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The simulation of eddy currents in laminated iron cores by the finite element method (FEM) is of great interest in the design of electrical devices. Modeling each laminate by finite elements leads to extremely large nonlinear systems of equations impossible to solve with present computer resources reasonably. The purpose of this study is to show that the multiscale finite element method (MSFEM) overcomes this difficulty.

A new MSFEM approach for eddy currents of laminated nonlinear iron cores in three dimensions based on the magnetic vector potential is presented. How to construct the MSFEM approach in principal is shown. The MSFEM with the Biot–Savart field in the frequency domain, a higher-order approach, the time stepping method and with the harmonic balance method are introduced and studied.

Various simulations demonstrate the feasibility, efficiency and versatility of the new MSFEM.

The novel MSFEM solves true three-dimensional eddy current problems in laminated iron cores taking into account of the edge effect.

This paper presents a comparative study of different stator-segmentation topologies of a permanent magnet synchronous machine (PMSM) used in traction drives and their effect on iron losses. Using stator segmentation allows one to achieve more significant copper fill factors, resulting in increased power densities and efficiencies. The segmentation of the stators creates additional air gaps and changes the soft magnetic material’s material properties due to the cut edge effect. The aim of this paper is to present an in-depth analysis of the influence of stator segmentation on iron losses. The main goal was to compare various segmentation methods under equal excitation conditions in terms of their influence on iron loss.

A transient finite element method analysis combined with an extended iron-loss model was used to evaluate discussed effects on the stator’s iron losses. The workflow to obtain a homogenized airgap length accounting for cut edge effects was established.

The paper concludes that the segmentation in most cases slightly decreases the iron losses in the stator because of the overall reduced magnetic flux density B due to the additional air gaps in the magnetic circuit. An increase of the individual components, as well as total power loss, was observed in the Pole Chain segmentation design. In general, segmentation did not change the total iron losses significantly. However, different segmentation methods resulted in the different distortion of the magnetic field and, consequently, in different iron loss compositions. The analysed segmentation methods exhibited different iron loss behaviour with respect to the operation points of the machine. The final finding is that analysed stator segmentations had a negligible influence on the total iron loss. Therefore, applying segmentation is an adequate measure to improve PMSMs as it enables, e.g. increase of the winding fill factor or simplifying the assembly processes, etc.

The influence of stator segmentation on iron losses was analysed. An in-depth evaluation was performed to determine how the discussed changes influence the individual iron loss components. A workflow was developed to achieve a computationally cheap homogenized model.

This work introduces an efficient and accurate technique to solve the eddy current problem in laminated iron cores considering vector hysteresis.

The mixed multiscale finite element method based on the based on the T,Φ-Φ formulation, with the current vector potential T and the magnetic scalar potential Φ allows the laminated core to be modelled as a single homogeneous block. This means that the individual sheets do not have to be resolved, which saves a lot of computing time and reduces the demands on the computer system enormously.

As a representative numerical example, a single-phase transformer with 4, 20 and 184 sheets is simulated with great success. The eddy current losses of the simulation using the standard finite element method and the simulation using the mixed multiscale finite element method agree very well and the required simulation time is tremendously reduced.

The vector Preisach model is used to account for vector hysteresis and is integrated into the mixed multiscale finite element method for the first time.

The practical purpose of this research is to propose a candidate for post-quantum signature standard that is free of significant drawback of the finalists of the NIST world competition, which consists in the large size of the signature and the public key. The practical purpose is to propose a fundamentally new method for development of algebraic digital signature algorithms.

The proposed method is distinguished by the use of two different finite commutative associative algebras as a single algebraic support of the digital signature scheme and setting two different verification equation for a single signature. A single public key is computed as the first and the second public keys, elements of which are computed exponentiating two different generators of cyclic groups in each of the algebras.

Additionally, a scalar multiplication by a private integer is performed as final step of calculation of every element of the public key. The same powers and the same scalar values are used to compute the first and the second public keys by the same mathematic formulas. Due to such design, the said generators are kept in secret, providing resistance to quantum attacks. Two new finite commutative associative algebras, multiplicative group of which possesses four-dimensional cyclicity, have been proposed as a suitable algebraic support.

The introduced method is novel and includes new techniques for designing algebraic signature schemes that resist quantum attacks. On its base, a new practical post-quantum signature scheme with relatively small size of signature and public key is developed.

The purpose of this work is to optimize the monitoring of vibrations on dynamometric test rigs for railway brakes. This is a quite demanding application considering the continuous increase of performances of high-speed trains that involve higher testing specifications for brake pads and disks.

In this work, authors propose a mixed approach in which relatively simple finite element models are used to support the optimization of a diagnostic system that is used to monitor vibration levels and rotor-dynamical behavior of the machine. The model is calibrated with experimental data recorded on the same rig that must be identified and monitored. The whole process is optimized to not interfere with normal operations of the rig, using common inertial sensor and tools and are available as standard instrumentation for this kind of applications. So at the end all the calibration activities can be performed normally without interrupting the activities of the rig introducing additional costs due to system unavailability.

Proposed approach was able to identify in a very simple and fast way the vibrational behavior of the investigated rig, also giving precious information concerning the anisotropic behavior of supports and their damping. All these data are quite difficult to be found in technical literature because they are quite sensitive to assembly tolerances and to many other factors. Dynamometric test rigs are an important application widely diffused for both road and rail vehicles. Also proposed procedure can be easily extended and generalized to a wide value of machine with horizontal rotors.

Most of the studies in literature are referred to electrical motors or turbomachines operating with relatively slow transients and constant inertial properties. For investigated machines both these conditions are not verified, making the proposed application quite unusual and original with respect to current application. At the same time, there is a wide variety of special machines that are usually marginally covered by standard testing methodologies to which the proposed approach can be successfully extended.

The purpose of this paper is to present the freely available finite element simulation software Pyrit.

In a first step, the design principles and the objective of the software project are defined. Then, the software’s structure is established: The software is organized in packages for which an overview is given. The structure is based on the typical steps of a simulation workflow, i.e., problem definition, problem-solving and post-processing. State-of-the-art software engineering principles are applied to ensure a high code quality at all times. Finally, the modeling and simulation workflow of Pyrit is demonstrated by three examples.

Pyrit is a field simulation software based on the finite element method written in Python to solve coupled systems of partial differential equations. It is designed as a modular software that is easily modifiable and extendable. The framework can, therefore, be adapted to various activities, i.e., research, education and industry collaboration.

The focus of Pyrit are static and quasistatic electromagnetic problems as well as (coupled) heat conduction problems. It allows for both time domain and frequency domain simulations.

In research, problem-specific modifications and direct access to the source code of simulation tools are essential. With Pyrit, the authors present a computationally efficient and platform-independent simulation software for various electromagnetic and thermal field problems.

The purpose of this article is to determine necessary and sufficient conditions in order that (D, K) to be an S-accr pair, where D is an integral domain and K is a field which contains D as a subring and S is a multiplicatively closed subset of D.

The methods used are from the topic multiplicative ideal theory from commutative ring theory.

Let S be a strongly multiplicatively closed subset of an integral domain D such that the ring of fractions of D with respect to S is not a field. Then it is shown that (D, K) is an S-accr pair if and only if K is algebraic over D and the integral closure of the ring of fractions of D with respect to S in K is a one-dimensional Prüfer domain. Let D, S, K be as above. If each intermediate domain between D and K satisfies S-strong accr*, then it is shown that K is algebraic over D and the integral closure of the ring of fractions of D with respect to S is a Dedekind domain; the separable degree of K over F is finite and K has finite exponent over F, where F is the quotient field of D.

Motivated by the work of some researchers on S-accr, the concept of S-strong accr* is introduced and we determine some necessary conditions in order that (D, K) to be an S-strong accr* pair. This study helps us to understand the behaviour of the rings between D and K.

The “chiralisation” of Euclidean polygonal tessellations is a novel, recent method which has been used to design new auxetic metamaterials with complex topologies and improved geometric versatility over traditional chiral honeycombs. This paper aims to design and manufacture chiral honeycombs representative of four distinct classes of 2D Euclidean tessellations with hexagonal rotational symmetry using fused-deposition additive manufacturing and experimentally analysed the mechanical properties and failure modes of these metamaterials.

Finite Element simulations were also used to study the high-strain compressive performance of these systems under both periodic boundary conditions and realistic, finite conditions. Experimental uniaxial compressive loading tests were applied to additively manufactured prototypes and digital image correlation was used to measure the Poisson’s ratio and analyse the deformation behaviour of these systems.

The results obtained demonstrate that these systems have the ability to exhibit a wide range of Poisson’s ratios (positive, quasi-zero and negative values) and stiffnesses as well as unusual failure modes characterised by a sequential layer-by-layer collapse of specific, non-adjacent ligaments. These findings provide useful insights on the mechanical properties and deformation behaviours of this new class of metamaterials and indicate that these chiral honeycombs could potentially possess anomalous characteristics which are not commonly found in traditional chiral metamaterials based on regular monohedral tilings.

To the best of the authors’ knowledge, the authors have analysed for the first time the high strain behaviour and failure modes of chiral metamaterials based on Euclidean multi-polygonal tessellations.

This paper focuses on the unconditionally optimal error estimates of a linearized second-order scheme for a nonlocal nonlinear parabolic problem. The first step of the scheme is based on Crank–Nicholson method while the second step is the second-order BDF method.

A rigorous error analysis is done, and optimal L² error estimates are derived using the error splitting technique. Some numerical simulations are presented to confirm the study’s theoretical analysis.

Optimal L² error estimates and energy norm.

The goal of this research article is to present and establish the unconditionally optimal error estimates of a linearized second-order BDF finite element scheme for the reaction-diffusion problem. An optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, that is, the temporal error and the spatial error. Since the spatial error is not dependent on the time step, the boundedness of the numerical solution in L∞-norm follows an inverse inequality immediately without any restriction on the grid mesh.