The aim of this paper is to deal with multi-physics simulation of micro-electro-mechanical systems (MEMS) based on an advanced numerical methodology. MEMS are very small devices in which electric as well as mechanical and fluid phenomena appear and interact. Because of their microscopic scale, strong coupling effects arise between the different physical fields, and some forces, which were negligible at macroscopic scale, have to be taken into account. In order to accurately design such micro-electro-mechanical systems, it is of primary importance to be able to handle the strong coupling between the electric and the mechanical fields. In this paper, the finite element method (FEM) is used to model the strong coupled electro-mechanical interactions and to perform static and transient analyses taking into account large mesh displacements. These analyses will be used to study the behaviour of electrostatically actuated micro-mirrors.
Classical methods used to simulate the coupling between electric and mechanical fields are usually based on staggered procedures, which consist in computing quasistatic configurations using two separate models: a structural model loaded by electrostatic forces predicted at the current iteration step and an electrostatic model defined on the current deformed structure. Staggered iterations then lead to the static equilibrium position. In this paper, a fully coupled electro-mechanical FE formulation is proposed, which allows static equilibrium positions to be computed in a non-staggered way, and which provides fully consistent tangent stiffness matrices that can be used for transient analyses.
In the classical approach, staggered coupling procedures are typically used to simulate electromechanically coupled systems. The electrostatic and the mechanical domains are discretised and solved independently in different analysis steps. Iterations are then performed: the electrostatic field is first computed using, for instance, the boundary element method or the finite element method and it provides the electrostatic forces acting on the mechanical structure. Then a mechanical FEM code computes the structure deformation under the effect of electrostatic forces. The deformed structure defines new boundaries for the electrostatic problem and the electric field has to be computed again. This method is commonly presented in the literature (see e.g. Lee and al. [1]) and is illustrated in figure 1. The staggered method can also be understood as block-Gauss Seidel procedure. The method proposed here consists in considering the monolithically coupled problem. The electric and mechanical fields are computed simultaneously in a unified formulation. Since the problem is non-linear, it must be solved by an iterative algorithm such as the Newton-Raphson method or Riks-Crisfield method. The solution strategy consists in the following steps. Given an electric potential applied on the boundaries of the structure, a first solution is obtained by considering the coupled problem linearised around an initial configuration. The resulting structural deformation then defines a modified electric domain and a new linearised problem is defined around the modified configuration. This process continues until the solution has converged, namely until the electric and mechanical equilibrium are satisfied up to a prescribed tolerance (Figure 2). The monolithic formulation has many advantages compared to the staggered method. Indeed the tangent stiffness matrix can be explicitly constructed for the entire system. Therefore, the stability and the natural frequency of the electrostatically coupled structure can easily be evaluated around a given equilibrium configuration [3]. Also the staggered coupled method becomes numerically unstable when the pull-in voltage is reached [2]. Finally, the coupled method permits to pass over the pull-in voltage. Strong coupled methods are also proposed by ANSYS and FEMLAB to obtain the tangent stiffness matrix.
The software ANSYS proposes different solutions to achieve the equilibrium between electrostatic forces and mechanical structure [2,4,5]. The first one is a sequential coupling electro-mechanical solver called ESSOLV which uses the staggered methodology as presented previously. The second possibility is to use TRANS126 element which is a strong coupled transducer formed by 2-nodes line [5]. The vacuum between the plates is discretised by one-dimensional condensers To simulate very strong coupling in two dimensions TRANS109 element is usually used [4]. This transducer is based on virtual principle. It is represented by a finite element 3-nodes triangle with electric and mechanical degrees of freedom. A standard Ritz-Galerkin approach is applied to compute the energy in the triangle. The forces in each node are obtained by deriving this energy by the nodes displacement. This methodology is used when the capacitance-stroke relation is difficult to obtain. The advantage of this element is its very accurate precision to compute the electrostatic forces and the electromechanical coupling, but the convergence is sensitive to the mesh discretisation and it is less robust for devices that experience large deformation [2]. An extension in three dimensions of this element has been performed by Avdeev [6]. It is called ``3D strongly coupled tetrahedral transducer’’. Displacement and voltage are the variables assigned to the nodes. The energy in one element is computed by the relation:
where V is the electric potential, Ω the volume, h i is the tetrahedral altitudes and n i the interface normal vector of the plane i as shown in figure 3. To compute electrostatic forces and tangent stiffness matrix, the altitudes and the normal vector are derived by the displacement of the nodes. This element as well as the TRANS109 element, has an automatic mesh morphing based on equilibrium considerations. The initial mesh is automati
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