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n-sided polygonal hybrid finite elements involving element boundary integrals only for anisotropic thermal analysis

Cao R. F., Zhao X. J., Lin W. Q., Wang H.

Archives of Mechanics

2020

Vol. 72, nr 2

109--137

As a combination of the traditional finite element method and boundary element method, the n-sided polygonal hybrid finite element method with fundamental solution kernels, named as HFS-FEM, is thoroughly studied in this work for two-dimensional heat conduction in fully anisotropic media. In this approach, the unknown temperature field within the polygon is represented by the linear combination of anisotropic fundamental solutions of problem to achieve the local satisfaction of the related governing equations, but not the specific boundary conditions and the continuity conditions across the element boundary. To tackle such a shortcoming, the frame temperature field is independently defined on the entire boundary of the polygonal element by means of the conventional one-dimensional shape function interpolation. Subsequently, by the hybrid functional with the assumed intra- and inter-element temperature fields, the stiffness equation can be obtained including the line integrals along the element boundary only, whose dimension is reduced by one compared to the domain integrals in the traditional finite elements. This means that the higher computing efficiency is expected. Moreover, any shaped polygonal elements can be constructed in a unified form with the same fundamental solution kernels, including convex and non-convex polygonal elements, to provide greater flexibility in meshing effort for complex geometries. Besides, the element boundary integrals endow the method higher versatility with a non-conforming mesh in the pre-processing stage of the analysis over the traditional FEM. No modification to the HFS-FEM formulation is needed for the non-conforming mesh and the element containing hanging nodes is treated normally as the one with more nodes. Finally, the accuracy, convergence, computing efficiency, stability of non-convex element, and straightforward treatment of non-conforming discretization are discussed for the present n-sided polygonal hybrid finite elements by a few applications in the context of anisotropic heat conduction.

Hybrid Finite Element - Wave Based Method for acoustic problems

van Hal B., Desmet W., Vandepitte D., Sas P.

Computer Assisted Mechanics and Engineering Sciences

2003

Vol. 10, No. 4

479-494

The finite element method (FEM) is widely accepted for the steady-state dynamic response analysis of acoustic systems. It exhibits almost no restrictions with respect to the geometrical features of these systems. However, its application is practically limited to the low-frequency range. An alternative method is the wave based method, which is an indirect Trefftz method. It exhibits better convergence properties than the FEM and therefore allows accurate predictions at higher frequencies. However, the applicability is limited to systems of moderate geometrical complexity. The coupling between both methods is proposed. Only the parts of the problem domain with a complex geometry are modelled using the FEM, while the remaining parts are described with a wave based model. The proposed hybrid method has the potential to cover the mid-frequency range, where it is still difficult for currently existing (deterministic) techniques to provide satisfactory prediction results within a reasonable computational time.