### Summary

Micro-electro-mechanical systems (MEMS) are micron-scale machines used in everything from accelerometers to digital projectors. Resonating MEMS are critical to many everyday gadgets – without MEMS resonators, your iPhone would not know when to switch from landscape to portrait mode! I work on simulation tools to help engineers design these types of devices.

For many applications of resonant MEMS, it is important to minimize damping, but it is often unclear which mechanisms most strongly affect damping. HiQLab is a finite element code for analyzing high-frequency resonant MEMS, with specialized models and solvers for dealing with loss mechanisms such as thermoelastic damping (TED) and anchor loss. I developed it as part of my PhD thesis work at Berkeley. With HiQLab, we provide designers with tools to understand damping in resonant devices, and to understand how changes to the design (or accidental changes due to processing variation) affect the device behavior.

### Papers

D. Bindel, “Structured and Parameter-Dependent Eigensolvers for Simulation-Based Design of Resonant MEMS,” PhD thesis, University of California, Berkeley, 2006. Appears as Tech Report EECS-2006-109.
2008 Householder Award (best NLA thesis over three years)
@phdthesis{2006-dissertation,
author = {Bindel, David},
title = {Structured and Parameter-Dependent Eigensolvers for
Simulation-Based Design of Resonant {MEMS}},
school = {University of California, Berkeley},
month = aug,
year = {2006},
status = {unrefereed},
submit = {Appears as Tech Report EECS-2006-109.},
notable = {2008 Householder Award (best NLA thesis over three years)}
}


#### Abstract:

This dissertation is about computational tools to aid in the design of resonant Micro-Electro-Mechanical Systems (MEMS), tiny vibrating devices built by processes like those used to make integrated circuits. Vibrating MEMS are used in accelerometers and gyroscopes, in sensors to detect chemicals and to measure pressure, and in communication devices such as cell phones. MEMS engineers can use computer simulations to design devices using fewer costly and time-consuming prototype tests, but these simulations are only as useful as the models on which they are built. In this work, we contribute new mathematical models, numerical methods, and software tools to simulate resonant MEMS, and apply these tools to analyze specific devices. We describe physical models of damped vibrations of MEMS, including anchor loss and thermoelastic effects which are widely recognized as important, but not modeled in generality by existing tools. Though the resulting systems of equations are large and non-Hermitian, and depend nonlinearly on frequency, we use the equation structure to develop efficient structured Krylov subspace projection methods for computing free vibrations and reduced-order models. We also provide efficient continuation methods for re-computing eigendecompositions under changes to design parameters or operating conditions. Our models and analysis methods are integrated into HiQLab, a new finite element tool with a particularly flexible architecture which we have designed. Using HiQLab, we simulate example resonator designs, and compare our results to laboratory measurements. Our simulations reveal a previously-unknown mode interference phenomenon, subsequently observed in experiments, which dramatically affects the amount of damping near certain critical values of geometric parameters.

T. Koyama, D. Bindel, W. He, E. Quevy, J. Demmel, S. Govindjee, and R. Howe, “Simulation Tools for Damping in High Frequency Resonators,” in Proceedings of IEEE SENSORS 2005, 2005.
@inproceedings{2005-sensors,
author = {Koyama, Tsuyoshi and Bindel, David and He, Wei and Quevy, Emmanuel and Demmel, James and Govindjee, Sanjay and Howe, Roger},
title = {Simulation Tools for Damping in High Frequency Resonators},
booktitle = {Proceedings of IEEE SENSORS 2005},
month = nov,
year = {2005},
doi = {10.1109/ICSENS.2005.1597708}
}


#### Abstract:

This paper presents the development of HiQLab, a simulation tool to compute the effect of damping in high frequency resonators. Existing simulation tools allow designers to compute resonant frequencies but few tools provide estimates of damping, which is crucial in evaluating the performance of such devices. In the current code, two damping mechanisms: thermoelastic damping and anchor loss, have been implemented. Thermoelastic damping results from irreversible heat flow due to mechanically-driven temperature gradients, while anchor loss occurs when high-frequency mechanical waves radiate away from the resonator and into the substrate. Our finite-element simulation tool discretizes PDE models of both phenomena, and evaluates the quality factor ($Q$), a measure of damping in the system, with specialized eigencomputations and model reduction techniques. The core functions of the tool are written in C++ for performance. Interfaces are in Lua and MATLAB, which give users access to powerful visualization and pre- and postprocessing capabilities.

D. Bindel and S. Govindjee, “Elastic PMLs for Resonator Anchor Loss Simulation,” International Journal for Numerical Methods in Engineering, vol. 64, no. 6, pp. 789–818, Oct. 2005.
@article{2005-ijnme,
author = {Bindel, David and Govindjee, Sanjay},
title = {Elastic {PMLs} for Resonator Anchor Loss Simulation},
journal = {International Journal for Numerical Methods in Engineering},
volume = {64},
number = {6},
pages = {789--818},
month = oct,
year = {2005},
doi = {10.1002/nme.1394}
}


#### Abstract:

Electromechanical resonators and filters, such as quartz, ceramic, and surface-acoustic wave devices, are important signal-processing elements in communication systems. Over the past decade, there has been substantial progress in developing new types of miniaturized electromechanical resonators using microfabrication processes. For these micro-resonators to be viable they must have high and predictable quality factors ($Q$). Depending on scale and geometry, the energy losses that lower $Q$ may come from material damping, thermoelastic damping, air damping, or radiation of elastic waves from an anchor. Of these factors, anchor losses are the least understood because such losses are due to a complex radiation phenomena in a semi-infinite elastic half-space. Here, we describe how anchor losses can be accurately computed using an absorbing boundary based on a perfectly matched layer (PML) which absorbs incoming waves over a wide frequency range for any non-zero angle of incidence. We exploit the interpretation of the PML as a complex-valued change of coordinates to illustrate how one can come to a simpler finite element implementation than was given in its original presentations. We also examine the convergence and accuracy of the method, and give guidelines for how to choose the parameters effectively. As an example application, we compute the anchor loss in a micro disk resonator and compare it to experimental data. Our analysis illustrates a surprising mode-mixing phenomenon which can substantially affect the quality of resonance.

D. Bindel and S. Govindjee, “Elastic PMLs for Resonator Anchor Loss Simulation,” Structural Engineering Mechanics and Materials, Department of Civil and Environmental Engineering, University of California, Berkeley, UCB/SEMM-2005/01, Feb. 2005. Appeared in International Journal on Numerical Methods in Engineering in 2005.
@techreport{2005-pml-tr,
author = {Bindel, David and Govindjee, Sanjay},
title = {Elastic {PML}s for Resonator Anchor Loss Simulation},
number = {UCB/SEMM-2005/01},
institution = {Structural Engineering Mechanics and Materials,
Department of Civil and Environmental Engineering,
University of California, Berkeley},
month = feb,
year = {2005},
status = {unrefereed},
submit = {Appeared in International Journal on Numerical Methods in Engineering in 2005.}
}


#### Abstract:

Electromechanical resonators and filters, such as quartz, ceramic, and surface-acoustic wave devices, are important signal-processing elements in communication systems. Over the past decade, there has been substantial progress in developing new types of miniaturized electromechanical resonators using microfabrication processes. For these micro-resonators to be viable they must have high and predictable quality factors ($Q$). Depending on scale and geometry, the energy losses that lower $Q$ may come from material damping, thermoelastic damping, air damping, or radiation of elastic waves from an anchor. Of these factors, anchor losses are the least understood because such losses are due to a complex radiation phenomena in a semi-infinite elastic half-space. Here, we describe how anchor losses can be accurately computed using an absorbing boundary based on a perfectly matched layer (PML) which absorbs incoming waves over a wide frequency range for any non-zero angle of incidence. We exploit the interpretation of the PML as a complex-valued change of coordinates to illustrate how one can come to a simpler finite element implementation than was given in its original presentations. We also examine the convergence and accuracy of the method, and give guidelines for how to choose the parameters effectively. As an example application, we compute the anchor loss in a micro disk resonator and compare it to experimental data. Our analysis illustrates a surprising mode-mixing phenomenon which can substantially affect the quality of resonance.

D. Bindel, E. Quevy, T. Koyama, J. Demmel, and R. Howe, “Anchor Loss Simulation in Resonators,” in Proceedings of MEMS 2005, 2005.
@inproceedings{2005-mems,
author = {Bindel, David and Quevy, Emmanuel and Koyama, Tsuyoshi and Demmel, James and Howe, Roger},
title = {Anchor Loss Simulation in Resonators},
booktitle = {Proceedings of MEMS 2005},
month = feb,
year = {2005},
doi = {10.1109/MEMSYS.2005.1453885}
}


#### Abstract:

Surface-micromachined resonators and filters are attractive for many RF applications. While existing simulation tools allow designers to compute resonant frequencies, few tools provide estimates of the damping in these devices. This paper reports on a new tool that allows designers, for the first time, to compute anchor losses in high-frequency resonators and account for sub-surface scatterers. By exercising the tool on a family of radially driven disk resonators, we show that the anchor loss mechanism for this class of devices involves a parasitic mixed-mode bending action that pumps energy into the substrate. Further, using the tool, we predict a large variation in resonator quality depending upon film thickness. Our simulation shows that the source of this variation is a complex radial-to-bending motion interaction, which we visualize with a root-locus diagram. We experimentally verify this predicted sensitivity using poly-SiGe disk resonators having $Q$’s ranging from 200 to 54,000.

D. Bindel, Z. Bai, and J. Demmel, “Model Reduction for RF MEMS Simulation,” in Proceedings of PARA 2004, 2004.
@inproceedings{2004-para,
author = {Bindel, David and Bai, Zhaojun and Demmel, James},
title = {Model Reduction for {RF MEMS} Simulation},
booktitle = {Proceedings of PARA 2004},
month = jun,
year = {2004},
doi = {10.1007/11558958_34}
}


#### Abstract:

Radio-frequency (RF) MEMS resonators, integrated into CMOS chips, are of great interest to engineers planning the next generation of communication systems. Fast simulations are necessary in order to gain insights into the behavior of these devices. In this paper, we discuss two structure-preserving model-reduction techniques and apply them to the frequency-domain analysis of two proposed MEMS resonator designs.

### Talks

#### Computer Aided Design of Micro-Electro-Mechanical Systems

Civil Engineering Seminar at Duke
gyro mems pml rf-mems rom tedseminar external invited

#### Computer Aided Design of Micro-Electro-Mechanical Systems

SCMS Workshop on Recent Advances in Scientific Computing, Fudan University
gyro mems pml rf-mems rom tedmeeting external invited

#### Computer Aided Design of Micro-Electro-Mechanical Systems

FIST Workshop at Shanghai Tech
gyro mems pml rf-mems rom tedmeeting external invited

#### Structure-Preserving Model Reduction for MEMS

SIAM CSE Meeting
gyro mems pml rf-mems rom tedminisymposium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Cornell MAE Colloquium
mems pml rf-mems rom tedcolloquium local

#### Structure-Preserving Model Reduction for MEMS Modeling

SIAM Annual Meeting
mems pml rf-mems rom tedminisymposium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Stanford LA/Opt Seminar
mems pml rf-mems rom tedseminar external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

MIT Applied Math Colloquium
mems pml rf-mems rom tedcolloquium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Householder Symposium (Householder Award Talk)
mems pml rf-mems rom tedmeeting external invited plenary

#### Damping Mechanisms in Resonant Microsystems

NYU Courant Materials Working Group
mems pml rf-mems rom tedseminar local

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

McGill NA Seminar
mems pml rf-mems rom tedseminar external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Rice University Applied Math Colloquium
mems pml rf-mems rom tedcolloquium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Temple University NA Seminar
mems pml rf-mems rom tedseminar external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Cornell CS Colloquium
mems pml rf-mems romcolloquium external invited

#### Numerical and Semi-Analytical Structure-Preserving Model Reduction for MEMS

ICIAM
mems pml rf-mems rom tedminisymposium external invited

#### Numerical and Semi-Analytical Structure-Preserving Model Reduction for MEMS

ICIAM
mems pml rf-mems rom tedminisymposium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Columbia Applied Math Colloquium
mems pml rf-mems rom tedcolloquium external invite

#### Model Reduction and Mode Computation for Damped Resonant MEMS

SIAM CSE
mems pml rf-mems rom tedminisymposium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Thesis talk, UC Berkeley CS
mems pml rf-mems romlocal

#### Modeling Resonant Microsystems

Abel Symposium
mems pml rf-mems rommeeting external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Purdue CS
mems pml rf-mems romseminar external invited

#### Eigenvaues, Resonance Poles, and Damping in MEMS

UC Berkeley LAPACK Seminar
mems pml resonance rf-memsseminar local

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

CU Boulder CS Colloquium
mems rf-mems romcolloquium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

UC Davis CS Colloquium
mems pml rf-mems romcolloquium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Bay Area Scientific Computing Day
mems pml rf-mems rommeeting invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Penn State University
mems pml rf-mems romseminar external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Caltech CS Colloquium
mems pml rf-mems romcolloquium external invited

#### Computer-Aided Design for Micro-Electro-Mechanical Systems

Sandia National Labs
mems pml rf-mems romseminar external invited

#### Simulating Losses in Resonant MEMS

UC Berkeley Applied Math Seminar
mems pml rf-memsseminar local

#### Elastic PMLs for Resonator Anchor Loss Simulation

US National Conference on Computational Mechanics
mems pml rf-memsminisymposium external invited

#### Eigenproblems in Resonant MEMS Design

SIAM Annual Meeting
mems pml rf-mems tedminisymposium external invited

#### Complex Symmetric Matrices

UC Berkeley LAPACK Seminar
mems pml rf-memsseminar local

#### HiQLab: Simulation of Resonant MEMS

Tutorial talk at UC Berkeley
mems rf-memstutorial local

#### Computer-Aided Design of MEMS

NYU Numerical Analysis Seminar
mems pml rf-mems rom tedseminar external invited