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Typus
Verschleierung
Bearbeiter
Graf Isolan
Gesichtet
No.png
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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