Curl Of Vector Research Articles

Numerical study on three-dimensional flow around a cylinder with perforated shrouds at different Reynolds numbers

The perforated shrouds have been proposed to control cylinder flows, while the effects and mechanisms at different Reynolds numbers (Res) remain unclear. Three-dimensional numerical simulations are conducted in this paper to compare the aerodynamic performance of the flow around a smooth cylinder and a shrouded cylinder at Re of 3900 and 1.4×105. The results indicate that the drag of the perforated shrouded cylinder is reduced by 30.8% at the high Re, while increased by 26% at Re of 3900 compared with the smooth cylinder. Differently, the lift oscillations of the cylinder are greatly weakened by 83.3% at the Re of 3900 and 98.5% at the Re of 1.4×105, which implies the wake oscillations are nearly eliminated at the Re of 1.4×105. Further analysis exhibits the near wake region is elongated along the mainstream, with significantly recovered pressure. Especially, the greater pressure loss owing to outer shrouds even leads to the negative drag of the inner cylinder at the high Re. In addition, the incoming flow is broken up by outer shrouds, and different flow patterns appear in the gap. The discernible vortex pairs occur in the gap at the Re of 3900, while as Re increases to 1.4×105, the quantities of small-scale vortex weaken the impact on the inner cylinder. The shear layer characteristics are elucidated by Lamb vector curl and Kelvin–Helmholtz instabilities. The vorticity stretching and tilting in the shear layer of the shrouded cylinder is much weaker at the high Re. Generally, the energy for shear layer instabilities at low frequencies is diminished with the presence of perforated shrouds. However, the energy at higher frequencies is strengthened at the low Re.

Limitations of Curl and Directional Filters in Elastography

In the approaches to elastography, two mathematical operations have been frequently applied to improve the final estimate of shear wave speed and shear modulus of tissues. The vector curl operator can separate out the transverse component of a complicated displacement field, and directional filters can separate distinct orientations of wave propagation. However, there are practical limitations that can prevent the intended improvement in elastography estimates. Some simple configurations of wavefields relevant to elastography are examined against theoretical models within the semi-infinite elastic medium and guided waves in a bounded medium. The Miller–Pursey solutions in simplified form are examined for the semi-infinite medium and the Lamb wave symmetric form is considered for the guided wave structure. In both cases, we examine simple but practical wave combinations that can prevent the curl and directional filter operations from directly providing an improved measure of shear wave speed and shear modulus. Additional factors including signal-to-noise and the support of filters also restrict the applicability of these strategies for improving elastographic measures. Thus, some implementations of shear wave excitations applied to the body and to bounded structures within the body are shown to involve waves that are not easily resolved by the vector curl operator and directional filters. These limitations may be overcome by more advanced strategies or simple improvements in baseline parameters including the size of the region of interest and the number of shear waves propagated within.

Second-Order Nédélec Curl-Conforming Hexahedral Element for Computational Electromagnetics

We follow a systematic approach to obtain mixed-order curl-conforming basis functions for the hexahedron that are compatible with basis functions for tetrahedra and triangular prisms previously published. The approach is mathematically sound since we obtain the functions as the dual basis with respect to properly discretized Nédélec degrees of freedom. Well-conditioned bases without the need for added orthogonalization procedures are obtained. We provide simple closed-form expressions for second-order basis functions in a reference hexahedron in terms of integer coefficients and monomials. The expressions are ready to use as long as the appropriate geometric mappings are made. We apply the Method of Manufactured Solutions (MMS) to a finite-element double curl vector wave formulation for verification purposes; specifically, we conduct a study of the non-symmetrical structure of the corresponding tensor product finite-element space. We also solve generalized eigenvalue problems for well-known cavities. We provide the open-source code for generating the coefficients, evaluating the basis functions, and computing the finite-element matrices involved in some of the numerical solutions shown in the article.

Large Eddy Simulation of the cavitating flow around a Clark-Y mini cascade with an insight on the cavitation-vortex interaction

In this research, the Large Eddy Simulation method is utilized to predict the cavitation-vortex interaction in the flow around a Clark-Y mini cascade along with the dynamical processes of the U-type cavity, analyzed using the Lamb vector divergence transportation. To provide a novel perception of the development of the U-type cavity, various fundamental mechanisms of the periodic evolution of the U-type cavity are investigated thoroughly. The dynamical progress explicates the mechanisms that generate the high-pressure zone and low-velocity zone. This mechanism, combined with the re-entrant jet and vortex motions plays a significant role in the development of the U-type cavity. Since the Lamb vector curl associates the vorticity transport equation and is closely related to the source of vorticity of the vortex evolution, the Lamb vector curl captures the vortical structures which show good agreement with the Ω-method results. In addition, the distributions of high- and low-momentum flow in the wake of the cascade are delineated by the Lamb vector divergence. The transition from the maximum to the minimum of the lamb vector divergence in a region illustrates a quasi-cyclic behavior that is closely related to the momentum transport mechanisms and the cyclical motions of the vortex.

Active Electric Dipole Energy Sources: Transduction via Electric Scalar and Vector Potentials.

The creation of electromagnetic energy may be realised by engineering a device with a method of transduction, which allows an external energy source, such as mechanical, chemical, nuclear, etc., to be impressed into the electromagnetic system through a mechanism that enables the separation of opposite polarity charges. For example, a voltage generator, such as a triboelectric nanogenerator, enables the separation of charges through the transduction of mechanical energy, creating an active physical dipole in the static case, or an active Hertzian dipole in the time-dependent case. The net result is the creation of a static or time-dependent permanent polarisation, respectively, without an applied electric field and with a non-zero vector curl. This system is the dual of a magnetic solenoid or permanent magnet excited by a circulating electrical current or fictitious bound current, respectively, which supplies a magnetomotive force described by a magnetic vector potential and a magnetic geometric phase proportional to the enclosed magnetic flux. Thus, the active electric dipole voltage generator has been described macroscopically by a circulating fictitious magnetic current boundary source and exhibits an electric vector potential with an electric geometric phase proportional to the enclosed electric flux density. This macroscopic description of an active dipole is a semi-classical average description of some underlying microscopic physics, which exhibits emergent nonconservative behaviour not found in classical closed-system laws of electrodynamics. We show that the electromotive force produced by an active dipole in general has both electric scalar and vector potential components to account for the magnitude of the electromotive force it produces. Independent of the electromagnetic gauge, we show that Faraday’s and Ampere’s law may be derived from the time rate of change of the magnetic and dual electric geometric phases. Finally, we analyse an active cylindrical dipole in terms of scalar and vector potential and confirm that the electromotive force produced, and hence potential difference across the terminals is a combination of vector and scalar potential difference depending on the aspect ratio (AR) of the dipole. For long thin active dipoles (AR approaches 0), the electric field is suppressed inside, and the voltage is determined mainly by the electric vector potential. For large flat active dipoles (AR approaches infinity), the electric flux density is suppressed inside, and the voltage is mainly determined by the scalar potential.

Divergence and Curl of COVID19 Spreading in the Lower Peninsula of Michigan

This paper explores the COVID19 transmission pattern and circulation dynamics in the Euclidean space at the lower peninsula of Michigan by using the divergence and curl concept in vector field. The COVID19 transmission volume flux can be calculated for each county by using vector divergence. The results shows Wayne county had the highest divergence (162660), the Kent county had the second highest divergence (152540), and the Saginaw county had the third highest divergence (103240), the divergence is positive which means the COVID19 virus was transmitted from these counties to other places. The results also shows Monroe county had the lowest divergence (-187843), the Allegan county had the second lowest number in divergence (-90824), the divergence is negative which means the COVID19 virus was transmitted from other places to these counties. The circulation of the virus is also calculated by using vector curl. The positive curl means that the virus has circulated in a counter-clockwise direction, and the negative curl means the virus has circulated in a clockwise direction.
 
 The divergence is an operator of the COVID19 transmission vector field, which produces a scalar field giving the quantity of the transmission vector field’s source at each location. The COVID19 spreading volume density of the outward flux of transmission field is represented by divergence around a given location. 
 
 The curl is an operator of the COVID19 transmission field, which describes the circulation of a transmission vector field. The curl at a location in COVID19 transmission field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. The curl of a transmission field is formally defined as the circulation density at each location of COVID19 transmission field.

Comparison of rigid and deformable coregistration between mpMRI and CT images in radiotherapy of prostate bed cancer recurrence.

To evaluate the accuracy of rigid coregistration between multiparametric magnetic resonance (mpMR) and computed tomography (CT) images for radiotherapy of prostate bed cancer recurrence. Fifty-three patients (59 nodules) accrued in a prospective study on salvage radiotherapy for prostatic bed recurrence were suitable for the analysis. Patients underwent a pre radiotherapy mpMR exam and a planning CT in the same treatment position and with control of organ filling. The site of recurrence was delineated on mpMR images and contours transferred on planning CT images using both rigid and deformable registrations. Coregistrations were evaluated by mathematical operators that quantify deformation (Jacobian determinant and vector curl) and similarity indices (Dice and Jaccard coefficients). Dose coverage was evaluated. Deformable registration did not change volumes, (p=0.92MW test). The Jacobian coefficient and the vector curl revealed no important image deformations. Dice and Jaccard coefficients indicated dislocation of the nodule volumes. Dislocation magnitude was d=(5.6±3.1)mm. Organ filling was not correlated with deformation or dislocation. Volumes were covered by the 95% isodose in 96% of cases when rigid registration was performed versus 75% of cases when deformed. Rigid image coregistration is sufficiently accurate in this setting. The results indicate that the deformable registration tends to shrink the voxels and to dislocate the ROI, the adopted expansion for the recurrence volume adequately accounts for the observed deformation and dislocation, provided that organ filling is controlled.

About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function

Abstract A simple algorithm for calculating Christoffel symbols, a covariant projection of the result of the Laplace operator's action on the vector, vector curl and other similar operations in an arbitrary oblique base are proposed. For an arbitrary base with ortho ei is found the expressions of vector projections (ΔA) i and (rot A) i , where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA) i and (rot A) i for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (ΔA) i and (rot A) i as an example are made in cases of spherical and cylindrical coordinate systems.

Quantitative assessment of finite-element models for magnetostatic field calculations

We present quantitative means for assessing the numerical accuracy of static magnetic field calculations in finite-element models. Our calculations use the three-dimensional Opera simulation software suite of Dassault Systèmes. Our need to assess the effects of fringe fields requires such a 3D algorithm. While we do discuss and compare our approach to a method of accuracy estimation used in the Opera post-processor, our methods are generally applicable to any model using relaxation techniques in finite-element systems. For purposes of illustration, we present modeling and analysis of two types of quadrupole electromagnets presently in operation in the south arc of the Cornell Electron Storage Ring (CESR). Calculations of field multipole expansion coefficients and numerical deviations from Maxwell’s equations in source-free regions are discussed, with emphasis on the dependence of their accuracy on changes to the finite-element model. Successive refinement steps in the finite-element model for the non-extraction type of CESR south arc quadrupole achieve a reduction in the RMS value of the longitudinal component of the curl vector on the magnet axis by a factor of nearly 70 from 4.38×10−2 T/m to 6.36×10−4 T/m, which is 0.0021% of the field gradient. An accuracy of 2.9% is achieved for a dodecapole coefficient of 2.4×10−4 of that of the quadrupole in the longitudinal integral of the vertical field gradient at a radius of 1 cm.

Effects of Differential Geometry Parameters on Grid Generation and Segmentation of MRI Brain Image

Accurate segmentation of brain tissue in magnetic resonance images (MRI) is a difficult task due to different types of brain abnormalities. In this paper, we review the deformation method focus on the construction of diffeomorphisms to generate the 2D MRI image grids and extend it to the 3D MRI images, address clearly a new formation of the deformation problem for moving domains and construction of diffeomorphisms through a completely different approach. The idea is to control directly the first order differential operators including the Jacobian determinant (JD), divergence (DIV), and curl vector (CV) generated by the 2D and 3D MRI image grids and use them as CNN channels with other modalities (T1-weighted, T1-IR, and T2-FLAIR) or single T1-weighted modality to improve the performance of brain segmentation. More importantly, we discuss the influence of three optimization parameters to the generation of the 2D and 3D MRI image grids by both theoretical analysis and the numerical experiments. We test this method on the IBSR dataset and MRBrainS18 dataset based on VoxResNet and verify the influence of three parameters on the accuracy of MRI brain segmentation. Finally, we also compare the segmentation performance of our method in three networks, including 2D U-Net, 3D U-Net, and VoxResNet. We believe the proposed method can advance the performance in brain segmentation and clinical diagnosis.

Transverse anomalies and Dyson–Schwinger equation in QED3 and QED2 theories

Based on the method of computing the curl of vector, axial-vector and the tensor current operator equations, the possible quantum anomaly for the transverse Ward–Takahashi relations in QED3 and QED2 theories is studied. The absence of transverse anomalies for both axial-vector current in QED2 theory and vector (tensor) current in QED3 theory are verified respectively. Further, it is found that when we ignore the contribution of the three integral-term involving the vertex function in the Wilson line, we can get a set of closed coupled integral Dyson–Schwinger equations in QED3.

On manufacturing constraints for tow-steered composite design optimization

Automatic fiber placement machines have made it viable to manufacture composites where fiber angles vary continuously—tow-steered composites. The additional freedom provided by tow-steered composites has the potential to increase structural performance; however, this approach comes with added design complexity. Numerical optimization can address this complexity, though it is critical that constraints are enforced to ensure that the resulting optimal tow-steered designs are manufacturable using current automatic fiber placement machines. In this work, we consider two manufacturing constraints: tow path curvature and gaps/overlaps. To develop these constraints, we consider a general tow-steered layer pattern as a 2D unit vector field, where the field streamlines represent the tow paths laid down by the automated fiber placing machine. This mathematical formulation provides a relationship between tow path curvature, gap/overlap propagation rate, vector curl, and divergence. These relationships also lead to a constraint on the minimum cut/add length of a tow for a given tow-steered pattern. We demonstrate the developed constraint formulations on two analytical examples, as well as on a structural optimization. We also explore the relationship between curl and divergence of rotated tow patterns. This analysis leads to the conclusion that layups featuring such patterns require strict constraints on bounds to ensure satisfaction of manufacturability requirements. Finally, we use these relationships to motivate a family of gap/overlap-free and curvature-free tow-steered patterns.

Large-eddy simulation of sonic coaxial jets with different total pressure ratios of the inner to outer nozzle

Large-eddy simulation of sonic coaxial jets issuing into a quiescent environment was carried for two total pressure ratios of the inner to outer nozzle, i.e. 5/3 and 3/5. The effects of the total pressure ratio on the flow characteristics were mainly investigated. Various fundamental mechanisms dictating the complex flow characteristics, including jet shear layer evolution, shock system formation, shock/shear-layer interaction, turbulence behavior, and mixing property, have been studied. It is found that the total pressure ratio has an important influence on the flow evolution of coaxial jets as well as turbulence behavior and jet mixing property. The fluid-dynamic shearing and compressing processes are analyzed based on the Lamb vector curl and divergence. The multi-layer structures of the shear layers and shock waves are reasonably captured and the relevant fluid-dynamic processes are clearly clarified. It is also identified that the turbulence behavior and mixing property of the coaxial jets are mainly associated with the shearing effect in the outer shear layer region and the shearing and compressing coupled effect in the jet core region.

A Low-Storage Discontinuous Galerkin Time-Domain Method

In this letter, a discontinuous Galerkin time-domain (DGTD) method with a low memory scheme for the solution of Maxwell’s equations has been proposed. By expressing curl vector basis functions in terms of barycentric coordinates, the matrices in the DGTD method including mass and flux matrices can be rewritten into the summation of a few universal matrices. With the use of the proposed scheme, the computer memory storage requirement of the DGTD method is greatly reduced with the increases of the order of spatial basis functions and the number of meshing elements, albeit a slightly increase of computational time. Numerical results including the slotted waveguide and Vivaldi antenna array are given to illustrate good computational performance.

Utilizing Deformable Image Registration and Quality Assurance to Automate Analysis of Anatomic Changes After Head and Neck Radiation Therapy

Deformable image registration (DIR) algorithms are clinically utilized in radiation therapy without quality assurance (QA) of the deformation vector fields (DVFs) produced. This study investigated the feasibility of simultaneous automated detection of anatomic changes and DIR-QA for patients who receive head and neck (H&N) radiation therapy. For 10 patients with H&N cancer, three DIR algorithms were used to map initial planning computed tomography (CT) images to posttreatment images and analyze anatomic changes in organs-at-risk (OARs) and gross tumor volumes (GTVs). Automated analysis of anatomic changes was carried out by computing the Jacobian determinant, |J|, of the deformation vector field (DVF) resulting from each of 3 different, commercially available DIRs, which map image intensity between images. |J| is the relative deformation of each mapped voxel, and we focused on the 50% deformation level, i.e., where voxels expand or contract by at least 50% (J50). The Jacobian volume histogram (JVH) for OARs and GTVs described the volume of each structure, which deformed at a specified level. In order to differentiate DVF error from true deformation in QA of the DVFs, we utilized the magnitude of vector curl, |C|, and knowledge of limitations in dense tissue deformation (e.g., in bone we expect J50 = 0). Deformation vector field regions with relatively large |C| and/or large deformation in dense tissue were considered erroneous. Analysis of |J| revealed >50% regression (J50) in up to 48% of GTV volumes (J50 >10% in 3/10 cases) and up to 70% volume of parotid glands (J50>10% in 6/10 cases); however, these relatively large volume changes were not consistent across the three DIR algorithms with differences in J50 up to 7 cm3 of GTV and up to 16 cm3 of parotids. Inconsistent deformations at J50 in up to 5 cm3 of tissue were also detected in the oral cavity (6/10) and larynx (3/10 cases), but not in the esophagus (maximum difference of 0.9 cm3) or spinal cord (maximum difference of 0.9 cm3). For mandible, volumes of large deformation up to 23 cm3 were detected implying DVF errors. Thresholding |C| at 0.8 (80% relative rotation) allowed rapid identification of errors in DVFs. For GTVs, at least 1 DVF per patient could be identified and disregarded due to non-physical, large rotations in each of the 3/10 cases with large differences in J50(>5 cm3 with |C|>0.8). No single algorithm was consistently superior to the others in terms of minimizing large, unrealistic deformations across the ten cases considered. Fully automated detection of anatomic response to radiation therapy is not currently feasible based on the presence of implausible deformations and rotations; however, by incorporating DVF-QA utilizing Jacobian and curl, it may be possible to rule-out DVFs and find realistic, patient-specific anatomic variations.

New development of non-rigid registration

We propose a new nonrigid registration algorithm which is based on the optimal control approach. In our previously proposed methods, the Jacobian determinant and the curl vector were used as control functions. In this algorithm, we use a new set of control functions. A main advantage of using the new controls is that the positivity and normalization of the Jacobian determinant are satisfied automatically. Numerical results on large deformation brain images are provided to show the accuracy and efficiency of the algorithm. doi:10.1017/S1446181114000091

NEW DEVELOPMENT OF NONRIGID REGISTRATION

AbstractWe propose a new nonrigid registration algorithm which is based on the optimal control approach. In our previously proposed methods, the Jacobian determinant and the curl vector were used as control functions. In this algorithm, we use a new set of control functions. A main advantage of using the new controls is that the positivity and normalization of the Jacobian determinant are satisfied automatically. Numerical results on large deformation brain images are provided to show the accuracy and efficiency of the algorithm.

A fully divergence-free method for generation of inhomogeneous and anisotropic turbulence with large spatial variation

A fully divergence-free method is proposed for generation of inhomogeneous/anisotropic turbulence with large spatial variation. The method is based on the method of Smirnov et al., which is known to violate the divergence-free constraint when spatial variation of turbulence is present. In the proposed method a vector potential field is introduced; by taking the vector curl of the potential field one can generate a strictly divergence-free flow field. A novel formulation for scaling the vector potential field, together with a coordinate transformation strategy, is proposed in this work. The result is a six-step procedure for the generation of inhomogeneous turbulence fields. The proposed formulation is proven to reproduce the prescribed velocity correlation with energy spectrum at large Reynolds numbers. Four test cases are considered to evaluate the new method. First, the statistical quantities introduced in the proposed method are verified numerically in a classical homogeneous turbulence case. Then, the performance of the new method is demonstrated in three different inhomogeneous turbulence cases: a confined turbulent flow in a “slip-wall” box, a planar channel flow and an annular flow. It is shown that the unphysical patterns in the fluctuation fields and divergence errors produced with Smirnovʼs method are absent in the results from the new method. The accuracy of the proposed methods is verified by comparing the velocity correlations with the prescribed scaling functions. The proposed method is efficient and simple to implement.

Three-dimensional deformation in curl vector field

Deformation is an important research topic in graphics. There are two key issues in mesh deformation: (1) self-intersection and (2) volume preserving. In this paper, we present a new method to construct a vector field for volume-preserving mesh deformation of free-form objects. Volume-preserving is an inherent feature of a curl vector field. Since the field lines of the curl vector field will never intersect with each other, a mesh deformed under a curl vector field can avoid self-intersection between field lines. Designing the vector field based on curl is useful in preserving graphic features and preventing self-intersection. Our proposed algorithm introduces distance field into vector field construction; as a result, the shape of the curl vector field is closely related to the object shape. We define the construction of the curl vector field for translation and rotation and provide some special effects such as twisting and bending. Taking into account the information of the object, this approach can provide easy and intuitive construction for free-form objects. Experimental results show that the approach works effectively in real-time animation.

Su1240 Magnetic Resonance Elastography (MRE) of the Pancreas, a Feasibility Study in Healthy Volunteers

Background: MRE is a non-invasive advanced imaging technique utilizing wave propagation to characterize the In Vivo mechanical properties of tissue, including stiffness, in organs such as the liver. MRE of the pancreas has not been thoroughly studied, but could provide a non-invasive tool for assessing pancreatic pathology such as pancreatitis, solid and cystic pancreatic masses. Aims: To determine the feasibility of MRE of the pancreas and establish normal ranges for tissue density of the normal pancreas. Methods: IRB approval was obtained. Stage 1: MRE of the pancreas was performed in 12 healthy volunteers with a 2-D gradientecho (GRE) sequence. An acoustic driver (AD) system delivered 40 HZ vibrations into the abdomen via a plastic drum secured via an elastic belt and powered pneumatically by a speaker located outside of the scan room. In an attempt to optimize wave propagation, the AD was placed over the liver, pancreas and left flank with the patient supine, and over the mid back with patient prone. MR elastograms were generated using an inversion algorithm to yield quantitative images of tissue stiffness in kilopascals (kPa). Stage 2: 3-D MRE was performed in 12 healthy volunteers using a flow-compensated single-shot spin-echo planar imaging (EPI) acquisition with motion encoding performed sequentially in the X, Y, and Z directions. Processing of the measured 3D displacement field was performed by first calculating the 3D vector curl of the displacement field and then using a 3D local frequency estimation (LFE) algorithm to calculate the tissue stiffness from the curl data. The pancreas was manually segmented using the MRE magnitude images and the mean stiffness in kilopascals (kPa) was recorded. Results: Stage 1: With the 2D MRE technique, there was poor pancreatic wave propagation at all AD locations including patient supine and prone for all 12 volunteers. This precluded accurate mean shear pancreatic stiffness calculation. Stage 2: With 3D MRE technique (image 1), 10 of 12 volunteers had adequate pancreatic wave propagation, with calculated mean shear hepatic stiffness of 1.04 ± 0.25 kPa (range 0.861.25 kPa). Of the 2 with poor wave propagation, one had BMI >30 kg/m2, and one had an atrophic pancreas. Conclusions: MRE of the pancreas is feasible, but requires 3D image sequencing to obtain adequate elastograms to calculate tissue stiffness. Further refinements in the technology are required to obtain consistent elastograms, including for patients with obesity or pancreatic atrophy. Ongoing investigations include determining the variation in stiffness within regions of the pancreas and to compare the normal pancreas with the spectrum of pancreatic diseases including pancreatic cancer, pancreatic cysts, acute and chronic pancreatitis.