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Field lines generated by a stranded inductor (half geometry): solution of a 2-D plane model in the XY plane ( , portion on the left) and of a 2-D axisymmetrical model in the YZ plane ( , portion on the right); the interface between the two portions is shown.
Source publication
Model refinements of magnetic circuits are per-formed via a subdomain finite element method based on a per-turbation technique. A complete problem is split into subprob-lems, some of lower dimensions, to allow a progression from 1-D to 3-D models. Its solution is then expressed as the sum of the subproblem solutions supported by different meshes. T...
Context in source publication
Context 1
... correction procedure is proposed for 3-D inductors with portions satisfying translational or rotational symmetries, that can be first studied via 2-D models. As an example, a 3-D stranded inductor is defined via the combination of a 2-D plane model for its portion with a translational symmetry and a 2-D axisymmetrical model for its end winding (Figs. 3 and 4). This consists in initially neglecting some end effects, zeroing on the portions caps. Besides, each field is forced to be zero out of each , which defines a discontinuity of through . With such assumptions, two subproblems 1 and 2 with adjacent non-overlapping sudomains and share a common interface through which a combination of field ...
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Citations
... III-D. This approach resembles the sub-domain FEM based on a perturbation technique, where a missing continuity at the interface of two sub-domains is also restored via a jump in the tangential magnetic field strength expressed as a surface current density source [13]. ...
The major advantage of reduced magnetic vector potential formulations (RMVPs) is that complicated coil structures do not need to be resolved by a computational mesh. Instead, they are modeled by thin wires, whose source field is included into the simulation model along Biot-Savart’s law. Such an approach has already been successfully employed in ROXIE for the simulation of superconducting Large Hadron Collider magnets at CERN. This work presents an updated RMVP approach, which significantly outperforms the original method. The updated formulation is postulated, implemented, verified, compared to the original formulation, and applied for the simulation of a quadrupole magnet. The promising results of this work encourage further investigation towards an updated simulation framework for next-generation accelerator magnets.
... By separating the field produced by the coils (H s ) from the field produced by the bulk (H r ), we can exploit the 2-D axisymmetric nature of the coils [22], as shown in Fig. 1b). In addition, according to the Biot-Savart law, the magnetic field produced by the coils normalized by the current in the coils is a constant, such that ...
Although the H-formulation has proven to be one of the most versatile formulations used to accurately model superconductors in the finite element method, the use of vector dependent variables in non-conducting regions leads to unnecessarily long computation times. Additionally, in some applications of interest, the combination of multiple magnetic components interacting with superconducting bulks and/or tapes leads to large domains of simulation. In this work, we separate the magnetic field into a source and reaction field and use the H-$\phi$ formulation to efficiently simulate a superconductor surrounded by magnetic bodies. We model a superconducting cube between a pair of Helmholtz coils and a permanent magnet levitating above a superconducting pellet. In both cases, we find excellent agreement with the H-formulation, while the computation times are reduced by factors of nearly three and four in 2-D and 3-D, respectively. Finally, we show that the H-$\phi$ formulation is more accurate and efficient than the H-A formulation in 2-D.
... This leads to negligible edges and corners of magnetic shells, increasing with thickness. In order to overcome this disadvantage, the Sub-Problem Method (SPM) for the magnetodynamic problem with dual formulation has been proposed for one-way coupling [3][4][5][6][7][8][9][10]. In this development, a subdomain technique based on the SPM is extended for the h-conformal magnetostatic finite element formulation in order to improve the local fields (magnetic scalar potential, magnetic flux density and magnetic field) appearing around the edges and corners of magnetic shells. ...
... A canonical magnetostatic problem q presented at step q is solved in a domain Ω , with boundary ߲Ω ൌ Γ ൌ Γ , ∪ Γ , . Maxwell's equations, constitutive laws and boundary conditions (BCs) of the problem q give [3][4][5][6][7][8][9][10][11]: ...
... The notation [•] ఊ = | ఊ శ − | ఊ ష is the discontinuity of a quantity across the negative and positive sides of any interface ߛ in Ω . The field , is a SS between subdomains [3][4][5][6][7][8][9][10]. In addition, the magnetic field in (1a) is split in two parts ௦, and , , i.e. = ௦, + , , where , is the reaction field and ௦, is a source magnetic field due to the imposed current density ௦, (curl ௦, = ௦, ). ...
This paper presents a subproblem approach with h-conformal magnetostatic finite element formulations for treating the errors of magnetic shell approximation, by replacing volume thin regions by surfaces with interface conditions. These approximations seem to neglect the curvature effects in the vicinity of corners and edges. The process from the surface-to-volume correction problem is presented as a sequence of several subdomains, which can be composed to the full domain, including inductors and thin magnetic regions. Each step of the process will be separately performed on its own subdomain and submesh instead of solving the problem in the full domain. This allows reducing the size of matrix and time computation.
... Based on the SPM strategy [3][4][5], the scenario of the perturbation method is herein considered in two steps: A problem attending with the stranded inductor and TS model is first solved on a simplified mesh. The inaccuracy on TS solution is then improved by the volume correction taken by a robust correction procedure in order to overcome the cancellation error [4][5][6][7]. The relationship between SPs is constrained by surface sources (SSs) or volume sources (VSs), where SSs show changes of ICs across surfaces from previous SPs, and VSs point out changes of material properties of volume thin regions. ...
... Stranded inductors belong to Ω , . The equations, material relations, and boundary conditions (BCs) of the SP p are [4][5][6][7]: ...
... The field ௦, is an imposed electric current density in inductors and presented changes of conductivity. For that, the changes from SP u (ߤ ௨ and ߪ ௨ to SP p (ߤ and ߪ ), ௦, and ௦, ) are defined [4][5][6][7]: ...
This research proposes a robust correction procedure to improve inaccuracies around edges and corners inherent to thin shell electromagnetic problems by means of perturbation technique. This proposal is developed with three processes: A classical thin shell approximation replaced with an impedance-type interface condition across a surface is first considered and then a volume correction is introduced to overcome the thin shell approximation. However, the volume correction is quite sensitive to cancellation errors, with dramatic effects in the calculation of the local fields near edges and corners. Therefore, a robust correction procedure is added to improve cancellation errors of the volume correction. Each step of the developed method is validated on the practical problem.
... In case of 2-D symmetries, (22) can be re-written as ...
... Reference [22] presents a way to correct problem q for higher values of µ q , i.e., H p approaches to −H q . This correction is important in the case of projections of curl A [23] between different meshes and that is not the case here because the H p field is calculated directly along the mesh used to solve problem q. ...
The Facet finite-element method (FFEM) requires a source field solution that is commonly obtained by the Biot-Savart (BS) equation that can be time-consuming. The subproblem modeling is applied to the FFEM in order to minimize the computational effort and time necessary to obtain the solution. The volume sources, which allow solving the BS equation only along the active regions, are presented, as well as the BS boundary condition correction.
... It is based on a finite element (FE) subproblem (SP) method (SPM) with magnetostatic and magnetodynamic problems solved in a sequence on different adapted meshes [1]- [5], from simple 1-D models up to accurate 3-D models, in a large frequency range, of the magnetic circuits and their windings (stranded or massive coils). Each step of the SPM aims at improving the solution obtained at previous steps via any coupling of the following changes, defining model refinements: change from ideal to real (with leakage flux) flux tubes [1], change from 1-D to 2-D to 3-D [2], change of material properties [1]- [3] (e.g., from linear to nonlinear), change from perfect to real materials [4], change from single wire to volume conductor windings [4], [5], and newly developed change from homogenized [6] to fine models (cores as lamination stacks and coils as wire or foil windings, with the details affecting their high frequency behaviors). The methodology involves and couples numerous techniques that have been developed by the authors and, up to now, only applied for simplified test problems [1]- [5]. ...
... A preliminary equivalent magnetic circuit can define a 0-D model. Some of its branches can then be progressively refined via the consideration of their actual geometries, in 1-D, 2-D and 3-D, via SPs of associated dimensions [2]. Such SPs consider changes of boundaries of domains that can be either extended or connected together. ...
... Each dimension fixes some boundaries through which some assumptions on magnetic flux are considered via some boundary conditions (BCs). A higher dimension modifies such BCs via interface condition (IC) surface sources (SSs) [2]. Changes of material properties can be considered via volume sources (VSs) or SSs when adding, removing, changing or moving some regions [1]- [3]. ...
Purpose
This paper aims to develop a methodology for progressive finite element (FE) modeling of transformers, from simple to complex models of both magnetic cores and windings.
Design/methodology/approach
The progressive modeling of transformers is performed via a subproblem (SP) FE method. A complete problem is split into SPs with different adapted overlapping meshes. Model refinements are performed from ideal to real flux tubes, one-dimensional to two-dimensional to three-dimensional models, linear to nonlinear materials, perfect to real materials, single wire to volume conductor windings and homogenized to fine models of cores and coils, with any coupling of these changes.
Findings
The proposed unified procedure efficiently feeds each SP via interface conditions (ICs), which lightens mesh-to-mesh sources transfers and quantifies the gain given by each refinement on both local fields and global quantities, with a clear view on its significance to justify its usefulness, if any. It can also help in education with a progressive understanding of the various aspects of transformer designs.
Originality/value
Models of different accuracy levels are sequenced with successive additive corrections supported by different adapted meshes. The way the sources act at each correction step, up to the full models with their actual geometries, is given a particular care and generalized, allowing the proposed unified procedure. For all the considered corrections, the sources are always of IC type, thus only needed in layers of FE along boundaries, which lightens the required mesh-to-mesh projections between subproblems.
... Instead of solving a complete magnetodynamic problem, including all conducting and magnetic regions, it is here proposed to perform successive finite element (FE) calculations via a subproblem (SP) method (SPM) [1]- [6], mainly by separating the regions, with the advantage of using a different mesh at each step, or no mesh when the Biot-Savart law is used. Source and reaction fields are considered but, at the difference with the common method that adds these fields in the whole domain to define the total field, the source fields are here to be defined only in the added regions [3]- [5]. Such a support reduction is of importance for efficient calculations, especially for source fields calculated via the Biot-Savart law. ...
... To allow a progression from simple to more elaborate models, a complete problem is split into a series of SPs that define a sequence of changes, with the complete or total solution given by the sum of the SP solutions [3]- [5]. Each SP is defined in its particular domain, generally distinct from the complete one and usually overlapping those of the other SPs. ...
... They can classically be remnant fields in magnets or fixed current densities in conductors. With the SPM, h s,p is also used for expressing changes of permeability and j s,p for changes of conductivity [3], [4]. For changes from µ q and σ q for previous SP q to µ p and σ p for SP p in some regions, the associated VSs h s,p and j s,p , nonzero only in these regions, are ...
... [2]). It is here proposed to perform successive FE refinements via a subproblem (SP) method (SPM) [3] to correct the models with approximate BCs. Accurate skin and proximity effects, i.e. distributions of fields and current densities, are to be obtained for accurate force and Joule loss density distributions as well as for accurate interactions with neighboring regions. ...
... To allow a natural progression from simple to more elaborate models, a complete problem is split into a series of SPs that defines a sequence of changes, with the complete solution replaced by the sum of the SP solutions [3]. Each SP is defined in its particular domain, generally distinct from the complete one and usually overlapping those of the other SPs. ...
... The source j s,p fixes the current density in inductors. With the SPM, h s,p is also used for expressing changes of permeability and j s,p for changes of conductivity [3], [5]. For changes from µ q and σ q for SP q to µ p and σ p for SP p in some regions, the associated VSs h s,p and j s,p in these regions are ...
A finite element subproblem method is developed to correct the inaccuracies proper to perfect conductor and impedance boundary condition models, particularly near edges and corners of the conductors, for a large range of conductivities and frequencies. Successive local corrections, supported by fine local meshes, can be obtained from each model to a more accurate one, allowing efficient extensions of their domains of validity.
... The solutions of electromagnetic scattering in both transverse electric and magnetic polarisation cases were got by a method of coupling of finite element and boundary integral equation methods, and the accuracy of this method was verified according to the numerical results (Li, 2010). Model refinements of magnetic circuits were performed based on a sub-domain finite element method, and meshing and solving processes were simplified (Dular et al., 2010). The cogging torque of a single-phase permanent-magnet brushless DC (SP PM BLDC) motor was analysed by the transient finite-element model, and the simulation time could be saved (Fazil and Rajagopal, 2011). ...
In order to analyse the electromagnetic field of the permanent magnet linear synchronous motor, the wavelet finite element method is applied to compute the electromagnetic field of it. First, the basic theory of Daubechies' wavelet is studied. Then the wavelet finite element model of the electromagnetic field of permanent magnet linear synchronous motor is constructed. Then simulation analysis of electromagnetic field is carried out by the wavelet finite element method, the traditional finite element method and actual test. And simulation results show that the wavelet finite element method has better convergence and can use fewer wavelet finite elements to get the higher computing precision for computing the electromagnetic field of the permanent magnet linear synchronous motor.
... The solution by means of subproblems provides clear advantages in repetitive analyses and can also help in improving the overall accuracy of the solution [1], [2]. In the case of thin shell (TS) problems the method allows to benefit from previous computations instead of starting a new complete finite element (FE) solution for any variation of geometrical or physical characteristics. ...
A subproblem technique is applied on dual formulations to the solution of thin shell finite element models. Both the magnetic vector potential and magnetic field formulations are considered. The subproblem approach developed herein couples three problems: a simplified model with inductors alone, a thin region problem using approximate interface conditions, and a correction problem to improve the accuracy of the thin shell approximation, in particular near their edges and corners. Each problem is solved on its own independently defined geometry and finite element mesh.
