In this paper, we present a semi-numerical method for determining the dynamics of micro-resonators with finite width immersed in incompressible viscous fluids. The micro-resonator is modeled using Kirchhoff plate theory, and the hydrodynamic force acting on the plate is determined from a boundary integral formulation of the Stokes equations. The resulting equation of motion is solved with a continuous/discontinuous finite element method in which an interior penalty term imposes $C^1$-continuity to the plate’s deflection. Numerical investigations show the method to be convergent with an exponent of the convergence rate equals 2. Examples demonstrate that the proposed method overcomes the limitations of existing semi-analytic methods, only applicable to beam geometries, considering arbitrary plate modes in the structure’s dynamics and their effects on the fluid flow. Different resonator geometries are investigated for which displacement spectrum, mode shapes, and quality factors are not determinable with existent semi-analytic methods. Moreover, we find excellent agreement between simulated and experimental data, which has not been achieved even with purely numerical methods. The present method will allow the understanding of high quality factors of wide micro-resonators in viscous fluids and facilitate new applications in liquid atomic force microscopy and gas sensing in ambient and low-pressure conditions.

在本文中，我们提出了一种半数值方法，用于确定浸入不可压缩粘性流体中的具有有限宽度的微谐振器的动力学。微谐振器使用基尔霍夫板理论建模，作用在板上的流体动力由斯托克斯方程的边界积分公式确定。由此产生的运动方程用连续/不连续有限元方法求解，其中内部惩罚项强加$C^1$- 板偏转的连续性。数值研究表明该方法收敛，收敛速度的指数等于 2。 实例表明，该方法克服了现有半解析方法的局限性，仅适用于梁几何，考虑结构动力学中的任意板模式及其对流体流动的影响。研究了不同的谐振器几何形状，对于这些几何形状，使用现有的半解析方法无法确定位移谱、模式形状和品质因数。此外，我们发现模拟数据和实验数据之间具有极好的一致性，即使使用纯数值方法也无法实现。