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Mapping technique

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Mapping techniques transform a parent region with regular grid spacing to a more arbitrarily shaped region. For a unit square, the region being mapped should have four logical sides and four rather natural corners (Fig.3).

 Fig.3. Mesh generation with mapping technique.

Suppose we want to generate a quadrilateral mesh inside a domain that has the shape of second-order quadrilateral. Mapping technique shown in Fig.3 can be used for this purpose. If each side of the curvilinear quadrilateral domain can be approximated by parabola then the domain looks like 8-node isoparametric element. The domain is mapped to a square in the local coordinate system ξ, η. The square in coordinates ξ, η is divided into rectangular elements then nodal coordinates are transformed back to the global coordinate system x, y. The algorithm of coordinate computation for the nodes inside the domain is given below:

 (1)

where and represents the number of elements in ξ and η direction respectively and where the shape functions Nk defined in local coordinates ξ, η (-1 ξ≤1; -1≤η≤1) are:

(a) for bi-linear elements:

 k=1,2,3,4 (2)

 (3)