International audienceThis work focuses on the modeling of mechanical contact applied to Nano-Electro-Mechanical-Systems (NEMS). A magneto-elastic formulation associated to the definition of unilateral contact in static case is presented. This model is discretised by finite element method and a numerical study of a magnetic nano switch is performed. FEM results are compared to those of a semi-analytical model developed for optimization processes

Coulomb, Jean-Louis

Delinchant, Benoît

Galopin, Nicolas

Pham Quang, Phuong

English

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Finite Element Modeling of deformable NEMS taking
into account mechanical contact
Nicolas Galopin, Phuong Pham Quang, Benoît Delinchant, Jean-Louis
Coulomb
To cite this version:
Nicolas Galopin, Phuong Pham Quang, Benoît Delinchant, Jean-Louis Coulomb. Finite Element
Modeling of deformable NEMS taking into account mechanical contact. COMPUMAG 2011, Jul
2011, Sydney, Australia. ￿hal-00610878￿
9. COUPLED MULTI-PHYSICS PROBLEMS
Finite Element Modeling of deformable NEMS taking into account
mechanical contact
N. Galopin, P. Pham-Quang, B. Delinchant and J.L. Coulomb
Grenoble Electrical Engineering Laboratory
G2ELab, UMR CNRS 5269 - INPG - UJF
BP 46, 38402 Saint Martin d’He`res Cedex, France
Email: nicolas.galopin@g2elab.grenoble-inp.fr
Abstract— This work focuses on the modeling of mechanical
contact applied to Nano-Electro-Mechanical-Systems (NEMS).
A magneto-elastic formulation associated to the definition of
unilateral contact in static case is presented. This model is
discretised by finite element method and a numerical study of a
magnetic nano switch is performed. FEM results are compared
to those of a semi-analytical model developed for optimization
processes.
I. INTRODUCTION
Magnetic NEMS based instruments and devices are at the
forefront of nanotechnology, enabling fundamental measure-
ments and standards and new types of devices related to mag-
netism. This is given by low voltage and power consumption
with large actuation distance which provides a number of
advantages compared to electrostatic NEMS [1]. In NEMS
technology, magnetic nano switches have many applications
as nanomechanical memory, power switches,. . . Their working
principle is based on the deflection of a beam submitted to the
influence of a magnetic field. The mechanical contact between
beam and substrate is an important parameter for the behavior
of these nano switches [2]. Indeed, contact quality strongly
depends on roughness of contact area and on surface and
contact forces. Nevertheless, their modelings are often strongly
approximated [3]. We propose in this paper a magneto-
elastic model included unilateral contact formulation and its
discretisation by the finite element method. The modelisation
is applied to a magnetic nano switch, and the numerical
results are compared with a semi-analytical model realizing
the magnetic structural coupling considering contact analysis.
II. FINITE ELEMENT MODELING
The problem under consideration involves the contact of an
elastic body, submitted to magnetic forces, with a rigid (or
elastic) body in static case (Fig. 1).
A. Contact problem description
Consider two elastic bodies occupying a bounded domain,
Ω1 and Ω2 (Fig. 1), of Rd with d = 2 or 3. The bodies are
subjected to body forces f and to prescribed tractions t and
displacements u0 on the part Γσ and Γu of the boundary
Γ respectively. For the two bodies, the potential contact
boundaries are Γc1 and Γc2 . For a classical linear elasticity
Ω1
Ω1
Ω2
Ω2
Γc1
Γc1
Γc2Γc2
Ω3
Γ3
Br
Js
Fig. 1. Study domain of magneto-mechanical contact problem (Ω = Ωm ∪
Ω3 and Ωm = Ω1 ∪ Ω2)
problem, the equations are:

∇ · σ + f = 0 in Ωm
σ = C : s in Ωm
σ(u) · n = t on Γσ
u = u0 on Γu
(1)
where n is the outward unit normal vector, C the stiffness
tensor and s the strain tensor defined in the assumption of
small displacement by:
s(u) =
1
2
(
∇u+∇ut
) (2)
To state the laws of contact and friction, the displacement
and tractions vectors are decomposed on Γc into normal and
tangential components:
σn = σij ni nj σti = σij nj − σn ni
un = ui ni uti = ui − un ni
(3)
For a frictionless contact, the unilateral contact conditions
on the boundaries Γc, which implies that no boundary point
of the first body may penetrate the other, are defined by :

gn = (u2 − u1) · n ≥ 0
σn = σn1 = −σn2 ≤ 0
σn · gn = 0
(4)
with gn the signed normal distance. The first condition states
that no penetration may occur. Hence, this is the form in
which the impenetrability constraint is cast. Using this, the
9. COUPLED MULTI-PHYSICS PROBLEMS
normal traction can be characterised. The second condition
states that the contact normal traction should be compressive.
Finally, the third condition states a complementarity condition.
If there is no contact, then no compressive tractions can occur.
Alternatively, if there are no compressive stresses, then the
distance must be positive.
B. Magneto-elastic contact problem formulation
In the static case, the formulation of the magneto-elastic
contact problem, without magnetostrictive phenomena [4], can
be established from a minimisation of the functional energy
F in terms of magnetic flux density b and strain s:
F(b, s) =W (b, s)− T (5)
where W (b, s) and T are respectively the magneto-elastic
energy and the work of magnetic and mechanical sources,
defined by:
W (b, s) =
∫
Ω


b∫
0
h(b′)db′ +
s∫
0
σ(s′)ds′

dΩ (6)
T =
∫
Ω
a · j dΩ+
∫
Γh
a · (h× n) dΓh
+
∫
Ωm
u · f dΩ+
∫
Γσ
u · (σ · n) dΓσ
(7)
with a the magnetic vector potential and j the current density.
Application of variational principles, and taking into account
the contact conditions (4), gives the following magnetic and
mechanical formulations associated to arbitrary variations δa
and δu:∫
Ω
∇×δa ν∇×a dΩ+
∫
Γh
δa ·(h×n) dΓ =
∫
Ω
δa ·j dΩ (8)
∫
Ωm
s(δu)Cs−δu·fdΩ−
∫
Γσ
δu·tdΓ−δu (W (b, 0)) ≥ 0 (9)
The inequality (9) is due to the contact definition and
define the variational inequality of the magneto-elastic contact
problem. The last term of the right-hand side of (9) are the
nodal magnetic forces determined by the local derivative of
magnetic energy [5]. Resolution are realizing by a penalty
method, the Lagrange multiplier method or an augmented
lagrangian method [7].
III. SEMI-ANALYTICAL MODELING
Thanks to basic geometries of NEMS, magnetic fields radi-
ated by permanent magnets and conductors are computed by
using pure algebraic equations through Coulombian equivalent
surface charge approach and Biot-Savart law. Surface and
volume numerical integrations are used to compute magnetic
forces and torques.
A semi-analytical structure model compute the deflection,
due to magnetic forces and torques, of the cantilever beam
in the presence of contact. It is based on assumptions of
small displacement, which allows reducing the cantilever beam
0 2 4 6 80
100
200
300
Co
n
ta
ct
le
n
gt
h
(n
m
)
Force on the beam (mN)
0 2 4 6 80
1
2
3
4
5
Co
n
ta
ct
fo
rc
e
(m
N
)
Force on the beam (mN)
Fig. 2. Results of the semi-analytical structural contact model
3D to 1D problem, on linear and isotropic materials and on
considering only one direction of deformation. To compute the
deformation, the 1D-beam is split into a set of segments, and
deformation of each segments is calculated by the following
equation:
E I
∂2s
∂x2
=M (10)
where E is the Young’s modulus, I the second moment of area
and M the bending moment. Adequate boundary conditions
and connection conditions are used to ensure continuous
deformation. In application of the superposition principle, the
total deformation is the sum of the deformations of each
segments created by each forces and torques.
Model of contact is inherently coupled with the structural
model by replacing the contact by a distribution of force on
contact. The contact force Fc is computed by:
Fc = Ft − Fr (11)
where Ft is the total force applied on the beam and Fr the
reaction force on fixed side of the beam. As the distribution
of forces, which replaces the contact, is unknown, an iterative
method is used. Figure 2 shows the variation of the contact
length and contact force as function of forces applied on the
beam.
IV. CONCLUSION
In this paper, the formulation of a magneto-mechanical
contact problem for a finite element analysis has been pre-
sented. In the full paper, this model will be developed and
particularly the finite element discretisation, the numerical
resolution thanks to regularization methods and the procedure
to improve the contact management. Comparisons with the
semi-analytical model will be made for a magnetic nano
switch.
REFERENCES
[1] O. Cugat, J. Delamare and G. Reyne, “Magnetic Micro-Actuators and
Systems (MAGMAS)”, IEEE Transactions on Magnetics, 39(5):3607-
3612, 2003.
[2] Y. P. Zhao, L. S. Wang and T. X. Yu, “Mechanics of adhesion in MEMS
- a review”, J. Adh. Sci. Technol., 17(4):519-546, 2003.
[3] G. D. Gray and P. A. Kohl, “Modeling and Performance of a Magnetic
MEMS Wiping Actuator”, Journal of Microelectromechanical Systems,
15(4):904-911, 2006.
[4] M. Besbes, Z. Ren and A. Razek, “A generalized finite element model
of magnetostriction phenomena”, IEEE Transactions on Magnetics,
37(5):3324-3328, 2001.
[5] Z. Ren and A. Razek, “Local force computation in deformable bodies
using edge elements”, IEEE Transactions on Magnetics, 28(2):1212-1215,
1992.
[6] G. Duvaut and J. L. Lions, Les ine´galite´s variationnelles en me´canique
et en physique, Dunod, Paris, 1972.
[7] P. Wriggers, Nonlinear finite element methods, Springer, London, 2008.


Finite Element Modeling of deformable NEMS taking into accountmechanical contact

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Finite Element Modeling of deformable NEMS taking into accountmechanical contact

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