Analyse:Jem/Fragment 124 01

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Bearbeiter Graf Isolan

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Untersuchte Arbeit: Seite: 124, Zeilen: 1-16

Quelle: Fan et al 2004 Seite(n): 8-9, Zeilen: 8:8-18; 9:3-4.6-14

[This linear system can be written in matrix form as $\mathbf A \mathbf x = \mathbf b\quad\quad$ (6.16) where the 2N x 2N coefficient matrix A = [A1A2] is defined by $\mathbf A_1 = \left[ \begin{matrix} 1 & \cdots & 1\\ 0 & \cdots & 0\\ -d_1^2 & \cdots & -d_N^2\\ \vdots & \ddots & \vdots\\ 2(1-N)d_1^{2N-1} & \cdots & 2(1-N)d_N^{2N-1} \end{matrix} \right]\quad\quad$ (6.17) and] $\mathbf A_2 = \left[ \begin{matrix} 0 & \cdots & 0\\ 1 & \cdots & 1\\ 2d_1 & \cdots & 2d_N\\ \vdots & \ddots & \vdots\\ 2(N-1)d_1^{2N-2} & \cdots & 2(N-1)d_N^{2N-2} \end{matrix} \right]\quad\quad$ (6.18) The 2N vector of unknowns x is defined by ${\mathbf x} = \left[S_{\omega_1} \cdots S_{\omega_N} \; S_{\delta_1} \cdots S_{\delta_N}\right]^T \quad\quad$ (6.19) and the known right-hand side is ${\mathbf b} = \left[ \overline{S}_{m_0} \cdots \overline{S}_{m_{2N-1}} \right]^T \quad\quad$ (6.20) As shown below, with the DQMOM approximation, the right-hand side of Eq. (6.10) is closed in terms of N weights and abscissas. As N increases, the quadrature approximation will approach the exact value, although at a higher computational cost. If the abscissas dq are unique, then A will be full rank. For this case, the source terms for the transport equations of the weights ωq and weighted diameters δq can be found by inverting A in Eq. (6.16): There are cases for which the matrix A is not full rank (the matrix is singular). These cases can occur when one or more of the abscisses dq are non-distinct.

[page 8] This linear system can be written in matrix form as: $\mathbf A \mathbf \alpha = \mathbf d,\quad\quad$ (13) where the 2N x 2N coefficient matrix A = [A1A2] is defined by $\mathbf A_1 = \left[ \begin{matrix} 1 & \cdots & 1\\ 0 & \cdots & 0\\ -L_1^2 & \cdots & -L_N^2\\ \vdots & \ddots & \vdots\\ 2(1-N)L_1^{2N-1} & \cdots & 2(1-N)L_N^{2N-1} \end{matrix} \right]\quad\quad$ (14) and $\mathbf A_2 = \left[ \begin{matrix} 0 & \cdots & 0\\ 1 & \cdots & 1\\ 2L_1 & \cdots & 2L_N\\ \vdots & \ddots & \vdots\\ 2(N-1)L_1^{2N-2} & \cdots & 2(N-1)L_N^{2N-2} \end{matrix} \right].\quad\quad$ (15) The 2N vector of unknowns α is defined by ${\mathbf \alpha} = \left[\alpha_1 \cdots \alpha_N \; b_1 \cdots b_N\right]^T = \left[ \begin{matrix} {\mathbf a}\\ {\mathbf b} \end{matrix} \right],\quad\quad$ (16) and the known right-hand side is ${\mathbf d} = \left[ \overline{S}^{(N)}_0 \cdots \overline{S}^{(N)}_{2N-1} \right]^T. \quad\quad$ (17) [page 9] As shown below, with the DQMOM approximation the right-hand side of Eq. 18 is closed in terms of the N weights and abscissas. [...] As N increases, the quadrature approximation will approach the exact value, albeit at a higher computational cost. If the abscissas Lα are unique, then A will be full rank. For this case, the source terms for the transport equations of the weights ωα and weighted lengths ${\mathcal L}_\alpha$ can be found simply by inverting A in Eq. 13: If at any point in the computational domain two abscissas are equal, then the matrix A is not full rank (or the matrix is singular), and therefore it is impossible to invert it.

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