A three dimensional adaptive multiscale method for crack growth in Silicon

Concurrently coupled solid shell-based adaptive multiscale method for fracture

This paper proposes an adaptive atomistic- continuum numerical method for quasi-static crack growth. The phantom node method is used to model the crack in the continuum region and a molecular statics model is used near the crack tip. To ensure self-consistency in the bulk, a virtual atom cluster is used to model the material of the coarse scale. The coupling between the coarse scale and fine scale is realized through ghost atoms. The ghost atom positions are interpolated from the coarse scale solution and enforced as boundary conditions on the fine scale. The fine scale region is adaptively enlarged as the crack propagates and the region behind the crack tip is adaptively coarsened. An energy criterion is used to detect the crack tip location. The triangular lattice in the fine scale region corresponds to the lattice structure of the (111) plane of an FCC crystal. The Lennard-Jones potential is used to model the atom–atom interactions. The method is implemented in two dimensions. The results are compared to pure atomistic simulations; they show excellent agreement.

The design, modeling, and experimental tests results of both a single-crystal silicon and a polycrystalline nickel micromechanical device developed to evaluate fracture and fatigue of very small fractures are described. The devices are cantilever beams, approximately 300- mu m long. Electrodes excite the devices at resonance. Fatigue crack propagation or time dependent structural change is measured by detecting the shift in resonant frequency. In the silicon device, the frequency change is caused by the extension of a preexisting crack introduced near the fixed end of the cantilever. Experimental data are presented demonstrating time-dependent crack growth in silicon. This study indicates the possibility of crack growth in silicon structures, given the correct environment. Given the greater dislocation mobilities in metals and additional failure modes available in polysilicon, long-term crack growth should be included as a design consideration in the design of micromechanical structures.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The association between invasive and native species varies across spatial scales and is affected by phylogenetic relatedness, but these issues have rarely been addressed in aquatic ecosystems. In this study, we used a non-native, highly invasive species of Poaceae (tropical signalgrass) to test the hypotheses that (i) tropical signalgrass success correlates negatively with success of most native species of macrophytes at fine spatial scales, but its success correlates positively or at random with natives at coarse spatial scales, and that (ii) tropical signalgrass is less associated with native species belonging to the family Poaceae than with species belonging to other families (Darwin’s naturalization hypothesis). We used a dataset obtained at fine (0.25 m2) and coarse (ca. 1,000 m2) scales. The presence/absence of all species was recorded at both scales, and their biomass was also measured at the fine scale. We tested the association between tropical signalgrass biomass and individual native species with logistic regressions at the fine scale, and using the T-score index between tropical signalgrass and each native species at both scales. The likelihood of the occurrence of six species (submersed and free-floating) was negatively affected by tropical signalgrass biomass at the fine scale. T-scores showed that three species were less associated with tropical signalgrass than expected by chance, but 22 species co-occurred more than expected by chance at the coarse scale. Associations between species of Poaceae and tropical signalgrass were null at the fine scale, but were positive or null at the coarse scale. In addition to showing that spatial scale affects the patterns of association among the non-native and individual native species, our results indicate that phylogeny did not explain associations between the invasive and native macrophytes, at both scales.

An efficient space-angle subgrid scale discretisation of the neutron transport equation

A meshless adaptive multiscale method for fracture

Integration of dynamic data typically requires the solution of an inverse problem that can be computationally intensive and practically infeasible for fine scale reservoir models. In this paper we present a new methodology to directly update fine scale geostatistically-based reservoir models by combining gradual deformation parameterization for the fine scale geostatistical model and an upscaling technique for the coarse scale flow simulation model. The proposed methodology includes: Perturbation of the fine scale geostatistical model using the gradual deformation parameterization. Gradual deformation ensures the preservation of the overall geostatistical properties of the fine model. Generation of the coarse scale flow simulation model by upscaling the fine scale geostatistical model. Sensitivity computation of the flow simulation results with respect to the fine scale parameterization. This sensitivity computation is analytical and takes into account the upscaling process. Direct updating of the fine scale geostatistical model using classical optimization process. Direct updating ensures consistency between the fine and coarse scale models. The accuracy of the proposed methodology was improved by calibrating the flow simulation model. The objective of this calibration is to reduce the error introduced by the upscaling step during the flow simulation. We applied successfully our methodology for fine scale reservoir description by integrating permanent down-hole gauge measurements directly into a three-dimensional geostatistical model containing about two million grid blocks. This test is designed to highlight several key issues of the proposed methodology: Efficiency of the upscaling step coupled with gradient-based optimization to speed up the history matching process. Usefulness of the calibration step for a correct integration of upscaling techniques in history matching. Capability of the methodology for maintaining consistency and coherency between fine scale and coarse scale models. Improvement of the reservoir characterization by integrating dynamic data at the fine geostatistical scale. We conclude that the proposed methodology can be used effectively and efficiently for reservoir characterization purposes.

In order to run reservoir simulation efficiently, a coarse scale (CS) dynamic model is created by upscaling of a fine scale (FS) static model. All history match (HM) changes usually done in the CS dynamic model need to be downscaled to FS for geological justifications and consistency maintenance between the FS static and CS dynamic models. This paper proposes a robust downscaling method for integration of FS static and CS dynamic models. The proposed method downscales a HMDM (dynamic model) to HMSM (static) in multiple steps. Scale-up the ISM (initial) to CS to create an IDM. Identify the cell changes between HMDM and IDM, and transfer the changes to FS to create a MSM (modified). Scale-up the MSM to CS to create to a MDM and calculate the ratios between HMDM and MDM for all cell properties. Transfer the ratios to FS to create a HMSM. Scale-up the HMSM to CS to confirm its identity to the HMDM. Selection of sampling and zone mapping methods is critical in all steps. The proposed method has been successfully applied in a giant carbonate oil field in the Caspian Sea that consists of a matrix dominated platform and a fracture/karst dominated rim. Due to the field's complex geology and high H2S content (15%), a dual porosity, dual permeability compositional model has been created to model compositional sour crude flow within/between matrix and fracture/karst. The FS static model contains a 236m × 236m horizontal grid with 593 layers while the CS dynamic model has the horizontal cell sizes in a range of 236m to 944m with 73 layers. Rock regions, permeability, and reservoir connectivity in the CS dynamic model were calibrated using the field historical production data (e.g., static pressure, PLT, interference test, and GOR/water-cut data) to create a HMDM. Since the HM process was performed only in the CS dynamic model, the FS static model and HMDM became inconsistent. Appling the proposed downscaling method has helped the HM team to resolve this issue and resulted in a seamless link between the FS static and CS dynamic models for current and future HM and model updates.

Calculated potential for water enhanced crack growth in silicon

Micromechanical fatigue testing

We study the convergence of multigrid schemes for the Helmholtz equation, focusing in particular on the choice of the coarse scale operators. Let $G_{\rm c}$ denote the number of points per wavelength at the coarse level. If the coarse scale solutions are to approximate the true solutions, then the oscillatory nature of the solutions implies the requirement $G_{\rm c} > 2$. However, in examples the requirement is more like $G_{\rm c} \gtrsim 10$, in a trade-off involving also the amount of damping present and the number of multigrid iterations. We conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators. Standard 5-point finite differences in two dimensions are our first example. A new coarse scale 9-point operator is constructed to match the fine scale phase speeds. We then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator. The required $G_{\rm c}$ is reduced from about 10 to about 3.5, with less damping present so that waves propagate over $>$ 100 wavelengths in the new scheme. Next, we consider extensions of the method to more general cases. In three dimensions, comparable results are obtained with standard 7-point differences and optimized 27-point coarse grid operators, leading to an order of magnitude reduction in the number of unknowns for the coarsest scale linear system. Finally, we show how to include perfectly matched layers at the boundary, using a regular grid finite element method. Matching coarse scale operators can easily be constructed for other discretizations. The method is therefore potentially useful for a large class of discretized high-frequency Helmholtz equations.

Fatigue crack growth in Silicon solar cells and hysteretic behaviour of busbars

Where land surface air temperature data are not available, satellite land surface temperature are used. However, the coarse spatial resolution of satellite‐derived products may yield errors at the local scale. This work shows the differences between the MODIS Land Surface Temperature and Emissivity (MOD11A1) product and ground measurements at two different scales. We used data from 21 SNOTEL stations across the northern Front Range of Colorado to represent the coarse scale and 17 iButton temperature sensors across the Colorado State University Mountain Campus to represent the fine scale. We found significant differences in the temperature and its changes with elevation for the two spatial scales. At the fine scale, cold air drainage can induce an inversion of the temperature gradient with elevation. A higher correlation was found during the nighttime at the fine scale, while, at the coarse scale, higher correlations were observed during the daytime. On windy nights, temperatures do not cool as much as on calmer nights, and the coarse scale near‐surface temperature gradient with elevation persists, though the fine scale inversions do not develop. The near‐surface temperature gradients with elevation based on the MODIS pixels are similar to the ground‐based data at the coarse scale but not at the fine scale. Thus, one must be cautious in selecting the near‐surface temperature gradients with elevation for mountainous terrain when different scales are considered, and a proper validation of satellite products is necessary prior to their use to avoid the propagation of uncertainties.

A micromachined specimen with a test section only 150-μm thick was developed for investigating subcritical crack growth in silicon. Crack growth rates in the range 10−4–10−10m/s were measured as a function of applied stress intensity (v–Kcurves) during tests in humid air and dry nitrogen lasting up to 24 h. The fracture toughness,KIcof {110} silicon was also measured at 1.15 ± 0.08 MPa m1/2. While some evidence MPa-m1/2of subcritical crack growth appeared to occur in the region 0.9KIc&lt;K&gt; 0.98KIc, the extremely high crack growth exponent (n100) and the high ratio of the apparent stress corrosion threshold,KIscc, to the fracture toughness,KIscc/KIc&gt; 0.9, suggests that no clear evidence exists for a stress corrosion process in silicon exposed to humid air.

Currently little data exists on the fracture and subcritical crack growth behavior of single crystal silicon, a common material for micromachined devices. In order to obtain stable crack growth in silicon, we utilize a decreasing stress intensity specimen geometry (Gonzales and Pantano, 1990). We present new data on crack growth rates in silicon as a function of applied stress intensity (K). We made direct measurements of these values via optical and electrical resistance techniques under monotonic loading conditions. We also present new data on the critical stress intensity (KIc) of silicon for a sample of micromachined proportions.