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2). The even- and odd-parity components can be treated separately, thereby obtaining Eqs. (26) and (27) where the matrices of Eq. (28) have been used. (p, = ATKTET= Y Y Y + @ D;Y (26) xy = ( M A ) ' A ~ K ~ T@S D;Y s* = p a y,(h)yT(h) (27) (28) Continuity can be imposed on Eq. (26), Eq. (27), or a linear combination of the two. Using weights of % a and bywhich are free parameters, Eqs. (26) and (27) can be linearly combined as shown in Eqs. (29) and (30). Combining Eqs. (24) and (30), Eq. (3 1) is obtained.
Introduction The variational nodal method implemented in VARIANT [ 1,2] cannot be applied directly to voided nodes because of the cross section appearing in the denominator of the second-order even-parity equation. Problems are also encountered when very low-density media occupy a node. A potential alternative for these situations is to form the nodal response matrix using the first-order form of the transport equation, for then the cross section no longer appears in the denominator. In this work a weighted residual approach is applied to the first-order form of the transport equation assuming the presence of isotropic scattering as well as an isotropic source.
The result is that the methods cannot be applied directly to void regions - that is to regions where nothing is present. In particular, the void region problem appears in variational nodal methods, such as those contained in the widely-used VARIANT code (Palmiotti, et. , 1995). These methods combine spherical harmonics expansions with hybrid finite elements to obtain solutions of the second-order transport equations in response matrix form. The challenge is thus to obtain void region response matrices that are compatible with spherical harmonics expansions and hybrid finite elements.