PRISM: NNSA Center for Prediction of Reliability, Integrity and Survivability of Microsystems

Numerical approach for quantification of epistemic uncertainty

John Jakeman et al. — Wed, 20 Jun 2012 12:53:51 PDT

In the field of uncertainty quantification, uncertainty in the governing equations may assume two forms: aleatory uncertainty and epistemic uncertainty. Aleatory uncertainty can be characterised by known probability distributions whilst epistemic uncertainty arises from a lack of knowledge of probabilistic information. While extensive research efforts have been devoted to the numerical treatment of aleatory uncertainty, little attention has been given to the quantification of epistemic uncertainty. In this paper, we propose a numerical framework for quantification of epistemic uncertainty. The proposed methodology does not require any probabilistic information on uncertain input parameters. The method only necessitates an estimate of the range of the uncertain variables that encapsulates the true range of the input variables with overwhelming probability. To quantify the epistemic uncertainty, we solve an encapsulation problem, which is a solution to the original governing equations defined on the estimated range of the input variables. We discuss solution strategies for solving the encapsulation problem and the sufficient conditions under which the numerical solution can serve as a good estimator for capturing the effects of the epistemic uncertainty. In the case where probability distributions of the epistemic variables become known a posteriori, we can use the information to post-process the solution and evaluate solution statistics. Convergence results are also established for such cases, along with strategies for dealing with mixed aleatory and epistemic uncertainty. Several numerical examples are presented to demonstrate the procedure and properties of the proposed methodology. (C) 2010 Elsevier Inc. All rights reserved.

Strain rate sensitivity of nanocrystalline Au films at room temperature

K Jonnalagadda et al. — Thu, 02 Dec 2010 11:36:02 PST

The effect of strain rate on the inelastic properties of nanocrystalline Au films was quantified with 0.85 and 1.76 lm free-standing microscale tension specimens tested over eight decades of strain rate, between 6 106 and 20 s1. The elastic modulus was independent of the strain rate, 66 ± 4.5 GPa, but the inelastic mechanical response was clearly rate sensitive. The yield strength and the ultimate tensile strength increased with the strain rate in the ranges 575–895 MPa and 675–940 MPa, respectively, with the yield strength reaching the tensile strength at strain rates faster than 101 s1. The activation volumes for the two film thicknesses were 4.5 and 8.1 b3, at strain rates smaller than 104 s1 and 12.5 and 14.6 b3 at strain rates higher than 104 s1, while the strain rate sensitivity factor and the ultimate tensile strain increased below 104 s1. The latter trends indicated that the strain rate regime 105–104 s1 is pivotal in the mechanical response of the particular nanocrystalline Au films. The increased rate sensitivity and the reduced activation volume at slow strain rates were attributed to grain boundary processes that also led to prolonged (5–6 h) and significant primary creep with initial strain rate of the order of 107 s1.

An Experimental Investigation on Viscoelastic Behavior in Tunable

Hao-Han Hsu et al. — Thu, 02 Dec 2010 11:25:19 PST

Abstract—In this paper, the viscoelastic behavior of a tunable RF-MEMS resonator and its impacts are studied by means of direct RF measurements for the first time. This tunable resonator consists of one λ/2 coplanar waveguide (CPW) resonator and two nanocrystalline-Ni RF-MEMS varactors. S-parameters of this tunable resonator have been measured for 80 hours under a bi-state bias condition of 0 and 40 V. It is demonstrated that the resonant frequency is shifted by 90 MHz and the varactor deformed by 0.12 μm over the 80 hour period. The gap of the loaded varactor is extracted from the measured S-parameters using finite-element analysis (FEA) tools. A generalized Voigt- Kelvin model is employed to verify the viscoelastic behavior of the resonator. The creep compliance extracted from the RF measurements is in excellent agreement with results in literature. Index Terms—creep, nickel, RF-MEMS, tunable resonator, viscoelastic.

A VISCOELASTIC-AWARE EXPERIMENTALLY-DERIVED MODEL FOR ANALOG RF MEMS VARACTORS

Hao-Han Hsu et al. — Thu, 02 Dec 2010 11:25:19 PST

In this paper we present, for the ¯rst time, an experimentally-extracted model for the spring con- stant and tuning range of an analog RF-MEMS var- actor that includes viscoelastic e®ects in RF-MEMS devices. By utilizing a bi-state bias condition with one state lasting 60 minutes and the other 1 minute, this model focuses on capturing the true electrome- chanical behavior of the varactor. An experimental setup with very high long-term accuracy is created to measure capacitance of the varactor up to 1,370 hours. The impact of these e®ects and the e®ective- ness of the model are demonstrated on a tunable- resonator loaded with RF-MEMS varactors.

An Unstructured Finite Volume Method For Incompressible Flows with Complex Immersed Boundaries

Lin Sun et al. — Thu, 02 Dec 2010 11:25:18 PST

A numerical method is developed for solving the 3D, unsteady, incompressible flows with immersed moving soldis of arbitrary geometrical complexity. A co-located (non-staggered) finite volume method is employed to solve the Navier-Stokes governing equeations for flow region using arbitrary convex polyhedral meshes. The solid region is represented by a set of material points with known position and velocity. Faces in the flow region located in the immediate vicinity of the solid body are marked as immersed boundary (IB) faces. At every instant in time, the influence of the body on the flowis accounted for by reconstructing implicitly the velocity the IB faces from a stencil of fluid cells and solid material points. Specific numerical issues related to the non-staggered formulation are addressed, including the specification of face mass fluxes, and corrections to the continuity equation to ensure overall mass balance. Incorporation of this immersed boundary technique within the framework of the SIMPLE algorithm is described. Canonical test cases of laminar flow around stationary and moving spheres adn cylinders are used to verify the implementation. Mesh convergence tests are carried out. The simulation results are shown to agree well with experiments for the case of micro-cantilevers vibrating in a viscous fluid.

Phase stability and transformations in NiTi from density

Karthik Guda Vishnu et al. — Thu, 02 Dec 2010 11:25:16 PST

We used density functional theory to characterize various crystalline phases of NiTi alloys: (i) high-temperature austenite phase B2; (ii) orthorhombic B19; (iii) the monoclinic martensite phase B190; and (iv) a body-centered orthorhombic phase (BCO), theoretically predicted to be the ground state. We also investigated possible transition pathways between the various phases and the energetics involved. We found B19 to be metastable with a 1 meV energy barrier separating it from B190. Interestingly, we predicted a new phase of NiTi, denoted B1900, that is involved in the transition between B190 and BCO. B1900 is monoclinic and can exhibit shape memory; furthermore, its presence reduces the internal stress required to stabilize the experimentally observed B190 structure, and it consequently plays a key role in NiTi’s properties.

Thermal conduction in molecular materials using coarse grain dynamics: Role of mass diffusion and quantum corrections for molecular dynamics simulations

Ya Zhou et al. — Thu, 02 Dec 2010 11:25:15 PST

We use a mesodynamical method, denoted dynamics with implicit degrees of freedom DID, to characterize thermal transport in a model molecular crystal below and above its melting temperature. DID represents groups of atoms molecules in this case using mesoparticles and the thermal role of the intramolecular degrees of freedom DoFs are described implicitly using their specific heat. We focus on the role of these intramolecular DoFs on thermal transport. We find that thermal conductivity is independent of intramolecular specific heat for solid samples and a linear relationship between the two quantities in liquid samples with the coefficient of proportionality being the mass diffusivity of the mesoparticles. As the temperature of the liquids is increased, thermal conductivity exhibits an increased sensitivity with respect to the specific heat of the internal DoFs due to the enhanced molecular mobility. Based on these results, we propose a simple method to incorporate quantum corrections to thermal conductivity obtained from nonequilibrium molecular dynamics simulations of molecular liquids. Our results also provide insight into the development of thermally accurate coarse grain models of soft materials.

Uncertainty propagation in a multiscale model of nanocrystalline plasticity

Marisol Koslowski et al. — Thu, 02 Dec 2010 11:25:14 PST

We characterize how uncertainties propagate across spatial and temporal scales in a physicsbased model of nanocrystalline plasticity of fcc metals. Our model combines molecular dynamics (MD) simulations to characterize atomic level processes that govern dislocation basedplastic deformation with a phase field approach to dislocation dynamics (PFDD) that describes how an ensemble of dislocations evolve and interact to determine the mechanical response of the material. We apply this approach to a nanocrystalline Ni specimen of interest in micro-electromechanical (MEMS) switches. Our approach enables us to quantify how internal stresses that result from the fabrication process affect the properties of dislocations (using MD) and how these properties, in turn, affect the yield stress of the metallic membrane (using the PFMM model). Our predictions show that, for a nanocrystalline sample with small grain size (4 nm), a variation in residual stress of 20 MPa (typical in today’s microfabrication techniques) would result in a variation on the critical resolved shear yield stress of approximately 15 MPa, a very small fraction of the nominal value of approximately 9 GPa.

Squeeze-film damping of flexible microcantilevers at low ambient pressures: theory and experiment

Jin Lee et al. — Tue, 30 Nov 2010 10:19:54 PST

An improved theoretical approach is proposed to predict the dynamic behavior of long, slender and flexible microcantilevers affected by squeeze-film damping at low ambient pressures. Our approach extends recent continuum gas damping models (Veijola 2004 J. Micromech. Microeng. 14 1109–18, Gallis and Torczynski 2004 J. Microelectromech. Syst. 13 653–9), which were originally derived for a rigid oscillating plate near a wall, to flexible microcantilevers for calculating and predicting squeeze-film damping ratios of higher order bending modes at reduced ambient pressures. Theoretical frequency response functions are derived for a flexible microcantilever beam excited both inertially and via external forcing. Experiments performed carefully at controlled gas pressures are used to validate our theoretical approach over five orders of the Knudsen number. In addition, we investigate the relative importance of theoretical assumptions made in the Reynolds-equation-based approach for flexible microelectromechanical systems.

NON-GRAY PHONON TRANSPORT USING A HYBRID BTE-FOURIER SOLVER

James Loy et al. — Tue, 30 Nov 2010 06:42:31 PST

Non-gray phonon transport solvers based on the Boltzmann transport equation (BTE) are frequently employed to simulate sub-micron thermal transport. Typical solution procedures using sequential solution schemes encounter numerical difficulties because of the large spread in scattering rates. For frequency bands with very low Knudsen numbers, strong coupling between the directional BTEs results in slow convergence for sequential solution procedures. In this paper, we present a hybrid BTE-Fourier model which addresses this issue. By establishing a phonon group cutoff (say Kn=0.1), phonon bands with low Knudsen numbers are solved using a modified Fourier equation which includes a scattering term as well as corrections to account for boundary temperature slip. Phonon bands with high Knudsen numbers are solved using a BTE solver. Once the governing equations are solved for each phonon group, their energies are then summed to find the total lattice energy and correspondingly, the lattice temperature. An iterative procedure combining the lattice temperature determination and the solutions to the modified Fourier and BTE equations is developed. The procedure is shown to work well across a range of Knudsen numbers.

SIMULATION OF SUB-MICRON THERMAL TRANSPORT IN A MOSFET USING A

James Loy et al. — Tue, 30 Nov 2010 06:42:30 PST

Self-heating has emerged as a critical bottleneck to scaling in modern transistors. In simulating heat conduction in these devices, it is important to account for the granularity of phonon transport since electron-phonon scattering occurs preferentially to select phonon groups. However, a complete accounting for phonon dispersion, polarization and scattering is very expensive if the Boltzmann transport equation (BTE) is used. Moreover, difficulties with convergence are encountered when the phonon Knudsen number becomes small. In this paper we simulate a two-dimensional bulk MOSFET hotspot problem using a partially-implicit hybrid BTE-Fourier solver which is significantly less expensive than a full BTE solution, and which shows excellent convergence characteristics. Volumetric heat generation from electron-phonon collisions is taken from a Monte Carlo simulation of electron transport and serves as a heat source term in the governing transport equations. The hybrid solver is shown to perform well in this highly non-equilibrium situation, matching the solutions obtained from a pure all-BTE solution, but at significantly lower computational cost. The paper establishes that this new model and solution methodology are viable for the simulation of thermal transport in other emerging transistor designs and in other nanotechnology applications as well.

Coarse grain modeling of spall failure in molecular crystals: role of intra-molecular degrees of freedom

Karen Lynch et al. — Mon, 29 Nov 2010 10:32:42 PST

We use a recently developed thermodynamically accurate mesodynamical method (Strachan and Holian 2005 Phys. Rev. Lett. 94 014301) where groups of atoms are represented by mesoparticles to characterize the shock compression and dynamical failure (spall) of a model molecular crystal. We characterize how the temperature rise caused by the shockwave depends on the specific heat of the degrees of freedom (DoFs) internal to the mesoparticles (Cint) and the strength of the coupling between the internal DoFs and the mesoparticles. We find that the temperature of the shocked material decreases with increasing Cint and decreasing coupling and quantify these effects. Our simulations also show that the threshold for plastic deformation (the Hugoniot elastic limit) depends on the properties of the internal DoFs while the threshold for failure is very insensitive to them. These results have implications on the results of all-atom MD simulations, whose classical nature leads to a significant overestimation of the specific heat of molecular materials.

AN UNSTRUCTURED FINITE VOLUME METHOD FOR INCOMPRESSIBLE FLOWS WITH COMPLEX IMMERSED BOUNDARIES

Lin Sun et al. — Mon, 29 Nov 2010 10:32:40 PST

A numerical method is developed for solving the 3D, unsteady, incompressible flows with immersed moving solids of arbitrary geometrical complexity. A co-located (non-staggered) finite volume method is employed to solve the Navier-Stokes governing equations for flow region using arbitrary convex polyhedral meshes. The solid region is represented by a set of material points with known position and velocity. Faces in the flow region located in the immediate vicinity of the solid body are marked as immersed boundary (IB) faces. At every instant in time, the influence of the body on the flow is accounted for by reconstructing implicitly the velocity the IB faces from a stencil of fluid cells and solid material points. Specific numerical issues related to the non-staggered formulation are addressed, including the specification of face mass fluxes, and corrections to the continuity equation to ensure overall mass balance. Incorporation of this immersed boundary technique within the framework of the SIMPLE algorithm is described. Canonical test cases of laminar flow around stationary and moving spheres and cylinders are used to verify the implementation. Mesh convergence tests are carried out. The simulation results are shown to agree well with experiments for the case of micro-cantilevers vibrating in a viscous fluid.

Entropy Considerations In Numerical Simulations Of Non-Equilibrium Rarefied Flows

Sruti Chigullapalli et al. — Fri, 05 Nov 2010 12:20:51 PDT

Non-equilibrium rarefied flows are encountered frequently in supersonic flight at high altitudes, vacuum technology and in microscale devices. Prediction of the onset of non-equilibrium is important for accurate numerical simulation of such flows. We formulate and apply the discrete version of Boltzmann’s H-theorem for analysis of non-equilibrium onset and accuracy of numerical modeling of rarefied gas flows. The numerical modeling approach is based on the deterministic solution of kinetic model equations. The numerical solution approach comprises the discrete velocity method in the velocity space and the finite volume method in the physical space with different numerical flux schemes: the first-order, the second-order minmod flux limiter and a third-order WENO schemes. The use of entropy considerations in rarefied flow simulations is illustrated for the normal shock, the Riemann and the two-dimensional shock tube problems. The entropy generation rate based on kinetic theory is shown to be a powerful indicator of the onset of non-equilibrium, accuracy of numerical solution as well as the compatibility of boundary conditions for both steady and unsteady problems.

A Parallel Spectral Element Method For Dynamic Three-Dimensional Nonlinear Elasticity Problems

S. Dong et al. — Fri, 05 Nov 2010 12:20:50 PDT

We present a high-order method employing Jacobi polynomial-based shape functions, as an alternative to the typical Legendre polynomial-based shape functions in solid mechanics, for solving dynamic three-dimensional geometrically nonlinear elasticity problems. We demonstrate that the method has an exponential convergence rate spatially and a second-order accuracy temporally for the four classes of problems of linear/geometrically nonlinear elastostatics/elastodynamics. The method is parallelized through domain decomposition and message passing interface (MPI), and is scaled to over 2000 processors with high parallel performance.

BDF-Like Methods For Nonlinear Dynamic Analysis

S. Dong — Fri, 05 Nov 2010 12:20:49 PDT

We present several time integration algorithms of second-order accuracy that are numerically simple and effective for nonlinear elastodynamic problems. These algorithms are based on a general four-step scheme that has a resemblance to the backward differentiation formulas. We also present an extension to the composite strategy of the Bathe method. Appropriate values for the algorithmic parameters are determined based on considerations of stability and dissipativity, and less dissipative members of each algorithm have been identified. We demonstrate the convergence characteristics of the proposed algorithms with a nonlinear dynamic problem having analytic solutions, and test these algorithms with several three-dimensional nonlinear elastodynamic problems involving large deformations and rotations, employing St. Venant-Kirchhoff and compressible Neo-Hookean hyperelastic material models. These tests show that stable computations are obtained with the proposed algorithms in nonlinear situations where the trapezoidal rule encounters a well-known instability.

Real-Time Monitoring Of Contact Behaviour Of RF MEMS Switches With A Very Low Power CMOS Capactive Sensor Interface

Adam Fruehling et al. — Thu, 04 Nov 2010 11:39:03 PDT

This paper presents the first ultra-low power, fully electronic methodology for real-time monitoring of the dynamic behavior of RF MEMS switches. The measurement is based on a capacitive readout circuit composed of 67 transistors with 105 µm x 105 µm footprint consuming as little as 60 µW. This is achieved by accurately sensing the capacitance change around the contact region at sampling rates from 10 kHz to 5 MHz. Experimental and simulation results show that times of not only the first contact event but also all subsequent contact bounces can be accurately measured with this technique without interfering with the switch performance. This demonstrates the potential of extending this technique to real-time on-chip dynamic monitoring of packaged RF MEMS switches through their entire lifetime and after their integration in their final system.

A Continuum Plasticity Model That Accounts For Hardening And Size Effects In Thin Films

Abigail Hunter et al. — Wed, 03 Nov 2010 05:20:39 PDT

We conducted three-dimensional finite element simulations of the mechanical response of passivated single crystal copper thin films with a continuum crystal plasticity model. The model introduces the formation of high density dislocation layers close to the substrate and passivation interfaces obtained from dislocation dynamics simulations. These dislocation structures are responsible for an increase in strain hardening as the film thickness decreases. The model predicts an increase in strain hardening as the film thickness decreases in agreement with experimental observation in films with thickness in the range 0.2 to 2μm.

Modeling Of Subcontinuum Thermal Transport Across Semiconductor-Gas Interfaces

Dhruv Singh et al. — Wed, 03 Nov 2010 05:20:37 PDT

A physically rigorous computational algorithm is developed and applied to calculate subcontinuum thermal transport in structures containing semiconductor-gas interfaces. The solution is based on a finite volume discretization of the Boltzmann equation for gas molecules in the gas phase and phonons in the semiconductor. A partial equilibrium is assumed between gas molecules and phonons at the interface of the two media, and the degree of this equilibrium is determined by the accommodation coefficients of gas molecules and phonons on either side of the interface. Energy balance is imposed to obtain a value of the interface temperature. The classic problem of temperature drop across a solid-gas interface is investigated with a simultaneous treatment of solid and gas phase properties for the first time. A range of transport regimes is studied, varying from ballistic phonon transport and free molecular flow to continuum heat transfer in both gas and solid. A reduced-order model is developed that captures the thermal resistance of the gas-solid interface. The formulation is then applied to the problem of combined gas-solid heat transfer in a two-dimensional nanoporous bed and the overall thermal resistance of the bed is characterize in terms of the governing parameters. These two examples exemplify the broad utility of the model in practical nanoscale heat transfer applications.

Uncertainty Quantification Models For Micro-Scale Squeeze-Film Damping

Xiaohui Guo et al. — Wed, 03 Nov 2010 05:20:36 PDT

Two squeeze-film gas damping models are proposed to quantify uncertainties associated with the gap size and the ambient pressure. Modeling of gas damping has become a subject of increased interest in recent years due to its importance in micro-electro-mechanical systems (MEMS). In addition to the need for gas damping models for design of MEMS with movable micro-structures, knowledge of parameter dependence in gas damping contributes to the understanding of device-level reliability. In this work, two damping models quantifying the uncertainty in parameters are generated based on rarefied flow simulations. One is a generalized polynomial chaos (gPC) model, which is a general strategy for uncertainty quantification, and the other is a compact model developed specifically for this problem in an early work. Convergence and statistical analysis have been conducted to verify both models. By taking the gap size and ambient pressure as random fields with known probability distribution functions (PDF), the output PDF for the damping coefficient can be obtained. The first four central moments are used in comparisons of the resulting non-parametric distributions. A good agreement has been found, within 1%, for the relative difference for damping coefficient mean values. In study of geometric uncertainty, it is found that the average damping coefficient can deviate up to 13% from the damping coefficient corresponding to the average gap size. The difference is significant at the nonlinear region where the flow is in slip or transitional rarefied regimes.