A general computer aided description environment for the treatment of discrete problems, based on a concise structure for both development and application levels, is described and applied to the finite element method. Its characteristics reveal its great ability to welcome in an evolutive way a wide range of physical models and numerical methods, with any coupling of them. It is therefore adapted to various activities, such as research, collaboration, education, training and industrial studies.

A general environment for the treatment of discrete problems and its application to the finite element method

We present a finite-element analysis of a diffraction problem involving a coated cylinder enabling the electromagnetic cloaking of a lossy object with sharp wedges located within its core. The coating consists of a heterogeneous anisotropic material deduced from a geometrical transformation as first proposed by Pendry [Science 312, 1780 (2006)]. We analyze the electromagnetic response of the cloak in the presence of an electric line source in p polarization and a loop of magnetic current in s polarization. We find that the electromagnetic field radiated by such a source located a fraction of a wavelength from the cloak is perturbed by less than 1%. When the source lies in the coating, it seems to radiate from a shifted location.

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Electromagnetic analysis of cylindrical invisibility cloaks and the mirage effect

We consider H (curl ;Ω)-elliptic problems that have been discretized by means of Nedelec's edge elements on tetrahedral meshes. Such problems occur in the numerical computation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.

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Residual based a posteriori error estimators for eddy current computation

Publisher Summary This chapter presents a critical survey on the approach to the solution of the general three-dimensional eddy current problem, which originally started from the necessity of studying the transient electromagnetic phenomena in tokamaks. The mathematical models are described, including the assumptions made on the field equations and the material properties. An excursion from full Maxwell equations to the stationary and quasistationary fields is briefly presented, with focus on the magneto-quasistationary model—that is, the eddy current problem. The chapter also presents vector and scalar potentials along with the corresponding gauge conditions. The emphasis is put on the benefits offered by the use of the vector potentials with the classic Coulomb and Lorentz gauges in dealing with material interfaces and the unified treatment of magnetostatic and eddy current problem. Other vector potentials such as, two-component vector potential are also discussed. The main features of the edge elements, whose degrees of freedom are associated with the tangential components or the line integrals of the vector field along the edges, are also covered. They give rise to a set of vector shape functions, for which the continuity of the tangential components is preserved, allowing for the discontinuity of the normal component between adjacent elements.

Finite Element Methods for the Solution of 3D Eddy Current Problems

Presents a prototype microrobot based on magnetic principles. Miniature items are to be transported and assembled in hazardous environments. A microrobot can be remotely operated with 3 DOF in an enclosed environment by transferring magnetic energy and optical signals from outside. The magnetic drive unit consists of 8 electromagnets (4 pairs), 2 permanent magnets, a return yoke and a pole piece. The microrobot is manipulated under the pole piece by regulating magnetic field. It consists of a magnetic head, a body (electronic circuit and batteries), and copper alloy ribbon ringers. A shape memory alloy actuator activates the fingers by illuminating/extinguishing several LED. PID controls were applied. To cope with uncertainties and variations in payload masses, an adaptive control law was also employed for positioning along the z axis to enable the controller parameters to be adjusted in real-time. Effectiveness of the control was verified by the results of several experiments. The microrobot has a net mass of 8.1 g and it can elevate and manipulate objects with masses up to 1.5 g within a volume of 29/spl times/29/spl times/26 mm/sup 3/ with a precision of 0.05 mm.

Design and control of a microrobotic system using magnetic levitation

A generalization of the Whitney complex is proposed, which is not now associated with simplices, i.e. tetrahedra in three dimensions, but with collections of three kinds of geometric elements: tetrahedra, hexahedra and prisms. Nodal, edge, facet and volume finite elements, i.e. mixed elements, associated with collections of those geometric elements, are defined. Base functions for approximation relative to these finite elements are defined and their properties are established. A geometric interpretation of these functions is given. >

Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms

The representation of electromagnetic quantities by differential forms allows the use of nonorthogonal coordinate systems. A judicious choice of coordinate system facilitates the finite element modeling of infinite or very thin domains.

Transformation methods in computational electromagnetism

A general method to compute source fields in magnetostatics or magnetodynamics is presented for inductors of any shape. That source field is not the physical one because the zero divergence condition is not satisfied. However, the freedom so obtained is exploited to minimize its support as well as to reduce the CPU time. The use of edge finite elements enables its rigorous construction. A test problem illustrates the method.

A generalized source magnetic field calculation method for inductors of any shape

A sequence of finite element spaces built on tetrahedra, hexahedra and prisms, is presented, and gauge condition for vector potential is shown to be well defined in these spaces. Results are presented for a modified vector potentials formulation for 3D eddy current problems. >

A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials

This paper presents a numerical modelling of an electromechanical relay connected with an electric excitation circuit. This transient modelling not only takes into account the classical electromagnetic equations of the device but also the movement and circuit equations. The use of the finite element-boundary element coupling method facilitates the computation of the movement while the actual coupling with circuit equations is necessary for an accurate and reliable representation of transient phenomena. >

A. Genon

Papers

Mixed finite elements associated with a collection of tetrahedra, hexahedra and prisms

Transformation methods in computational electromagnetism

A generalized source magnetic field calculation method for inductors of any shape

A discrete sequence associated with mixed finite elements and its gauge condition for vector potentials

Modelling of electromechanical relays taking into account movement and electric circuits