A geometrical approach to build a locally structured orthogonal grid

This work presents a geometric technique to build orthogonal or nearly orthogonal grids on planar regions which have a special shape: are elongated and may have varying width. We refer to them as meander-like regions 1 . The process entails splitting the domain into blocks which satisfy certain convexity constraints, building an orthogonal grid on each of them and then assembling them to conform a global grid. We construct a data base of lemniscatic and elliptic curve segments, which are used to build automatically an orthogonal grid on each block, which is fitted to the block's boundary. The process of approximation involves geometric comparisons between the polygonal boundaries of the blocks and those of the elements of the database. The endpoints of a segment of the database determine a normalized lemniscatic or elliptic region. Similarly, each block is also normalized, which then allows the comparisons that drive the fitting. The orthogonal grid on a lemniscatic or elliptic region is created via conformal mapping with a disc sector, which is then transferred to the block. Our method allows for grid refinement without any further computation beyond quadratic function evaluation and solving quadratic equations.