Applications of Muller’s method and the argument principle to eigenvalue problems in solid mechanics

Numerical implementation of the problems of solid mechanics leads to an algebraic problem of real and complex eigenvalues. Using discrete numerical methods, in particular, the finite element technique, it makes sense to solve only the partial eigenvalue problem both from the standpoint of errors of the corresponding numerical method and the mechanical content of the problems under study. This determines the requirement for an algorithm which implies that eigenvalues must be sought in ascending order. In the present paper, we propose an algorithm to solve an algebraic problem of complex eigenvalues using the Muller method. It is shown that even though the algorithm is an efficient one, it has a drawback related to the condition for evaluating roots in ascending order. In order to eliminate this disadvantage, we apply an additional procedure based on the argument principle to the developed algorithm. A technique for calculating eigenvalues that is built upon the Muller method and the argument principle is described. Relevant studies that demonstrate the use of the proposed algorithm for solving the problems of solid mechanics problems are discussed.