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In this paper, the usage of the Parareal method is proposed for the time-parallel solution of the eddy current problem. The method is adapted to the particular challenges of the problem that are related to the differential algebraic character due to non-conducting regions. It is shown how the necessary modification can be automatically incorporated by using a suitable time-stepping method. This paper closes with the first demonstration of a simulation of a realistic four-pole induction machine model using Parareal. [ABSTRACT FROM AUTHOR]

Electro-quasi-static (EQS) field problems involving nonlinear materials are commonly discretized in space using finite elements. In this paper, it is proposed to solve the resulting system of ordinary differential equations (ODEs) by an explicit Runge–Kutta–Chebyshev time-integration scheme. This mitigates the need for Newton–Raphson iterations, as they are necessary within fully implicit time integration schemes. However, the EQS system of ODE has a Laplace-type mass matrix such that parts of the explicit time-integration scheme remain implicit. An iterative solver with constant preconditioner is shown to efficiently solve the resulting multiple right-hand side problem. This approach allows an efficient parallel implementation on a system featuring multiple graphic processing units. [ABSTRACT FROM AUTHOR]

This paper focuses on the acceleration of the numerical linear algebra for solving large nonlinear electro-quasi-static problems in time domain as they occur in high-voltage applications utilizing nonlinear electric field stress grading materials. An algebraic multigrid (AMG) scheme is executed on multiple graphic processor units (GPUs) as a preconditioner for the conjugate gradients method. The speedup is further increased by exploiting the fact that the repeated solutions of almost similar systems must be obtained in the Newton–Raphson iterations of the implicit time-stepping scheme. A tailored update algorithm for the AMG preconditioner is proposed that significantly reduces the computational time due to a reduction of communication between GPUs and host. [ABSTRACT FROM AUTHOR]

In this contribution, a multi-graphic processing unit (GPU) implementation of Krylov sub-space methods with algebraic multi-grid preconditioners is proposed. It is used to solve large linear systems stemming from finite element or finite difference discretizations of elliptic problems as they occur, e.g., in electrostatics. The distribution of data across multiple GPUs and the effects on memory and speed are discussed when using an approach that preserves the effects of fine-grained parallelism with shared memory on the GPU while distributing data across multiple GPUs with minimal communication effort. [ABSTRACT FROM PUBLISHER]

The design of electrical machines and high-voltage devices requires the simulation of non-linear low frequency electromagnetic problems in time domain. Typically magneto- and electro-quasistatic problems in magnetic vector/electric scalar potential formulation lead after space discretization to differential–algebraic or ordinary differential equations, respectively. Therefore, huge systems of equations have to be solved in both cases. Typically a large number of degrees of freedom (DoF) is in domains with constant material parameters, i.e., linear subdomains, e.g., air or vacuum in exterior domains. In this paper, we present a method for low frequency simulations based on the proper orthogonal decomposition to reduce the DoF in these linear subspaces. The application of the method will be shown for a simple transformer model and within a global sensitivity analysis (uncertainty quantification) of the switching point in a non-linear resistive material used in a 11 kV standard insulator model. [ABSTRACT FROM AUTHOR]

The simulation of coupled electromagnetic/thermal problems with high resolution requires efficient numerical schemes. High-performance computing languages like CUDA help in unlocking the massively parallel capabilities of graphic processor units (GPUs) to accelerate those calculations. This reduces the time needed to solve real-world problems. In this paper, the speedup is discussed, which is obtained using NVIDIA's recently presented Kepler architecture as well as by GPU-accelerated algebraic multigrid preconditioners. In particular, extended memory allows for the solving of larger problems with more degrees of freedom without swapping. We discuss a new host-based multigrid setup for GPU-accelerated iterative solvers. [ABSTRACT FROM AUTHOR]

In this paper, we study the magnetoquasistatic problem with uncertainties in the nonlinear magnetic material characteristic. In the case of small input uncertainties, adjoint techniques can be used to efficiently approximate the statistics of the quantities of interest. We carry out the corresponding sensitivity analysis and investigate the method's approximation properties. Numerical results discussing the approximation error are given for an electrical transformer. [ABSTRACT FROM AUTHOR]

The solution procedure of coupled electromagnetic-/thermal-simulations with high resolution requires efficient solvers. High performance computing libraries and languages like Nvidia's CUDA help in unlocking the massively parallel capabilities of GPUs to accelerate calculations. They reduce the time needed to solve real world problems. In this paper, the speed-up is discussed, which is obtained by using GPUs for coupled time domain simulations with finite difference schemes. A tailor-made implementation of the time consuming sparse matrix vector multiplication is shown to have advantages over standard CUDA-libraries like cuSparse. [ABSTRACT FROM PUBLISHER]

A method is presented that reduces the degrees of freedom (DoFs) in linear subdomains in transient non-linear electro-quasistatic (EQS) field finite-element method (FEM) simulations. The electro-quasistatic field model yields a suitable approximation to simulate high-voltage devices such as insulators or surge arresters featuring non-linear resistive field grading materials. These materials are usually applied as thin layers, i.e., they represent only a very small volume part in the overall model. Despite the application of unstructured FEM meshes, commonly most of the DoFs are located in the domain with constant material parameters. The non-linear subdomain is much smaller with respect to the number of DoFs than the part with constant materials. The application of model order reduction techniques, in particular proper orthogonal decomposition (POD), is proposed to minimize the DoFs in the linear subdomain of the simulation model. POD captures the dynamic in the linear subdomain. Large reduction factors can be achieved for low dynamic exterior domains, thus considerably reducing the computational costs. Numerical results are presented for an IEC norm surge arrester and a typical 11 kV insulator design with a field grading inlay. [ABSTRACT FROM AUTHOR]

This paper discusses the weak coupling of nonlinear magnetoquasistatic field models (e.g., of induction motors) to an external electric network model (e.g., of frequency converters). A piece-wise linear inductance (as lumped field model) is not sufficiently accurate if the magnetic field exhibits strong eddy current effects. Here, we propose to represent the field model in the circuit equations by a more complex surrogate model. It is fitted by dynamic iteration (‘waveform relaxation’) during the cosimulation procedure. We demonstrate the link between dynamic iterations and the achievable convergence order of the time integrator numerically. [ABSTRACT FROM PUBLISHER]

In this paper domain substructuring is adapted to the nonlinear transient eddy current problem: conductive and nonconductive domains are separately treated for a more efficient time integration. A matrix factorization of the linear (nonconductive) subproblem, e.g., air, is executed on beforehand and used throughout the simulation. For a general 3D problem these non-sparse factors must be replaced by sparse approximations or explicit time integration must be carried out to increase the efficiency of the solution process. This approach is validated using implicit and half-explicit time integration methods. Numerical results underline the feasibility and the possible gain obtained by higher order half-explicit methods. [ABSTRACT FROM PUBLISHER]

In this work, isogeometric mortaring is used for the simulation of a six-pole permanent magnet synchronous machine. Isogeometric mortaring is especially well suited for the efficient computation of rotating electric machines, as it allows for an exact geometry representation for arbitrary rotation angles without the need for remeshing. The appropriate B-spline spaces needed for the solution of Maxwell’s equations and the corresponding mortar spaces are introduced. Unlike classical finite-element methods (FEMs), their construction is straightforward in the isogeometric case. The torque in the machine is computed using two different methods, that is, Arkkio’s method and using the Lagrange multipliers from the mortaring. [ABSTRACT FROM AUTHOR]

Conventional magneto-static finite element (FE) analysis of electrical machine design is time-consuming and computationally expensive. Since each machine topology has a distinct set of parameters, design optimization is commonly performed independently. This article presents a novel method for predicting key performance indicators (KPIs) of differently parameterized electrical machine topologies at the same time by mapping a high-dimensional integrated design parameters in a lower-dimensional latent space using a variational autoencoder (VAE). After training, via a latent space, the decoder and multi-layer neural network will function as meta-models for sampling new designs and predicting associated KPIs, respectively. This enables parameter-based concurrent multi-topology optimization. [ABSTRACT FROM AUTHOR]

This work deals with shape optimization of electric machines using isogeometric analysis. Isogeometric analysis is particularly well suited for shape optimization as it allows to easily modify the geometry without remeshing the domain. A 6-pole interior permanent magnet synchronous machine (IPMSM) is modeled using a multipatch isogeometric approach and rotation of the machine is realized by modeling the stator and rotor domain separately and coupling them at the interface using harmonic basis functions. Shape sensitivity analysis is used to find the shape derivative of the optimization goal functional, i.e., the total harmonic distortion of the electromotive force. This method allows for a freeform shape optimization which is not restricted by a choice of a set of optimization parameters. This freeform shape optimization is applied for the first time to an isogeometric model of an IPMSM minimizing the total harmonic distortion of the electromotive force as a goal functional, where a reduction of 75 % is achieved. [ABSTRACT FROM AUTHOR]

We discuss numerical experiments to compare an isogeometric discretization of the electric field integral equation and a parametric Raviart–Thomas approach. Therein, we focus on accuracy with respect to degrees of freedom, briefly commenting on the conditioning of the system as well. Due to the utilization of parametric mappings even in the Raviart–Thomas approach, our investigation disregards any errors induced by meshing, which commonly favors the isogeometric approach when compared to curvilinear higher order Raviart–Thomas elements. [ABSTRACT FROM AUTHOR]

We apply the proper generalized decomposition (PGD) to a static electrothermal model subject to uncertainties. The quadratic electrothermal coupling term is non-standard and requires the introduction of a trilinear form. We use an existing finite integration technique (FIT)-based solver to demonstrate the opportunities of integrating the PGD in existing codes. The obtained reduced model approximates the temperature distribution in a voltage-excited microelectronic chip package that exhibits 12 wires of uncertain length. [ABSTRACT FROM AUTHOR]

The spatially discretized magnetic vector potential formulation of magnetoquasistatic field problems is transformed from an infinitely stiff differential-algebraic equation system into a finitely stiff ordinary differential equation (ODE) system by applying a generalized Schur complement for nonconducting parts. The ODE can be integrated in time using explicit time-integration schemes, such as the explicit Euler method. This requires the repeated evaluation of a pseudo-inverse of the discrete curl–curl matrix in nonconducting material by the preconditioned conjugate gradient (PCG) method, which forms a multiple right-hand side problem. The subspace projection extrapolation method and proper orthogonal decomposition are compared for the computation of suitable start vectors in each time step for the PCG method, which reduce the number of iterations and the overall computational costs. [ABSTRACT FROM AUTHOR]

This paper deals with the modeling and simulation of the effect of rotor eccentricity in permanent magnet synchronous machines. Static eccentricity is analyzed in a 2-D setting. The 3-D effect of an inclined rotor shaft is accounted by considering 2-D slices and interpolating on a grid constructed from finite-element simulations [response surface model (RSM)]. Common tools of uncertainty quantification, i.e., generalized polynomial chaos and Monte Carlo, are used to study the effect on the electromotive force. The novelty of this paper is the twofold use of the RSM: 1) to speed up the calculations in the 2-D setting and 2) to mimic a multislice model for inclined eccentricity. [ABSTRACT FROM AUTHOR]

Electromagneto-quasistatic (EMQS) field models exhibit capacitive, resistive, and inductive effects, in the absence of wave propagation, while the associated initial boundary value problems are ill-posed. A framework that is based on penalizing a Coulomb-type differential constraint that recovers well-posedness is proposed here. The advocated strategy yields two classes of EMQS formulations that implicitly gauge the magnetic vector potential. Comparative numerical tests demonstrate the performance of the resulting formulations relative to the recently introduced two-step Maxwell scheme. [ABSTRACT FROM AUTHOR]

This paper introduces a new matrix assembly algorithm for nonlinear problems that decomposes the problem into parts that are well suited for parallel computation on GPGPUs. It has the strength that recomputations in nonlinear entries of the stiffness matrices are embarrassingly parallel. The algorithm is used to solve electro-quasistatic problems. The proposed discretization includes the handling of Dirichlet and homogeneous Neumann boundary conditions. [ABSTRACT FROM AUTHOR]

In this article, we present and analyze the numerical stability of two coupled finite element formulations. The first one is the $h$ - $a$ -formulation and is well suited for modeling systems with superconductors and ferromagnetic materials. The second one, the so-called $t$ - $a$ -formulation with thin-shell approximation, applies for systems with thin superconducting domains. Both formulations involve two coupled unknown fields and are mixed on the coupling interfaces. Function spaces in mixed formulations must satisfy compatibility conditions to ensure stability of the problem and reliability of the numerical solution. We propose stable choices of function spaces using hierarchical basis functions and demonstrate the effectiveness of the approach on simple 2D examples. [ABSTRACT FROM AUTHOR]

The application of high-temperature superconductors to accelerator magnets for future particle colliders is under study. Numerical methods are crucial for an accurate evaluation of the complex dynamical behavior of the magnets, especially concerning the magnetic field quality and thermal behavior. We present a coupled ${{\mathbf {A}}}\!-\!{{\mathbf {H}}}$ field formulation for the analysis of magneto-thermal transients in accelerator magnets. The magnetic field strength ${{\mathbf {H}}}$ accounts for the eddy current problem in the source regions containing the superconducting domains, while the magnetic vector potential ${{\mathbf {A}}}$ represents the magnetoquasistatic problem in the normal and nonconducting domains. Furthermore, we include a thin-shell approximation for the source regions, making the formulation suitable for large-scale models composed of thousands of tapes. In this article, the relevant equations are derived and discussed, with emphasis on the coupling conditions. The weak formulation is derived, and numerical results are provided in order to both verify the formulation and scale it to the size of an accelerator magnet. [ABSTRACT FROM AUTHOR]

This paper addresses uncertainty quantification of electromagnetic devices determined by the eddy current problem. The multilevel Monte Carlo (MLMC) method is used for the treatment of uncertain parameters while the devices are discretized in space by the finite element method. Both methods yield numerical approximations such that the total error is split into stochastic and spatial contributions. We propose a particular implementation where the spatial error is controlled based on a Richardson extrapolation-based error indicator. The stochastic error, in turn, is efficiently reduced in the MLMC approach by distributing the samples on multiple grids. The method is applied to a toy problem with closed-form solution and to a permanent magnet synchronous machine with uncertainties. The uncertainties under consideration are related to the material properties in the stator and the magnets in the rotor. The examples show that the error indicator works reliably, the meshes used for the different levels do not have to be nested, and, most importantly, MLMC reduces the computational cost by at least one order of magnitude compared to standard Monte Carlo. [ABSTRACT FROM AUTHOR]

In this paper, the concept of multirate partial differential equations (MPDEs) is applied to obtain an efficient solution for nonlinear low-frequency electrical circuits with pulsed excitation. The MPDEs are solved by a Galerkin approach and a conventional time discretization. Nonlinearities are efficiently accounted for by neglecting the high-frequency components (ripples) of the state variables and using only their envelope for the evaluation. It is shown that the impact of this approximation on the solution becomes increasingly negligible for rising frequency and leads to significant performance gains. [ABSTRACT FROM AUTHOR]

Uncertainty quantification plays an important role in biomedical engineering as measurement data are often unavailable and literature data show a wide variability. Using the state-of-the-art methods, one encounters difficulties when the number of random inputs is large. This is the case, e.g., when using composite Cole–Cole equations to model random electrical properties. It is shown that how the number of parameters can be significantly reduced by the Karhunen–Loève expansion. The low-dimensional random model is used to quantify uncertainties in the axon activation during deep brain stimulation. Numerical results for a Medtronic 3387 electrode design are given. [ABSTRACT FROM AUTHOR]

Stern–Gerlach magnets are used to magnetically separate a beam of atoms or atom clusters. The design is difficult, since both the magnetic field gradient and its homogeneity should be maximized. This paper proposes the numerical optimization of the device’s pole-shoe shapes, starting from a reference geometry given in the literature. The main contributions of this paper are the generalization of a magnetic field–magnetic circuit coupling and the application of isogeometric analysis (IGA), which are shown to reduce the computational complexity and increase the accuracy. The field–circuit coupling significantly reduces the size of the field model part, in which the IGA’s spline-based framework enables a highly accurate evaluation of local field quantities, even across elements. [ABSTRACT FROM PUBLISHER]

Inrush currents are an important factor in the design of electrical machines. The uncertainties in measurements of magnetic materials parameters propagate via fitted material laws into the simulation results. In this paper, generalized polynomial chaos is used to compute the solution statistics and perform an analysis of variance; by this, it is revealed which measurement points are the most influential. As fitting strategies, we compare the result for the Brauer model and a smoothing spline approach. This is an essential step forward compared with considering only the uncertainty quantification for analytical material curve models. [ABSTRACT FROM AUTHOR]

A challenge in accelerator magnet design is the strong nonlinear behavior due to magnetic saturation. In practice, the underlying nonlinear saturation curve is modeled according to measurement data that typically contain uncertainties. The electromagnetic fields and in particular the multipole coefficients that heavily affect the particle beam dynamics inherit this uncertainty. In this paper, we propose a stochastic model to describe the uncertainties and we demonstrate the use of generalized polynomial chaos for the uncertainty quantification of the multipole coefficients. In contrast to previous works we propose to start the stochastic analysis from uncertain measurement data instead of uncertain material properties and we propose to determine the sensitivities by a Sobol decomposition. [ABSTRACT FROM AUTHOR]