Eigenvalue Analysis of Timoshenko Beams and Mindlin Plates with Unfitted Finite Element Methods

This paper focuses on finite element methods for eigenvalue analysis of arbitrarily shaped domains with multimaterial or material–void interfaces that do not follow element boundaries. Such configurations are found in problems with evolving discontinuities and interfaces as in, for example, fluid–structure interaction, topology optimization, and crack propagation problems. The differential equations considered include the elliptic operator, Timoshenko beam, and Mindlin plate. The compatibility conditions at the interface are weakly imposed using either Nitsche’s method or Lagrange multipliers, and results are benchmarked against interface-fitted discretizations. Nitsche’s method shows a direct dependence on a penalty parameter, and the Lagrange multiplier method requires additional degrees of freedom but yields robustness. The convergence rate of the discretized forms is computationally determined and shown to be optimal for eigenvalue analysis of both Timoshenko beams and Mindlin plates.