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In this paper, a finite element (FE) procedure for modeling electrostatically actuated MEMS is presented. It concerns a perturbation method for computing electrostatic field distortions due to moving conductors. The computation is split in two steps. First, an unperturbed problem (in the absence of certain conductors) is solved with the conventional FE method in the complete domain. Second, a perturbation problem is solved in a reduced region with an additional conductor using the solution of the unperturbed problem as a source. When the perturbing region is close to the original source field, an iterative computation may be required. The developed procedure offers the advantage of solving sub-problems in reduced domains and consequently of benefiting from different problem-adapted meshes. This approach allows for computational efficiency by decreasing the size of the problem.

Increased functionality of MEMS has lead to the development of micro-structures that are more and more complex. Besides, modeling tools have not kept the pace with this growth. Indeed, the simulation of a device allows to optimize its design, to improve its performance, and to minimize development time and cost by avoiding unnecessary design cycles and foundry runs. To achieve these objectives, the development of new and more efficient modeling techniques adapted to the requirements of MEMS, has to be carried out [

Several numerical methods have been proposed for the simulation of MEMS. Lumped or reduced order models and semi-analytical methods [

The scope of this work is to introduce a perturbation method for the FE modeling of electrostatically actuated MEMS. An unperturbed problem is first solved in a large mesh taking advantage of any symmetry and excluding additional regions and thus avoiding their mesh. Its solution is applied as a source to the further computations of the perturbed problems when conductive regions are added [

As test case, we consider a micro-beam subjected to an electrostatic field created by a micro-capacitor. The micro-beam is meshed independently of the complete domain between the two electrodes of the device. The electrostatic field is computed in the vicinity of the corners of the micro-beam by means of the perturbation method. For the sake of validation, results are compared to those calculated by the conventional FE approach. Furthermore, the accuracy of the perturbation method is discussed as a function of the extension of the reduced domain.

We consider an electrostatic problem in a domain Ω, with boundary ∂Ω, of the 2-D or 3-D Euclidean space. The conductive parts of Ω are denoted Ω_{c}. The governing differential equations and constitutive law of the electrostatic problem in Ω are [

The electrostatic problem can be calculated as a solution of the electric scalar potential formulation obtained from the weak form of the Laplace _{Ω} and 〈·,·〉_{Γd} respectively denote a volume integral in Ω and a surface integral on Γ of products of scalar or vector fields. The surface integral term in (3) is used for fixing a natural boundary condition (usually homogeneous for a tangent field constraint) on a portion Γ of the boundary of Ω;

Hereafter, the subscripts _{c,p}_{c,p}_{p}_{c,p}_{p}

Particularizing

_{p}_{u}_{p}_{u}_{p}_{u}_{p}_{u}_{u}_{p}_{u}

For added perfect conductors, carrying floating potentials, one must have _{p}_{∂}_{Ωc,p}= 0 and consequently _{∂Ωc,p}= −_{∂Ωc,p}. This leads to the following condition on the perturbation electric scalar potential

This way, _{u}

Two independent meshes are used. A mesh of the whole domain without any additional conductive regions and a mesh of the perturbing regions. A projection of the results between one mesh and the other is then required.

Particularizing (_{u}

The source of the perturbation problem, _{s}_{c}_{,}_{p}_{u}_{c}_{,}_{p}_{c}_{,}_{p}_{c}_{,}_{p}_{s}_{s}_{c}_{,}_{p}

In case of a dielectric perturbing region, the projection should be extended to the whole domain Ω_{c}_{,}_{p}_{u}

The perturbation problem is completely characterised by (3) applied to the perturbation potential υ as follows
_{s}_{∂}_{Ω}_{c}_{,}_{p}

For a micro-beam subjected to a floating potential and placed inside a parallel-plate capacitor (_{p}

When the perturbing region Ω_{c,p}

For each iteration _{2i} in Ω, with _{0}= _{u}_{c,p}_{s,2i+1} for a perturbation problem. This way, we obtain a potential _{2i}_{+1} in ∂Ω_{c}_{,}_{p}_{c}_{s,}_{2i+2} for the initial configuration has then to be calculated. This is done by projecting _{2i+}_{1} from its support mesh to that of Ω as follows

A new perturbation electric scalar potential problem is defined in Ω as
_{2i+}_{2} = −_{s,2i+}_{2}∣_{∂Ω}_{c}

This iterative process is repeated until convergence for a given tolerance.

A parallel-plate capacitor (_{c}_{c}_{,}_{p}_{1} of the electrode at 1V.

First, we study the accuracy of the perturbation method as a function of the size of the perturbing domain. In this case, _{1} = 75

The electrostatic field between the plates of the capacitor is first calculated in the absence of the micro-beam. The solution of this problem is then evaluated on the added micro-beam and used as a source for the so-called perturbation problem.

In

Comparing with the conventional FE solution, we observe that the relative error of the local electric field is under 1.2% when the perturbation domain is extended to infinity through a shell transformation (

The relative error of the electric potential and the electric field near the micro-beam increases when the latter is close to electrode at 1V (

To illustrate the iterative perturbation process, the distance _{1} = 3

At iteration 0, the unperturbed electric potential scalar is computed in the whole domain Ω. Projecting this quantity in the domain Ω_{p}_{p}_{p}_{p}_{p}

In order to highlight the relationship between the distance separating the micro-beam and electrode at 1V and the number of iterations required to achieve the convergence without and with Aitken ac-celeration [_{1} of the micro-device are considered (

As expected, several iterations are needed to obtain an accurate solution when the micro-beam is close to the considered electrode. When the Aitken accelaration is used, the number of iterations is reduced.

A perturbation method for computing electrostatic field distortions due to the presence of conductive micro-structure has been presented. First, an unperturbed problem (in the absence of certain conductors) is solved with the conventional FE method in the complete domain. Second, a perturbation problem is solved in a reduced region with an additional conductor using the solution of the unperturbed problem as a source.

In order to illustrate and validate this method, we considered a 2-D FE model of a capacitor and a moving micro-beam. Results are compared to those obtained by the conventional FE method. When the moving region is close to the electrostatic field source, several iterations are required to obtain an accurate solution. Successive perturbations in each region are thus calculated not only from the original source region to the added conductive perturbing domain, but also from the latter to the former. The Aitken acceleration has been applied to improve the convergence of the iterative process.

This work was supported by the Belgian French Community (ARC 03/08-298) and the Belgian Sci-ence Policy (IAP P6/21).

^{2}process, of Wynn's epsilon algorithm and of Brezinski's iterated theta algorithm

A moving micro-beam carrying a floating potential inside a parrallel-plate capacitor

Mesh of Ω (1); distribution of the unperturbed electric potential _{u}_{u}_{p}_{p}_{p}

Meshes for the perturbation problems without

_{p}_{p}

Relative error of _{p}(left)_{p}

Relative error of _{p}_{p}

Iteration numbers to achieve the convergence versus the distance separating electrode at 1V and the micro-beam