A Finite Element Formulation for Unbounded Homogeneous Continua

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Abstract
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.

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