Analytical finite element matrix elements and global matrix assembly for hierarchical 3-D vector basis functions within the hybrid finite element boundary integral method

Abstract

Abstract. A hybrid higher-order finite element boundary integral (FE-BI) technique is discussed where the higher-order FE matrix elements are computed by a fully analytical procedure and where the gobal matrix assembly is organized by a self-identifying procedure of the local to global transformation. This assembly procedure applys to both, the FE part as well as the BI part of the algorithm. The geometry is meshed into three-dimensional tetrahedra as finite elements and nearly orthogonal hierarchical basis functions are employed. The boundary conditions are implemented in a strong sense such that the boundary values of the volume basis functions are directly utilized within the BI, either for the tangential electric and magnetic fields or for the asssociated equivalent surface current densities by applying a cross product with the unit surface normals. The self-identified method for the global matrix assembly automatically discerns the global order of the basis functions for generating the matrix elements. Higher order basis functions do need more unknowns for each single FE, however, fewer FEs are needed to achieve the same satisfiable accuracy. This improvement provides a lot more flexibility for meshing and allows the mesh size to raise up to λ/3. The performance of the implemented system is evaluated in terms of computation time, accuracy and memory occupation, where excellent results with respect to precision and computation times of large scale simulations are found.

Accuracy Analysis of Div-Conforming Hierarchical Higher-Order Discretization Schemes for the Magnetic Field Integral Equation

“…In this communication, the WF identity operator discretization scheme is introduced and investigated together with HO expansion functions up to order 1.5. In particalur, we demonstrate that a hierarchical HO basis [28]- [30] is only capable of curing the MFIE inaccuracies in part and additional measures seem to be required to boost the MFIE accuracy to really good levels. Preliminary results regarding the accurate full first-order discretization of the MFIE only are found in [34].…”

“…Subsets of the div-conforming hierarchical HO functions complete up to 𝑝th polynomial order of the expansion functions are described as follows [28]- [30]. The LO part, also called firstorder "quasi"-gradient space, contains just the RWG functions and exhibits 𝑝 = 0.5 order.…”

“…The full-first order functions with 𝑝 = 1 supplement the first-order "quasi"-gradient space by a first-order rotational space. Furthermore, we consider the case 𝑝 = 1.5, which includes the second-order "quasi"-gradient space in addition [28]- [30]. Examples of the surface current distributions on (pairs of, where appropriate) triangular facets are depicted in Fig.…”

Improved Discretization of the Full First-Order Magnetic Field Integral Equation

“…The 𝜷 expansion functions of the electric currents may be just 𝑁 RWG functions defined on all neighboring pairs of triangular mesh facets. For a full first-order discretization, we split these functions into an RWG part with 𝑁/2 functions 𝜷 LO and the complementary extension to full first order [20], [25] with 𝑁/2 functions 𝜷 HO . The following investigations are based on this hierarchical HO approach.…”

“…In this paper, we show that -unlike sometimes claimed in literature -higher-order functions may not fully solve the MFIE accuracy issues. To do so, we consider a full first-order discretization which was considered in [20], [24], [25] and should basically be similar to other full-first-order approaches involving Trintinalia-Ling functions [26]. In order to arrive at an accurate discretization, it may still be necessary to cure the RWG part of the identity operator discretization just as this is necessary for the LO discretization alone.…”