Topology optimization of electrostatic MEMS including stability constraints

Among actuation techniques available for MEMS devices, electrostatic actuation is often used as it provides a short response time and is relatively easy to implement. However, these actuators possess a limit voltage called pull-in voltage beyond which they are unstable. The pull-in effect, can eventually damage the device since it can be impossible to separate the electrodes afterward. Consequently, pull-in phenomenon should be taken into account during the design process of electromechanical microdevices to ensure that it is avoided within utilization range. In this thesis, a topology optimization procedure which allows controlling pull-in phenomenon during the design process is developed.

A first approach is based on a simplified optimization problem where the optimization domain is separated from the electrical domain by a perfectly conducting material layer making the optimization domain purely mechanical. This assumption reduces the difficulty of the optimization problem as the location of the electrostatic forces is then independent from the design. However, it allows us to develop and validate a design function based on pull-in voltage in the framework of a topology optimization problem.

Nevertheless, in some applications, the developed pull-in voltage optimization procedure suffers from design oscillations that prevent from reaching solution. In order to solve this issue, we propose to investigate an alternative approach consisting in formulating a linear eigenproblem approximation for the nonlinear stability problem. The first eigenmode of the proposed stability eigenproblem corresponds to the actual pull-in mode while higher order modes allow estimating upcoming instability modes. By including several instability modes into a multiobjective formulation, it is possible to circumvent the oscillations encountered with pull-in voltage design function.

Next, the possibility to generalize the pull-in optimization problem by removing the separation between optimization and electrostatic domains is studied. Unlike the original method, the dielectric permittivity has then to depend on the pseudo-density like the Young Modulus to represent the different electrostatic behavior of void and solid. Additionally, in order to render perfect conducting behavior for the structural part of the optimization domain, a fictitious permittivity is also introduced into the material model. Difficulties caused by non-physical local instability modes could be solved by using a force filtering technique which removes electrostatic forces originating from numerical inaccuracies of the modeling method. Thanks to these improvements, the optimization problem based on the pull-in design function can be generalized. As a result, the optimizer is able to adapt the electrostatic force distribution applied on the structure which leads to a higher efficiency of the optimal device.

In order to illustrate the interest of the pull-in voltage design function, the pull-in voltage optimization problem is merged with the electrostatic actuator optimization problem. In this new optimization problem, the pull-in voltage does not appear anymore in the objective function but in a constraint which prevents the pull-in voltage to decrease below a given minimal value. Firstly, the new optimization problem is compared to the basic electrostatic actuator design procedure on basis of a numerical application. The pull-in voltage constraint proved to be very useful since it prevents the pull-in voltage of the mechanism to decrease below the driving voltage during the optimization process. Finally, the effect of geometric nonlinearity modeling is also tested on numerical applications of our optimization procedure.