An asymmetric indirect Trefftz method for solving free-vibration problems

In this paper, a new asymmetric indirect Trefftz method (AITM) has been developed to solve free-vibration problems. The proposed method is categorized into a regular type of boundary element methods (BEMs) such that no singular or hypersingular integration is necessary. However, like other regular BEMs the proposed approach encounters the numerical instability as the number of elements increases. To deal with such an ill-posed behavior, Tikhonovʹs regularization method in conjunction with the generalized singular-value decomposition (GSVD) is adopted. It is proved that the degeneracy of the proposed indirect Trefftz method has the same mathematical structure as the direct Trefftz method. Thus, no special effort should be paid in programming. Besides, such an equivalency indicates that the current method does not have spurious eigensolutions. Furthermore, the proposed approach can easily treat a multiply connected domain of genus 1 as shown in Fig. 1. Due to its indirect nature, the present approach can also represent the mode shapes within its own mathematical formulations. Several numerical examples are given to show the validity of the proposed approach.