Abstract
In this paper, six-node hybrid-Trefftz triangular finite element models which can readily be incorporated into the standard finite element program framework in the form of additional element subroutines are devised via a hybrid variational principle for Helmholtz problem. In these elements, domain and boundary variables are independently assumed. The former is truncated from the Trefftz solution sets and the latter is obtained by the standard polynomial-based nodal interpolation. The equality of the two variables are enforced along the element boundary. Both the plane-wave solutions and Bessel solutions are employed to construct the domain variable. For full rankness of the element matrix, a minimal of six domain modes are required. By using local coordinates and directions, rank sufficient and invariant elements with six plane-wave modes, six Bessel solution modes and seven Bessel solution modes are devised. Numerical studies indicate that the hybrid-Trefftz elements are typically 50% less erroneous than their continuous Galerkin element counterpart.
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Acknowledgments
The present work is supported by the Hong Kong Research Grant Council GRF Grant HKU 7167 08E. The authors are grateful to Professor Clark Kimberling for pointing out that the points C in Figure 3(d) and 3(e) are commonly known as Fermat points (also known as Torricelli points and the 1st isogonic centers) of the triangles formed by the corner and side nodes, respectively [42].
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Sze, K.Y., Liu, G.H. Hybrid-Trefftz six-node triangular finite element models for Helmholtz problem. Comput Mech 46, 455–470 (2010). https://doi.org/10.1007/s00466-010-0494-0
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DOI: https://doi.org/10.1007/s00466-010-0494-0