Residual and equilibrated error estimators for magnetostatic problems solved by finite element method

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In this paper, an a posteriori residual error estimator is presented for the 3-D eddy current problem modeled by the space-time A − ϕ potential formulation. It is solved by the finite element method in space and the backward Euler scheme in time. Once the reliability as well as the efficiency of the estimator is established, two numerical tests are proposed: 1) an analytical one to validate the theoretical results and 2) a physical one to illustrate the performance for a real eddy current problem.Index Terms-Eddy current, finite element (FE) analysis, potential formulation, space-time a posteriori estimators.

Section: B Efficiencysupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Space-Time Residual-Based <italic>a posteriori</italic> Estimator for the <inline-formula> <tex-math notation="LaTeX">$A-\varphi$ </tex-math></inline-formula> Formulation in Eddy Current Problems

Tittarelli¹,

Menach²,

Creusé³

et al. 2015

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“…Concerning this issue, it is well known that the use of FEM in electromagnetics (as in other subjects) introduces errors related to the discretization and numerical problems can also arise because of the large size of the algebraic systems that have to be solved. In relation to the reliability of the FEM simulations regarding this topic, various indices of fitness have been proposed [6]- [12], but the problem of the precision required does not make this method suitable for performing accurate evaluations of the magnetic field relative to unconventional configurations of electrical machines. Furthermore, the option to increase the FEM meshing has its limits [13], because an excessive discretization greatly increases the computation time and can cause a numerical ill-conditioning that produces errors greater than those associated with a less dense mesh.…”

Section: Introductionmentioning

confidence: 99%

Symmetry of Magnetostatic Fields Generated by Toroidal Helicoidal Magnets

Muscia

2015

In this paper the axial symmetry of the magnetic field generated by a permanent magnet of helicoidal toroidal kind is shown. In the first part of the paper we illustrate the shape of the magnet and the number of areas where the field is calculated to demonstrate the symmetry. We define quantitatively the size of the toroidal helical magnet and the regions where the magnetostatic field is evaluated. The field is carried out for each angular sector that represents the regions where the magnetic flux density is computed. This calculation is performed with reference to a matrix of points belonging to each sector. Two sets of evaluations are performed. The first one is referred to a less dense matrix of points relative to all the regions. The aim of this computation is to demonstrate the axial symmetry of the field. The second set of calculations concerns the field evaluation by using a much higher dense matrix of points. By using this data we are able to interpolate the same field with a high precision. This second evaluation of the field is carried out with reference only to the flat region facing the first coil of the helical toroidal magnet. The use of an interpolation surface through the final points of the magnetic induction vectors previously computed allows a very fast evaluation of the field virtually in all the infinite points of the angular sector. The symmetry enables us to drastically reduce the time computation of the magnetostatic field in the points of interest

“…The REE and HEE can be applied to the solutions of various FE problems [10], [13], [14], [16]. They are here particularized to the eddy current FE problem in a domain , based on the magnetic vector potential a-formulation [16].…”

Section: Eddy Current Problems and Eesmentioning

confidence: 99%

“…Some are based on heuristic methods [3], [4], [7], evaluating only a part of the error, whereas others are developed from mathematical considerations [1], [5], [7], [13], [16], evaluating the whole error. Among these EEs are: 1) Zienkiewicz and Zhu's EEs, currently used due to their easy implementation [7], [11], [12]; 2) equilibrium EEs based on the verification of constitutive relations, but restricted to statics [5], [14]; 3) residual EE (REE) [13], [14], evaluating all the weakly verified terms in the FE formulations; and 4) hierarchical EE (HEE) [6], [10], evaluating local higher order solutions obtained using hierarchical higher order test functions. A comparison of REE and HEE has shown their equivalent behaviors [16].…”

Section: Introductionmentioning

confidence: 99%

Finite Element Mesh Adaptation Strategy From Residual and Hierarchical Error Estimators in Eddy Current Problems

Dular

Menach²,

Tang

et al. 2015

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A strategy of mesh adaptation in eddy current finite element modeling is developed from both residual and hierarchical error estimators. Wished distributions of element sizes of adapted meshes are determined from the element-wise local contributions to the estimators and define constraints for the mesh generator. Uniform distributions of the local error are searched.Index Terms-Eddy currents, error estimation, finite element (FE) method, mesh adaptation.