Visualizing Edge-Conforming Discrete Field Quantities in Electromagnetic Field Problems with Interfaces

Finite element-based electromagnetic field simulation strongly benefits from using edge-conforming representations of the electric field. In this paper we address the visualization of discrete field data resulting from such simulations on 10-node quadratic tetrahedral grids. The use of higher-order grids enables, on the one hand, the accurate approximation of curved interfaces between electromagnetic materials, and on the other hand, it allows for a more accurate computation of derived quantities such as Coulomb forces and related surface tensions. However, a major drawback so far has been the lack of appropriate visualization techniques—common visualization systems do not support this type of data—necessitating a resampling step with all the involved drawbacks, including artifacts in the form of imposed continuity across material boundaries. We introduce a visualization framework implemented as a set of ParaView plugins that evaluates edge-conforming data by means of vector basis functions. Based on this framework we present different visualization approaches for the investigation of the electric field at material boundaries. We demonstrate their utility using electrohydrodynamics simulations of a water droplet on the surface of a high voltage insulator, representing a twophase flow problem driven by strong electric fields. Introduction The investigation of electromagnetic fields on material boundaries is of special interest in the development of structures that are exposed to strong electromagnetic fields. While the electric field at the interfaces is continuous tangential to the material surface, it commonly exhibits discontinuities in normal direction. These discontinuities pose problems in numerical simulation with finite element methods. The traditional approach, based on nodebased elements, imposes continuity at material boundaries due to the component-wise interpolation, resulting in an undesired smooth modeling of quantities across the interface. Therefore, edge-based representation of vector quantities was introduced in finite element methods. In the resulting approach, the simulated quantity is interpolated by means of vector shape functions that provide several characteristics important for the representation of the electric field at material boundaries. Since neighboring elements share tangential components on the edges, the tangential field is continuous across those elements. On the other hand, the shape functions cause the tangential component to vanish at the element faces opposite to an edge (see Figures 2a) and 2b) for an illustration), thus allowing for discontinuities in the normal component, which in turn enables correct representation of the electric field at interfaces. Additionally, the introduction of nonlinear elements, such as quadratic tetrahedra, enables a more accurate approximation of curved surfaces, which has particular importance for the computation of the electric field, as the sharp, piecewise linear representation would artificially amplify the electric field magnitude. Consequently, the nonlinear elements significantly reduce the required number of tetrahedra for the approximation of curved boundaries. Current visualization frameworks, however, do not allow for correct representation of the electric field resulting from edge-based elements—due to their component-wise interpolation they do not provide accurate representation, and more important, they miss field discontinuities across the interfaces. Thus, in this paper we present a framework that is able to correctly visualize the simulated field, also at material boundaries. The field is directly evaluated from the vector shape functions, taking higher-order elements (quadratic tetrahedra) into account, thus providing visualizations consistent with the simulation model. We exemplify the utility of our approach using electro-hydrodynamics simulations of a water droplet on the surface of a high voltage insulator. Related Work Edge elements were introduced into the finite element method in [10] and [3]. Bossavit [3] identified Whitney elements [20], which are widely used in finite element simulations, as a natural discretization method for eddy currents. The basis functions of edge elements were generalized to arbitrary order for the finite element method, and the resulting convergence was analyzed in [19]. The advantages and disadvantages of edge elements as well as their applications are discussed in [18] and [9]. Isoparametric elements described in [4] represent an alternative that allows for efficient and accurate higher-order computations. In the field of visualization, several techniques have been developed to address appropriate representation of higher-order elements. Wiley et al. [21] developed a technique for direct ray casting of curved 25 ILASS – Europe 2013 Paper title here 2 quadratic elements without prior tessellation into linear elements. For cell-based polynomial fields, isosurface extraction from higher-order finite elements was presented in [13], where adaptive mesh refinement is employed for accurate representation. A technique for isosurface extraction providing a trade-off between rendering speed and quality was suggested by Pagot et al. [12], based on a particle transport along the gradient field, and ray casting in the neighborhood of the final location of the particles. Schroeder et al. [15] addressed the complexity of higher-order basis functions from pand hp-adaptive methods by employing an automatic tessellation technique with recursive edge-based subdivision. Direct visualization of discontinuous Galerkin simulations was presented in [17], where an adaptive sampling technique is used for high quality volume rendering, and utilization of a GPU cluster allows for interactivity. Feature extraction from discontinuous Galerkin simulations based on the parallel vectors operator was proposed in [11]. A solution for the visualization of non-conforming meshes, based on point-based rendering, was developed by Zhou and Garland [22]. Isosurfaces from higherorder elements can be also visualized in a point-based manner [8], in which case the costly inverse mapping to evaluate the basis functions is avoided. Most recent works in the field of higher-order finite element visualization include a ray casting method with pre-computation of world-element space transformation [2], as well as ray casting with depth peeling [6]. Interface reconstruction for multiphase flow simulation was visualized in [5]. For visualization of electromagnetic fields, examples include the visualization of the field in Tokamak reactors [14], visualization of the coronal field [7], and topological analysis of magnetic fields [1]. It is worth noting that edge elements have not been introduced into visualization so far, and it is thus the aim of this paper to do so. Quadratic Tetrahedra Quadratic tetrahedra are particularly suitable for finite element-based simulations involving electromagnetic field problems with interfaces, since they allow for curved edges and faces. Considering that the elements can adapt to relatively strong deformations, mesh refinement can be avoided for many problems. This at the same time allows for a significant reduction of the required number of elements—curved parts of the simulation domain no longer require strong refinement for accurate computation of the electric field. These elements, however, are more difficult to implement, and the mesh formation takes significantly longer than for linear elements [23]. Quadratic tetrahedra have variable metric, i.e., the Jacobian determinant is not constant over tetrahedron. This further complicates computations, e.g., point location inside tetrahedra. A quadratic tetrahedron is illustrated in Figure 1d) (cf. Figure 1c) for comparison with a linear tetrahedron). The element is defined by 10 nodes, each carrying three components of the vector field, resulting in 30 degrees of freedom. The shape functions corresponding to the nodes of quadratic tetrahedra are defined as: ) 1 2 ( − = l l l L L N , for 4 ,..., 1 = l (1) 2 1 5 4 L L N = , 3 2 6 4 L L N = , 3 1 7 4 L L N = , 4 1 8 4 L L N = , 4 2 9 4 L L N = , 4 3 10 4 L L N = . (2) The world coordinates ( 1 x , 2 x , 3 x ) of a point inside a quadratic tetrahedron can be obtained from barycentric coordinates ( 1 L , 2 L , 3 L , 4 L ) using: l l i l i N x x , 10 1 =  = (3) Figure 1. Data representation on tetrahedral elements. a) Node-based data representation, b) edge-based data representation, c) linear tetrahedron, and d) quadratic tetrahedron with additional mid-edge points. a) b) c) d)