Finite Element Tearing and Interconnecting Method

Let us start with derivation of basic equations. The energy functional of elastic body has the form

where the integrals are taken over all domain. Domain decomposition algorithms split original domain into non-overlapping subdomains with boundaries . At this stage the energy functional can be rewritten into the form

The latest term in (11) enforces the compatibility of displacements via Lagrange multipliers. The discretized form of the energy functional is

There are unknowns vectors in this functional. The extreme value of the energy functional is reached in the stationary point which can be obtained by solving this system

With the usual notation

(15) |

for the stiffness matrix and

(16) |

for the load vector the system of equations has the form

The solution of Eqs (17) and (18) is not straightforward because of the singularity of the matrix . As it was mentioned above the original domain is split into independent subdomains. the continuity on the boundaries is enforced by the Lagrange multipliers. The independence of the subdomains leads to the singularity, if there are not enough supports on the subdomain. Due to the possible singularity of the matrix one cannot write

(19) |

(20) |

because the matrix does not exist. This matrix is replaced by so called pseudoinverse matrix which is defined

The solvability of the system (17) is guaranteed by the condition

where stands for the kernel of the matrix. The kernel is defined

The resulting system of equations can be written with defined pseudoinverse matrix and with the condition (22) as

where the matrix contains rigid body motions of the subdomain as columns. The vector contains the coefficients of linear combinations. After combining relations (24) and (18) one obtains

The solvability condition (22) can be expressed by matrix

For the next purpose these quantities will be defined

The resulting system of equations can be written with quantities from the Eqs (27), (28), (29), (30) and (31) as

The solution of the matrix Eq. (32) will be done by the modified conjugate gradient method. The original conjugate gradient method must be modified because of presence of the additional condition. When the resulting vector of Lagrange multipliers is obtained it is possible to compute the vector of coefficients of the linear combination from equation

The matrix is not a square matrix that means that the inverse matrix does not exist. After multiplying the relation (33) from left by the invertible matrix will be on the left hand side. The vector of the coefficients is expressed

Now there are all necessary data for computing the resulting displacements. The displacement vector is the sum of two parts, the first contribution is vector of the generalized residual

(35) |

and the second one is created by the linear combination of rigid body motions

(36) |

For enough supported subdomain vector is equal to the zero vector. The resulting displacement can be written in the form

*Daniel Rypl
2005-12-03*