In terms of usage, there is no doubt that the three methods discussed in subsections 1.3.1 through 1.3.3 far outweigh any other method for solving PDEs. It is probably fair to state, however, that these “other” methods are either still in their infancy – relatively speaking – or have been used only for a special class of problems for which they were designed. Outside of the three volume discretization based methods discussed in the preceding three subsections, a myriad of other methods for solving PDEs exist and continue to be used. Sandip Mazumder, in Numerical Methods for Partial Differential Equations, 2016 1.3.4 Other deterministic methods It works very well for T-meshes that can be generated automatically for complicated geometries. Figure 10.10 shows an ES-FEM for 2D problems and an FS-FEM for 3D problems. Therefore, there is a family of models called ES-FEM NS-FEM CS-FEM FS-FEM and αFEM. The currently proven strain smoothing operations are based on node-based, edge-based, cell-based, and face-based smoothing domains, or even their combinations. The key to the S-FEM lies in how one performs the strain operations. The formulation of S-FEM can be based either on smoothed weakform or the W2 form. In S-FEM, the field function interpolation is based on elements (as in the FEM), but it uses smoothed strains to construct numerical models, and hence no integration is needed for the weakform. This has led to the recent development of the S-FEM that combines the existing standard FEM and the existing strain smoothing techniques used in meshfree methods ( Liu and Trung, 2010). The idea of combining the meshfree techniques with the FEM therefore seems like an attractive proposition. The meshfree method can be computationally more expensive compared to the well-established FEM.
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