A new formulation of the magnetic vector potential method for three dimensional magnetostatic field problems

A novel formulation of the magnetic vector potential method for three dimensional magnetostatic field calculations is derived. Rigorously defining the interface and boundary conditions of the gauge of the vector potential, the new method gives a unique solution to the problem. The new field equation does not contain the gauge condition against the usual formulations[1], [2], [3], and takes the form of the diffusion equation. Computed results are favorably compared with the analytic solution of a test problem. This formulation is directly applicable to three dimensional eddy current problems.     

A three dimensional eddy current formulation using two potentials: The magnetic vector potential and total magnetic scalar potential

A formulation of the three dimensional eddy current problem is presented. The magnetic vector potential is used in regions with source currents and conducting material and the total magnetic scalar potential is employed elsewhere. The continuity of the normal component of flux density and tangential component of field intensity are used to couple the two potentials on the interface between regions. The formulation leads to a symmetric system amenable to traditional solution techniques. The formulation is also valid for static problems with modification that are easily implemented.     

3-D eddy current calculations using the magnetic vector potential

Two methods for using the magnetic vector potential for 3-D eddy current calculation are treated. One method uses the magnetic vector potential that is continuous over the entire region and generally accompanies the electric scalar potential. It has the advantage that no cutting is necessary for the multiply-connected-region problem. The other method uses the magnetic vector potential that is discontinuous across the interface surface between different media. This magnetic vector potential can be arranged so that the electric scalar potential does not appear in the equations when the conductivity is constant. It has the disadvantage that cutting is necessary for the multiply-connected-region problem. New boundary value problem formulations are given for both methods, precisely defining the interface and boundary conditions     

Performance of different vector potential formulations in solvingmultiply connected 3-D eddy current problems

The authors describe their numerical experiences in applying FEM (finite-element method) solution techniques to a 3-D (three-dimensional) eddy-current problem with a coil-driven multiply connected conductor, the benchmark problem No.7 of the International TEAM Workshops. Several formulations have been tried using a magnetic vector and electric scalar potential or an electric vector and a magnetic scalar in the conductor and a magnetic vector or scalar potential outside. The problem has been solved at two frequencies. The authors briefly describe the formulations used and compare the performance. Magnetic field and current density plots are also compared. The advantages and disadvantages of the various versions are pointed out. The use of a magnetic scalar potential H rather than a magnetic vector potential A outside the conductor and the hole substantially reduces the number of degrees of freedom and thus the computational effort. The versions using it in the conductor yield relatively ill-conditioned systems. Also, at the higher frequency, the conditioning deteriorates considerably     

On the boundary conditions for the vector potential formulation in electrostatics

The vector potential formulation is a promising solution method for nonlinear electromechanically coupled boundary value problems. However, one of the drawbacks of this formulation is the non‐uniqueness of the (electric) vector potential in three dimensions. The present paper focuses on the Coulomb gauging method to overcome this problem. In particular, the corresponding gauging boundary conditions and their consistency with the physical boundary conditions are examined in detail. Furthermore, certain topological features like cavities and multiply connectedness of the domain of analysis are taken into account. Different variational/weak formulations being appropriate for finite element implementation are described. Finally, the suitability of these formulations is demonstrated in several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Mapped infinite elements for 3-D vector potential magnetic problems

Numerous engineering problems, especially those in electromagnetics, often require the treatment of the unbounded continua. Mapped infinite elements have been developed for the solution of 3-D magnetic vector potential equations in infinite domain that may be used in conjunction with the standard finite elements. The electromagnetic field equations are written in terms of the magnetic vector potential for the infinite domain, and 3-D mapped infinite eiement formulation based on these equations is presented in detail. A series of magnetostatics and eddy current problems are solved to demonstrate the validity and efficiency of the procedure. These numerical results indicate that the combined finite–infinite element procedure is computationally much more economical for the solution of unbounded electromagnetic problems, especially when using the vector potential formulation, as the number of system equations decreases substantially compared to the finite element only procedure. The present procedure shows promise for the treatment of large practical industrial 3-D eddy current problems with manageable computer resources.     

Nonlinear magnetic field solutions based on a vector magnetic potential

The earlier successful techniques for numerical solution of magnetic field problems required defining a field quantity and magnetic permeability over the problem grid model. Later improved but limited techniques were introduced whereby it was necessary to define only a field quantity, usually a potentia function, over the grid because the magnetic material characteristics were incorporated directly into the equations. The work presented in this paper represents a culmination of these latter efforts in that technique is presented whereby the general three-dimensional magnetic field boundary value problem with current regions can be solved numerically.     

Generalized potential formulation for 3-D magnetostatic problems

Scalar potential formulations as applied to the solution of magnetostatic problems are reviewed. The concept of a generalized potential formulation is introduced. Based on this concept, six methods that can be used for a wider range of problems are developed. It is shown that the other existing formulations and methods are special cases of the generalized potential     

Application of the general potential formulation

The authors discuss an efficient method to calculate magnetostatic problems involving multiply connected iron regions by standard finite-element method (FEM) code without cancellation errors. This method has been successfully tested using the latest version of the ANSYS finite element analysis software program. The authors summarize the most important steps of its implementation     

Application of the complex variable boundary element method to solving potential problems in doubly connected domains

The complex variable boundary element method (CVBEM) developed by Hromadka for the solution of potential problems in simply connected domains is extended to the solution of heat conduction problems in doubly connected domains. A cut is made in the doubly connected domain, and it was found that the complex potentials along the cut do not cancel out but result in a complex stream function that plays the role of perturbation in the nodal equations. Cauchy–Riemann conditions are used to derive additional equations which relate the stream functions and the boundary heat fluxes and potentials when Neumann and Robin conditions are imposed on the boundaries. The resulting nodal equations are expressed in matrix form, and coding rules and methods for checking the matrix elements are developed. Three solution methods (implicit, explicit and hybrid) are described, and by means of examples, the efficacy of these methods is discussed and compared.     

Automatic implementation of cuts in multiply connected magneticscalar regions for 3D eddy current models

A novel scheme for automatically generating cuts in the magnetic scalar region of a finite element mesh is presented. Cuts are generated allowing multiply connected eddy current problems to be solved. The scheme has no topological limitations; it has been tested for knotted conductors, interlinked conductors, and hollow conductors. The scheme is fully automatic, requiring just a standard well-formed finite element mesh and appropriate boundary conditions. It has been implemented and tested in the finite element package MEGA for the Aψ formulation     

Some features of the variational formulation of the general criterion of evolution in the local potential method

The variational formulation of nonlinear problems of the thermodynamics of irreversible processes in the local potential method is subjected to a comparative analysis. The particular example of the solution obtained by the method of successive approximations is examined and the possibility of using other methods is discussed.     

A method for rendering 3D finite element vector field solutionsnondivergent

The use of the method of constraints for enforcing the zero divergence condition in vectorial finite-element schemes is discussed. An earlier implementation of the method was shown to produce the correct solution for a 3-D resonant cavity problem modeled by a single finite element. Partial success in extending the method to multielement cases is reported. The reduction in matrix size alone would justify the development of the technique for general multielement grids, but it will require the implementation of a global approach to the method of constraints     

Polyethylenimine functionalized magnetic nanoparticles as a potential non-viral vector for gene delivery

Polyethylenimine (PEI) functionalized magnetic nanoparticles were synthesized as a potential non-viral vector for gene delivery. The nanoparticles could provide the magnetic-targeting, and the cationic polymer PEI could condense DNA and avoid in vitro barriers. The magnetic nanoparticles were characterized by Fourier transform infrared spectroscopy, X-ray powder diffraction, dynamic light scattering measurements, transmission electron microscopy, vibrating sample magnetometer and atomic force microscopy. Agarose gel electrophoresis was used to asses DNA binding and perform a DNase I protection assay. The Alamar blue assay was used to evaluate negative effects on the metabolic activity of cells incubated with PEI modified magnetic nanoparticles and their complexes with DNA both in the presence or absence of an external magnetic field. Flow cytometry and fluorescent microscopy were also performed to investigate the transfection efficiency of the DNA-loaded magnetic nanoparticles in A549 and B16-F10 tumor cells with (+M) or without (?M) the magnetic field. The in vitro transfection efficiency of magnetic nanoparticles was improved obviously in a permanent magnetic field. Therefore, the magnetic nanoparticles show considerable potential as nanocarriers for gene delivery.     

Further discussion on magnetic vector potential finite-elementformulation for three-dimensional magnetostatic field analysis

A linear soft-iron and current model called the IEE Japan model using a novel vector potential finite-element formulation is examined. Calculated and measured results are in close agreement. For comparison, the same model was calculated by the conventional variational formulation. The divergence of magnetic vector potential equals zero at the boundary of different materials and the values themselves are small enough at the Gaussian quadratural points, which means that uniqueness of the solution is guaranteed. The gauge condition is determined by the formulation, not by the boundary conditions. The new formulation requires less computing time and memory than the conventional variational formulation     

Alternative vector potential formulations of 3-D magnetostatic field problems

A unified theory of three-dimensional vector potential formulations of magnetostatic field problems is presented. It is shown that existing formulations are based on one of two equivalent boundary value problems. A new formulation is derived, using the concept of projection between the space of arbitrary vector finite elements and the space of non-divergent vector finite elements. The merits of the different approaches are examined.     

A point-iterative algorithm for three-dimensional magnetic vector problems

Magnetostatic field problems are solved in three dimensions by applying a variational method that employs finite elements. Formulation through a partial differential equation allows solution for the magnetic vector potential given an inhomogeneous, orthotropic medium and a distributed current source. Three vector boundary conditions are discussed and interior sheet currents are allowed within the medium. In addition, the Lorentz condition is enforced by including a penalty term in the energy functional. A point-iterative algorithm is used to solve the set of equations resulting from finite element discretization. This method is particularily suitable for regions with regular geometry and a moderate (1,000 to 10,000) number of unknowns.     

On the formulation of diffusive mixture theories for two-phase regions

This paper deals with the modelling of solidifying two-phase alloys and is primarily concerned with aspects of phase conversion. It briefly reviews existing theories and shows, by considering mass conservation and dissipation, how modelling proceeds when there is a clearly identifiable dominant rate process associated with the phase change.