Formulation of a computational asymptotic bifurcation theory applicable to hill-top branching and multiple bifurcation analyses

To diagnose hill-top branching and multiple bifurcation, which exhibit two critical eigenvalues of the tangent stiffness matrix in stability problems, a sophisticated computational asymptotic bifurcation theory is developed. The theory generally uses three modes which are composed of two homogeneous solutions (critical eigenvectors) and one particular solution of the singular stiffness equations. The first- and second-order derivatives of the stiffness matrix with respect to nodal degrees-of-freedom (DoF) are required to formulate the proposed computational asymptotic bifurcation theory. In two benchmark problems of hill-top branching and multiple bifurcation, the validation and performance of the proposed theory are discussed.