N.A. Golias and T.D. Tsiboukis
Two a‐posteriori error estimators are presented for adaptive mesh generation in eddy current finite element computation : the discontinuity of the magnetic flux density on the…
Abstract
Two a‐posteriori error estimators are presented for adaptive mesh generation in eddy current finite element computation : the discontinuity of the magnetic flux density on the interface between neighboring elements and the energy perturbation between 1st and 2nd order finite element meshes. Validation of the results is being accomplished by application of the adaptive refinement in the case of a T‐shaped slot embedded conductor, a problem several times approached in the literature.
N.A. Golias, C.S. Antonopoulos, T.D. Tsiboukis and E.E. Kriezis
A finite element formulation for the solution of 3D eddy current problems in terms of the electric intensity E is presented. A weak formulation, based on a Galerkin weighted…
Abstract
A finite element formulation for the solution of 3D eddy current problems in terms of the electric intensity E is presented. A weak formulation, based on a Galerkin weighted residual procedure, is presented and edge elements, that impose only tangential continuity across element interfaces of the approximated field, are employed for the discretization of the problem with the finite element method. The reliability and validity of the suggested method is verified by its application to the calculation of the 3D eddy current distribution in two conducting systems.
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J.K. Wilson and B.H.V. Topping
A new h‐refinement adaptive tetrahedral mesh generation algorithm is presented. Three‐dimensional domains, to be analysed by the finite element method, are initially modelled by a…
Abstract
A new h‐refinement adaptive tetrahedral mesh generation algorithm is presented. Three‐dimensional domains, to be analysed by the finite element method, are initially modelled by a coarse background mesh of tetrahedral elements. This mesh forms the input for finite element analysis and error estimation by the Zienkiewicz‐Zhu simple error estimator. Adaptive mesh refinement proceeds by selecting an element for remeshing whose longest edge is shared by elements that also require refinement. This group of elements is refined by inserting a new node at the mid‐point of the shared edge thereby bisecting all elements within the group. Adaptive parameters are calculated for the new node and elements. Refinement then proceeds until no further group of elements can be found for refinement or no elements within the current mesh require further refinement. The shape quality of the current mesh is then enhanced by the iterative application of nodal relaxation plus three topological transformations. The entire refinement process is repeated iteratively until the required degree of mesh refinement is reached. Ten‐noded linear strain tetrahedral finite element meshes have been used for the finite element and error estimation analyses. Four examples of adaptive tetrahedral mesh generation for linear elastic stress/displacement analysis are presented which show that this algorithm is robust and efficient in terms of reduction of the domain error with a minimum number of degrees of freedom being generated, number of iterations, and therefore finite element analyses, required and computational time for refinement when compared to the advancing front method and Delaunay triangulation.
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The v×B term in eddy current equations for conducting fluids is an instance of contraction of a differential form by a vector field. We search for a natural way to discretize such…
Abstract
The v×B term in eddy current equations for conducting fluids is an instance of contraction of a differential form by a vector field. We search for a natural way to discretize such contractions. Looking at the operation of extrusion of a manifold, which is dual to contraction, provides the main clue. Two example applications, Carpenter's gauge and Eulerian computations in MHD problems, are suggested.
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Haoyu Huang, Julin Shan, S.H. Lo, Fei Yu, Jie Cao, Jihai Chang and Z.Q. Guan
In this study, we propose a tetrahedral mesh generation and adaptive refinement method for multi-chamber, multi-facet, multiscale and surface-solid mesh coupling with extremely…
Abstract
Purpose
In this study, we propose a tetrahedral mesh generation and adaptive refinement method for multi-chamber, multi-facet, multiscale and surface-solid mesh coupling with extremely thin layers, solving the two challenges of mesh generation and refinement in current electromagnetic simulation models.
Design/methodology/approach
Utilizing innovative topology transformation techniques, high-precision intersection judgment algorithms and highly reliable boundary recovery algorithms to reduce the number of Steiner locking points. The feasible space for the reposition of Steiner points is determined by using the linear programming. During mesh refinement, an edge-split method based on geometric center and boundary facets node size is devised. Solving the problem of difficult insertion of nodes in narrow geometric spaces, capable of filtering the longest and boundary edges of tetrahedrons, refining the mesh layer by layer through multiple iterations, and achieving collaborative optimization of surface and tetrahedral mesh. Simultaneously, utilizing a surface-facet preserving mesh topology optimization algorithm to improve the fit degree between the mesh and geometry.
Findings
Initial mesh generation for electromagnetic models, compared to commercial software, the method proposed in this paper has a higher pass rate and better mesh quality. For the adaptive refinement performance of high-frequency computing, this method can generate an average of 50% fewer meshes compared to commercial software while meeting simulation accuracy.
Originality/value
This paper proposes a complete set of mesh generation and adaptive refinement theories and methods designed for the structural characteristics of electromagnetic simulation models, which meet the needs of real-world industrial applications.
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Jinlong Dong, Luca Di Rienzo, Olivier Chadebec and Jianhua Wang
This paper aims to present the mathematical formulations of a magnetic inverse problem for the electric arc current density reconstruction in a simplified arc chamber of a…
Abstract
Purpose
This paper aims to present the mathematical formulations of a magnetic inverse problem for the electric arc current density reconstruction in a simplified arc chamber of a low-voltage circuit breaker.
Design/methodology/approach
Considering that electric arc current density is a zero divergence vector field, the inverse problem can be solved in Whitney space W2 in terms of electric current density J with the zero divergence condition as a constraint or can be solved in Whitney space W1 in terms of electric vector potential T where the zero divergence condition naturally holds. Moreover, the tree gauging condition is applied to ensure a unique solution when solving for the vector potential in space W1. Tikhonov regularization is used to treat the ill-posedness of the inverse problem complemented with L-curve method for the selection of regularization parameters. A common mode approach is proposed, which solves for the reduced electric vector potential representing the internal current loops instead of solving for the total electric vector potential. The proposed inversion approaches are numerically tested starting from simulated magnetic field values.
Findings
With the common mode approach, the reconstruction of current density is significantly improved for both formulations using face elements in space W2 and using edge elements in space W1. When solving the inverse problem in space W1, the choice of the regularization operator has a key role to obtain a good reconstruction, where the discrete curl operator is a good option. The standard Tikhonov regularization obtains a good reconstruction with J-formulation, but fails in the case of T-formulation. The use of edge elements requires a tree-cotree gauging to ensure the uniqueness of T. Moreover, additional efforts have to be taken to find an optimal regularization operator and an optimal tree when using edge elements. In conclusion, the J-formulation is to be preferred.
Originality/value
The proposed approaches are able to reconstruct the three-dimensional electric arc current density from its magnetic field in a non-intrusive manner. The formulations enable us to incorporate a priori knowledge of the unknown current density into the solution of the inverse problem, including the zero divergence condition and the boundary conditions. A common mode approach is proposed, which can significantly improve the current density reconstruction.
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Marc Schober and Manfred Kasper
This paper aims to show that simple geometry‐based hp‐algorithms using an explicit a posteriori error estimator are efficient in wave propagation computation of complex structures…
Abstract
Purpose
This paper aims to show that simple geometry‐based hp‐algorithms using an explicit a posteriori error estimator are efficient in wave propagation computation of complex structures containing geometric singularities.
Design/methodology/approach
Four different hp‐algorithms are compared with common h‐ and p‐adaptation in electrostatic and time‐harmonic problems regarding efficiency in number of degrees of freedom and runtime. An explicit a posteriori error estimator in energy norm is used for adaptive algorithms.
Findings
Residual‐based error estimation is sufficient to control the adaptation process. A geometry‐based hp‐algorithm produces the smallest number of degrees of freedom and results in shortest runtime. Predicted error algorithms may choose inappropriate kind of refinement method depending on p‐enrichment threshold value. Achieving exponential error convergence is sensitive to the element‐wise decision on h‐refinement or p‐enrichment.
Research limitations/implications
Initial mesh size must be sufficiently small to confine influence of phase lag error.
Practical implications
Information on implementation of hp‐algorithm and use of explicit error estimator in electromagnetic wave propagation is provided.
Originality/value
The paper is a resource for developing efficient finite element software for high‐frequency electromagnetic field computation providing guaranteed error bound.
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Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the…
Abstract
Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.
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Gives a bibliographical review of the finite element meshing and remeshing from the theoretical as well as practical points of view. Topics such as adaptive techniques for meshing…
Abstract
Gives a bibliographical review of the finite element meshing and remeshing from the theoretical as well as practical points of view. Topics such as adaptive techniques for meshing and remeshing, parallel processing in the finite element modelling, etc. are also included. The bibliography at the end of this paper contains 1,727 references to papers, conference proceedings and theses/dissertations dealing with presented subjects that were published between 1990 and 2001.
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The geometric properties of finite element mesh are of great importance. The refinement algorithm leads often to very irregular elements. The problem of optimization of 3D‐meshes…
Abstract
The geometric properties of finite element mesh are of great importance. The refinement algorithm leads often to very irregular elements. The problem of optimization of 3D‐meshes is unsolved at this time. Only fragmentary methods are known. This paper presents a new concept of optimization algorithm and compares three different strategies for subdivision of tetrahedral elements.