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Extra resources for Mattiussi - The Finite Volume, Finite Difference, and Finite Elements Methods as Numerical Methods for Physical Field Problems - FDTD
Conversely, given an oriented p-cell, we can use this definition to propagate its orientation to neighboring p-cells [on orientable n-dimensional domains it is always possible to propagate the orientation of an n-cell to all the n-cells of the complex (Schutz, 1980)l. FIGURE 24. Two adjacent cells have compatible orientation if they induce on the common face opposite orientations. The concept of induced orientation can be used to propagate the orientation of a p-cell to neighboring p-cells. 190 CLAUD10 MATTIUSSI 4.
123)l of the coboundary operator may seem abstract. However, it has a very intuitive meaning that can be exemplified as follows (Tonti, 1975). Equation (124) can be rewritten by substituting 873 with its expression in terms of incidence numbers. Exploiting the linearity of the chaincochain pairing, after some reordering of terms, gives More generally, Eq. (122) becomes This means that the coboundary operator operates on ap-cochain cP and builds ~ global a ( p 1)-cochain SCP, which assumes on each ( p 1)-cell t P +the + + CLAUD10 MATTIUSSI FIGURE 28.
Previously, we derived the topological laws of electromagnetism in discrete form [Eqs. (56) through (62)l. The fundamental property of all these relations-shared by all topological laws-is that they equate a global physical quantity associated with a geometric object to 200 CLAUD10 MATTIUSSI another global quantity associated with its boundary. This appears even more clearly in the space-time formulation of the same laws [Eqs. (63) through (66), repeated next for easy reference]: If the domain is meshed with the primary and secondary cell complexes we will have to substitute the generic geometric objects appearing in Eqs.