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Substructuring method
The substructuring method is based on decomposition of the original structure
into several substructures.
Let us assume the system of linear algebraic equations

(1) 
where the matrix of system is
and vectors
.
It is possible to split original system (1) into blocks

(2) 
If the block
is regular, the vector
can be expressed
from the first matrix equation and has a form

(3) 
The second equation of the original system (2) has modified form
after substitution of (3)

(4) 
Important result of this simple modification is reduction of the number of
unknowns because there is only a part of the original vector
in the
Eq. (4). The matrix
is called Schur's complement in mathematical community and the modification
process expressed by Eq. (3) is called static condensation among
engineers.
Let the original domain is decomposed into subdomains. The stiffness matrix
has a special form thanks to appropriate ordering of the unknowns

(5) 
The following notations is used in Eq. (5):

, where
 vectors of displacements of
internal nodes on the th subdomain,

 vector of boundary displacements,

, where
 vectors of loads acting on
the internal nodes on the th subdomain,

 vector of loads on the boundaries.
Matrix
contains contributions from the subdomains

(6) 
similarly, vector
.
Matrices
and
are transpose because of symmetry of
the stiffness matrix.
Each of the vectors
can be expressed in the form

(7) 
and after substitution to the last Eq. in (5) it follows

(8) 
where the th row is missing. The reduced system of equations is obtained
after substitution of all vectors of internal unknowns to the Eq. (5)
and has a form

(9) 
Only the vector of boundary unknowns occurs in the previous equation. Eq.
(9) is called the resulting system of equations or the reduced system.
Next: Finite Element Tearing and Interconnecting Method
Up: Implicit Algorithms for Solution of Large
Previous: Implicit Algorithms for Solution of Large
Daniel Rypl
20051203