A Model Order Reduced Finite Element Approach for the Multiscale Simulation of Ferroelectric Structures
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Ferroelectric materials are widely used as smart structures in electromechanical devices, e.g., actuators, sensors, etc. The computational analysis of their multiscale constitutive behavior in technical applications is both a relevant and a demanding task. Contemporary multiscale approaches, like the FE2-method, provide a powerful basis for that purpose, however, at the expense of high computational cost. In [1] an effective scale–bridging technique, the so–called Condensed Method (CM), is introduced to investigate the polycrystalline ferroelectric behavior of a mesoscopic volume element (MVE) at the corresponding macroscopic material point (MMP) without any kind of discretization scheme. Besides the constitutive modeling of ferroelectrics, other fields of application of the CM have been exploited, e.g., dissipation heating, ferromagnetic and multiferroic behavior. Since just the behavior at a single MMP is represented by the CM, the method itself is unable to solve arbitrary boundary value problems. In this contribution a methodology is presented which integrates the CM into a Finite Element (FE) environment [2], thus accounting for domain switching on the microscale, grain interactions on the mesoscale and homogenized mechanical and electrical fields on the macroscale. This approach finally allows for a computational investigation of arbitrary ferroelectric structures, whereupon a spatial discretization is efficiently restricted to the macroscale. In a more sophisticated concept, the mesostructure is decoupled from the FE discretization by introducing an independent MVE grid, providing a basis of reduced order modeling, thus saving even more computational cost. Numerical examples are finally presented in order to verify the approach, including energetic consistency of the MVE grid. Reference [1] Lange S., Ricoeur A., A condensed microelectromechanical approach for modeling tetragonal ferroelectrics, International Journal of Solids and Structures, Vol. 54, pp. 100 – 110, 2015. [2] Wakili R., Lange S., Ricoeur A., FEM-CM as a hybrid approach for multiscale modeling and simulation of ferroelectric boundary value problems, Computational Mechanics, Vol. 72, pp. 1295 – 1313, 2023.
