N.A. Golias

Constitutive inconsistency: rigorous solution of Maxwell equations based on a dual approach

A dual scheme is proposed, that correctly represents the electromagnetic fields as differential forms. A rigorous solution of Maxwells equations is obtained that satisfies both Amperes and Faradays laws. The solution of Maxwells equations derived from the numerical discretization of two complementary formulations is inconsistent with the constitutive laws. This inconsistency is used as an error estimator in a 3D adaptive refinement procedure resulting in very accurate solutions and reduced computational cost. A new error criterion, the dual constitutive error, consisting of two components: the electric error and the magnetic error, is introduced. Edge elements with tangential continuity are used giving no spurious solutions. The validity of the proposed technique is illustrated by an application to a loaded cavity. >

Magnetostatics with edge elements: a numerical investigation in the choice of the tree

On solving the Magnetostatic problem with edge elements a spanning tree technique must be employed so that uniqueness of the magnetic vector potential is ensured and solution is possible. It is shown that the choice of the tree affects very much the accuracy of the approximation. The use of an arbitrary tree results in poor convergence with reduced accuracy or even in inability to solve the problem. On the other hand employment of optimal tree structures results in very good convergence of the ICCG and increased accuracy. An algorithm for constructing nearly optimal tree structures is developed and applied in the solution of various problems. >

Three-dimensional automatic adaptive mesh generation

A technique for three-dimensional mesh refinement and adaptive mesh generation is presented. A procedure for refining three-dimensional tetrahedral meshes, based on Delaunay criteria, is developed. Additional nodes are inserted on an existing mesh and the tetrahedra produced are transformed, so that an optimum mesh is formed. Solution of a problem with an initial coarse mesh is followed by successive refinements. Furthermore, a posteriori error analysis is employed to estimate local errors and refine the mesh at those regions. A criterion of error estimation, the discontinuity of the normal field gradient on common interfaces, is proposed. Several examples using the proposed technique are presented to illustrate the usefulness of the method. >

Adaptive refinement strategies in three dimensions

3-D adaptive mesh refinement seems to be the answer towards full automation in the analysis of electromagnetic devices, as well as to controlling the computational borden that 3-D models present. The strategy employed in the self-adaptive refinement procedure can play a great role in the efficiency of the refinement process. In this paper some strategies for adaptive refinement are presented and discussed. Application to a test problem shows the effectiveness of the self-adaptive procedure.

3D eddy current computation with edge elements in terms of the electric intensity

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.

Adaptive methods in computational magnetics

A review of adaptive methods in 2-D and 3-D for the efficient solution of electromagnetic problems with the finite element method, developed by the authors, is presented in the paper. The adaptive methods presented consist mainly of two processes: mesh refinement and error estimation. A highly efficient technique for the refinement of arbitrary unstructured triangular and tetrahedral meshes, based on Delaunay triangulation, directly applicable and suitable in an automatic adaptive procedure, is described. The use of various error estimation criteria for electromagnetic problems is presented. The implementation of an automatic adaptive procedure with the finite element method, incorporating the above techniques, is provided and applied to the efficient solution of various 2-D and 3-D electromagnetic field problems.

Automatic finite element analysis: Application to the shielding by a spherical shell

ContentsAn automatic finite element analysis procedure is developed with the employment of a self-adaptive refinement technique. The process of mesh generation, being usually a highly interactive process, is transformed into a batch—oriented process. A characteristic of the algorithm is that the generated tetrahedra have a very high aspect ratio (nearly equilateral), while the presence of sliver elements is completely avoided. The generation of high quality tetrahedra makes possible the approximation of curved surfaces directly in the refinement process. The inconsistence of forms in the numerical presentation of the magnetic inductionB is used as an error estimator, so that high error regions are recognized and refined. The proposed procedure presents reduced computational requirements, making it very well suited for the solution of 3-D problems. Application to the shielding by a magnetic permeable spherical shell shows the ability to tackle problems with complicated geometry and solve them with high accuracy.ÜbersichtIn der vorliegenden Arbeit wird eine automatische Prozedur zur Berechnung mit finiten Elementen unter Anwendung einer adaptiven Netzgenerierung entwickelt. Der meistens stark interaktive Prozeß der Netzgenerierung wird automatisiert. Eine charakteristische Eigenschaft des Algorithmus ist, daß die Tetraeder ein gutes Qualitätsverhältnis (falls gleiche Seiten) aufweisen, degenerierte Elemente werden ganz vermieden. Die Generierung von Tetracdern hoher Qualität macht die Annäherung von gekrümmten Oberflächen direkt durch den Verfeinerungsprozeß möglich. Die Unstetigkeit der InduktionB wird zur Fehlerabschätzung benutzt. Dadurch werden Bereiche, die starke Fehler aufweisen, erkannt und weiter verfeinert. Der vorgeschlagene Prozeß betet eine Reduzierung der erforderlichen Computer-Ressourcen und ist gut für 3-D Probleme geeignet. Die Anwendung auf das Problem einer Abschirmung durch eine permeable Kugelschale zeigt die Fähigkeit zur Behandlung solcher komplizierten Geometrien mit hoher Genauigkeit.