of ocean surface waves , 1983 .