Modified stiffness iteration to pinpoint multiple bifurcation points Original Research Article

A modified stiffness iteration to precisely compute a bifurcation point with multiple zero eigenvalues is presented. The iteration combines the direct pinpointing iteration for simple bifurcation points with a procedure to modify the tangent stiffness matrix based on pertinent congruent transformation to reduce the multiple eigenvalues to single ones. Two different transformation matrices are employed to modify the stiffness matrix: One is a non-orthogonal transformation matrix amplifying the values of the entries in a certain row and a column of the stiffness matrix. The other is an orthogonal one, which is chosen with reference to the symmetry of the system under consideration. The use of the non-orthogonal transformation matrix is a key idea termed “stiffness amplification method” in this paper, whereas the orthogonal matrix is based on block-diagonalization, which is becoming popular in group-theoretic analysis of symmetric structures. Extensive numerical examples show the robustness and usefulness of the proposed iteration method for pinpointing multiple bifurcation points.