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. As problems increase in size, the computation can be made more efficient by employing local RBF methods such that the solution at a particular collocation point, say, x i, is approximated by a weighted set of local RBFs which are close neighbors, rather than by all the RBFs within the solution domain (see Section 7.5).The meshfree method can be computationally more expensive compared to the well-established FEM. Consequently, the class of compactly supported RBFs has been developed whereby their value is truncated at a certain distance from the associated collocation point (see Section 7.2.2).
#FASSHAUER MESHFREE METHODS CHAPER 14 SERIES#
This definition of the basis is applicable because each Fourier series can be formed from a linear combination of these basis functions.įor problems having complex solutions it may be advantageous to restrict the influence of distant RBFs. This is called a Fourier series, and the set of Fourier series functions has as its basis, where ω is the fundamental frequency of the process represented by f ( t) and nω the nth harmonic. For example, a representation of a periodic function f ( t) can be formed using a weighted sum of sines and cosines. In other words, if we wish to describe a function mathematically over a continuous domain, we can form a representation from a weighted sum of appropriate basis functions. The field of radial basis functions (RBFs) was initiated by Hardy, who proposed the now widely used multiquadratic RBF for use in a multidimensional scattered interpolation method.Įvery continuous function can generally be represented as a linear combination of basis functions (also called blending functions). The elements referred to here are a set of linearly independent basis functions (with radial symmetry or that are radially invariant) that apply to a function space. A basis set is a set of elements that are used to describe a particular system of interest. Here the main focus of our discussion will be interpolation of scattered data and the numerical solution of partial differential equations (PDEs) using radial basis functions (RBFs). It will, of necessity, be brief as this is a very wide subject that is still developing and that has many application areas. In this chapter we provide an introduction to meshless methods.
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