3.1.3 Finite Element Method 有限元法[cfd-3-1-3]

The finite element method was originally employed for structural analysis only. It was first introduced by Turner et al. [45] in 1956. About ten years later, researchers started to use the finite element method also for the numerical solution of field equations in continuous media. However, only with the beginning of the 90's, did the finite element method gain popularity in the solution of the Euler and the Navier-Stokes equations. A good introduction into the classical finite element methodology can be found in [46]. Applications to flow problems are described in [47], [48], and more recently in [49].

The finite element method, as it is in general applied to the solution of the Euler/Navier-Stokes equations, starts with a subdivision of the physical space into triangular (in 2-D) or into tetrahedral (in 3-D) elements. Thus, an unstructured grid has to be generated. Depending on the element type and the required accuracy, a certain number of points at the boundaries and/or inside an element is specified, where the solution of the flow problem has to be found. The total number of points multiplied with the number of unknowns determines the number of degrees of freedom. Furthermore, the so-called shape functions have to be defined, which represent the variation of the solution inside an element. In practical implementations, linear elements are usually employed, which use the grid nodes exclusively. The shape functions are then linear distributions, whose value is zero outside the corresponding element. This results in a second-order accurate representation of the solution on smooth grids.

Within the finite element method, it is necessary to transform the governing equations from the differential into an equivalent integral form. This can be accomplished in two different ways. The first one is based on the variational principle, i.e., a physical solution is sought, for which a certain functional possesses an extremum. The second possibility is known as the method of weighted residuals or the weak formulation. Here, it is required that the weighted average of the residuals is identically zero over the physical domain. The residuals can be viewed as the errors of the approximation of the solution. The weak formulation has the same advantage as the finite volume discretisation of the conservation laws - it allows the treatment of discontinuous solutions such as shocks. Therefore, the weak formulation is preferred over the variational methodology.

The finite element method is attractive because of its integral formulation and the use of unstructured grids, which are both preferable for flows in or around complex geometries. The method is also particularly suitable for the treatment of non-Newtonian fluids. The finite element method has a very rigorous mathematical foundation, particularly for elliptic and parabolic problems. Although it can be shown in certain cases that the method is mathematically equivalent to the finite volume discretisation, the numerical effort is noticeably higher. This may explain why the finite volume method became more popular. However, both methods are sometimes combined - particularly on unstructured grids. So for example, the treatment of the boundaries and the discretisation of the viscous fluxes is usually "borrowed" from the finite element method.