Innovative Numerical Methods for Nonlinear MEMS:\\ the Non-Incremental FEM vs. the Discrete Geometric Approach

Electrostatic microactuator is a paradigm of MEMS. Cantilever and double clamped microbeams are often used in microswitches, microresonators and varactors. An efficient numerical prediction of their mechanical behaviour is af- fected by the nonlinearity of the electromechanical coupling. Sometimes an addi- tional nonlinearity is due to the large displacement or to the axial-flexural coupling exhibitedin bending. To overcome thecomputationallimitsoftheavailablenumer- ical methods two new formulations are here proposed and compared. Modifying the classical beam element in the Finite Element Method to allow the implemen- tation of a Non incremental sequential approach is firstly proposed. The so-called Discrete Geometric Approach (DGA), already successfully used in the numerical analysis of electromagnetic problems, is then applied. These two methods are here formulated, for the first time, in the case of strongly nonlinear electromechanical coupling. Numerical investigations are performed to find the pull-in of microbeam actuators experimentally tested. The non incremental approach is implemented by discretizing both the structure and the dielectric region by means of the FEM, then by meshing the electric domain by the Boundary Element Method (BEM). A preliminary experimental validationis finally presented in the case ofplanar micro- cantilever actuators.

[1]  A new consistent way to build symmetric constitutive matrices on general 2-D grids , 2004, IEEE Transactions on Magnetics.

[2]  Ruben Specogna,et al.  Discrete Constitutive Equations in - Geometric Eddy-Current Formulation , 2005 .

[3]  T. Steinmetz,et al.  Electro-quasistatic field simulations based on a discrete electromagnetism formulation , 2006, IEEE Transactions on Magnetics.

[4]  E. Tonti.  Finite Formulation of the Electromagnetic Field , 2001 .

[5]  Antonio Gugliotta,et al.  Large deflections of microbeams under electrostatic loads , 2004 .

[6]  P. Bettini,et al.  Static Behavior Prediction of Microelectrostatic Actuators by Discrete Geometric Approaches , 2008, IEEE Transactions on Magnetics.

[7]  G. Novati,et al.  Weak Coupling of the Symmetric Galerkin BEM with FEM for Potential and Elastostatic Problems , 2006 .

[8]  G. E. Bridges,et al.  A Geometrical Comparison between Cell Method and Finite Element Method in Electrostatics , 2007 .

[9]  A Special Finite Element for Static and Dynamic Study of Mechanical Systems under Large Motion, Part 2 , 2002 .

[10]  Y. Hon,et al.  Geometrically Nonlinear Analysis of Reissner-Mindlin Plate by Meshless Computation , 2007 .

[11]  Gabriel M. Rebeiz.  RF MEMS: Theory, Design and Technology , 2003 .

[12]  Francesca Cosmi,et al.  Numerical Solution of Plane Elasticity Problems with the Cell Method , 2001 .

[13]  R. Specogna,et al.  Symmetric Positive-Definite Constitutive Matrices for Discrete Eddy-Current Problems , 2007, IEEE Transactions on Magnetics.

[14]  R. A. Adey,et al.  Simulation and Design of Microsystems and Microstructures , 1996 .

[15]  Enzo Tonti,et al.  A Direct Discrete Formulation of Field Laws: The Cell Method , 2001 .

[16]  Eugenio Brusa,et al.  Validation of compact models of microcantilever actuators for RF-MEMS application , 2008, 2008 Symposium on Design, Test, Integration and Packaging of MEMS/MOEMS.

[17]  F. Trevisan.  3-D eddy current analysis with the cell method for NDE problems , 2004, IEEE Transactions on Magnetics.

[19]  Eugenio Brusa,et al.  Geometrical nonlinearities of electrostatically actuated microbeams , 2004 .

[20]  D. Ostergaard,et al.  Electro-Mechanical Transducer for MEMS Analysis in ANSYS , 1999 .

[21]  Eniko T. Enikov,et al.  Microsystems mechanical design , 2006 .

[22]  Rolf Schuhmann,et al.  解説 Discrete Electromagnetism by the Finite Integration Technique , 2002 .

[23]  Xiangyang Cui,et al.  A Smoothed Finite Element Method (SFEM) for Linear and Geometrically Nonlinear Analysis of Plates and Shells , 2008 .

[24]  Alain Bossavit,et al.  Yee-like schemes on staggered cellular grids: a synthesis between FIT and FEM approaches , 2000 .

[25]  A. Bossavit.  How weak is the "weak solution" in finite element methods? , 1998 .

[26]  Satya N. Atluri,et al.  On Simple Formulations of Weakly-Singular Traction {\&} Displacement BIE, and Their Solutions through Petrov-Galerkin Approaches , 2003 .

[27]  E. Brusa,et al.  Experimental characterization of electrostatically actuated in-plane bending of microcantilevers , 2008 .

[28]  P. Bettini,et al.  Electrostatic analysis for plane problems with finite formulation , 2003 .

[29]  G. Benderskaya,et al.  Transient electro-quasistatic adaptive simulation schemes , 2004, IEEE transactions on magnetics.