Heat conduction in anisotropic composite materials has received wide
attention in material engineering. In this chapter, the n-sided polygonal hybrid finite
element method with fundamental solution kernels, named as HFS-FEM, is thoroughly
studied for two-dimensional heat conduction in fully anisotropic media, which can be
regarded as a combination of the traditional finite element method and boundary element
method. In this approach, the unknown temperature field within an n-sided 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 shortcoming, a frame temperature field is independently defined over the
entire boundary of 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 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 integration
strategy endows the method with higher versatility and with non-conforming mesh in the
pre-processing stage of the analysis over the traditional finite elements. 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.
Keywords: Hybrid finite element, Fundamental solution, n-sided polygon, Nonconforming
mesh, Anisotropic material, Heat conduction.
