a Monte Carlo Investigation M. Bagherinia, S. Mariani, A. Corigliano - - PowerPoint PPT Presentation

▶

May 27, 2023 133 likes •313 views

Dipartimento di Ingegneria Civile e Ambientale Stochastic Effects on the Dynamics of a Resonant MEMS Magnetometer: a Monte Carlo Investigation M. Bagherinia, S. Mariani, A. Corigliano Dipartimento di Ingegneria Civile e Ambientale Politecnico

SLIDE 1

Dipartimento di Ingegneria Civile e Ambientale

Stochastic Effects on the Dynamics of a Resonant MEMS Magnetometer: a Monte Carlo Investigation

M. Bagherinia, S. Mariani, A. Corigliano

Dipartimento di Ingegneria Civile e Ambientale

Politecnico di Milano, Italy

SLIDE 2

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

2 2

Magnetometers: engineering motivation

The earth magnetic field as a vector quantity

X,Y, Z components Orientation determination

SLIDE 3

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

3

Working principle

The external magnetic fields Change of the resonating system configuration ( displacement, eigen-frequency)

ELECTRICAL MEASUREMENT

The magnetic field component is defined as a function of the configuration change

Mems Structure

Static Plate

Mems Structure

Static Plate

SLIDE 4

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

4

Design requirements

High sensitivity Process limitations Mechanical acceleration filtering

Design Demands The goal of

ur designs

1 2 3

Some Designs In The Literature

Herrera-May et al 1 1 1 2 2 3 3 3

VTT technical research center Behraad Bahreyni

SLIDE 5

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

5 5

New design ideas (Capacitive parallel plates)

Mechanical acceleration filtered out

Absence of acceleration Presence of acceleration

Displacement due to magnetic field Displacement due to acceleration i

v v

acceleration

g g

c

Bz

SLIDE 6

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

Multi-physics modeling

6 i

Thermo electro magneto mechanical problem

Joule effect Lorentz force and eddy current Hamilton’s principle

SLIDE 7

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

Multi-physics modeling

7

Applying Galerkin method to Hamilton’s principle One degree of freedom equivalent system (Duffing nonlinear equation)

Clamped - Clamped First Eigen mode

SLIDE 8

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

8 Maximum amplitude of oscillation is given by the solution of

Current frequency

Multi-physics modeling

SLIDE 9

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

9

Multi-physics simulation

To check the model, Ansys multi-physics simulations were performed Static thermo-electro-structural analysis

SLIDE 10

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

10

Multi-physics simulation

,

SLIDE 11

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

Parameter optimization

11

To have an optimal device, we perform a multi objective

ptimization procedure, consisting of a structural objective function

(ZMax) and an electrical objective function (Relec)

l h

Sensor’s performance Sensitivity (Maximum amplitude) Power consumption (Minimum electrical resistance) Optimal h, l of the multi-physics solution for the fundamental component

SLIDE 12

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

12

Optimal design for dynamic compliance

Constrained objective function (red line), and optimal path followed by the minimization algorithm (black dotted line) Unconstrained objective function

Except for the first iterations that move from a point violating the prescribed equality constraint, the optimizer provides a set of feasible solutions to the arising sub-problems. It finally ends with the expected global minimum h=2μm, l=580.9 μm

SLIDE 13

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

13

Optimal design for power consumption

Constrained objective function (red line), and optimal path followed by the minimization algorithm (black dotted line) Unconstrained objective function

Except for the first iterations that move from a point violating the prescribed equality constraint, the optimizer provides a set of feasible solutions to the arising sub-problems. It finally ends with the expected global minimum h=3.8μm, l=798.1 μm

SLIDE 14

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

Uncertainty assessment of sensors performance

14

Sources of uncertainty in the model and probability density functions (PDFs) Young’s modulus Over etching uncertainty in beam width due to process (uniform PDF)

1.4 1.42 1.44 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 x 10

0.5 1 1.5 x 10

Data Density E data Normal Lognormal Inverse gaussian Weibull Rician

Maximum likelihood

E distribution

PDF of E

RVE of polysilicon

SLIDE 15

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

15

2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 x 10

100 200 h distribution 1.35 1.4 1.45 1.5 1.55 1.6 1.65 x 10

200 400 E distribution 1 1.5 2 2.5 x 10

500 amplitude histogram amplitude number of occurrence 2 2.05 2.1 2.15 2.2 2.25 2.3 2.35 x 10

1 2 3 x 10

h amplitude 1.35 1.4 1.45 1.5 1.55 1.6 1.65 x 10

1 2 3 x 10

E amplitude 2 2.1 2.2 2.3 2.4 x 10

1.2 1.4 1.6 1.8 x 10

1 2 3 x 10

h E amplitude

Monte Carlo simulation: effect of uncertainties on ZMax around the optimal values Uncertainty assessment of sensors performance

SLIDE 16

Eurosime2013 - A. Corigliano, M. Bagherinia, M. Bruggi, S. Mariani, E. Lasalandra. www.stru.polimi.it/mems

Planned activities

16

 Validate the multi-physics model by a commercial FEM code (either Ansys or Comsol)  Adopt a topology optimization approach to find the

ptimal shape of the base component (to search for the
ptimal shape, not only beam shaped structures, but also

Stochastic Effects on the Dynamics of a Resonant MEMS Magnetometer: a Monte Carlo Investigation

Politecnico di Milano, Italy

2 2

Magnetometers: engineering motivation

The earth magnetic field as a vector quantity

X,Y, Z components Orientation determination

3

Working principle

The external magnetic fields Change of the resonating system configuration ( displacement, eigen-frequency)

The magnetic field component is defined as a function of the configuration change

4

Design requirements

High sensitivity Process limitations Mechanical acceleration filtering

Design Demands The goal of

Some Designs In The Literature

VTT technical research center Behraad Bahreyni

5 5

New design ideas (Capacitive parallel plates)

Mechanical acceleration filtered out

Absence of acceleration Presence of acceleration

Displacement due to magnetic field Displacement due to acceleration i

v v

acceleration

g g

c

Bz

Multi-physics modeling

6 i

Thermo electro magneto mechanical problem

Joule effect Lorentz force and eddy current Hamilton’s principle

Multi-physics modeling

7

Applying Galerkin method to Hamilton’s principle One degree of freedom equivalent system (Duffing nonlinear equation)

Clamped - Clamped First Eigen mode

8

Maximum amplitude of oscillation is given by the solution of

Current frequency

Multi-physics modeling

9

Multi-physics simulation

To check the model, Ansys multi-physics simulations were performed Static thermo-electro-structural analysis

10

Multi-physics simulation

,

Parameter optimization

11

To have an optimal device, we perform a multi objective

(ZMax) and an electrical objective function (Relec)

l h

Sensor’s performance Sensitivity (Maximum amplitude) Power consumption (Minimum electrical resistance) Optimal h, l of the multi-physics solution for the fundamental component

12

Optimal design for dynamic compliance

Constrained objective function (red line), and optimal path followed by the minimization algorithm (black dotted line) Unconstrained objective function

Except for the first iterations that move from a point violating the prescribed equality constraint, the optimizer provides a set of feasible solutions to the arising sub-problems. It finally ends with the expected global minimum h=2μm, l=580.9 μm

13

Optimal design for power consumption

Constrained objective function (red line), and optimal path followed by the minimization algorithm (black dotted line) Unconstrained objective function

Except for the first iterations that move from a point violating the prescribed equality constraint, the optimizer provides a set of feasible solutions to the arising sub-problems. It finally ends with the expected global minimum h=3.8μm, l=798.1 μm

Uncertainty assessment of sensors performance

14

Sources of uncertainty in the model and probability density functions (PDFs) Young’s modulus Over etching uncertainty in beam width due to process (uniform PDF)

Maximum likelihood

E distribution

PDF of E

RVE of polysilicon

15

Monte Carlo simulation: effect of uncertainties on ZMax around the optimal values Uncertainty assessment of sensors performance

Planned activities

16

 Validate the multi-physics model by a commercial FEM code (either Ansys or Comsol)  Adopt a topology optimization approach to find the

tapered, curved and any other arbitrary shape)  Performing the experimental tests on the devices