Numerical Approaches for Recovering the Deformable Membrane Profile of Electrostatic Microdevices for Biomedical Applications

Recently, a circular symmetrical nonlinear stationary 2D differential model for biomedical micropumps, where the amplitude of the electrostatic field is locally proportional to the curvature of the membrane, was studied in detail. Starting from this, in this work, we first introduce a positive and limited function to model the dielectric properties of the material constituting the membrane according to experimental evidence which highlights that electrostatic capacitance variation occurs when the membrane deforms. Therefore, we present and discuss algebraic conditions of existence, uniqueness, and stability, even with the fringing field formulated according to the Pelesko–Driskoll theory, which is known to take these effects into account with terms characterized by reduced computational loads. These conditions, using “gold standard” numerical approaches, allow the optimal numerical recovery of the membrane profile to be achieved under different load conditions and also provide an important criterion for choosing the intended use of the device starting from the choice of the material constituting the membrane and vice versa. Finally, important insights are discussed regarding the pull-in voltage and electrostatic pressure.