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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
2005-12-03