Zhuoxiang Ren

The movement in field modeling

The different ways of considering the movement in field equations are first presented. Then the proposed methods and the accuracy of the obtained results are illustrated by simulation examples.

Comparison of different force calculation methods in 3D finite element modelling

Different methods for the local and total force calculation in 3D are examined. The accuracy and efficiency of these methods when applied to two dual formulations are compared through numerical examples. The local application of virtual work principle using edge elements permits calculation of local and total force in conductors as well as in magnetic materials. Concerning the equivalent sources methods (magnetising current method and magnetic charge method), they are theoretically equivalent to the Maxwell stress tensor method for total force calculation, while the numerical accuracy depends on the accuracy of the normal and tangential components of the field. >

Local force computation in deformable bodies using edge elements

A local force calculation method based on the virtual work principle and the use of the edge element is presented. In the edge elements, the magnetic coenergy or magnetic energy is expressed by the circulation of magnetic field along edges or by the magnetic flux density across facets according to the use of a magnetic formulation or an electric formulation. The magnetic force is obtained by the derivative of magnetic coenergy or energy with respect to configuration parameters while keeping the circulation of magnetic field or the magnetic flux constant. An eddy current problem is taken as a numerical example. The magnetic force is calculated by the present method and by the j*b method. Results are compared with analytical ones. It is shown that a better accuracy is obtained by the present method. >

Degenerated Whitney prism elements-general nodal and edge shell elements for field computation in thin structures

Nodal and edge shell elements are derived from a degeneration of Whitney prism elements. They can be easily applied to a variational formulation to solve thin structure problems. It is just needed to replace the volume integral by surface integral and the field discontinuity across the thin structure is correctly taken into account. A thin shell eddy current problem is solved using the present elements and dual formulations.

A new hybrid model using electric field formulation for 3-D eddy current problems

A hybrid finite-element-boundary integral method using an electric variational formulation (3-D code Trifou-e) is presented. Whitneys edge elements are used in conducting regions and the boundary element technique is used for exterior regions. The electric field is taken as the state variable for both of the regions, whether modeled by the finite-element or boundary-integral techniques, so that the problem of multiply connected regions can be treated in a convenient way. >

A generalized finite element model of magnetostriction phenomena

We present a generalized finite element model of magnetostriction phenomena where the direct and inverse effects are taken into account. The variational formulation in terms of magnetic vector potential and displacement is used to solve this coupled problem. Different computing steps in 3D and 2D cases are reported.

Calculation of mechanical deformation of magnetic materials in electromagnetic devices

A coupled model is developed to calculate mechanical deformation of magnetic materials in a magnetic field. The two equations governing magnetic and mechanical phenomena are solved simultaneously using the finite element method. The magnetic force distribution is calculated through a local application of the virtual work principle. The saturation of the material is considered. The nonlinear coupled system is solved by the Newton-Raphson method and the whole matrix is symmetrized. An example is given showing the effectiveness of the model. >

Computation of 3‐D electromagnetic field using differential forms based elements and dual formulations

The vector and scalar variables describing electromagnetic fields with different requirements of continuity can be identified to four different degrees of differential forms. The association of differential forms with finite elements leads to a set of differential forms based elements (Whitney elements); they are naturally adapted to the discretization of different vector and scalar variables. With the help of a Tonti diagram, Maxwell equations can be classified by two dual sequences together with the constitutive laws of materials. The application of Whitney elements to the two dual sequences leads to two dual approximation schemes. As an example, two dual formulations for eddy current computation using potential variables and the hybrid finite element—boundary element method are derived, where Whitney 3-D and 2-D elements are employed. A numerical application is given at the end of the paper, where the dual features of the two formulations are reported.

High order differential form-based elements for the computation of electromagnetic field

The Whitney elements, discrete spaces based on differential forms, have proven their efficiency in electromagnetic field computation. However, they are built only in first order. This paper gives a general description of high order p-form (nodal, edge, facet and volume) elements. Their function spaces and the assignment of degrees of freedom on the simplexes are analyzed. General expressions of the basis functions are given. A comparison of several 2nd order elements is carried out. A procedure for the generation of the hierarchical basis of p-form elements is provided.

A coupled electromagnetic-mechanical model for thin conductive plate deflection analysis

A thin conducting shell eddy current model and its coupling with a mechanical model for analyzing thin-plate elastic deflection due to the Laplace force are described. The eddy current model is based on the use of a boundary integral method in which the surface current density is introduced as a working variable. A triangular normal continuity edge element together with a spanning tree technique is used for numerical discretization. The mechanical model consists of the finite-element discretization of the thin-plate bending equation. The coupling between the eddy current model and the mechanical model is ensured by a time step-by-step procedure. A spherical shell eddy current problem is used to test the thin plate eddy current model. A TEAM Workshop benchmark problem (the beam deflection problem) is taken as a coupled eddy current-mechanical example. >