Continuous Velocity Field Research Articles

A 4D continuous representation of myocardial velocity fields from tissue phase mapping magnetic resonance imaging.

Myocardial velocities carry important diagnostic information in a range of cardiac diseases, and play an important role in diagnosing and grading left ventricular diastolic dysfunction. Tissue Phase Mapping (TPM) Magnetic Resonance Imaging (MRI) enables discrete sampling of the myocardium’s underlying smooth and continuous velocity field. This paper presents a post-processing framework for constructing a spatially and temporally smooth and continuous representation of the myocardium’s velocity field from TPM data. In the proposed scheme, the velocity field is represented through either linear or cubic B-spline basis functions. The framework facilitates both interpolation and noise reducing approximation. As a proof-of-concept, the framework was evaluated using artificially noisy (i.e., synthetic) velocity fields created by adding different levels of noise to an original TPM data. The framework’s ability to restore the original velocity field was investigated using Bland-Altman statistics. Moreover, we calculated myocardial material point trajectories through temporal integration of the original and synthetic fields. The effect of noise reduction on the calculated trajectories was investigated by assessing the distance between the start and end position of material points after one complete cardiac cycle (end point error). We found that the Bland-Altman limits of agreement between the original and the synthetic velocity fields were reduced after application of the framework. Furthermore, the integrated trajectories exhibited consistently lower end point error. These results suggest that the proposed method generates a realistic continuous representation of myocardial velocity fields from noisy and discrete TPM data. Linear B-splines resulted in narrower limits of agreement between the original and synthetic fields, compared to Cubic B-splines. The end point errors were also consistently lower for Linear B-splines than for cubic. Linear B-splines therefore appear to be more suitable for TPM data.

Modelling and simulation of H2-H2O bubbly flow through a stack of three cells in a pre-pilot filter press electrocoagulation reactor

Computational fluid dynamics simulations were carried out to describe the hydrodynamic characteristics of a two-phase bubbly flow in a filter-press flow reactor stack of three cells, which is typically used in electrocoagulation (EC). The hydrogen evolution reaction (HER) took place at the cathode; dissolution of aluminium occurred at the anode. The fundamental transport equations of momentum and electrical potential were simultaneously solved to simulate the H2-H2O flow. Continuous (H2O) and dispersed phase (H2) velocity fields were modelled via the Euler-Eulerian approach, using the biphasic Reynolds Averaged Navier-Stokes (RANS) equations and the standard k − ε turbulence model. The influence of volumetric flow rate (1.7 ≤ Q ≤ 15 cm3 s−1) and applied current density (–28 ≤ j ≤ –5 mA cm−2) was systematically addressed to calculate the fraction of dispersed phase and current distribution along the electrodes. The evolved H2 bubbles were transported away from the electrode by the liquid flow. The dispersion of H2 through the electrode gap showed a modest bubble curtain profile due to the liquid flow rate. A homogeneous current distribution along the electrode length was experienced due to the geometrical design of the electrochemical cell and the low degree of H2 dispersion. The velocity profiles of the H2-H2O mixture were different in each cell due to the change of flow direction. H2 bubbles increased the velocity of the liquid phase but the gas fraction of such bubbles resulted in a higher pressure drop. Good agreement between theoretical and experimental residence time distribution curves was achieved; the experimental aluminium dose released by the anode agreed well with the simulations.

Global graphene market to reach $1.08 bn by 2027

We present continuum-scale modeling of multiphase compressible flow in porous media with applications to hydrocarbon reservoir engineering. A new black-oil model is developed and compared with a fully compositional simulator to model the thermodynamic phase behavior. In the context of black-oil modeling, where components are lumped into a gas and liquid pseudocomponent with only the gas transferring between liquid and gas phases, we allow for a variable bubble point pressure (e.g., when gas enters an undersaturated zone). Traditionally, a primary variable switching strategy has been used, which is known to be prone to convergence and phase identification issues. Instead, we adopt an overall molar composition-based framework that can robustly model phase appearance or disappearance. Phase properties across a broad range of pressures for different black-oil compositions are constructed from compositional phase split calculations to correctly model the phase transition. Mass transport is updated explicitly by a locally mass conserving multilinear discontinuous Galerkin method. Globally continuous pressure and velocity fields are obtained through an implicit mixed hybrid FE scheme. The robustness and accuracy of our FE simulator are demonstrated in several problems, where we have attained considerable speed-up and maintained the accuracy with the new black-oil model.

Finite-Time Singularity Formation for Strong Solutions to the Boussinesq System

As a follow up to our work [27], we give examples of finite-energy and Lipschitz continuous velocity field and density $$(u_0,\rho _0)$$ which are $$C^\infty $$-smooth away from the origin and belong to a natural local well-posedness class for the Boussinesq equation whose corresponding local solution becomes singular in finite time. That is, while the sup norm of the gradient of the velocity field and the density remain finite on the time interval $$t\in [0,1)$$, both quantities become infinite as $$t\rightarrow 1$$. The key is to use scale-invariant solutions similar to those introduced in [27]. The proof consists of three parts: local well-posedness for the Boussinesq equation in critical spaces, the analysis of certain special infinite-energy solutions belonging to those critical spaces, and finally a cut-off argument to ensure finiteness of energy. All of this is done on spatial domains $$\{(x_1,x_2): x_1 \ge \gamma |x_2|\}$$ for any $$\gamma > 0$$ so that we can get arbitrarily close to the half-space case. We show that the 2D Euler equation is globally well-posed in all of the situations we look at, so that the singularity is not coming from the domain or the lack of smoothness on the data but from the vorticity amplification due to the presence of a density gradient. It is conceivable that our methods can be adapted to produce finite-energy $$C^\infty $$ solutions on $${\mathbb {R}}^2_+$$ which become singular in finite time.

Application of T-EMSD based p-y curves in the three-dimensional analysis of laterally loaded pile in undrained clay

To apply the p-y curve method in laterally loaded pile analyses, three-dimensional effects in component such as the ultimate pile bearing capacity, pile compatibility factor and initial subgrade modulus must be incorporated. A framework to consider these factors in the theoretical p-y curve derived by the Total-displacement-loading Extended Mobilisable Strength Design (T-EMSD) approach is presented. The bearing capacity design equation can be used to estimate the continuous velocity fields based pile compatibility factor profile. To verify the pile bearing capacity design equation fitted from an upper-bound solution, centrifuge experiments of rigidly translated piles were performed. Corresponding finite element analyses were also performed. The pile compatibility factor profile was obtained through the verified bearing capacity equation and experimental results. Finally, the applications of plane-strain p-y curves to three-dimensional problems were demonstrated via experiment simulation and parametric study.

Integration of GPS and InSAR Data for Resolving 3‐Dimensional Crustal Deformation

AbstractWe develop an algorithm to integrate GPS and InSAR data for a 3‐dimensional crustal deformation field at the Earth's surface. In the algorithm discrete GPS data points are interpolated to obtain a 3‐dimensional continuous velocity field, which is then combined with the InSAR line‐of‐sight (LOS) velocity data pixel by pixel using the least‐squares method. Advantages of our method over previous ones are that: 1) The GPS data points are optimally interpolated by balancing a trade‐off between spatial resolution and solution stability. 2) A new algorithm is developed to estimate realistic uncertainties for the interpolated GPS velocities, to be used as weights for GPS data in GPS‐InSAR combination. 3) Realistic uncertainties for the InSAR LOS rate data are estimated and used as weights for InSAR data in GPS‐InSAR combination. 4) The ramps and/or offsets of the InSAR data are globally estimated for all the images to minimize data misfit, particularly at regions where the data overlaps. Application of this method to real data from southern California shows its capability of successfully restoring 3‐dimensional continuous deformation field from spatially limited GPS and dimensionally limited InSAR data. The deformation field reveals water withdrawal induced subsidence and drought caused uplift at various regions in southern California.

Face stability analysis of a shallow horseshoe-shaped shield tunnel in clay with a linearly increasing shear strength with depth

The face stability of a horseshoe-shaped shield tunnel in undrained clays with a linearly increasing strength is numerically and analytically investigated in this paper. Based on the kinematic approach of limit analysis, a continuous velocity field for a horseshoe-shaped shield tunnel is proposed to evaluate the limit support pressure of both the collapse and blow-out of the tunnel face. Moreover, the analytical results are compared with those obtained from the numerical simulation and other existing literatures. The results show that the horseshoe-shaped shield tunnel can improve the face stability compared to the circular shield tunnel. Finally, an improved velocity field is proposed, which provides a more realistic failure pattern and a better upper bound solution of the limit support pressure for the horseshoe-shaped shield tunnel.

Effect of Longitudinal Gradient on 3D Face Stability of Circular Tunnel in Undrained Clay

The longitudinal gradient existed in shield‐driven tunnel crossing river or channel has a longitudinal gradient, which is often ignored in most stability analyses of the tunnel face. Considering the influence of the longitudinal gradient into A(a) continuous velocity field, the present paper, conducting a limit analysis of the tunnel face in undrained clay, adopted to yield the upper‐bound solutions of the limit pressure supporting on a three‐dimensional tunnel face. The least upper bounds of the collapse and blow‐out pressures can be obtained by conducting an optimization procedure. These upper‐bound solutions are given in the design charts, which provide a simple way to assess the range of the limit pressure in practice. The influence of the longitudinal gradient becomes more significant with the increase of γD/su and C/D. The blow‐out pressure for tunneling in a downward movement could be overestimated and the collapse pressure for tunneling in an upward movement could be conversely underestimated, with ignoring the influence of the longitudinal gradient.

Simulation of velocity field changes in a backwater area of the Kama reservoir in different water regime phases

The area of variable backwater of any reservoir is a complex natural and anthropogenic system that changes significantly during a year. The changes are primarily determined by water balance dynamics during a year that is a correlation between input and output water masses into a reservoir or at any part of the water object. Secondly, the input and output of sediment load changes constantly during a year. Transition of sediments from one site to another significantly changes the morphometrical features and morphology of the latter. Finally, the cyclical nature of the changes mentioned above is caused by daily, weekly, and annual discharge regulation that is conducted according to the Kama reservoir operating regulations. The simulation results of the flow velocity changes at the site Tyul’kino village – Berezniki town during different phases of water regime for high water year and average water years are presented in the article. The calculation of continuous velocity field at the site under investigation (a hydrodynamic model) is based on a digital elevation model of the bottom (DEM), and water level, water surface gradient and bottom roughness have been also considered in the model.

Face stability analysis for a longitudinally inclined tunnel in anisotropic cohesive soils

A stability analysis approach of tunnel face considering a longitudinally inclined tunnel angle and anisotropic purely cohesive soils based on a continuous velocity field (CVF) is investigated in this study. Based on the kinematic approach of limit analysis and the discretization technique, an improved three-dimensional CVF model for longitudinally inclined tunnels driven by pressurized shields is proposed. With the proposed model, the critical support pressure acted on tunnel face is determined by the work-balance equation. A serial of finite element numerical models are conducted to validate the proposed model. Finally, the effects of tunnel inclination angles, several dimensionless parameters as well as soil anisotropy on the critical support pressure are investigated. The numerical results show that the effects of the soil anisotropy and the tunnel inclination angle on tunnel faces should be considered in the actual design of tunneling engineering.

Upper-bound solutions for the face stability of a non-circular NATM tunnel in clays with a linearly increasing undrained shear strength with depth

The face stability of the NATM tunnels (the New Austrian Tunneling Method) in undrained clays with a linearly increasing strength is numerically and analytically investigated by defining two types of the safety factors F in this paper. Based on the kinematic approach of the limit analysis, a continuous velocity field for an open-face tunnel is proposed. Moreover, the calculated solutions are compared with the results obtained from the numerical simulation and other existing literatures to validate its effectivity and superiority. The stability charts for different influence factors are also provided. Finally, an improved velocity field is presented, which provides a closer result to the real situation of the face collapse.

A tool for pathline creation using TOUGH simulation results and fully unstructured 3D Voronoi grids

Pathline or streamline computation is used today to improve the understanding of fluid flow behavior in groundwater modelling. In particular, pathline computation on unstructured grids requires an accurate velocity interpolation method when the flow velocity field is given by numerical simulators only at the faces of the grid blocks; as for example in the case of the Integral Finite Difference method is used for spatial discretization. In the case of 3D Voronoi unstructured grids, a corner velocity interpolation (CVI) method allows continuous velocity field interpolation while preserving the mass conservation principle. We have developed a computer code, named TOUGH2Path, able to compute the velocity field based on the CVI method and generalized barycentric coordinates for 3D Voronoi polytope grid blocks tailored for the TOUGH family of codes. Results for the classical quarter five spot problem show that the coded algorithm correctly reproduces the pathlines computed with semi-analytical methods.

Upper-bound solutions for face stability of circular tunnel in purely cohesive soil using continuous rotational velocity field

This paper presents a 2D analysis model based on limit analysis and slip-line theories for the face stability of a circular tunnel in purely cohesive soil. The analysis model depends on the ratio of cover depth C to tunnel diameter D. When C/D is equal to 0.5, the mechanism consists of three blocks, namely, Zone I, Zone II, and Zone III. When C/D is greater than 0.5, the mechanism consists of four blocks, namely, Zone I, Zone II, Zone III, and a possible Zone IV. Zone II is a transition zone satisfying the normality conditions. The possible Zone IV is a Rankine zone that is subjected to the influence of the vertical soil arching effect appearing at the top of Zone III. The criterion for the collapse thickness limit of the tunnel is proposed based on Terzaghi’s theory of relative soil pressure. The results show that significant improvements have been made to the existing solutions using the proposed failure mechanism for the face stability of circular tunnels in purely cohesive soil.

Modeling total-nitrogen dynamics during electrocoagulation process with stainless-steel electrodes

The efficiency of the electrocoagulation (EC) process for the removal of total N (TN) is analyzed using a pilot device (reactor) with stainless-steel electrodes. Additionally, a three-dimensional (3D) numerical model for TN removal via the EC process is developed. The effects of the flow rate through the reactor and the initial NaCl concentration were the focus of the research. The concentrations of TN, nitrate (NO3-N), dissolved O, Fe (Fe2+/Fe3+), and chloride (Cl-), as well as the water temperature and pH, were measured in the EC reactor. By using the 3D numerical model, spatially continuous velocity fields and the TN concentration were determined. The results of this study show that the TN removal efficiency is dependent primarily on the initial concentration of NaCl and secondarily on the flow rate through the reactor. The initial TN concentration is reduced by 35% after 240 min of reactor operation when the flow rate is set as 0.15 L/s and the initial NaCl concentration is 2 g/L, while the current density is maintained at 150 A/m2. The NO3-N concentrations and pH are reduced by 5% with respect to the initial values.

Face stability analysis of large-diameter slurry shield-driven tunnels with linearly increasing undrained strength

The kinematic approach of limit analysis is explored in three-dimensional face stability of large-diameter tunnels in soils with linearly increasing undrained strength with depth. Due to the slurry density by a shield machine, the support pressure is not uniformly distributed on the face. Such a non-uniform distribution cannot be ignored, especially for the large-diameter shield tunnels. This paper includes it into the stability analysis of the tunnel face subjected to local and global failures. The failure mechanism with a spherical cap is adopted to obtain the least upper-bound solutions of local stability of the tunnel face. To evaluate its global stability, a continuous velocity field with a toric envelope is employed and yields the critical pressures on tunnel face against collapse and blow-out. The calculated results are compared with the solutions derived from other well-established methods for verification of the presented approach. An approximation formula based on the derived upper-bound solutions, is given to directly calculate the necessary collapse and blow-out pressures, which can be used for preliminary design in practice. An example is given to illustrate its convenient use.

Higher-order black-oil and compositional modeling of multiphase compressible flow in porous media

We present continuum-scale modeling of multiphase compressible flow in porous media with applications to hydrocarbon reservoir engineering. A new black-oil model is developed and compared with a fully compositional simulator to model the thermodynamic phase behavior. In the context of black-oil modeling, where components are lumped into a gas and liquid pseudocomponent with only the gas transferring between liquid and gas phases, we allow for a variable bubble point pressure (e.g., when gas enters an undersaturated zone). Traditionally, a primary variable switching strategy has been used, which is known to be prone to convergence and phase identification issues. Instead, we adopt an overall molar composition-based framework that can robustly model phase appearance or disappearance. Phase properties across a broad range of pressures for different black-oil compositions are constructed from compositional phase split calculations to correctly model the phase transition. Mass transport is updated explicitly by a locally mass conserving multilinear discontinuous Galerkin method. Globally continuous pressure and velocity fields are obtained through an implicit mixed hybrid FE scheme. The robustness and accuracy of our FE simulator are demonstrated in several problems, where we have attained considerable speed-up and maintained the accuracy with the new black-oil model.

Upper bound of tunnel face stability using asymmetric yielding

Although significant advancement has been made over recent years with respect to three-dimensional upper bound calculations of tunnel facing, a considerable difference still exists between analytically and empirically based stability values. The current work suggests that the difference may well be the outcome of the traditional use of Tresca yield criterion for the upper bound calculations, which, by definition, does not distinguish among the shearing modes (compression, extension, plane strain). Consequently, this paper suggests and discusses a new yield function, which allows for asymmetric yielding. Such yielding is only beneficial in the case of three-dimensional and continuous velocity fields, and therefore a numerical procedure that generates relevant kinematically admissible fields for classical upper bound calculation is suggested. The procedure involves conversion from a load controlled boundary value problem to a velocity controlled problem at the limit state of collapse. The analysis results in significantly lower upper bound values than those presented earlier (for Tresca material), and the values are much closer to the stability curves of Kimura and Mair (1981), which are commonly used in design.

Upper-bound solutions for face stability of circular tunnels in undrained clays

The aim of this paper is to determine the critical supporting pressure on the circular face of an advancing tunnel in undrained clay in which undrained strength increases linearly with depth. Based on the kinematic approach of limit analysis, a continuous velocity field with a toric envelope is adopted to yield upper bounds of the face pressure for both collapse and blow-out. Combining with a closed-form expression describing the overall velocity field, an optimisation procedure is carried out to find the lowest upper-bound solutions. The calculated results are presented in terms of a dimensionless coefficient N (called the stability number) and then stability charts for use are produced. Alternatively, an approximation formula is given to estimate easily the collapse and blow-out pressure in practice.

Crustal Deformation in the India‐Eurasia Collision Zone From 25 Years of GPS Measurements

AbstractThe India‐Eurasia collision zone is the largest deforming region on the planet; direct measurements of present‐day deformation from Global Positioning System (GPS) have the potential to discriminate between competing models of continental tectonics. But the increasing spatial resolution and accuracy of observations have only led to increasingly complex realizations of competing models. Here we present the most complete, accurate, and up‐to‐date velocity field for India‐Eurasia available, comprising 2576 velocities measured during 1991–2015. The core of our velocity field is from the Crustal Movement Observation Network of China‐I/II: 27 continuous stations observed since 1999; 56 campaign stations observed annually during 1998–2007; 1000 campaign stations observed in 1999, 2001, 2004, and 2007; 260 continuous stations operating since late 2010; and 2000 campaign stations observed in 2009, 2011, 2013, and 2015. We process these data and combine the solutions in a consistent reference frame with stations from the Global Strain Rate Model compilation, then invert for continuous velocity and strain rate fields. We update geodetic slip rates for the major faults (some vary along strike), and find that those along the major Tibetan strike‐slip faults are in good agreement with recent geological estimates. The velocity field shows several large undeforming areas, strain focused around some major faults, areas of diffuse strain, and dilation of the high plateau. We suggest that a new generation of dynamic models incorporating strength variations and strain‐weakening mechanisms is required to explain the key observations. Seismic hazard in much of the region is elevated, not just near the major faults.

Efficient triple-grid multiscale finite element method for 3D groundwater flow simulation in heterogeneous porous media

In this paper, an efficient triple-grid multiscale finite element method (ETMSFEM) is proposed for 3D groundwater simulation in heterogeneous porous media. The main idea of this method is to employ new 3D linear base functions and the domain decomposition technique to solve the local reduced elliptical problem, thereby simplifying the base function construction process and improving the efficiency. Furthermore, by using the ETMSFEM base functions, this method can solve Darcy’s equation with high efficiency to obtain a continuous velocity field. Therefore, this method can considerably reduce the computational cost of solving for heads and velocities, which is crucial for large-scale 3D groundwater simulations. In the application section, we present numerical examples to compare the ETMSFEM with several classical methods to demonstrate its efficiency and effectiveness.