Procedia Engineering 87 ( 2014 ) 480 – 483

1877-7058 © 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Peer-review under responsibility of the scientific committee of Eurosensors 2014doi: 10.1016/j.proeng.2014.11.400

ScienceDirectAvailable online at www.sciencedirect.com

EUROSENSORS 2014, the XXVIII edition of the conference series

Perforated plates of inertial sensors – modeling by effective material properties

S. Michaela,*, A. Franka, G.Hölzerb, G. Lorenzc

aInstitut für Mikroelektronik- und Mechatronik-Systeme gemeinnützige GmbH, Ehrenbergstr. 27, 98693 Ilmenau, Germany bXFAB-AG, Haarbergstr. 67, 99097 Erfurt, Germany

cCoventor, 3 avenue de Quebec, 91140 Villebon sur Yvette, France

Abstract

In this paper a method for the modeling of flexible perforated plates by continuous ones is presented using the example of an accelerometer. This model order reduction (MOR) by effective material properties enables the accurate simulation of out-of-plane modes needed for drop tests e.g. based on the system level design tool MEMS+ library building blocks.

© 2014 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of the scientific committee of Eurosensors 2014.

Keywords: modeling, inertial sensors, effective material properties, perforated plates

1. Introduction

A commonly used technology for the processing of inertial sensors is based on SOI wafers. Inertial sensors processed on SOI technology show the characteristic release-etch perforation of movable masses to release the device layer by dry etching from the handle layer. The modeling of such flexible perforated structures requires in case of standard FE programs like ANSYS a huge amount of finite elements leading often to unmanageable memory demands and long simulation times. Furthermore, an accurate modeling of flexible perforated plates is not yet available in library based system level design tools such as MEMS+. System level model relies either on the

* Corresponding author. Tel.: +49-3677-6955; fax: +49-3677-695515. E-mail address: steffen.michael@imms.de

© 2014 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).Peer-review under responsibility of the scientific committee of Eurosensors 2014

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481 S. Michael et al. / Procedia Engineering 87 ( 2014 ) 480 – 483

0 5 10 15 2046

48

50

52

54

56

58

60

62

64

nbcells

Ex/

y [MP

a]

perf. elements + edge (type 1)perf. elements (type 2)

assumption of rigid perforated plates or flexible plates without perforation. Both assumptions have proven to be inadequate to more large thin plates.

In [1,2] the replacement of circular perforated structures by continuous ones of the same overall dimensions with similar load-displacement characteristics is described. The perforated structure is subdivided into representative unit cells. A similar load-displacement characteristic of a continuous unit cell by comparison to the perforated one is obtained by the introduction of effective material parameters (density and compliance respectively stiffness matrix).

2. Modeling approach

A MOR by effective material properties is done for the mechanical simulation of a 100g accelerometer (Fig. 1). Structures with periodic elements like the perforated mass and the comb structures are replaced by rectangles respectively cuboids with effective material properties. a b

Fig. 1: Quarter model of the accelerometer (a) before ROM; (b) after ROM

The perforated mass has a perforation width of 4µm. Due to an edge width of also 4µm two different unit cell types can be introduced – one unit cell including the edge (type 1) and one without edge (type 2). a b

c

Fig. 2:( a) Effective Young’s Modulus versus number of UC; (b) 2x2 UC matrix type 1;( c) 2x2 UC matrix type 2

Fig. 2a shows the influence of unit cell numbers towards the effective Young’s Modulus in lateral direction. The Young’s Modulus of the type 2 unit cell plotted as reference value is per definition constant, the edge including unit

482 S. Michael et al. / Procedia Engineering 87 ( 2014 ) 480 – 483

cell type 1 shows the expected reciprocally proportional behavior at increasing unit cell number respectively inertial sensor dimension. With respect to calculation time efficiency unit cell type 2 without edge is selected for the replacement of the perforated structures. Hence the unit cell is independent of inertial sensor dimensions, and the perforated mass is replaced by a polygon with the unit cell material surrounded by an edge of base material silicon (width 2µm).

3. Calculation of effective material parameters

The corresponding properties of a continuous unit cell are calculated within a separate static simulation run. The perforation causes beside a lower effective density weaker elastic constants and a directional dependency as well. The calculation of the orthotropic material parameters requires a static simulation with four different load steps for the identification of the shear modulus components Gxy and Gxz as well as the Young’s Modulus components Ex and Ez and the corresponding Poisson’s ratio. The remaining components Gyz and Ey are covered by the symmetry of the quadratic perforation. The modulus components are obtained by a constrained deformation on one hand and the post-processing of the nodal reaction forces on the other hand. The nodal solution is used to prevent numerical errors in case of using directly the element solutions for strain and stress respectively.

Using isotropic Silicon as material of the perforated mass with

GPaGGPaE 69;22.0;169 === ν (1)

results in an orthotropic material for the continuous plate with the following elasticity parameters:

GPaGGPaE

GPaGGPaE

GPaGGPaE

xzxzz

yzyzy

xyxyx

6914.0169

6914.02.48

2107.02.48

=========

ννν

(2)

To get the effective elasticity parameters of the substituted comb structures the base elasticity parameters are corrected by the fill factor of the comb structure (ratio of comb width to comb width plus gap).

4. Results

Modal reference simulations for the base accelerometer model are done with ANSYS. The given accelerometer model is then transformed automatically in a model with continuous mass and comb structures by means a MATLAB function which reduces the number of FE model nodes by factor of five.

Fig. 3: Modal analysis of first out-of-plane frequency with ANSYS (f=65.1kHz)

483 S. Michael et al. / Procedia Engineering 87 ( 2014 ) 480 – 483

In MEMS+ perforated plates can be modelled by the library element rigid plate. The transformation of the rigid perforated plates into flexible plates with the effective material parameters and the surrounding edge with the base material parameter is done by a MATLAB function with respect to usability.

Fig. 4: Modal analysis of first out-of-plane frequency after MOR with MEMS+ (f=69 kHz, quarter model).

The comparison of the modal analysis simulation results of the base accelerometer model (Fig. 3) and the reduced model with continuous flexible elements simulated in MEMS+ (Fig. 4) shows a good concordance.

5. Summary

The presented method permits an efficient and accurate mechanical modeling of inertial sensors with perforated masses. In case of using standard FE programs like ANSYS the number of model nodes can be reduced by the factor of five and results in a significant reduction of simulation time. The method expands the library based system level design tool MEMS+ by the modeling of out-of-plane modes of perforated structures.

Acknowledgements

The work has been carried out in the research project MEMS2015 (support code 01M3093G) funded by the German Federal Ministry of Education and Research (BMBF).

References

[1] V. Rabinovich, R. Gupta and S. Senturia, The Effect of Release-Etch Holes on the Electromechanical Behavior of MEMS Structures, Solid State Sensors and Actuators, Vol. 2 (1997), p. 1125-1128.

[2] W. Yu, A variational-asymptotic cell method for periodically heterogeneous materials. In Proceedings of the 2005 ASME International Mechanical Engineering Congress and Exposition, Orlando, Florida, Nov. 5-11 2005. ASME.