Magnetic Diffusion Problem Research Articles

An exact solution for the magnetic diffusion problem with a step-function resistivity model

An exact solution for the magnetic diffusion problem with a step-function resistivity model

Stability analysis of a class of nonlinear magnetic diffusion equations and its fully implicit scheme

We studied a class of nonlinear magnetic diffusion problems with step-function resistivity $ \eta(e) $ in electromagnetically driven high-energy-density physics experiments. The stability of the nonlinear magnetic diffusion equation and its fully implicit scheme, based on the step-function resistivity approximation model $ \eta_\delta(e) $ with smoothing, were studied. A rigorous theoretical analysis was established for the approximate model of one-dimensional continuous equations using Gronwall's theorem. Following this, the stability of the fully implicit scheme was proved using bootstrapping and other methods. The correctness of the theoretical proof was verified through one-dimensional numerical experiments.

End-to-end GPU acceleration of low-order-refined preconditioning for high-order finite element discretizations

In this article, we present algorithms and implementations for the end-to-end GPU acceleration of matrix-free low-order-refined preconditioning of high-order finite element problems. The methods described here allow for the construction of effective preconditioners for high-order problems with optimal memory usage and computational complexity. The preconditioners are based on the construction of a spectrally equivalent low-order discretization on a refined mesh, which is then amenable to, for example, algebraic multigrid preconditioning. The constants of equivalence are independent of mesh size and polynomial degree. For vector finite element problems in H(curl) and H(div) (e.g., for electromagnetic or radiation diffusion problems), a specially constructed interpolation–histopolation basis is used to ensure fast convergence. Detailed performance studies are carried out to analyze the efficiency of the GPU algorithms. The kernel throughput of each of the main algorithmic components is measured, and the strong and weak parallel scalability of the methods is demonstrated. The different relative weighting and significance of the algorithmic components on GPUs and CPUs is discussed. Results on problems involving adaptively refined nonconforming meshes are shown, and the use of the preconditioners on a large-scale magnetic diffusion problem using all spaces of the finite element de Rham complex is illustrated.

Higher order corrections to the sharp-front magnetic diffusion wave formulas

An analytical solution to the sharp-front magnetic diffusion problem with a step-like resistance model is an interesting mathematical physics problem. It also relates to the plasma production at the surface of a metal under strong magnetic loading measured in experiments. In this paper, we make an advance on an earlier work of Xiao et al. [Phys. Plasmas 23, 082104 (2016)], by deriving formulas for the threshold magnetic field strength and the diffusion velocity of the magnetic diffusion wave under a step-like resistance model to the higher order accuracy.

Numerical Analysis of a Transposed Multiwired Armature in Electromagnetic Rail Launchers

Solid armatures in electromagnetic rail launchers have to undergo severe electromagnetic, mechanical, and thermal stresses. These stresses are unevenly distributed in the armature mainly due to the velocity skin effect. Contrasting this effect reduces the peak to average ratio of the stresses and allows better performance of the device. In this article, the behavior of a transposed multiconductor solid armature is numerically investigated by the research code electric network for electromagnetics (EN4EM) developed at the Department of Energy, System, Territory and Construction Engineering (DESTEC), University of Pisa, Pisa, Italy. The code is based on an integral formulation that reduces the magnetic diffusion problem in the analysis of a time-varying electric network. The code enables a strong electromechanical coupling and allows accounting for currents’ distribution in the rails and in the armature due to the proximity and “ordinary” skin effects, as well as the velocity skin effect. Preliminary computations have shown an improved (more regular) current density distribution in the armature. This allows increasing the total current on the armature without reaching the melting temperature in the armature. Improved performance of the launcher with transposed armature is predicted.

An optimal Galerkin scheme to solve the kinematic dynamo eigenvalue problem in a full sphere

The kinematic dynamo approximation describes the generation of magnetic field in a prescribed flow of electrically-conducting liquid. One of its main uses is as a proof-of-concept tool to test hypotheses about self-exciting dynamo action. Indeed, it provided the very first quantitative evidence for the possibility of the geodynamo. Despite its utility, due to the requirement of resolving fine structures, historically, numerical work has proven difficult and reported solutions were often plagued by poor convergence. In this paper, we demonstrate the numerical superiority of a Galerkin scheme in solving the kinematic dynamo eigenvalue problem in a full sphere. After adopting a poloidal–toroidal decomposition and expanding in spherical harmonics, we express the radial dependence in terms of a basis of exponentially convergent orthogonal polynomials. Each basis function is constructed from a terse sum of one-sided Jacobi polynomials that not only satisfies the boundary conditions of matching to an electrically insulating exterior, but is everywhere infinitely differentiable, including at the origin. This Galerkin method exhibits more rapid convergence, for a given problem size, than any other scheme hitherto reported, as demonstrated by a benchmark of the magnetic diffusion problem and by comparison to numerous kinematic dynamos from the literature. In the axisymmetric flows we consider in this paper, at a magnetic Reynolds number of O(100), a convergence of 9 significant figures in the most unstable eigenvalue requires only 40 radial basis functions; alternatively, 4 significant figures requires 20 radial functions. The terse radial discretization becomes particularly advantageous when considering flows whose associated numerical solution requires a large number of coupled spherical harmonics. We exploit this new method to confirm the tentatively proposed positive growth rate of the planar flow of Bachtiar et al. [4], thereby verifying a counter-example to the Zel’dovich anti-dynamo theorem in a spherical geometry.

A dynamic model for a displacement amplified magnetostrictive driver for activemounts

A magnetostrictive actuator with a hydraulic displacement amplification mechanism isdesigned to be used as a driver in active engine mounts. The dynamic response of theactuator is quantified in terms of the output displacement and the magnetostriction. Eddycurrent losses are modeled as a one-dimensional magnetic diffusion problem in cylindricalcoordinates. The Jiles–Atherton model is used to describe the magnetization state of thematerial as a function of applied magnetic fields. Magnetostriction, which is modeled as asingle-valued function of magnetization, provides an input to the mechanical modeldescribing the system vibrations. Friction at the elastomeric seals is modeled using theLuGre (Lund–Grenoble) friction model for lubricated contacts. Results show that themodel accurately describes the dynamic behavior of the actuator up to 500 Hz. An orderanalysis of the data and calculated responses shows that the model describesthe fundamental and higher-order spectral components generated by the device.

A dynamic actuation model for magnetostrictive materials

We quantify the dependence of strain on dynamic magnetic field in magnetostrictivetransducers. Dynamic eddy current losses are modeled as a one-dimensional (1D) magneticdiffusion problem in cylindrical coordinates. The constitutive magnetostrictive response toan average diffused magnetic field is quantified with the Jiles–Atherton model.The transducer is represented as a lumped-parameter, single-degree-of-freedomresonator with force input dictated by the magnetostriction. This equivalent force isexpressed as a summation of Fourier series terms. The total dynamic strain output isobtained by superposition of strain solutions due to each harmonic of the force input.

Frequency Dependent Strain-Field Hysteresis Model for Ferromagnetic Shape Memory Ni–Mn–Ga

We quantify the relationship between magnetic fields and strains in dynamic Ni-Mn-Ga actuators. As a result of magnetic field diffusion and structural actuator dynamics, the strain-field relationship changes significantly relative to the quasistatic response as the magnetic field frequency increases. We model the magnitude and phase of the magnetic field inside a Ni-Mn-Ga sample as a 1-D magnetic diffusion problem with applied dynamic fields known on the surface of the sample, from which we calculate an averaged or effective field. We use a continuum thermodynamics constitutive model to quantify the hysteretic response of the martensite volume fraction due to this effective magnetic field. We postulate that the evolution of volume fractions with effective field exhibits a zero-order response. To quantify the dynamic strain output, we represent the actuator as a lumped-parameter, single-degree-of-freedom resonator with force input dictated by the twin-variant volume fraction. This results in a second-order, linear ordinary differential equation whose periodic force input is expressed as a summation of Fourier series terms. The total dynamic strain output is obtained by superposition of strain solutions due to each harmonic force input. The model accurately describes experimental measurements at frequencies up to 250 Hz.

Verification of high-order mixed finite-element solution of transient magnetic diffusion problems

We develop and present high-order mixed finite-element discretizations of the time-dependent electromagnetic diffusion equations for solving eddy-current problems on three-dimensional unstructured grids. The discretizations are based on high-order H(Grad), H(Curl), and H(Div) conforming finite-element spaces combined with an implicit and unconditionally stable generalized Crank-Nicholson time differencing method. We develop three separate electromagnetic diffusion formulations, namely the E (electric field), H(magnetic field), and the A-/spl phi/ (potential) formulations. For each formulation, we also provide a consistent procedure for computing the secondary variables J(current flux density) and B(magnetic flux density), as these fields are required for the computation of electromagnetic force and heating terms. We verify the error convergence properties of each formulation via a series of numerical experiments on canonical problems with known analytic solutions. The key result is that the different formulations are equally accurate, even for the secondary variables J and B, and hence the choice of which formulation to use depends mostly on relevance of the natural and essential boundary conditions to the problem of interest. In addition, we highlight issues with numerical verification of finite-element methods that can lead to false conclusions on the accuracy of the methods.

An integro-differential formulation for magnetic induction in bounded domains: boundary element–finite volume method

Simulations of magnetohydrodynamic (MHD) flows in bounded domains using spectral methods suffer from a number of serious limitations. Alternative methods based on local discretization raise the problem of how to implement non-local boundary conditions for the magnetic field. We have developed a new strategy for the numerical solution of MHD problems in bounded domains, which combines the flexibility of a local discretization with a rigorous formulation of magnetic boundary conditions next to an insulator in arbitrary geometries. In accordance with the character of underlying equations we apply a global integral approach at the boundary and a differential approach inside the conducting domain. The formulation of the boundary problem in terms of primitive variables allows us to combine these approaches and propose a mixed finite volume and boundary element method. We illustrate its efficiency on magnetic diffusion problems in a sphere and in a finite cylinder.

Magnetoquasistatic response of conducting and permeable prolate spheroid under axial excitation

An analytical solution is presented for the problem of magnetic diffusion into and scattering from a permeable, highly but not perfectly conducting prolate spheroid under axial excitation, expressed in terms of an infinite matrix equation. The spheroid is assumed to be embedded in a homogeneous nonconducting medium as appropriate for low-frequency, high-contrast scattering governed by magnetoquasistatics. The solution is based on separation of variables and matching boundary conditions where the prolate spheroidal wavefunctions with complex wavenumber parameter are expanded in terms of spherical harmonics. For small skin depths, an approximate solution is developed that avoids any reference to the spheroidal wavefunctions. The problem of long spheroids and long circular cylinders is solved by using an infinite cylinder approximation. In some cases, our ability to evaluate the spheroidal wavefunctions breaks down at intermediate frequencies. To deal with this, a general broadband rational function approximation technique is developed and demonstrated. We treat special cases and provide numerical reference data for the induced magnetic dipole moment or, equivalently, the magnetic polarizability factor.

An application of first integrals method in magnetic diffusion problem

In this paper a rigorous approach to the magnetic diffusion problem is presented under the assumption that the random fluid velocity field is independent of x and completely determined by a prescribed probability measure on the space of symmetric velocity field u(t)=u(−t). By adopting the first integrals method developed by Visik and Fursikov [Mat. Sb. 92, 347 (1973)], the space characteristic functional associated with the random induction equation has been expressed by a functional power series. The explicit formula for this functional has been obtained, without deriving and solving a suitable functional equation for the [u, B] field.

Formulation and implementation of the current filament method for the analysis of current diffusion and heating in conductors in railguns and homopolar generators

The authors present the formulation of the current filament model and show how the method was applied to the diffusion problems in railguns and homopolar generators (HPGs). Some computational results are included. The method was used to investigate railgun current diffusion and heating at the gun breech as well as the proximity effect eddy current in a self-excited air-core HPG. It is concluded that the current filament method is a conceptually simple and useful method for analyzing two-dimensional linear magnetic diffusion problems in which the direction of the current density can be assumed. The method is easy to program and incorporate into lumped-parameter system models on the computer.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>