A new set of basis functions for the discrete geometric approach
Introduction
Maxwell equations formulated in terms of partial differential equations are commonly and conveniently discretized by means of finite elements techniques which produce algebraic relations through a sound mathematical machinery based, for example, on the Galerkin’s method [2].
However, in the recent years, a less conventional method has gained interest within the computational electromagnetic community, developed by Yee [1] with a FDTD method, by Clemens and Weiland [3] with the Finite Integration Technique (FIT), by Tonti [5] with the cell method (CM), by Bossavit [6] with a reinterpretation of finite element method (FEM) and by present authors [7] with the Discrete Geometric Approach (DGA).
In the DGA approach emphasis is put on the geometric structure behind Maxwell’s equations. The physical laws of electromagnetism are recognized to be balance equations and they are exactly translated into algebraic relations involving circulations and fluxes (of the electromagnetic field quantities) associated with geometric elements (nodes, edges, faces and volumes) of a pair of interlocked grids (primal grid-dual grid). Discrete counterparts of the constitutive relations between field quantities are also introduced; they are approximate algebraic operators (matrices) which map circulations along edges of the primal grid onto fluxes through faces of the dual grid or viceversa, and involve the material properties and the metric notions related to the geometry of the grids.
As a known result [8], [9] in order to ensure the consistency and the stability of the overall final system of algebraic equations, the discrete constitutive relations are required to satisfy stability and consistency properties. The stability requirement prescribes that the constitutive matrices are symmetric and positive definite. The consistency requirement prescribes that constitutive matrices exactly map circulations onto fluxes or viceversa, at least for element-wise uniform fields.
Stable and consistent discrete constitutive equations can be naturally constructed for pairs of orthogonal Cartesian grids as shown by Yee [1] and Clemens and Weiland [3]. Besides, as shown by Bossavit [11], the mass matrices constructed by means of edge and face elements introduced by Whitney and generalized by Nedelec [18], [19], satisfy both the stability and consistency properties required by DGA [11], for pairs of grids in which the primal is composed of tetrahedra and the dual grid is obtained according to the barycentric subdivision of the primal. This result unfortunately does not hold in general for edge and face elements relative to different geometries. For instance, present authors have proven in [14] that Whitney’s elements for generic hexahedral primal grids do not satisfy the consistency property required by DGA, for any choice of the dual grid.
By the introduction of a novel set of edge and face vector functions combined with an energetic approach [12], the present authors were able in [7] to derive novel constitutive matrices satisfying both the consistency and stability properties required by DGA, not only for tetrahedra but also for (oblique) prisms with triangular base.
However, for pairs of grids where the primal grid is based on general polyhedra, useful in many applications, no constitutive matrices satisfying the consistency and stability properties required by DGA were reported in literature, as far as the authors know. It is here noted that some approaches can be found in literature for generating discrete counterparts of constitutive relations over polyhedral grids, such the mimetic finite differences [20], [21], [22] or the mixed finite elements [23]. However all these methods do not lead in general to discrete constitutive relations satisfying the consistency property required by DGA. The present authors did first attempts to fill in this gap with papers [13], [14].
The novelty content of this work is the introduction of four new general sets of vector functions for polyhedral primal grids associated with edges and faces of both the primal and of the dual grids. They are constructed directly in terms of the geometric elements (edges and faces) of the primal and of the dual grids. These vector functions are designed in such a way to comply with the requirements of the energetic approach introduced by the authors [12] for deriving discrete constitutive equations which ensure the consistency and stability properties required by DGA.
The functions here proposed belong to a class which, as recently shown by some of the present authors [24], theoretically ensures the convergence of the solution of discrete equations to the exact solution of the continuous problem. It is here noted that this class of functions, unlike Whitney’s and Nedelec’s basis functions, do not satisfy any curl-conforming or div-conforming properties. Thus in DGA, by using a pair of dual grids instead of a single grid, convergence can be guaranteed by using basis functions which do not satisfy all the regularity conditions of Whitney’s and Nedelec’s basis functions.
Numerical experiments will demonstrate that the novel discrete constitutive matrices can be computed easily and in a very efficient way leading to accurate approximations of the solution of a magnetostatic problem proposed as an example.
Section snippets
Pair of interlocked grids and geometric properties
Without losing generality, we will focus on a primal grid consisting of a single polyhedron v, Fig. 1.
The geometric elements of the primal grid are nodes, edges, faces and the volume v. We denote a primal edge with ei, where i = 1, … , L, L being the number of edges of v and a primal face with fj, where j = 1, … , F, F being the number of faces of v. The geometric entities of the primal grid like ei, fj are provided with an inner orientation [4], [10]; For example in Fig. 1 the arrows indicate a possible
Construction of the basis vector functions
We consider a vector field x(p) in v; For example, an electric field E or a current density vector J within the polyhedron. The integral quantityrepresents either a circulation or a flux of the field x(p) provided that the geometric entity ri is an edge or a face respectively, is often referred to as Degree of Freedom; Symbol “dr” stands for dl or ds according to a line or surface integration is performed respectively. For example, (9) yields the usual electro-motive force
Constitutive matrix
We consider a single polyhedron v, where a pair of vector fields x, y exists, related by a constitutive relationm being a double tensor, representing the material property, assumed to be symmetric positive definite and homogeneous in v.
Now, we focus on the pairs of geometric elements ri, , one dual of the other, with i = 1, … , R and we introduce the corresponding pair of Degrees of freedom , . We denote in boldface type the arrays Xr, , of dimension R, formed by
Numerical results
The proposed constitutive matrices can be conveniently used to solve various typologies of problems arising in computational physics. In order to test the proposed constitutive matrices, we focus on reference magnetostatic problems which are solved by using a pair of complementary geometric formulations. The two formulations are based on circulation on primal edges of a magnetic vector potential A and on a magnetic scalar potential Ω defined in primal nodes respectively; see for example [16]
Conclusion
New vector basis functions, which allow to construct stable and consistent discrete constitutive equations for the Discrete Geometric Approach, have been introduced. The computation of the resulting constitutive matrices is computationally efficient, being based on a closed-form geometric construction. The results obtained by such constitutive matrices, considering a magnetostatic benchmark problem, are in good agreement with the analytical solution.
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2019, Journal of Computational PhysicsCitation Excerpt :Consequently, considerable effort has been devoted to construct the constitutive matrices in order to mitigate the error [25,26]. For example, researchers have been concentrating on constitutive matrices that are constructed in different types of grid, e.g., tetrahedron [27–29], hexahedron [30], and polyhedron [31–35], etc. It has been demonstrated that highly accurate results can be obtained by increasing the grid density [31].