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The purpose of this paper is to present new basis functions suitable to parameterize two‐dimensional static potentials or (magnetic) fields and to show their application in practical cases.

Regular multipole solutions of the potential equation in plane elliptic coordinates are found by separation. The resulting set of functions is reduced to complete subsets suitable for expanding regular potentials or irrotational source‐free fields. Approximate regular plane solutions of the potential equation in local toroidal coordinates are computed by R‐separation and power series expansions in the inverse aspect ratio. The harmonic signals induced in a coil rotating in such a toroidal multipole field are computed from the induction law by similar expansions.

The elliptic expansions are useful in a larger area than circular multipole expansions and give better fits. The toroidal expansions permit one to estimate the effect of the curvature of magnets on the field and give better adapted expansions. However, while the scalar multipoles for the potential are orthogonal the vector fields derived for the two‐dimensional field are not.

Derivations presuppose analytical fields.

Field data obtained from numeric field calculations or measurements do not represent exactly analytic fields. Application of the expansion requires care and checks.

The paper presents novel approaches for parameterizing longitudinally uniform cylindrical or toroidally uniform potentials and fields.

The purpose of this paper is to present new expressions in Cartesian coordinates for the potential and magnetic field of prolate and oblate spheroids with arbitrary direction of the symmetry axis in a homogeneous field.

The potentials found in prolate or oblate spheroidal coordinates are transformed to Cartesian coordinates. These results are represented in such a form that they depend only on expressions, which are invariant under rotations around the symmetry axis. Thus, it is easy to change to arbitrary directions of both the symmetry axis and of that of the primary field. The gradients of the potentials are calculated and transformed exactly to the simplest form possible.

The paper presents simple expressions for the magnetic perturbations due to homogeneous prolate or oblate spheroids in a homogeneous magnetic field.

Results are exact for single non‐ferromagnetic spheroids in a homogeneous field.

Superposition of these perturbations presupposes small values of the magnetic susceptibilities of both the spheroids and their environment as in biological tissues.

The paper presents novel formulas for fields of homogeneous spheroids in a homogeneous magnetic field which are very useful for modelling biological tissues in studies of magnetic resonance imaging and magnetic resonance spectroscopy.

The reliable transcranial imaging of brain inner structures for diagnostic purposes is deemed crucial owing to the decisive importance and contribution of the brain in human life. The purpose of this paper is to investigate the potential application of medical ultrasounds to transcranial imaging using advanced techniques, such as the total focussing method.

Initially, the fundamental details of the total focussing method are presented, while the skull properties, such as the increased acoustic velocity and scattering, are thoroughly examined. Although, these skull characteristics constitute the main drawback of typical transcranial ultrasonic propagation algorithms, they are exploited to focus the acoustic waves towards the brain. To this goal, a virtual source is designed, considering the wave refraction, to efficiently correct the reconstructed brain image. Finally, the verification of the novel method is conducted through numerical simulations of various realistic setups.

The theoretically designed virtual source resembles a focussed sensor; therefore, the directivity increment, owing to the propagation through the skull, is confirmed. Moreover, numerical simulations of real-world scenarios indicate that the typical artifacts of the conventional total focussing method are fully overcome because of the increased directivity of the proposed technique, while the reconstructed image is efficiently corrected when the proposed virtual source is used.

A new systematic methodology along with the design of a flexible virtual source is developed in this paper for the reliable and precise transcranial ultrasonic image reconstruction of the brain. Despite the slight degradation owing to the skull scattering, the combined application of the total focussing method and the featured virtual source can successfully detect arbitrary anomalies in the brain that cannot be spotted by conventional techniques.

A new finite element method (FEM) approach is used to evaluate transient electric field based on a surge wave in a dissipative open region. Instead of the solution of the basic equations in the time domain, the Laplace transform inversion is used to model planar and axisymmetrical transient fields. The main advantage of this approach consists in the possibility of modeling unbounded regions. Current‐driven electrodes, including electrodes with floating potential, are taken into consideration.