Finite Element Method - Mapped elements and numerical integration - infinite and singulrity elements _09 The description of the laws of physics for space- and time-dependent problems are usually expressed in terms of partial differential equations (PDEs). For the vast majority of geometries and problems, these PDEs cannot be solved with analytical methods. Instead, an approximation of the equations can be constructed, typically based upon different types of discretizations. These discretization methods approximate the PDEs with numerical model equations, which can be solved using numerical methods. The solution to the numerical model equations are, in turn, an approximation of the real solution to the PDEs. The finite element method (FEM) is used to compute such approximations.
Mapped elements and numerical integration - 'infinite' and 'singularity' elements 9.1 Introduction In the previous chapter we have shown how some general families of finite elements can be obtained for Co interpolations A progressively increasing number of nodes and hence improved accuracy characterizes each new member of the family and presumably the number of such elements required to obtain an adequate solution decreases rapidly To ensure that a small number of elements can represent a relatively complex form of the type that is liable to occur in real, rather than academic, problems, simple rectangles and triangles no longer suffice This chapter is therefore concerned with the subject of distorting such simple forms into others of more arbitrary shape Elements of the basic one-, two-, or three-dimensional types will be 'mapped' into distorted forms in the manner indicated in Figs 9.1 and 9.2 In these figures it is shown that the