The FV, FD, & FE Methods as Numerical Methods for Physical by Mattiussi

By Mattiussi

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72) appears as a result of a driving source term, with the parameter u derived from an "external" problem. This is an example of how the information about interacting phenomena is carried by terms appearing in the form of constitutive relations. Another example is given by boundary conditions describing a convective heat exchange through a part a D, of the domain boundary. If 6, is the external ambient temperature, h is the coefficient of convective heat exchange, and we denote with q , and 6, the convective heat flow density and the temperature at a generic point of aD,, we can write An alternative approach is to consider this as an example of coupledproblems, where the phenomena that originate the external driving terms are treated as separate interacting problems, which must also be discretized and solved.

We must also include information concerning the relative orientation of cells. This can be done as follows. Each oriented geometric object induces an orientation on its boundary (Figs. 4 and 6); therefore, each p-cell of an oriented cell complex induces an orientation on its (p - 1)-faces. We can compare this induced orientation with the default orientation of the faces as ( p - 1)-cells in K. Given the ith p-cell ti, and the jth ( p - 1)-cell t;, of a complex K, we define an incidence as follows (Fig.

We can, therefore, thanks to the definition of the operator 6, write the topological law [Eq. (1 16)] in terms of cochains only, as follows: 6G2 = Q3 ( 126) The definition [Eq. (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.

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