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13 \langle \phi \rangle(x,y) = \frac{1}{L_z + (T_f - T_0)}\int_0^{L_z} \int_{T_0}^{T_f} \phi(x, y, z, …
22 The function $\langle \phi \rangle (x,y)$ is represented on a 2-D finite element grid, taken from t…
26 Define a function space on the parent grid as $\mathcal{V}_p^\mathrm{parent} = \{ \bm v(\bm x) \in …
34 \langle \phi \rangle_z(x,y,t) = \frac{1}{L_z} \int_0^{L_z} \phi(x, y, z, t) \mathrm{d}z
41 \bm M u_N = \int_0^{L_x} \int_0^{L_y} u \psi^\mathrm{parent}_N \mathrm{d}y \mathrm{d}x
47 …ft [\frac{1}{L_z} \int_0^{L_z} \phi(x,y,z,t) \mathrm{d}z \right ] \psi^\mathrm{parent}_N(x,y) \mat…
53 \bm M [\langle \phi \rangle_z]_N = \frac{1}{L_z} \int_\Omega \phi(x,y,z,t) \psi^\mathrm{parent}_N(x…
58 This assumption means quadrature points in the full domain have the same $(x,y)$ coordinate locatio…
73 \bm M [\langle \phi \rangle]_N = \frac{1}{L_z + (T_f - T_0)} \int_\Omega \int_{T_0}^{T_f} \phi(x,y,…
75 where the integral $\int_{T_0}^{T_f} \phi(x,y,z,t) \mathrm{d}t$ is calculated on a running basis.
209 …d by $\bm{\Delta} = \bm{g}^{-1/2}$ where $\bm g = \nabla_{\bm x} \bm{X} \cdot \nabla_{\bm x} \bm{X…