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We establish an asymptotic profile that sharply describes the behavior as $t\to\infty$ for solutions to a non-solenoidal approximation of the incompressible Navier–Stokes equations introduced by Temam. The solutions of Temam's model are known to converge to the corresponding solutions of the classical Navier–Stokes, e.g., in $L^3_{\rm loc} (R^+ \times R^3)$, provided $\epsilon\to0$, where $\epsilon>0$ is the physical parameter related to the artificial compressibility term. However, we show that such model is no longer a good approximation of Navier–Stokes for large times: indeed, its solutions can decay much slower as $t\to+\infty$ than the corresponding solutions of Navier–Stokes.

Consider the Stokes equations in a sector-like C3 domain Ω⊂R2. It is shown that the Stokes operator generates an analytic semigroup in Lσp(Ω) for p∈[2,∞). This includes domains where the Lp-Helmholtz decomposition fails to hold. To show our result we interpolate results of the Stokes semigroup in VMO and L2 by constructing a suitable non-Helmholtz projection to solenoidal spaces.

The Stokes equations in a layer domain are investigated. For the case of a two-dimensional layer it is shown by resolvent estimates that there exists a semigroup of angle π/2 on solenoidal subspaces of L∞ and L1. Furthermore, the stationary problem for dimension two is solvable in these spaces. It is also shown that both results cannot be extended to three and higher dimensions in the same setting.

Full-wave electromagnetic simulations have been employed to analyze the design approaches that allow for solving the problems preventing inductive performance of micro-solenoidal RF probes and formation of strong and homogeneous magnetic fields inside probes at the studies of millimeter-range sized objects in 14 T MRI scanners. In particular, the effects of non-uniform coil wrapping on field homogeneity inside extended coils and of partitioning the coils by dielectric separators on coil self-resonances have been investigated. The possibility to utilize tunable C-C matching circuits with the coils and to mitigate the effects of sample insertion on the probe resonance frequency has been demonstrated. Challenges of coil fabrication have been also addressed.

We provide precise estimates of the Poincare constants firstly for scalar functions and secondly for solenoidal (i.e. divergence free) vector fields (in both cases with vanishing Dirichlet traces on the boundary) on 2d-annuli by the use of the first eigenvalues of the scalar Laplacian and the Stokes operator, respectively. In our non-dimensional setting each annulus ΩA is defined via two concentrical circles with radii A/2 and A/2+1. Additionally, corresponding problems on domains Ωσ*, the 2d-annuli from [7], are investigated - for comparison but also to provide limits for A→0. In particular, the Green's function of the Laplacian on Ωσ* with vanishing Dirichlet traces on ∂Ωσ* is used to show that for σ→0 the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit circle. On the other hand, we take advantage of the so-called small-gap limit for A→∞.

Summary We have developed a new divergence preserving method for the reconstruction of the Cartesian components of a vector field from the orthogonal projection of a vector field to the normals to edges in two dimensional. In this method, discrete divergences computed from the nodal components and from the normal ones are exactly the same. Our new method consists of two stages. At the first stage, we use an extended version of the local procedure described in [J. Comput. Phys., 139:406–409, 1998] to obtain a ‘reference’ nodal vector. This local procedure is exact for linear vector fields; however, the discrete divergence is not preserved. Then, we formulate a constrained optimization problem, in which this reference vector plays the role of a target, and the divergence constraints are enforced by using Lagrange multipliers. It leads to the solution of ‘elliptic’ like discrete equations for the cell-centered Lagrange multipliers. The new global divergence preserving method is exact for linear vector fields. We describe all details of our new method and present numerical results, which confirm our theory. Copyright © 2016 John Wiley & Sons, Ltd.

We propose a framework for the spectral processing of tangential vector fields on surfaces. The basis is a Fourier-type representation of tangential vector fields that associates frequencies with tangential vector fields. To implement the representation for piecewise constant tangential vector fields on triangle meshes, we introduce a discrete Hodge–Laplace operator that fits conceptually to the prominent cotan discretization of the Laplace–Beltrami operator. Based on the Fourier representation, we introduce schemes for spectral analysis, filtering and compression of tangential vector fields. Moreover, we introduce a spline-type editor for modelling of tangential vector fields with interpolation constraints for the field itself and its divergence and curl. Using the spectral representation, we propose a numerical scheme that allows for real-time modelling of tangential vector fields.

We propose the reconstruction of the solenoidal part of a vector field supported in the unit ball in 3 dimensions by using cone beam data from a curve surrounding it, and this curve satisfies the Tuy's condition of order 3. We use the quaternionic inversion formula to decompose the solenoidal part of a vector field into 2 parts. To recover the first one, which is the main part of the solenoidal component, another definition of a cone beam transform containing both Doppler and transverse data will be introduced. The second part will be reconstructed by using information from the first part as in Katsevich and Schuster's work with less data.

The effect of an inclined homogeneous magnetic field on thermal convection between rigid plates heated from below under the influence of gravity is numerically simulated in a computational domain with periodic horizontal extent. The numerical technique is based on solenoidal (divergence-free) basis functions satisfying the boundary conditions for both the velocity and the induced magnetic field. Thus, the divergence-free conditions for both velocity and magnetic field are satisfied exactly. The expansion bases for the thermal field are also constructed to satisfy the boundary conditions. The governing partial differential equations are reduced to a system of ordinary differential equations under Galerkin projection and subsequently integrated in time numerically. The projection is performed by using a dual solenoidal bases set such that the pressure term is eliminated in the process. The quasi-steady relationship between the velocity and the induced magnetic field corresponding to the liquid metals or melts is used to generate the solenoidal bases for the magnetic field from those for the velocity field. The technique is validated in the linear case for both oblique and vertical case by reproducing the marginal stability curves for varying Chandrasekhar number. Some numerical simulations are performed for either case in the nonlinear regime for Prandtl numbers Pr = 0.05 and Pr = 0.1. Copyright © 2014 John Wiley & Sons, Ltd.

A new potential vorticity is derived by using a specific entropy formulation expressed in terms of a moist-air entropy potential temperature. The new formulation is compared with Ertel's version and with others based on virtual and equivalent potential temperatures. The new potential vorticity is subject to conservative properties ensured by the Second Law applied to the moist-air material derivatives. It is shown that the upper tropospheric and stratospheric (dry) structures are nearly the same as those obtained with Ertel's component. Moreover, new structures are observed in the low troposphere, with negative values associated with moist frontal regions. The negative values are observed in the frontal regions where slantwise convection instabilities may take place, but they are smaller than those observed with the equivalent potential vorticity. The main purpose of the article is to diagnose the behaviour of the new potential vorticity from numerical output generated by the ARPEGE NWP model, with the help of isobaric charts and vertical cross-sections. Two inversion methods are suggested. The first method could be based on the invertibility principle verified by the virtual potential vorticity, with a possibility to control and modify separately potential vorticity components in the (dry) upper and (moist) lower atmospheric levels. The other method may consist of an inversion process directly applied to the new moist-air entropy potential vorticity, because the negative values and the solenoidal term are smaller than those observed with equivalent potential vorticity, as shown by numerical evaluations.

Recent progresses in the second generation REBa2Cu3O7 − x (RE123) coated conductor (CC) have paved a way for the development of superconducting solenoids capable of generating fields well above 23.5 T, i.e. the limit of NbTi−Nb3Sn-based magnets. However, the RE123 magnet still poses several fundamental and engineering challenges. In this work we review the state-of-the-art of conductor and magnet technologies. The goal is to illustrate a close synergetic relationship between evolution of high-field magnets and advancement in superconductor technology. The paper is organized in three parts: (1) the basics of RE123 CC fabrication technique, including latest developments to improve conductor performance and production throughput; (2) critical issues and innovative design concepts for the RE123-based magnet; and (3) an overview of noteworthy ongoing magnet projects.

In the framework of magnetohydrodynamics, the generation of magnetic fields by the prescribed motion of a liquid conductor in a bounded region is described by the induction equation, a linear system of parabolic equations for the magnetic field components. Outside G, the solution matches continuously to some harmonic field that vanishes at spatial infinity. The kinematic dynamo problem seeks to identify those motions, which lead to nondecaying (in time) solutions of this evolution problem. In this paper, the existence problem of classical (decaying or not) solutions of the evolution problem is considered for the case that G is a ball and for sufficiently regular data. The existence proof is based on the poloidal/toroidal representation of solenoidal fields in spherical domains and on the construction of appropriate basis functions for a Galerkin procedure. Copyright © 2012 John Wiley & Sons, Ltd.

SUMMARY The subject of this paper is a defect in the approximation of the pressure on dynamically changing spatial meshes in the computation of nonstationary incompressible flows. The observed behavior is due to the fact that discrete solenoidal fields lose this property under changes of the spatial discretization. This phenomenon is analyzed for DG finite element discretizations in time, and possible ways are considered to circumvent this problem. Copyright © 2011 John Wiley & Sons, Ltd.

We present the construction and performance of a 20-μL active volume probe that utilizes zero-susceptibility wire for the detection transceiver coil and a 3.5 mm outer diameter thin-wall bubble flow cell to contain the sample. The probe shows good rf homogeneity, resolution, line shape and sensitivity. The sensitivity and resolution of the 20-μL probe was compared to those for several other coil configurations, including smaller detection volumes, a thin wire copper coil immersed in susceptibility matching perfluorocarbon FC-43 (fluorinert) fluid, and a standard 5 mm probe. In particular, the (1)H mass sensitivity, S(m) (SNR per micromole), was 3-4 fold higher than that for the standard 5 mm probe. Finally, the use of the zero-susceptibility wire in smaller volume probes is discussed along with potential future improvements and applications.

This paper presents the linear stability analysis of a round jet in a radially unbounded domain using a spectral Petrov–Galerkin scheme coped with exponential coordinate transformation based on Fornberg's treatment. A Fourier–Chebyshev Petrov–Galerkin spectral method is described for the computation of the linear stability equations based on half a Gauss–Lobatto mesh. Complex basis functions presented here are exponentially mapped as Chebyshev functions, which satisfy the pole condition exactly at the origin, and can be used to expand vector functions efficiently by using the solenoidal condition. The mathematical formulation is presented in detail focusing on the solenoidal vector field used for the approximation of the flow. The scheme provides spectral accuracy in the present cases and the numerical results are in agreement with former works. Copyright © 2008 John Wiley & Sons, Ltd.

Field-cycled MRI provides a noninvasive quantification of the concentra- tion of bound (activated) targeted contrast agents. For field-cycled imaging to be per- formed within a clinical full-body superconducting MRI magnet, an auxiliary insertable magnet is required that can dynamically change the main field strength during the MR pulse sequence. Herein, the design and construction of a solenoidal insert coil is dis- cussed. Design aspects including active shielding, thermal performance, forces, and ramping times are examined. Proper materials, construction methods, and interfacing for the insert coil with a clinical MRI are also covered. Finally, a field-cycled experiment in a 1.5 T clinical MRI system is described, demonstrating the feasibility of this method and then future novel contrasts available with field-cycled MRI. 2009 Wiley Periodicals

Current applied to wire coils wound at the tip of an endovascular catheter can be used to remotely steer a catheter under magnetic resonance imaging guidance. In this study, we derive and validate an equation that characterizes the relationship between deflection and a number of physical factors: θ ∕ sin ( γ − θ ) = n I A B L ∕ E I A , where θ is the deflection angle, n is the number of solenoidal turns, I is the current, A is the cross-sectional area of the catheter tip, B is the magnetic resonance(MR)scanner main magnetic field, L is the unconstrained catheter length, E is Young’s Modulus for the catheter material, and I A is the area moment of inertia, and γ is the initial angle between the catheter tip and B . Solenoids of 50, 100, or 150 turns were wound on 1.8 F and 5 F catheters. Varying currents were applied remotely using a DC power supply in the MRI control room. The distal catheter tip was suspended within a phantom at varying lengths. Images were obtained with a 1.5 T or a 3 T MRscanner using “real-time” MR pulse sequences. Deflection angles were measured on acquired images. Catheter bending stiffess was determined using a tensile testing apparatus and a stereomicroscope. Predicted relationships between deflection and various physical factors were observed ( R 2 = 0.98 − 0.99 ) . The derived equation provides a framework for modeling of the behavior of the specialized catheter tip. Each physical factor studied has implications for catheter design and device implementation.

A new set of hierarchical, divergence-conforming, vector basis functions on curvilinear tetrahedrons is presented. The basis can model both mixed- and full-order polynomial spaces to arbitrary order, as defined by Raviart and Thomas, and Nedelec. Solenoidal- and non-solenoidal components are separately represented on the element, except in the case of the mixed first-order space, for which a decomposition procedure on the global, mesh-wide level is presented. Therefore, the hierarchical aspect of the basis can be made to extend down to zero polynomial order. The basis can be used to model divergence-conforming quantities, such as electromagnetic flux- and current density, fluid velocity, etc., within numerical methods such as the finite element method (FEM) or integral equation-based methods. The basis is ideally suited to p-adaptive analysis. The paper concludes with two example applications. The first is the FEM-based solution of the linearized acoustic vector wave equation, where it is shown how the decomposition into solenoidal components and their complements can be used to stabilize the method at low frequencies. The second is the solution of the electric field, volume integral equation for electromagnetic scattering analysis, where the benefits of the decomposition are again demonstrated.

An alternative variational framework suitable for pure advection is obtained by discarding the Galerkin part of stabilized methods. The resulting scheme is similar to the least-squares approach, but with the adjoint operator in the weighting slot. This formulation is not restricted to solenoidal (i.e. divergence free) velocities. Initial numerical results for such problems show that the method is promising.

That vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the A-B (Aharonov-Bohm) effect. We study scattering by two solenoidal magnetic fields (point-like magnetic fields) in two dimensions and analyze the asymptotic behavior of the scattering amplitude in the semiclassical limit. The corresponding classical mechanical system has a trajectory oscillating between the centers of the two fields. We derive the asymptotic formula with the first three terms and make clear how such a trapping trajectory is reflected in the asymptotic formula through the A-B effect. We also make a brief comment on an extension to the scattering by many solenoidal fields. The result depends on the location of the centers of the fields. In particular, the A-B effect appears strongly when the centers are on an even line.

Given a domain Ω in ℝ3 with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ⊂ ℝ3 into a sum u = u++u− were u± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd.

This paper considers the effects of conductor geometry on the performance of small solenoidal coils for high-field NMR. First, a simple analytical model is presented for investigating the effects of conductor geometry on the current distribution in such coils. The model was used to derive optimum parameters for coils constructed from wire with either rectangular or circular cross-sections as a function of the length-to-diameter ratio. Second, a commercial software package utilizing full three-dimensional finite-element solutions to Maxwell's equations was used to confirm the basic findings of the simple analytical model, and also to compare simulated S/N estimations with experimental NMR spectra acquired with 2.5 mm and 1.0 mm-diameter solenoid coils: reasonable agreement was found. Third, as a demonstration of the usefulness of such coils for mass-limited samples, multidimensional experiments were performed at 750 MHz on ∼4.7 nmol (41 µg) of PF1061, a protein from Pyrococcus furiosus. Copyright © 2006 John Wiley & Sons, Ltd.

The development of a composite resonator assembly for electron paramagnetic resonance (EPR)/nuclear magnetic resonance (NMR) imaging and its first application to EPR/NMR coregistration imaging are described. Anatomic registration would be useful for the meaningful interpretation of images by EPR of free radical distributions in vivo. The composite resonator assembly consists of a parallel coil resonator, a field modulation coil for EPR imaging, and a solenoidal detection coil for NMR imaging. Radio-frequency (RF) shielding was used in addition for the EPR imaging. The homogeneity of the RF magnetic field of the EPR resonator was measured experimentally using a small-loop antenna; it was reasonably homogeneous along the long axis and in the radial plane. EPR and NMR images were obtained with both phantom and animal tests. The presence of the parallel coil resonator did not significantly affect NMR images. The superposition of the distribution of free radicals and protons was computed using three positional markers. The assembly performs the registration of two multimodal images. It is compatible with an orthogonal crossed-coil configuration for open-type clinical NMR imagers. © 2005 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 25B: 1–11, 2005

SUMMARY We present a projection scheme whose end-of-step velocity is locally pointwise divergence free, using a continuous P1 approximation for the velocity in the momentum equation, arst-order Crouzeix- Raviart approximation at the projection step, and a P0 approximation for the pressure in both steps. The analysis of the scheme is done only for grids that guarantee the existence of a divergence free conforming P1 interpolant for the velocity. Optimal estimates for the velocity error in L 2 - and H 1 - norms are deduced. The numerical results demonstrate that these estimates should also hold on grids on which the continuous P1 approximation for the velocity locks. Since the end-of-step velocity is locally solenoidal, the scheme is recommendable for problems requiring good mass conservation. Copyright ? 2005 John Wiley & Sons, Ltd.

A new type of cable-in-conduit conductor composed of HTS tapes was proposed as a winding for high-performance HTS coils with high current capacities and low AC losses. In the fabrication of the conductor, the twist of stacked tapes around their axes was made before inserting them inside conduits. The twist angle should be changed continuously along the axis to reduce face-on oriented magnetic fields applied to the tape in the winding conductor during coil operation. In order to confirm the high current capacity of this type conductor, two single-layered solenoidal coils wound with sample conductors composed of five stacked Bi-2223 tapes with bias angles of 20 and 0° were fabricated and tested in liquid nitrogen. A copper magnet system was used in this experiment to generate the spread magnetic field with a spread angle of 20° from the coil axis to the radial direction, which simulates the profile of magnetic fields near the edge windings of practical coils. A large improvement on critical-current degradation affected by spread magnetic fields was successfully observed for the test coil wound with the 20° bias conductor. © 2005 Wiley Periodicals, Inc. Electr Eng Jpn, 153(4): 12–19, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20218

NMR investigation of volume-limited chemical and biological samples requires a radio frequency (RF) microcoil with high signal-to-noise ratio (SNR) over the region of interest. Conventional approaches using solenoidal microcoils provide high sensitivity (RF field strength per unit current) and spectral resolution but require the sample and sample container to be enclosed by the coil. Planar surface microcoils provide an alternative configuration that allows direct access to the sample, but with the sacrifice of RF field uniformity. In this study we evaluate a family of planar RF microcoils (500 MHz, Archimedean geometry, 1–6 turns, inner radius 0.75 to 4 mm, and conductor widths of 75, 100, and 200 μm). The design, fabrication, and performance (electrical and NMR) of the coils are described. This coil configuration exhibits a high local SNR, easy fabrication, good electrical properties, and moderate spectral uniformity (suitable for imaging and low resolution NMR spectroscopy). This design can also be scaled to smaller dimensions. These results suggest that planar spiral RF microcoils will find applications in multimodality microscopy and microfludic devices where sample manipulation and coil integration with the microanalysis systems is necessary. © 2003 Wiley Periodicals, Inc. Concepts Magn Reson Part B (Magn Reson Engineering) 17B: 1–10, 2003.

The homogeneity of the radio frequency (RF) magnetic field in the region of the NMR sample is prerequisite for good performance in many NMR experiments. When calculating such fields, the quasistationary approximation is valid for wavelengths that are large compared to the physical dimensions of the NMR coil, but it no longer applies for the geometrical dimensions of NMR coils at high frequencies. The present study investigates electromagnetic fields in solenoidal coils. Considering the electromagnetic field derivable from first principles allows us to systematically study the effects that originate from the finiteness of the wavelength compared to the physical size of solenoidal coils employed in NMR probes. There are three major effects that occur: the wavelength in the coil is generally different from the wavelength in free space, even for an absent sample; the dielectric properties of the NMR sample have an additional influence on the electromagnetic field distribution inside the coil; and the circuit surrounding the coil leads to a wave reflection that affects the RF magnetic field distribution originating from the superposition of forward and backward traveling waves. The results allow the derivation of principal conclusions for the characterization of NMR solenoidal coils. © 2002 Wiley Periodicals, Inc. Concepts in Magnetic Resonance (Magn Reson Engineering) 15: 129–155, 2002

A method for modelling of flow discontinuities based on deconvolution with a relaxation regularization (DDM) is compared with an essentially non-oscillatory scheme (ENO) without further modelling and with a dynamic Smagorinsky subgrid-scale model (DSM). For the DDM approach, a sufficiently accurate representation of the filtered non-linear combination of discontinuous solution components which arise from the convection term is obtained by regularized deconvolution applied to the filtered solution. For stable integration the evolution equations are supplemented by a relaxation regularization based on a secondary filter operation and a relaxation parameter. We apply the above method to the generic test case of a two-dimensional solenoidal velocity fluctuation field interacting with a normal shock. The results demonstrate a good agreement of DDM with a reference computation, while DDM significantly reduces computational cost. Predictions with DDM are found to provide a more accurate representation of the shock–vorticity–wave interaction phenomena than that which can be accomplished using the ENO scheme with or without subgrid-scale model. Copyright © 2002 John Wiley & Sons, Ltd.

The very weak solution for the Stokes, Oseen and Navier-Stokes equations has been studied by several authors in the last decades in domains of Rn, n ≥ 2. The authors studied the Oseen and Navier-Stokes problems assuming a solenoidal convective velocity in a bounded domain Ω ⊂ R3 of class C1,1 for v ∈ Ls (Ω) for s ≥ 3 in some previous papers. The results for the Navier-Stokes equations were obtained by using a fixed-point argument over the Oseen problem. These results improve those of Galdi et al. , Farwig et al. and Kim for the Navier-Stokes equations, because a less regular domain Ω ⊂ R3 and more general hypothesis on the data are considered. In particular, the external forces must not be small. In this work, existence of weak, strong, regularised and very weak solution for the Oseen problem are proved, mainly assuming that v ∈ L3(Ω) and its divergence ∇ · v is sufficiently small in the W−1,3(Ω)-norm. In this sense, one extends the analysis made by the authors for a given solenoidal v in some previous papers. As a consequence, the existence of very weak solution for the Navier-Stokes problem (u, π) ∈ L3(Ω) × W−1,3(Ω)/R for a non-zero divergence condition is obtained in the 3D case. Ministerio de Ciencia e Innovación (España) MTM2009-12927 Ministerio de Economía y Competitividad MTM2012-32325

In high-field proton NMR, the signal-to-noise ratio (SNR) achieved with a close-fitting solenoidal microcoil is adversely affected by radio frequency (RF) losses in the coil, its leads, the capacitor used to tune it, and finally, the sample. In Part II, a rigorous description of these various losses is presented, and their severity is related to the details of coil design. Results not only provide a rational basis for defining a microcoil's optimal wire diameter and the number of turns, but also for evaluating how the SNR varies with coil size and NMR frequency in high-field proton NMR studies involving either conducting or non-conducting samples. © 2001 John Wiley & Sons, Inc. Concepts Magn Reson 13: 190–210, 2001

High-resolution nuclear magnetic resonance (NMR) spectroscopy is one of the most powerful analytical tools used to probe details of molecular structure and dynamics. The study of large molecules such as proteins requires high sensitivity and high spectral resolution, which are both achieved with strong magnetic fields. These fields are generated by huge superconducting magnets, which are made stronger and bigger each year to tackle larger and larger molecules. The results of this amazing technological effort are bulky and static magnets permanently installed in dedicated NMR laboratories. The size of the superconducting magnets, their sensitivity to harsh environments, and the cost of maintenance and operation keep this technology away from fume hoods and production sites, where simpler devices that provide access to medium-size molecules would be needed. Robust NMR magnets can be made from permanent magnets like those used for NMR spectroscopy in the 1960s and 1970s. But to achieve high resolution for standard sample volumes, the permanent magnets then were as big as superconducting magnets today and weighed several hundred kilograms. Considering that the magnetic field strength remains constant when the volume of the magnet is scaled down to gain portability and that fields of up to 2 T are generated by permanent magnets, small magnets would offer a sensitivity only a factor of three smaller than that achieved in a standard (7 T) superconducting magnet (see the dotteddashed lines in Figure 1). This limitation is an affordable price to be paid for a small and portable system. However, a second factor seriously compromises the signal-to-noise ratio in the miniaturization process: the reduction of sample volume. For each magnet geometry the ratio between the magnet size and the size of the sensitive volume is a constant. When the size of the magnet is reduced, a smaller volume of highfield homogeneity is generated. For example, if the oldfashioned Varian T-60 magnet, with a volume of about 1 m, is reduced to palm-size dimensions, a sensitivity loss of about three orders of magnitude is expected (circle in Figure 1). Although this approach is valid in cases where the amount of sample is limited (capillary NMR), this sensitivity loss is simply unacceptable for most applications. We report herein on the construction of a small permanent magnet with an extraordinarily homogeneous magnetic field B0 suitable for measuring H NMR spectra of solutions in standard 5 mm NMR sample tubes (Figure 2). Weighing only 500 grams, the magnet can be transported along with the spectrometer, and NMR measurements can be performed on demand with this robust device at minimal maintenance cost. To efficiently reduce the sensor volume by three orders of magnitude over that of typical C-magnet designs, individual magnet blocks were compactly arranged in a cylindrical array based on the design by Halbach. This array provides a generous volume for sample positioning (large bore/magnet size ratio), and generates a magnetic field perpendicular to its cylinder axis (Figure 2), which allows the use of sensitive solenoidal radio-frequency (rf) coils to detect the NMR signals. In theory, the magnetic field generated by an infinitely long magnet built from perfect magnet blocks would be highly homogeneous along the length of the sample tube with almost zero stray field. However, in practice, the finite length of the magnet and the statistical imperfections of the sintered magnet blocks deteriorate the predicted homogeneity by several orders of magnitude. The new design presented herein combines three Halbach rings with different geometric proportions optimized to account for the field distortions along the cylinder axis due to the finite magnet length. To tackle the important source of inhomogeneity introduced by the variability of the pieces, each ring is composed of fixed trapezoidal elements with parallel gaps between them that guide the movement of rectangular magnet blocks (Figure 2). These pieces can be moved radially in and out to mechanically shim the magnetic field with highly efficiency and accuracy. By displacing the rectangular blocks in each ring with defined angular modulations and amplitudes, it is possible to independently Figure 1. Signal-to-noise ratio (SNR) for permanent and superconducting magnets as a function of the field strength B0. Squares show the SNR for water in a 5 mm NMR tube. Dashed and dotted lines correspond to solenoidal and birdcage rf coils used with permanent and superconducting magnets, respectively. The circle indicates the SNR value for a reduced sample volume in a capillary with a diameter of 0.3 mm.

This paper is a review discussing properties of field solutions produced by various finite difference schemes in the time domain. Considered algorithms include standard FDTD as well as its modifications based on different sets of electromagnetic equations and/or different discretization in space. By developing their complete dispersion relations, conclusions are drawn regarding divergence, curl, and energy conserva tion by each of the emulated eigenmodes separately. This is in extension to previous works which were concerned with the total solution, and permits to formulate conditions for restricting the total solutions to physical uncoupled solenoidal and potential modes. Original classification of spurious modes for the time domain finite difference modelling is also proposed. It is shown that spurious compensating modes can propagate over the mesh in the case of penalty schemes with p≠0, and that spurious degenerate modes at high- and low-frequencies appear in the case of condensed node discretization. Copyright © 1999 John Wiley & Sons, Ltd.

SUMMARY Solving the incompressible Navier‐Stokes equations requires special care if the velocity field is not discretely divergence-free. Approximate projection methods and many pressure Poisson equation methods fall into this category. The approximate projection operator does not dampen high frequency modes that represent a local decoupling of the velocity field. For robust behavior, filtering is necessary. This is especially true in two instances that were studied: long-term integrations and large density jumps. Projection-based filters and velocity-based filters are derived and discussed. A cell-centered velocity filter, in conjunction with a vertex-projection filter, was found to be the most effective in the widest range of cases. © 1998 John Wiley & Sons, Ltd.

The solenoidal basis functions defined on tetrahedrons are modified to permit the analysis of the problem of scattering of electromagnetic waves by inhomogeneous dielectric objects. In the analysis of the scattering of EM plane waves by an inhomogeneous dielectric cube, we compare our results with those obtained using pulse basis functions. We also apply our method in the analysis of the scattering of EM plane waves by an inhomogeneous dielectric sphere, and compare the results with those obtained using the Mie's series. Finally, we apply our method in the analysis of the electromagnetic fields induced inside human tissue. We compare these results with those obtained using pulse basis functions. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 23: 42–46, 1999.

Nuclear magnetic resonance (NMR) spectroscopy is one of the most information-rich analytical techniques available. However, it is also inherently insensitive, and this drawback precludes the application of NMR spectroscopy to mass- and volume-limited samples. We review a particular approach to increase the sensitivity of NMR experiments, namely the use of miniaturized coils. When the size of the coil is reduced, the sample volume can be brought down to the nanoliter range. We compare the main coil geometries (solenoidal, planar, and microslot/stripline) and discuss their applications to the analysis of mass-limited samples. We also provide an overview of the hyphenation of microcoil NMR spectroscopy to separation techniques and of the integration with lab-on-a-chip devices and microreactors.

Usually, the Poynting vector is considered to be a vector field expressing the power flow. However, in many textbooks on electromagnetics the following statements are given which cause confusion: (a) since an arbitrary vector field with zero divergence can be added without violating the energy conservation law, the Poynting vector is not a unique expression for the power flow; and (b) the Poynting vector has nonzero value near a charged magnet. Since it cannot be accepted that a power flow exists in such a space, the Poynting vector may not represent a power flow. Such statements which cause confusion come from the education in which the relativistic aspects of electromagnetic theory have not been considered seriously. This paper is intended to contribute to the electromagnetic education and presents: (a) a clear explanation for the fact that the electromagnetic theory is essentially relativistic; (b) an explanation why the Nukiyama vector does not replace the Poynting vector; (c) general conditions to be satisfied by a vector which can be added to the Poynting vector; and (d) the basis on which we take the Poynting vector as the only expression for the power flow density.

An innovative idea for the solution of viscous incompressible flows, in which the equation for conservation of mass is satisfied at the element level, is termed the solenoidal finite element approach. The term ‘solenoidal’ derives from the fact that the velocity components need to be solenoidal, i.e. to have zero divergence. The difficulty with this idea centres on the construction of a specialized element in which the velocity components are constrained to be solenoidal by the nature of their interpolation functions. If such an element can be constructed, then the pressure is suppressed from the prime solution. This has obvious attractions, although recourse to another novel idea is needed for its eventual retrieval. The validity of these ideas is demonstrated herein by the results for some classical benchmark problems. Where possible, comparisons are made with other results, both from other codes and from the literature.

General information is given about solenoidal superconducting magnets and an overview is given of the design principles and theories for superconducting magnets. Issues relevant to NMR are reviewed in detail. Advantages and disadvantages of different superconducting wire, construction methods, and mathematical concepts necessary for the main components of a superconducting magnet applicable to high-resolution NMR are surveyed. Major components such as joints, dump resistors, heat shields, and magnet dewars are illustrated and described.

Understanding how the heart functions as a whole and its functional changes in disease states is important in diagnosing and treating heart disease, the nation''s number one cause of mortality. To reconstruct the complex deformations of the heart, accurate global 3-D strain data is needed. MRI in conjunction with cardiac tagging has evolved as a means to study 3-D wall motion non-invasively throughout the entire heart. However, conventional GE Signa MRI scanners do not achieve the gradient field strengths and switching times needed to encode marked locations with the precision (on the order of .1mm), accuracy and imaging times achieved with previous studies that use bi-plane cine radiography of implanted metallic beads. Toward this goal, a special purpose gradient set with a Cos(2 Theta) current distribution and an inductively coupled solenoidal rf probe have been constructed. For use in an isolated canine heart preparation, the X, Y and Z gradients can attain gradient field strengths of 11.3, 4.7 and 15.5 G/cm, respectively, with switching times of 150(mu)sec. Tagged images at 10cm FOV, 5mm slice thickness, 256x128, TE 2.8ms, TR 8.9ms, 3.5mm tag separation, and SNR =0.05mm tag location accuracy.

The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts: 1) the so-called transverse part represented by a divergenceless vector; and 2) the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when the Helmholtz theorem is used are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, the Aharonov-Bohm effect is chosen here. As for the Aharonov-Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z-axis. It is shown that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is determined uniquely by the magnetic field and does not alter by a gauge transformation. On the other hand, the longitudinal part depends on the choice of special gauge. It is shown that the Aharonov-Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.

Among non-linear vector fields, the simplest that can be studied are those which are continuous and piecewise linear. Among these non-linear vector fields a large and important subset are those vector fields which are odd symmetric. Associated with these types of vector fields are partitions of the state space into a finite number of regions. In each region the vector field is linear. On the boundary between regions it is required that the vector field be continuous from both regions in which it is linear. This presentation is devoted to the analysis in two dimensions of the simplest possible types of continuous piecewise linear vector fields with odd symmetry, namely those vector fields possessing a pair of symmetric boundary conditions. As a practical concern the analysis will attempt to ask and answer questions raised about the existence of steady state solutions. Since the local theory of fixed points in a linear vector field is sufficient to determine the stability of fixed points in a piecewise linear vector field, most of the steady state behaviour to be studied will be towards limit cycles. the results will present sufficient conditions for the existence, or non-existence as the case may be, of limit cycles. Particular attention will be paid to the domain of attraction whenever possible. With these results qualitative statements may be made for piecewise linear models of physical systems which have odd symmetry.

Conventional calculation methods of aircore inductance utilized Neumann's equation for solenoidal coils and Fourier series for saddle coils. However, inductance formulas for arbitrarily shaped coils on a cylinder have not been well known. In this paper, mutual and self-inductances generated by any currents on a cylinder are expressed using Fourier-Bessel expansion of the vector potential. An experiment shows the validity of the proposed formulas.

An alternative derivation of the vector addition theorem is presented using the completeness of vector wave functions and integration by parts. The advantage of this derivation is that it leads directly to the simplified results of Bruning and Lo and of Stein. Moreover, the dichotomous results of the addition theorem when a spherical Hankel function is involved can be derived by contour integration.

The editing pulse sequence DEPT (D.T. Pegg, D.M. Doddrell, and M.R. Bendall, J. Chem. Phys. 77, 2745 (1982)) was modified using a scheme of various composite pulses and a 16-step phase cycling to obtain proton-decoupled natural-abundance 13C edited subspectra of solid tumors. A solenoidal probe including a Faraday shield and an orthogonal saddle decoupling coil was built for this purpose.