Optimal Analysis of Structures

In this paper optimal analysis of structures is presented and recent additions to this fieldare discussed. Analysis of a structure is called optimal if the main structural matrix is sparse,well-structured, and well-conditioned [1-3].A matrix is called sparse if the number of its zero entries is considerably more than itsnon-zero entries, Figure 1. The main method for creating sparse matrices is to form bases ofspecial properties. Such bases can be constructed by guided expansion utilizing optimizationmethods. Guided expansion can also be used in topology optimization of structures. Wellstructuredmatrix is a sparse matrix where its non-zero entries have a special pattern, Figure2. This pattern differs with the algorithm to be utilized for the solution. These patterns can becreated by suitable nodal or/and element ordering. Common example of these patterns isbanded matrices. A well-conditioned structural matrix has its dominant entries in thediagonal, Fig. 3. Dominant off-diagonal entries are the main cause of the ill-conditioning [1-3].

Keywords:

Optimal structural analysis , graph theory , force method , displacement method , ordering , decomposition , finite elements