Isogeometric methods for computational electromagnetics: B-spline and T-spline discretizations

Abstract

In this paper we introduce methods for electromagnetic wave propagation, based on splines and on T-splines. We define spline spaces which form a De Rham complex and, following the isogeometric paradigm, we map them on domains which are (piecewise) spline or NURBS geometries. We analyse their geometric structure, as related to the connectivity of the underlying mesh, and we give a physical interpretation of the fields degrees-of-freedom through the concept of control fields. The theory is then extended to the case of meshes with T-junctions, leveraging on the recent theory of T-splines. The use of T-splines enhance our spline methods with local refinement capability and numerical tests show the efficiency and the accuracy of the techniques we propose.

An Introduction to the Numerical Analysis of Isogeometric Methods

“…This section is devoted to the definition and the analysis of isogeometric methods for the approximation of vector fields, for which we will mainly follow the two papers [15] and [16]. We focus our attention on the construction of the so-called spline complex, i.e., spline approximation spaces for the De Rham diagram.…”

“…In the paper [16], the authors have introduced the concept of control field, which are Whitney forms defined on the control mesh. In view of the properties listed above, of Proposition 5.1, and the properties of the pull-back operators, the Whitney vector fields can be interpreted as control fields, i.e., exactly in the spirit of control points, fields which "carry" the degrees of freedom for the spline complex.…”

Shape Error Analysis of Functional Surface Based on Isogeometrical Approach

“…The necessary continuities between elements, C 1 continuity for Kirchhoff-Love element as an example, can be easily achieved by the high-order geometric basis functions. The method is now further developed in many areas including structural analysis [19][20][21], fluid-structure interaction [22], shape optimization [23,24], topology optimization [25,26], electromagnetic analysis [27], etc.. New IGA elements [28,29] are developed by researches. However, the non-interpolatory nature of the geometric basis functions makes the imposition of even the simple boundary conditions more difficult.…”

Characterization of analysis-suitable T-splines

“…For example, the isogeometric vector fields proposed in [7] for electromagnetic problems are based on AST-splines and their properties depends on the characterization. Furthermore, a complete theory of approximation properties of T-spline spaces in the context if isogeometric analysis will benefit from the characterization proposed here, see [3].…”