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Numerical Grid Generation
Published in M. Necati Özisik, Finite Difference Methods in Heat Transfer, 2017
A variety of approaches have been reported in the literature for the transformation of irregularly shaped regions into simple regular regions such as a square, rectangle, etc. The basic theory behind such transformations is quite old. For example, conformai transformations have been widely used in classical analysis, the Schwarz-Christoffel transformation is well known for conformai mapping of regions with polynomial boundaries onto upper-half plane. A comprehensive study of constructing conformai mapping using the Schwarz-Christoffel formula is given by Trefethen (1980) and a dictionary of conformai transformations is compiled by Kober (1957). Details of application of conformai transformation with complex variable technique can be found in the standard texts by Milne-Thompson (1950), Churchill (1948) and Davies (1979).
A General Formulation For Inverse Heat Conduction
Published in M. Necati Özisik, Helcio R. B. Orlande, Inverse Heat Transfer, 2018
M. Necati Özisik, Helcio R. B. Orlande
A variety of approaches has been reported in the literature for the transformation of irregularly shaped regions into simple regular regions such as a square, rectangle, etc. The basic theory behind such transformations is quite old. For example, conformal transformation has been widely used in classical analysis. Schwarz-Christoffel transformation is well known for conformal mapping of regions with polynomial boundaries onto an upper-half plane. A dictionary of conformal transformations was compiled by Kober [13]. Details of application of conformal transformation with complex variable technique can be found in the standard texts by Milne-Thompson [14] and Churchill [15].
Applications
Published in Vladimir Eiderman, An Introduction to Complex Analysis and the Laplace Transform, 2021
The Schwarz-Christoffel transformation is a mapping of the upper half-plane onto a polygon. First, we consider a bounded polygon whose nonadjacent sides do not intersect except the case when adjacent sides form exterior angles of −π (see Fig. 63). Then later we will consider the case when one or more vertices are at infinity.
Equilibrium locations of defects in two-dimensional configurations of the NLC director field
Published in Liquid Crystals, 2023
O. S. Tarnavskyy, M. F. Ledney
Using the function , let’s conformally map the region of the complex plane onto the upper half-plane of the complex plane . As a result of this mapping, the boundary points of the region become points of the real axis of the complex plane . At the same time, the boundary of the region is mapped onto the real axis . The derivative of the function at the point of the curve will be equal to
Exponential Time Differencing Schemes for Fuel Depletion and Transport in Molten Salt Reactors: Theory and Implementation
Published in Nuclear Science and Engineering, 2022
Zack Taylor, Benjamin S. Collins, G. Ivan Maldonado
This method involves evaluating the contour integral by choosing an analytic function that maps the real line onto the contour. Because the function decreases exponentially as , the approximation can be truncated to a finite number of quadrature points. When the spectrum of the transition matrix falls on the left-hand side of the complex plane close to the real axis, the contour denotes a Hankel-like contour that winds from on the lower half-plane and on the upper half-plane.31 This allows for definition of a general contour function that will enclose the eigenvalues on the left-hand side of the complex plane around the negative real axis.
Two-dimensional director configurations in a nematic-filled cylindrical capillary with the hybrid director alignment on its surface
Published in Liquid Crystals, 2020
O. S. Tarnavskyy, A. M. Savchenko, M. F. Ledney
maps the region , i.e. the semi-circle of unit radius, from the lower half-plane of the complex plane (see Figure 2) onto the upper half plane , where [24]. The boundary of the region is mapped onto the real -axis of the -plane. The point belonging to is mapped to the infinitely distant point of the -plane. Hence, the values of that correspond to the points of are real. We traverse the curve in the anticlockwise direction, the region being to the left. At , the angles which the director and the tangent to the curve make with the -axis are and , respectively.