The Mortar Edge Element Method in Three Dimensions: Application to Magnetostatics

Abstract

The subject presented in this paper concerns the approximation of a magnetostatic problem, where the primary unknown is the magnetic vector potential in a three-dimensional system composed of two solid parts separated by an interface. The approximation of the problem here is based on the mortar element method combined with edge elements on tetrahedral meshes that do not match at the interface. In this paper we describe the proposed method and its implementation aspects together with some numerical results that illustrate how the method works.Introduction. The flexibility and performance of a numerical method for the simulation of electromagnetic field distributions in electrotechnical devices relies, in several cases, on the possibility of working with nonmatching grids at the interface between adjacent subdomains. One example is given by the treatment of moving structures. We can choose to work in Eulerian coordinates, adding a convective term in the equations, or in Lagrangian coordinates, dealing with nonconforming discretizations. The second choice can be computationally more convenient whenever it is possible to avoid remeshing or interpolation procedures. Another example is the optimization of the structure shape. We can remesh either the whole domain or only a region containing the shape to be optimized: in the second case it can be useful to work with nonmatching grids to simplify the local remeshing task and successive solution of the problem. A third example consists of the possibility of coupling variational methods of different orders and types.In this paper we deal with the edge element approximation on nonmatching grids of a given magnetostatic problem in three dimensions. We will focus our attention on the description of the magnetic field distribution in a domain composed of two solid parts separated by an interface. We aim at describing the three-dimensional mortar edge element method and its main implementation aspects.The general problem of dealing with nonmatching grids in magneto-mechanical applications has been faced for a long time (see [15,16,17,20,24,29,31,32] for the main arguments that have been proposed since 1980 and [25] for further references on the subject). None of the existing methods is simultaneously characterized by involving positive definite matrices, avoiding remeshing and difficult intersection

Eddy-current calculations in three-dimensional moving structures

“…As now a standard procedure for nonconforming domain decomposition methods, the mortar element method leads to impose the transmission condition in a weak form by means of a suitable space of Lagrange multipliers . Once discretized, both domains by two independently created meshes of tetrahedra ( is the maximum size of all mesh tetrahedra), we introduce two edge element spaces as in [4] and the space of tangential components on of functions in . We choose as a proper subset of , such that as in [4].…”

“…Once discretized, both domains by two independently created meshes of tetrahedra ( is the maximum size of all mesh tetrahedra), we introduce two edge element spaces as in [4] and the space of tangential components on of functions in . We choose as a proper subset of , such that as in [4]. The definition of involving the edge element space gives a different but similar mortar method (in the mortar terminology, this means that we choose as the master and as the slave).…”

“…, find such that (4) To write the matrix form of the fully discretized problem at time , we need to construct a basis of the approximation space . The elements of the chosen basis are built from the edge element basis functions and, at the edges lying on , they are linked through the matching condition stated in .…”

“…The problem of dealing with nonmatching grids has been faced for a long time (see [6] for a short overview on the subject) and the existing methods are difficult to be applied for 3-D modeling. The mortar element method (see [2] for its mathematical analysis in the Maxwell's equations framework and [4] for its first application to magnetostatics in 3-D) is a nonconforming nonoverlapping domain decomposition technique which allows for independent meshes in adjacent subdomains. The idea of the method is to weakly impose the transmission conditions at the interfaces by means of Lagrange multipliers suitably chosen to ensure optimal properties on the discrete problem.…”

The three‐field formulation for elasticity problems

“…Here again, if one wants to avoid a local remeshing near the transition interface, possibly having to use rather distorted elements, the problem of non matching grids pops out again. There are obviously other applications where you cannot avoid non matching grids: for instance when studying sliding pieces [14], or contact problems [25], or several other types of problems (see e.g. [3], [34], [32], [39], [33] and the references therein).…”