BandGeneral SOE: Difference between revisions
No edit summary |
No edit summary |
||
| Line 11: | Line 11: | ||
NOTES: | NOTES: | ||
---- | ---- | ||
| Line 18: | Line 16: | ||
THEORY: | THEORY: | ||
An ''n''×''n'' matrix ''A''=(''a''<sub>''i,j'' </sub>) is a '''band matrix''' if all matrix elements are zero outside a diagonally bordered band whose range is determined by constants ''k''<sub>1</sub> and ''k''<sub>2</sub>: | |||
:<math>a_{i,j}=0 \quad\mbox{if}\quad j<i-k_1 \quad\mbox{ or }\quad j>i+k_2; \quad k_1, k_2 \ge 0.\,</math> | :<math>a_{i,j}=0 \quad\mbox{if}\quad j<i-k_1 \quad\mbox{ or }\quad j>i+k_2; \quad k_1, k_2 \ge 0.\,</math> | ||
Revision as of 01:12, 9 March 2010
- Command_Manual
- Tcl Commands
- Modeling_Commands
- model
- uniaxialMaterial
- ndMaterial
- frictionModel
- section
- geometricTransf
- element
- node
- sp commands
- mp commands
- timeSeries
- pattern
- mass
- block commands
- region
- rayleigh
- Analysis Commands
- Output Commands
- Misc Commands
- DataBase Commands
This command is used to construct a BandGeneralSOE linear system of equation object. As the name implies, this class is used for matrix systems which have a banded profile. The matrix is stored as shown below in a 1dimensional array of size equal to the bandwidth times the number of unknowns. When a solution is required, the Lapack routines are used. To following command is used to construct such a system:
| system BandGeneral> |
NOTES:
THEORY:
An n×n matrix A=(ai,j ) is a band matrix if all matrix elements are zero outside a diagonally bordered band whose range is determined by constants k1 and k2:
- <math>a_{i,j}=0 \quad\mbox{if}\quad j<i-k_1 \quad\mbox{ or }\quad j>i+k_2; \quad k_1, k_2 \ge 0.\,</math>
The quantities k1 and k2 are the left and right half-bandwidth, respectively. The bandwidth of the matrix is k1 + k2 + 1 (in other words, the smallest number of adjacent diagonals to which the non-zero elements are confined).
and matrices are usually stored by storing the diagonals in the band; the rest is implicitly zero.
For example, 6-by-6 a matrix with bandwidth 3:
- <math>
\begin{bmatrix}
B_{11} & B_{12} & 0 & \cdots & \cdots & 0 \\
B_{21} & B_{22} & B_{23} & \ddots & \ddots & \vdots \\
0 & B_{32} & B_{33} & B_{34} & \ddots & \vdots \\
\vdots & \ddots & B_{43} & B_{44} & B_{45} & 0 \\
\vdots & \ddots & \ddots & B_{54} & B_{55} & B_{56} \\
0 & \cdots & \cdots & 0 & B_{65} & B_{66}
\end{bmatrix} </math> is stored as the 6-by-3 matrix
- <math>
\begin{bmatrix}
0 & B_{11} & B_{12}\\
B_{21} & B_{22} & B_{23} \\
B_{32} & B_{33} & B_{34} \\
B_{43} & B_{44} & B_{45} \\
B_{54} & B_{55} & B_{56} \\
B_{65} & B_{66} & 0
\end{bmatrix}. </math>
Code Developed by: fmk