A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity

In this paper, we formulate a finite-element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. We approximate the flow variables by a mixed finite-element space and the displacement by a family of discontinuous Galerkin methods. Theoretical convergence error estimates are derived and, in particular, are shown to be independent of the constrained specific storage coefficient, c _o . This suggests that our proposed algorithm is a potentially effective way to combat locking, or the nonphysical pressure oscillations, which sometimes arise in numerical algorithms for poroelasticity.     

Carbon isotope fractionation in wood during carbonization

A significant uncertainty exists as to whether δ¹³C values in charcoal meaningfully represent the stable isotopic content of the original material, with studies suggesting variable responses to both natural and laboratory heating. An extensive study was undertaken using fully homogenised samples of wood taken from Eucalyptus spp., Quercus robur and Pinus radiata. The results demonstrate that the duration of heating had no tangible effect on the final composition of the charred material, with the δ¹³C and carbon content of wood fixed after 30 min of heating. Furthermore, all three wood types become progressively depleted in ¹³C with increasing temperature. The results demonstrate that even at temperatures commonly reached in natural fires (<450 °C) isotopic fractionation of up to 1.3‰ can take place indicating that the absolute values obtained from charcoal extracted for paleoenvironmental reconstruction must be interpreted with caution.     

Multicollinearity and correlation among local regression coefficients in geographically weighted regression

Present methodological research on geographically weighted regression (GWR) focuses primarily on extensions of the basic GWR model, while ignoring well-established diagnostics tests commonly used in standard global regression analysis. This paper investigates multicollinearity issues surrounding the local GWR coefficients at a single location and the overall correlation between GWR coefficients associated with two different exogenous variables. Results indicate that the local regression coefficients are potentially collinear even if the underlying exogenous variables in the data generating process are uncorrelated. Based on these findings, applied GWR research should practice caution in substantively interpreting the spatial patterns of local GWR coefficients. An empirical disease-mapping example is used to motivate the GWR multicollinearity problem. Controlled experiments are performed to systematically explore coefficient dependency issues in GWR. These experiments specify global models that use eigenvectors from a spatial link matrix as exogenous variables.This study was supported by grant number 1 R1 CA95982-01, Geographic-Based Research in Cancer Control and Epidermiology, from the National Cancer Institute. The author thank the anonymous reviewers and the editor for their helpful comments.     

From geometry to dynamics of microstructure: using boundary lengths to quantify boundary misorientations and anisotropy

The microstructure of a quartzite experimentally deformed and partially recrystallised at 900 °C, 1.2 GPa confining pressure and strain rate 10⁻⁶/s was investigated using orientation contrast and electron backscatter diffraction (EBSD). Boundaries between misoriented domains (grains or subgrains) were determined by image analysis of orientation contrast images. In each domain, EBSD measurements gave the complete quartz lattice orientation and enabled calculation of misorientation angles across every domain boundary. Results are analysed in terms of the boundary density, which for any range of misorientations is the boundary length for that range divided by image area. This allows a more direct comparison of misorientation statistics between different parts of a sample than does a treatment in terms of boundary number.The strain in the quartzite sample is heterogeneous. A 100×150 μm low-strain partially recrystallised subarea C was compared with a high-strain completely recrystallised subarea E. The density of high-angle (>10°) boundaries in E is roughly double that in C, reflecting the greater degree of recrystallisation. Low-angle boundaries in C and E are produced by subgrain rotation. In the low-angle range 0–10° boundary densities in both C and E show an exponential decrease with increasing misorientation. The densities scale with exp(−θ/λ) where λ is approximately 2° in C and 1° in E; in other words, E has a comparative dearth of boundaries in the 8–10° range. We explain this dearth in terms of mobile high-angle boundaries sweeping through and consuming low-angle boundaries as the latter increase misorientation through time. In E, the density of high-angle boundaries is larger than in C, so this sweeping would have been more efficient and could explain the relative paucity of 8–10° boundaries.The boundary density can be generalised to a directional property that gives the degree of anisotropy of the boundary network and its preferred orientation. Despite the imposed strain, the analysed samples show that boundaries are not, on average, strongly aligned. This is a function of the strong sinuosity of high-angle boundaries, caused by grain boundary migration. Low-angle boundaries might be expected, on average, to be aligned in relation to imposed strain but this is not found.Boundary densities and their generalisation in terms of directional properties provide objective measures of microstructure. In this study the patterns they show are interpreted in terms of combined subgrain rotation and migration recrystallisation, but it may be that other microstructural processes give distinctive patterns when analysed in this fashion.     

Convergence analysis of fixed stress split iterative scheme for anisotropic poroelasticity with tensor Biot parameter

We perform a convergence analysis of the fixed stress split iterative scheme for the Biot system modeling coupled flow and deformation in anisotropic poroelastic media with tensor Biot parameter. The fixed stress split iterative scheme solves the flow subproblem with all components of the stress tensor frozen using a multipoint flux mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the fixed stress split iterative scheme for anisotropic poroelasticity with Biot tensor is contractive. We also demonstrate that the scheme is numerically convergent using the classical Mandel’s problem solution for transverse isotropy.     

Testing for similarity in area‐based spatial patterns: Alternative methods to Andresen's spatial point pattern test

Andresen's spatial point pattern test (SPPT) compares two spatial point patterns on defined areal units; it identifies areas where the spatial point patterns diverge and aggregates these local (dis)similarities to one global measure. We discuss the limitations of the SPPT and provide two alternative methods to calculate differences in the point patterns. In the first approach we use differences in proportions tests corrected for multiple comparisons. We show how the size of differences matters, as with large point patterns many areas will be identified by SPPT as statistically different, even if those differences are substantively trivial. The second approach uses multinomial logistic regression, which can be extended to identify differences in proportions over continuous time. We demonstrate these methods by identifying areas where pedestrian stops by the New York City Police Department are different from violent crimes for 2006–2016.     

Comparing spatially varying coefficient models: a case study examining violent crime rates and their relationships to alcohol outlets and illegal drug arrests

In this paper, we compare and contrast a Bayesian spatially varying coefficient process (SVCP) model with a geographically weighted regression (GWR) model for the estimation of the potentially spatially varying regression effects of alcohol outlets and illegal drug activity on violent crime in Houston, Texas. In addition, we focus on the inherent coefficient shrinkage properties of the Bayesian SVCP model as a way to address increased coefficient variance that follows from collinearity in GWR models. We outline the advantages of the Bayesian model in terms of reducing inflated coefficient variance, enhanced model flexibility, and more formal measuring of model uncertainty for prediction. We find spatially varying effects for alcohol outlets and drug violations, but the amount of variation depends on the type of model used. For the Bayesian model, this variation is controllable through the amount of prior influence placed on the variance of the coefficients. For example, the spatial pattern of coefficients is similar for the GWR and Bayesian models when a relatively large prior variance is used in the Bayesian model.      

The Location of Office Space in the Metropolitan Service Economy of the United States, 1985–1990

As the economic base of most American metropolitan areas relies increasingly on services, office industries have shifted to a more pivotal role in the new metropolitan service economy. Using primary office-space usage data of the time period 1985–1990, the changing spatial patterns of this new metropolitan service economy are investigated. It is found that the primary office activities are predominantly concentrated in a relatively small number of the largest metropolitan areas. However, an emerging spatial dispersion in primary office activities is also exhibited. Such dispersion is extremely strong at the upper end of the urban hierarchy. The Sunbelt-Snowbelt dichotomy fails to reveal any differences in the growth pattern of the new metropolitan service economy. A more diversified growth pattern of primary office activities is observed. Concentrated dispersion is the major spatial characteristic of this new metropolitan service economy.