Continuous Velocity Field Research Articles

A sensitivity analysis on 3D velocity reconstruction from multiple registered echo Doppler views

We present a new method for reconstructing a 3D+t velocity field from multiple 3D+t colour Doppler images. Our technique reconstructs 3D velocity vectors from registered multiple standard 3D colour Doppler views, each of which contains a 1D projection of the blood velocity. Reconstruction is based on a scalable patch-wise Least Mean Squares approach, and a continuous velocity field is achieved by using a B-spline grid. We carry out a sensitivity analysis of clinically relevant parameters which affect the accuracy of the reconstruction, including the impact of noise, view angles and registration errors, using simulated data. A realistic simulation framework is achieved by a novel noise model to represent variations in colour Doppler images based on multiscale additive Gaussian noise. Simulations show that, if the Target Registration Error <2.5mm, view angles are >20° and the standard deviation of noise in the input data is <10 cm/s, the reconstructed velocity field presents visually plausible flow patterns and mean error in flow rate is approximately 10% compared to 2D+t Flow MRI. These results are verified by reconstructing 3D velocity on three healthy volunteers. The technique is applied to reconstruct 3D flow on three paediatric patients showing promising results for clinical application.

Modelling the Velocity Field in a Regular Grid in the Area of Poland on the Basis of the Velocities of European Permanent Stations

The paper presents the results of testing the various methods of permanent stations’ velocity residua interpolation in a regular grid, which constitutes a continuous model of the velocity field in the territory of Poland. Three packages of software were used in the research from the point of view of interpolation: GMT (The Generic Mapping Tools), Surfer and ArcGIS. The following methods were tested in the softwares: the Nearest Neighbor, Triangulation (TIN), Spline Interpolation, Surface, Inverse Distance to a Power, Minimum Curvature and Kriging. The presented research used the absolute velocities’ values expressed in the ITRF2005 reference frame and the intraplate velocities related to the NUVEL model of over 300 permanent reference stations of the EPN and ASG-EUPOS networks covering the area of Europe. Interpolation for the area of Poland was done using data from the whole area of Europe to make the results at the borders of the interpolation area reliable. As a result of this research, an optimum method of such data interpolation was developed. All the mentioned methods were tested for being local or global, for the possibility to compute errors of the interpolated values, for explicitness and fidelity of the interpolation functions or the smoothing mode. In the authors’ opinion, the best data interpolation method is Kriging with the linear semivariogram model run in the Surfer programme because it allows for the computation of errors in the interpolated values and it is a global method (it distorts the results in the least way). Alternately, it is acceptable to use the Minimum Curvature method. Empirical analysis of the interpolation results obtained by means of the two methods showed that the results are identical. The tests were conducted using the intraplate velocities of the European sites. Statistics in the form of computing the minimum, maximum and mean values of the interpolated North and East components of the velocity residuum were prepared for all the tested methods, and each of the resulting continuous velocity fields was visualized by means of the GMT programme. The interpolated components of the velocities and their residua are presented in the form of tables and bar diagrams.

Optimization of the Magnetic Dynamo

In stars and planets, magnetic fields are believed to originate from the motion of electrically conducting fluids in their interior, through a process known as the dynamo mechanism. In this Letter, an optimization procedure is used to simultaneously address two fundamental questions of dynamo theory: "Which velocity field leads to the most magnetic energy growth?" and "How large does the velocity need to be relative to magnetic diffusion?" In general, this requires optimization over the full space of continuous solenoidal velocity fields possible within the geometry. Here the case of a periodic box is considered. Measuring the strength of the flow with the root-mean-square amplitude, an optimal velocity field is shown to exist, but without limitation on the strain rate, optimization is prone to divergence. Measuring the flow in terms of its associated dissipation leads to the identification of a single optimal at the critical magnetic Reynolds number necessary for a dynamo. This magnetic Reynolds number is found to be only 15% higher than that necessary for transient growth of the magnetic field.

A continuous velocity field for Norway

In Norway, as in the rest of Fennoscandia, the process of Glacial Isostatic Adjustment causes ongoing crustal deformation. The vertical and horizontal movements of the Earth can be measured to a high degree of precision using GNSS. The Norwegian GNSS network has gradually been established since the early 1990s and today contains approximately 140 stations. The stations are established both for navigation purposes and for studies of geophysical processes. Only a few of these stations have been analyzed previously. We present new velocity estimates for the Norwegian GNSS network using the processing package GAMIT. We examine the relation between time-series length and precision. With approximately 3.5 years of data, we are able to reproduce the secular vertical rate with a precision of 0.5 mm/year. To establish a continuous crustal velocity field in areas where we have no GNSS receivers or the observation period is too short to obtain reliable results, either interpolation or modeling is required. We experiment with both approaches in this analysis by using (i) a statistical interpolation method called Kriging and (ii) a GIA forward model. In addition, we examine how our vertical velocity field solution is affected by the inclusion of data from repeated leveling. Results from our geophysical model give better estimates on the edge of the network, but inside the network the statistical interpolation method performs better. In general, we find that if we have less than 3.5 years of data for a GNSS station, the interpolated value is better than the velocity estimate based on a single time-series.

Bridging onshore and offshore present‐day kinematics of central and eastern Mediterranean: Implications for crustal dynamics and mantle flow

We present a new kinematic and strain model of an area encompassing the Calabrian and Hellenic subduction zones, western Anatolia and the Balkans. Using Haines and Holt's (1993) method, we derive continuous velocity and strain rate fields by interpolating geodetic velocities, including recent GPS data in the Balkans. Relative motion between stable Eurasia and the western Aegean Sea is gradually accommodated by distributed N‐S extension from Southern Balkans to the Eastern Corinth Gulf, so that the westward propagation of the North Anatolian Fault (NAF) throughout continental Greece or Peloponnesus is not required. We thus propose that the NAF terminates in north Aegean and that N‐S extension localized in the Corinth Gulf and distributed in Southern Balkans is due to the retreat of the Hellenic slab. The motion of the Hyblean plateau, Apulia Peninsula, south Adriatic Sea, Ionian Basin and Sirte plain can be minimized by a single rigid rotation around a pole located in the Sirte plain, compatible with the opening the Pelagian rifts (2–2.5 mm/yr) and seismotectonics in Libya. We interpret the trenchward ultraslow motion of the Calabrian arc (2–2.5 mm/yr) as pure collapse, the Calabrian subduction being now inactive. In the absolute plate motion reference frame, our modeled velocity field depicts two toroïdal crustal patterns located at both ends of the Hellenic subduction zone, clockwise in NW Greece and counter‐clockwise in western Anatolia. We suggest the NW Greece toroïdal pattern is the surface expression of a slab tear and consequent toroïdal asthenospheric flow.

Continuous velocity fields for collapse and blowout of a pressurized tunnel face in purely cohesive soil

SUMMARYFace stability analysis of tunnels excavated under pressurized shields is a major issue in real tunnelling projects. Most of the failure mechanisms used for the stability analysis of tunnels in purely cohesive soils were derived from rigid block failure mechanisms that were developed for frictional soils, by imposing a null friction angle. For a purely cohesive soil, this kind of mechanism is quite far from the actual velocity field. This paper aims at proposing two new continuous velocity fields for both collapse and blowout of an air‐pressurized tunnel face. These velocity fields are much more consistent with the actual failures observed in undrained clays. They are based on the normality condition, which states that any plastic deformation in a purely cohesive soil develops without any volume change. The numerical results have shown that the proposed velocity fields significantly improve the best existing bounds for collapse pressures and that their results compare reasonably well with the collapse and blowout pressures provided by a commercial finite difference software, for a much smaller computational cost. A design chart is provided for practical use in geotechnical engineering. Copyright © 2012 John Wiley &amp; Sons, Ltd.

Pathline tracing on fully unstructured control-volume grids

The trend toward unstructured grids in subsurface flow modeling has prompted interest in the issue of streamline or pathline tracing on unstructured grids. Streamline tracing on unstructured grids is problematic because a continuous velocity field is required for the calculation, while numerical solutions to the groundwater flow equations provide velocity in discretized form only. A method for calculating flow streamlines or pathlines from a finite-volume flow solution is presented. The method uses an unconstrained least squares method on interior cells and a constrained least squares method on boundary cells to approximate cell-centered velocities, which can then be continuously interpolated to any point in the domain of interest. Two-dimensional tests demonstrate that the method correctly reproduces uniform and corner-to-corner flow on fully unstructured grids. In three dimensions using regular hexahedral grids, the method agrees well with established semianalytical methods. Tests also demonstrate that the method produces physically realistic results on fully unstructured three-dimensional grids.

3-D crustal structure in the Agadir region (SW High Atlas, Morocco)

The 1960 Agadir earthquake (Mw 6.0) constitutes the most damaging earthquake event in Morocco. With the expansion of seismic networks during the last decade in Morocco, new seismic data have been collected in this region. The P and S arrivals at 19 stations located in Southern Morocco are used to investigate the lithosphere in the Agadir region. In this study, we use a linearized inversion procedure comprising two steps: (1) finding the minimal 1-D model and simultaneous relocation of hypocentres and (2) determination of local velocity structure using linearized inversion. The model parameterization in this method assumes a continuous velocity field. The resolution tests indicate that the calculated images give near true structure for the studied region from 0- to 45-km depth. The results show that the total crust thickness varies from 30 to 40 km in SW High Atlas and confirm the modest crustal tectonic shortening and thickening in the Atlas Mountains of Morocco. The inferred geological structure reconstructed from the calculated image illustrates the existence of fault-related folding. The evidence for coseismic ruptures in 1960 on the Kasbah anticline combined with the 1960 earthquake hypocentre located in the tomographic image determines the seismic potential of the active fault and related fold. The resulting tomographic image shows a high-velocity anomalies that could be associated with the location of deep active fault (10–30 km) associated with the fold structure. In the South Atlas, theses anomalies could be associated with the South atlas thrust front structure.

Soliton solutions of a Calogero model in a harmonic potential

A classical Calogero model in an external harmonic potential is known to be integrable for any number of particles. We consider here reductions which play a role of the ‘soliton’ solutions of the model. We obtain these solutions both for the model with finite number of particles and in a hydrodynamic limit. In the latter limit, the model is described by hydrodynamic equations on continuous density and velocity fields. Soliton solutions in this case are finite-dimensional reductions of the hydrodynamic model and describe the propagation of lumps of density and velocity in the nontrivial background.

Glacier surface velocity estimation using SAR interferometry technique applying ascending and descending passes in Himalayas

In this study ascending and descending passes interferometric synthetic aperture radar (InSAR) techniques are used for glacier surface velocity estimation in the Himalaya. Single-track interferometric measurements are sensitive to only a single component of the three dimensional (3-D) velocity vectors. European Remote Sensing satellites (ERS-1/2) tandem mission data in ascending and descending tracks provide an opportunity to resolve the three velocity components under the assumption that glacier flow is parallel to its surface. Using the surface slope as an essential input in this technique the velocity pattern of Siachen glacier in Himalaya has been modelled. Glaciers in the Himalayan region maintain excellent coherence of SAR return signals in one-day temporal difference. As a result we could obtain spatially continuous surface velocity field with a precision of fraction of radar wavelength. The results covering the main course of glacier are analysed in terms of spatial and temporal variations. A maximum velocity of 43cm/day has been observed in the upper middle portion of the glacier. This technique has been found accurate for monitoring the flow rates in this region, suggesting that routine monitoring of diurnal movement Himalayan glaciers would be immensely useful in the present day context of climate change.

A Connection Between Scott–Vogelius and Grad-Div Stabilized Taylor–Hood FE Approximations of the Navier–Stokes Equations

This article studies two methods for obtaining excellent mass conservation in finite element computations of the Navier–Stokes equations using continuous velocity fields. With a particular mesh construction, the Scott–Vogelius element pair has recently been shown to be inf-sup stable and have optimal approximation properties, while also providing pointwise mass conservation. We present herein the first numerical tests of this element pair for the time dependent Navier–Stokes equations. We also prove that the limit of the grad-div stabilized Taylor–Hood solutions to the Navier–Stokes problem converges to the Scott–Vogelius solution as the stabilization parameter tends to infinity. That is, we provide theoretical justification that choosing the grad-div parameter large does not destroy the solution. Numerical tests are provided which verify the theory and show how both Scott–Vogelius and grad-div stabilized Taylor–Hood (with large stabilization parameter) elements can provide accurate results with excellent mass conservation for Navier–Stokes approximations.

Study of Non-Monotonity of Forming Processes Using Finite Element Analysis

The method of severe plastic deformation is often used to produce bulk ultra-fine grained materials. Numerous methods exist, which are able to produce ultra fine grained materials. Investigated the mechanical schema of these procedures, we searched for such attributes, which characterize the mentioned techniques. By our previous researches can be established, that for these techniques shear deformation as well as the so-called non-monotonic deformation are characteristic. In this paper a degree of non-monotonity is presented, which is an appropriate measure to characterize the forming processes. Besides this the combined Euler – Lagrange method is showed, which is appropriate to obtain a continuous velocity field from finite element analysis. Eventually the mechanical analysis of the extrusion, conventional and asymmetric rolling is showed with respect to the non-monotonity.

Stability of splay states in globally coupled rotators

The stability of dynamical states characterized by a uniform firing rate (splay states) is analyzed in a network of N globally pulse-coupled rotators (neurons) subject to a generic velocity field. In particular, we analyze short-wavelength modes that were known to be marginally stable in the infinite N limit and show that the corresponding Floquet exponent scale as 1/N2. Moreover, we find that the sign, and thereby the stability, of this spectral component is determined by the sign of the average derivative of the velocity field. For leaky-integrate-and-fire neurons, an analytic expression for the whole spectrum is obtained. In the intermediate case of continuous velocity fields, the Floquet exponents scale faster than 1/N2 (namely, as 1/N4) and we even find strictly neutral directions in a wider class than the sinusoidal velocity fields considered by Watanabe and Strogatz [Physica D 74, 197 (1994)].

Estimation of sheet-rolling parameters on the basis of discontinuous and continuous velocity fields

Estimation of sheet-rolling parameters on the basis of discontinuous and continuous velocity fields

Adaptive Fup multi-resolution approach to flow and advective transport in highly heterogeneous porous media: Methodology, accuracy and convergence

In this paper, we present a new Monte-Carlo methodology referred to as Adaptive Fup Monte-Carlo Method (AFMCM) based on compactly supported Fup basis functions and a multi-resolution approach. We consider for illustration 2-D steady, linear and unidirectional flow and advective transport defined on a domain of size 64 I Y ∗ 32 I Y with isotropic exponential correlation heterogeneity structure and σ Y 2 up to 8. Accuracy and convergence issues are rigorously analyzed for each realization as well as for the ensemble. Log-conductivity is presented by continuous function at high resolution level ( n Y = 4–32 points per integral scale) reproducing very accurately prescribed statistics. The flow problem is the most demanding Monte-Carlo step due to satisfying detailed log-conductivity properties. Presented methodology inherently gives continuous and mesh-free velocity fields, which enables the construction of a new efficient and accurate particle tracking algorithm. Results indicate that resolutions n Y = 8 and n h = 32 enable very accurate flow solutions in each realization with mass balance error less than 3% and accurate ensemble velocity statistics. Results show that the proposed AFMCM enables tracking of an unlimited number of injected particles and calculates required transport variables as continuous functions with desired relative accuracy (0.1%) in each realization. Furthermore, we show that the resolution n Y = 8 yields a quite accurate pdf of the transverse displacement and travel time. All required flow and transport variables require 500 Monte-Carlo realizations in order to stabilize fluctuations of the higher-order moments and the probability density functions.

Solving the steady‐state groundwater flow equation for finite linear aquifers using a generalized Fourier series approach in two‐dimensional domains

AbstractA method to solve steady linear groundwater flow problems using generalized Fourier Series is developed and particularized for multiple Fourier series in two‐dimensional domains. It leads to a linear vector equation whose solution provides a finite number of generalized Fourier coefficients approximating the hydraulic head field. Its implementation is shown and two relevant properties are found for the system matrix. It is always symmetric and, once computed, if additional Fourier terms are needed for a better approximation of the hydraulic head field, previously computed matrix elements remain invariant, i.e. only new rows and columns are added to the system matrix. The method is demonstrated in three simple cases with different geometries and transmissivity fields, where solutions are compared with analytical and finite element method results. Thus, the method is verified as an alternative to other flow solvers. Additionally, it provides a direct way to obtain the spectral form of the flow equation solution, given a spectral representation of transmissivity, and can be easily extended to obtain continuous velocity fields and their approximated spectral expressions. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Finite element analysis for thermoforming process of continuous fiber reinforced thermoplastic composites

AbstractAn implicit finite element method is presented to model thermoforming process of thermoplastic materials reinforced with unidirectional fibers or woven fabric. The composite materials are regarded as a homogenized anisotropic viscous fluid restricted by kinematical constrains of material incompressibility and fiber inextensibility at the forming temperature. Unlike previous numerical approaches considering quasi‐static assumptions during each increment, a high order polynomial expression is used to define continuous displacement and velocity fields. The material parameters are also updated during each increment to determine the displacement and velocity fields satisfying the equilibrium equations as well as the kinematical constrains both in the rate and total deformation forms. An iterative solution method is presented to solve the finite element formulation. The numerical results are compared with analytical solutions and the results of previous implicit finite element approaches to demonstrate the computational accuracy and efficiency of the present method. POLYM. COMPOS., 2009. © 2008 Society of Plastics Engineers

Mixed method and convex optimization for limit analysis of homogeneous Gurson materials: a kinematical approach

A fully kinematical, mixed finite element approach based on a recent interior point method for convex optimization is proposed to solve the limit analysis problem involving homogeneous Gurson materials. It uses continuous or discontinuous quadratic velocity fields as virtual variables, with no hypothesis on a stress field. Its modus operandi is deduced from the Karush-Kuhn-Tucker optimality conditions of the mathematical problem, providing an example of cross-fertilization between mechanics and mathematical programming. This method is used to solve two classical problems for the von Mises plasticity criterion as a test case, and for the Gurson criterion for which analytical solutions do not exist. Using only the original plasticity criterion as material data, the method proposed appears robust and efficient. providing very tight bounds on the limit loadings investigated. (C) 2008 Elsevier Masson SAS. All rights reserved.

Application of geometric midline yield criterion to analysis of three-dimensional forging

A kinematically admissible continuous velocity field was proposed for the analysis of three-dimensional forging. The linear yield criterion expressed by geometric midline of error triangle between Tresca and Twin shear stress yield loci on the π-plane, called GM yield criterion for short, was firstly applied to analysis of the velocity field for the forging. The analytical solution of the forging force with the effects of external zone and bulging parameter is obtained by strain rate inner product. Compression tests of pure lead are performed to compare the calculated results with the measured ones. The results show that the calculated total pressures are higher than the measured ones whilst the relative error is no more than 9.5%. It is implied that the velocity field is reasonable and the geometric midline yield criterion is available. The solution is still an upper-bound one.

Dynamics of a Closed Rod with Twist and Bend in Fluid

We investigate the instability and subsequent dynamics of a closed rod with twist and bend in a viscous, incompressible fluid. A new version of the immersed boundary (IB) method is used in which the immersed boundary applies torque as well as force to the surrounding fluid and in which the equations of motion of the immersed boundary involve the local angular velocity as well as the local linear velocity of the fluid. An important feature of the IB method in this context is that self-crossing of the rod is automatically avoided because the rod moves in a continuous (interpolated) velocity field. A rod with a uniformly distributed twist that has been slightly perturbed away from its circular equilibrium configuration is used as an initial condition, with the fluid initially at rest. If the twist in the rod is sufficiently small, the rod simply returns to its circular equilibrium configuration, but for larger twists that equilibrium configuration becomes unstable, and the rod undergoes large excursions before relaxing to a stable coiled configuration.