As mentioned above the accuracy of the

As mentioned above, the accuracy of the analytical deflection model depends not only on the diaphragm’s center deflection ω0, but also on the shape of the deformed membrane ω′(x, y). This paper focuses on the determination of the deformed cannabinoid receptor shapes. We will study a much more general case of membranes and formulate highly accurate deflection shape functions for CMUTs with rigidly clamped rectangular membranes. A data fitting technique is applied to identify the parameters in the deflection shape function by using MATLAB. The effectiveness of the deflection shape function will be illustrated by comparing the predicted deflection profiles with FEA results (using ANSYS 15.0 software, ANSYS Inc.). For various clamped CMUT membranes with different geometry dimensions and loading conditions, the new deflection shape function shows excellent agreement with corresponding FEA results.
Finite element analysis of CMUT The 3D FEA model is chosen to simulate the deformation of the membrane. In the finite element simulation, a DC bias voltage (expressed as U) is applied between the electrodes to exert an electrostatic force on the diaphragm. As the bias voltage increases from zero across the membrane and the fixed back plate, the distance between them would decrease until the two plates suddenly snap into contact. This behavior is called the pull-in effect, and the transition voltage is called pull-in voltage. The loading condition on the diaphragm which causes pull-in effect is neglected in the simulation. The device specifications in our simulations are listed in Table 1. The membrane is modeled by SOLID 186 element while TRANS 126 is employed to apply the electrostatic force on the diaphragm. Considering the symmetrical characteristic of the diaphragm, we use only 1/4 of the diaphragm in the simulation so as to improve the computational efficiency. For boundary conditions, the edges of the membrane are strictly clamped. As shown in Table 2, comparison of center deflections between our simulation results and those in [15] has been conducted to verify the effectiveness of our simulation method. Note that the top electrode is ignored at first to keep consistent with simulations in [15]. It can be observed from Table 2 that our simulation results are nearly the same as those in [15], thus demonstrating that our simplified simulation method insures the accuracy while improving the operational speed.
New deflection shape functions In this section, the analytical deflection model for CMUTs with rectangular diaphragms is established. The rectangular diaphragms of interest have a thickness range of 0.6–1.5μm and a side length range of 100–1000μm. Each edge of the rectangular diaphragm is rigidly clamped and the deflection profile is determined by inner strain in the x and y directions as well as external pressure from the z direction. In the analytical model, a is half of the long side, b is half of the short side, h is the thickness of membrane, and denotes the aspect ratio. Assuming that the deflection shape function for the rectangular membranes is similar to that for the square ones in [15] and has the following formwhere the parameters c1, c2, c3 and c4 are determined by comparing the function with the FEA results. ω′(x, 0) and ω′(0, y) represent the deflection shape of the diaphragm along x-axis and y-axis, respectively. Fig. 2 shows the deviation error from the FEA results with the aspect ratio n against the standardized coordinate of x-axis and y-axis when the membrane thickness is 1μm. Negligible deviation error can be observed from Fig. 2(b) as n varies from 1 to 10, which reveals that function (6) can well predict the deformation profile along y-axis. However, Fig. 2(a) shows that the deviation error along x-axis is significant when the aspect ratio n is greater than four, while it is negligible when n is less than 4. Therefore, the function should be modified to improve the accuracy, especially in the cases when .