Keywords: FETI-DP method, fixing nodes, interface edges.

This paper deals with a selection algorithm of fixing nodes for the finite element tearing and interconnecting dual-primal (FETI-DP) method. The FETI-DP method is one of non-overlapping domain decomposition methods which combines the finite element tearing and interconnecting (FETI) method and the Schur complement method. The FETI-DP uses special nodes on the interface, called fixing nodes, where the interface continuity is enforced in primal variables similarly to the Schur complement method. The continuity conditions in all remaining interface nodes are enforced using Lagrange multipliers.

If the fixing nodes are selected properly, the subdomain matrices and the matrix of the coarse problem are nonsingular and can be easily factorized. It is a significant difference in comparison with the FETI method where the subdomain matrices can be singular. In order to obtain nonsingular subdomain matrices, a minimum number of fixing nodes has to be used. The number depends on problem solved. For example, at least three nodes have to be selected in the case of three-dimensional elasticity.

Many different software packages are available for mesh decomposition into submeshes. Unfortunately, the authors do not know of software for selection of fixing nodes in the case of irregular subdomains and irregular finite element meshes. The FETI-DP method cannot be used without the fixing nodes.

A selection algorithm for fixing nodes is proposed in this contribution. Nodal multiplicity, which is the number of subdomains which share a node, is determined for each node. The nodal multiplicity is greater than one for the interface nodes. These nodes are mapped to vertices of a graph and the set of edges is constructed based on mesh connectivity. Interface edges and surfaces can be constructed with the help of the graph.

The proposed algorithm can deal with information based only on mesh connectivity or it can exploit geometrical properties, i.e. nodal coordinates. The algorithm selects end nodes of the interface edges first. In the second step, it selects some nodes belonging to an interface edge. The number of nodes selected on each edge can be defined by the user. If the minimum number of fixing nodes should be used, a recursive approach is suitable. It selects randomly one interface node on the first subdomain. The two most farthest nodes from the first one are then selected. The selected nodes are marked on neighbouring subdomains and the algorithm continues to a subdomain, where at least one node is marked.

Behaviour of the proposed selection algorithm is studied on two examples. One is academic example and the second one represents a containment for a nuclear power plant.

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