Peter Gangl, Stefan Köthe, Christiane Mellak, Alessio Cesarano and Annette Mütze
This paper aims to deal with the design optimization of a synchronous reluctance machine to be used in an X-ray tube, where the goal is to maximize the torque while keeping low…
Abstract
Purpose
This paper aims to deal with the design optimization of a synchronous reluctance machine to be used in an X-ray tube, where the goal is to maximize the torque while keeping low the amount of material used, by means of gradient-based free-form shape optimization.
Design/methodology/approach
The presented approach is based on the mathematical concept of shape derivatives and allows to obtain new motor designs without the need to introduce a geometric parametrization. This paper presents an extension of a standard gradient-based free-form shape optimization algorithm to the case of multiple objective functions by determining updates, which represent a descent of all involved criteria. Moreover, this paper illustrates a way to obtain an approximate Pareto front.
Findings
The presented method allows to obtain optimal designs of arbitrary, non-parametric shape with very low computational cost. This paper validates the results by comparing them to a parametric geometry optimization in JMAG by means of a stochastic optimization algorithm. While the obtained designs are of similar shape, the computational time used by the gradient-based algorithm is in the order of minutes, compared to several hours taken by the stochastic optimization algorithm.
Originality/value
This paper applies the presented gradient-based multi-objective optimization algorithm in the context of free-form shape optimization using the mathematical concept of shape derivatives. The authors obtain a set of Pareto-optimal designs, each of which is a shape that is not represented by a fixed set of parameters. To the best of the authors’ knowledge, this approach to multi-objective free-form shape optimization is novel in the context of electric machines.
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Valentin Hanser, Markus Schöbinger and Karl Hollaus
This work introduces an efficient and accurate technique to solve the eddy current problem in laminated iron cores considering vector hysteresis.
Abstract
Purpose
This work introduces an efficient and accurate technique to solve the eddy current problem in laminated iron cores considering vector hysteresis.
Design/methodology/approach
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.
Findings
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.
Originality/value
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.
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Markus Schöbinger, Karl Hollaus and Joachim Schöberl
This paper aims to improve the efficiency of a numerical method to treat the eddy current problem on a laminated material, where using a mesh that resolves each individual…
Abstract
Purpose
This paper aims to improve the efficiency of a numerical method to treat the eddy current problem on a laminated material, where using a mesh that resolves each individual laminate would be too computationally expensive.
Design/methodology/approach
The domain is modeled using a coarse mesh that treats the laminated material as a bulk with averaged properties. The fine-structured behavior is recovered by introducing micro-shape functions in the ansatz space. One such method is analyzed to find further model restrictions.
Findings
By using a special reformulation, it is possible to eliminate the additional degrees of freedom introduced by the multiscale ansatz at the cost of an additional modeling error that decreases with the laminate thickness.
Originality/value
The paper gives a computationally more efficient approximate variant to a known multiscale method.
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Michael Leumüller, Karl Hollaus and Joachim Schöberl
This paper aims to consider a multiscale electromagnetic wave problem for a housing with a ventilation grill. Using the standard finite element method to discretise the apertures…
Abstract
Purpose
This paper aims to consider a multiscale electromagnetic wave problem for a housing with a ventilation grill. Using the standard finite element method to discretise the apertures leads to an unduly large number of unknowns. An efficient approach to simulate the multiple scales is introduced. The aim is to significantly reduce the computational costs.
Design/methodology/approach
A domain decomposition technique with upscaling is applied to cope with the different scales. The idea is to split the domain of computation into an exterior domain and multiple non-overlapping sub-domains. Each sub-domain represents a single aperture and uses the same finite element mesh. The identical mesh of the sub-domains is efficiently exploited by the hybrid discontinuous Galerkin method and a Schur complement which facilitates the transition from fine meshes in the sub-domains to a coarse mesh in the exterior domain. A coarse skeleton grid is used on the interface between the exterior domain and the individual sub-domains to avoid large dense blocks in the finite element discretisation matrix.
Findings
Applying a Schur complement to the identical discretisation of the sub-domains leads to a method that scales very well with respect to the number of apertures.
Originality/value
The error compared to the standard finite element method is negligible and the computational costs are significantly reduced.
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Joachim Schöberl and Sabine Zaglmayr
The goal of the presented work is the efficient computation of Maxwell boundary and eigenvalue problems using high order H(curl) finite elements.
Abstract
Purpose
The goal of the presented work is the efficient computation of Maxwell boundary and eigenvalue problems using high order H(curl) finite elements.
Design/methodology/approach
Discusses a systematic strategy for the realization of arbitrary order hierarchic H(curl)‐conforming finite elements for triangular and tetrahedral element geometries. The shape functions are classified as lowest order Nédélec, higher‐order edge‐based, face‐based (only in 3D) and element‐based ones.
Findings
Our new shape functions provide not only the global complete sequence property but also local complete sequence properties for each edge‐, face‐, and element‐block. This local property allows an arbitrary variable choice of the polynomial degree for each edge, face, and element. A second advantage of this construction is that simple block‐diagonal preconditioning gets efficient. Our high order shape functions contain gradient shape functions explicitly. In the case of a magnetostatic boundary value problem, the gradient basis functions can be skipped, which reduces the problem size, and improves the condition number.
Originality/value
Successfully applies the new high order elements for a 3D magnetostatic boundary value problem, and a Maxwell eigenvalue problem showing severe edge and corner singularities.
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James Elgy, Paul D. Ledger, John L. Davidson, Toykan Özdeğer and Anthony J. Peyton
The ability to characterise highly conducting objects, that may also be highly magnetic, by the complex symmetric rank–2 magnetic polarizability tensor (MPT) is important for…
Abstract
Purpose
The ability to characterise highly conducting objects, that may also be highly magnetic, by the complex symmetric rank–2 magnetic polarizability tensor (MPT) is important for metal detection applications including discriminating between threat and non-threat objects in security screening, identifying unexploded anti-personnel landmines and ordnance and identifying metals of high commercial value in scrap sorting. Many everyday non-threat items have both a large electrical conductivity and a magnetic behaviour, which, for sufficiently weak fields and the frequencies of interest, can be modelled by a high relative magnetic permeability. This paper aims to discuss the aforementioned idea.
Design/methodology/approach
The numerical simulation of the MPT for everyday non-threat highly conducting magnetic objects over a broad range of frequencies is challenging due to the resulting thin skin depths. The authors address this by employing higher order edge finite element discretisations based on unstructured meshes of tetrahedral elements with the addition of thin layers of prismatic elements. Furthermore, computer aided design (CAD) geometrical models of the non-threat and threat object are often not available and, instead, the authors extract the geometrical features of an object from an imaging procedure.
Findings
The authors obtain accurate numerical MPT characterisations that are in close agreement with experimental measurements for realistic physical objects. The assessment of uncertainty shows the impact of geometrical and material parameter uncertainties on the computational results.
Originality/value
The authors present novel computations and measurements of MPT characterisations of realistic objects made of magnetic materials. A novel assessment of uncertainty in the numerical predictions of MPT characterisations for uncertain geometry and material parameters is included.
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James Elgy and Paul David Ledger
Magnetic polarizability tensors (MPTs) provide an economical characterisation of conducting magnetic metallic objects and their spectral signature can aid in the solution of metal…
Abstract
Purpose
Magnetic polarizability tensors (MPTs) provide an economical characterisation of conducting magnetic metallic objects and their spectral signature can aid in the solution of metal detection inverse problems, such as scrap metal sorting, searching for unexploded ordnance in areas of former conflict and security screening at event venues and transport hubs. In this work, the authors aim to discuss methods for efficiently building large dictionaries for classification approaches.
Design/methodology/approach
Previous work has established explicit formulae for MPT coefficients, underpinned by a rigorous mathematical theory. To assist with the efficient computation of MPTs at differing parameters and objects of interest, this work applies new observations about the way the MPT coefficients can be computed. Furthermore, the authors discuss discretisation strategies for hp-finite elements on meshes of unstructured tetrahedra combined with prismatic boundary layer elements for resolving thin skin depths and using an adaptive proper orthogonal decomposition (POD) reduced-order modelling methodology to accelerate computations for varying parameters.
Findings
The success of the proposed methodologies is demonstrated using a series of examples. A significant reduction in computational effort is observed across all examples. The authors identify and recommend a simple discretisation strategy and improved accuracy is obtained using adaptive POD.
Originality/value
The authors present novel computations, timings and error certificates of MPT characterisations of realistic objects made of magnetic materials. A novel postprocessing implementation is introduced and an adaptive POD algorithm is demonstrated.
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M. Clemens, S. Feigh, M. Wilke and T. Weiland
The simulation of magnetic fields with geometric discretization schemes using magnetic vector potentials involves the solution of very large discrete consistently singular…
Abstract
The simulation of magnetic fields with geometric discretization schemes using magnetic vector potentials involves the solution of very large discrete consistently singular curl‐curl systems of equations. Geometric and algebraic multigrid schemes for their solution require intergrid transfer operators of restriction and prolongation that achieve the discrete conservation of integral quantities serving as state‐variables of geometric discretization methods. For non‐conservative restriction operations, a consistency error correction operator related to an algebraic filtering is proposed. Numerical results show the effects of the consistency correction for a non‐nested geometric multigrid method.
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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…
Abstract
Purpose
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.
Design/methodology/approach
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.
Findings
Various simulations demonstrate the feasibility, efficiency and versatility of the new MSFEM.
Originality/value
The novel MSFEM solves true three-dimensional eddy current problems in laminated iron cores taking into account of the edge effect.
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Andreas Hauck, Michael Ertl, Joachim Schöberl and Manfred Kaltenbacher
The purpose of this paper is to propose a solution strategy for both accurate and efficient simulation of nonlinear magnetostatic problems in thin structures using higher order…
Abstract
Purpose
The purpose of this paper is to propose a solution strategy for both accurate and efficient simulation of nonlinear magnetostatic problems in thin structures using higher order finite element methods. Special interest is put in the investigation of the step-lap joints of transformer cores, with a focus on the spatial resolution of the field quantities.
Design/methodology/approach
The usage of hierarchical finite elements of higher order makes it possible to adapt the local accuracy in different spatial directions in thin steel sheets. Due to explicit representation of gradients in the basis functions, a simple Schwarz-type block preconditioner with a conjugate gradient solver can efficiently solve the arising algebraic system. By adapting the block size automatically according to the aspect ratio, deterioration of convergence in case of thin elements can be prevented. The resulting Newton scheme is accelerated utilizing the hierarchical splitting in a two-level scheme, where an initial guess is computed on a coarse sub-space.
Findings
Compared to an isotropic choice of polynomial order for the basis functions, significant runtime and memory can be saved in the simulation of thin structures without losing accuracy. The iterative solution scheme proves to be robust with respect to the polynomial order, even for aspect ratios of 1:1000 and anisotropies in two directions. An additional saving in runtime and Newton iterations can be achieved by solving the nonlinear problem initially on the lowest order basis functions only and projecting the solution to the complete space as starting value, analogous to a full multigrid scheme.
Originality/value
Within the presented solution strategy, especially the anisotropic block preconditioner and the accelerated Newton scheme based on the two-level splitting constitute a novel contribution. They provide building blocks, which can be utilized for other types of magnetic field problems like transient nonlinear problems or hysteresis modeling as well.