Electrostatic simulation

[izyk]

Hi
I'm trying to develop an analytical model for membranes under electrostatic pressure. The classical law governing membrane's behavior is (developed by Timoshenko, "theory of plates and shells"):

laplacian^2 (w(x,y)) = P / ( D*h^3 )

w(x,y) - membrane's deflection
P - applied pressure
D - rigidity
h - membrane's thickness

In case of electrostaic pressure:

Pe = eps * V^2 / (2* (d-w(x,y))^2 )

eps - permittivity
V - applied voltage
d - gap

I've solved this equation numerically (using energy minimization method) and obtained the same results in ANSYS (for square and rectangular membrane). The assumption was that the electrode had the same dimensions as membrane.
After that i tried to change the membrane shape. In equation i used coefficient which represents the elecrode shape:

P = Pe * S(x,y)
S(x,y) = 1 under the electrode
S(x,y) = 1 where there is no electrode

In this case i'm taking into account only perpendicular electrostatic field. I've compared with Ansys and i've obtained difference. The higher applied voltage is, the bigger difference is. Furthermore, the smaller electrode is, the bigger difference is (bigger difference occurs when electrode is on the membrane's edge than in the centre).

Why there is the difference? Does ansys takes into account only perpendicular electrostatic field?

here is the figure showing how the system looks like:

|-----------------------|
|-----------------------| membrane


|-------------|
|-------------| electrode

perpendicular field - only above the electrode, neglecting the field which is under the membrane edge

thanks a lot