2019/07/10 by Benjamin Jurgelucks, Jurgelucks, Benjamin, Tom Lahmer +3
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1907.04555
openalex publication_date 2019/07/10 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
Piezoelectric appliances have become hugely important in the past century and\ncomputer simulations play an essential part in the modern design process\nthereof. While much work has been invested into the practical simulation of\npiezoelectric ceramics there still remain open questions regarding the partial\ndifferential equations governing the piezoceramics. The piezoelectric behavior\nof many piezoceramics can be described by a second order coupled partial\ndifferential equation system. This consists of an equation of motion for the\nmechanical displacement in three dimensions and a coupled electrostatic\nequation for the electric potential. Furthermore, an additional Rayleigh\ndamping approach makes sure that a more realistic model is considered. In this\nwork we analyze existence, uniqueness and regularity of solutions to theses\nequations and give a result concerning the long-term behavior. The\nwell-posedness of the initial boundary value problem in a bounded domain with\nsufficiently smooth boundary is proved by Galerkin approximation in the\ndiscretized weak version, followed by an energy estimation using Gronwall\ninequality and using the weak limit to show the results in the infinite\ndimensional space. Initial conditions are given for the mechanical displacement\nand the velocity.\n