A Finite Element Formulation for Unbounded Homogeneous Continua

Currently available finite element formulations to model dynamic responses of unbounded continua cannot accommodate the Sommerfeld radiation condition. Discrete numerical techniques explicitly rely on analytical expressions of frequency-dependent field variables to account for the energy loss associated with outgoing waves. In the proposed method, such a dependence is eliminated. For steady dynamic responses, the analog of the quadratic eigenvalue problem (occurring in continuum mechanics) is herein constructed in the form of a quadratic matrix equation. The unknown relates to the desired stiffness matrix pertaining to the infinite or semi-infinite domain. The matrix coefficients are obtained from the conventional mass and stiffness matrices of a suitably chosen, bounded finite element. A benchmark example is included in this paper to demonstrate the very high numerical accuracy and significant computational efficiency of the proposed cloning algorithm.