Subsections

• 2.1.1 Domain Discretization
• 2.1.2 Topological Equations
• 2.1.3 Constitutive Relation Discretization
  • 2.1.3.1 Field Function Reconstruction and Projection

2.1 Generic Discretization Concepts

This section gives an overview of the basic concepts for discretization schemes in scientific computing. The most important mechanisms are introduced and are then later converted into applicable and well known discretization schemes, such as the above mentioned finite volumes, finite elements, and finite differences. The purpose of this section is to unify different discretization schemes to a common kernel which is then directly converted to software design concepts. The concepts for discretization are mostly derived from an abstract reference discretization scheme [35], which is not a full discretization scheme by itself. Here only the necessary concepts for time-dependent electromagnetic problems which are discussed in subsequent chapters, are introduced, with some additional concepts from literature [32,24,33,36]

2.1.1 Domain Discretization

The simulation domain, either a simple space or a more complex space-time, is discretized by means of two dual oriented cell complexes for the primary and secondary mesh. The two cell complexes and therefore the primary and secondary mesh are not required to be physically or geometrically distinct. The logical separation is required to distinguish the different cochains/differential forms only. With dual cell complexes, the attribution of boundary conditions and the treatment of material discontinuities can be greatly simplified [36]. The use of non-dual cell complexes, e.g., operator adjointness, is more complex to obtain. The primary $p$ -cells are identified by $\ensuremath{\tau_{p}^{}}$ whereas the corresponding secondary $n-p$ -cells are identified by $\ensuremath{\tau_{n-p}^{}}$ and the default orientation, as given in Figure 2.1 for three dimensions.

Figure 2.1: Discretization concept for the primary and secondary cell complex with consistent orientation.

Time is also subdivided into two dual cell complexes. Here the 0-cells are time instants indexed with increasing time. The time interval between $t^n$ to $t^{n+1}$ is indexed as $t^{n+1/2}$ , similar to the formulation used by Yee (see Section 2.4.2). The secondary cells are given by $\tilde t_{1-p}^{n}$ and depicted in Figure 2.2.

Figure 2.2: Time stepping for the discretization concepts.

The time-dependent equation

$$\mathrm{curl} \left( \ensuremath{\mathbf{E}} \right) = - \partial_t \ensuremath{\mathbf{B}}$$ (2.2)

can then be interpreted as a higher dimensional object of the spatial domain $\times$ the time domain:

$$E(\partial A \times I) + B(A \times \partial I) = 0(A \times I)$$ (2.3)

A graphical representation is given in Figure 2.3 where two 0-cells are used for the corresponding time steps $t_0^n$ and $t_0^{n+1}$ and one 1-cell for the time interval $t_1^{n+1/2}$ . The corresponding quantity $B$ is attached to the time interval $t_1^{n+1/2}$ , whereas the quantity $E$ corresponds to the $t_0$ cells.

Figure 2.3: Space-time stepping for the discretization concepts.

This topological time stepping was proven [35] to be consistent with the conservation of charge as well as to preserve the absence of sources for the Faraday and Amperé laws. Based on this unique topological time stepping and on various discrete constitutive links, a variety of numerical time-stepping procedures, including explicit and implicit ones, can be derived.

2.1.2 Topological Equations

Balance or conservation equations can be written in local or global forms [24,33], e.g.:

$$\mathrm{div} \left( \ensuremath{\mathbf{D}} \right) = \varrho \quad \quad Q_{src}(V) = Q_{flow}(\partial V)$$ (2.4)

where the left equation represents a local or differential formulation and the right equation represents a global or discrete form. The discretization of the given relation means that the equation must be written for domains of finite extension. For topological equations this transition takes place without any error of approximation [33]. The balance equation by itself does not require uniform fields, homogeneous materials or any other conditions. Various laws which can be described by topological equations are intrinsically discrete. They state a balance of global quantities whose validity does not depend on metrical or material properties and is therefore invariant for very general transformations. This gives them topological significance and the name topological equation was used [33] to express the idea of invariance under arbitrary homeomorphic transformations. For a topological equation, the discrete or global version appears as the fundamental one, with the differential statement resulting from it if additional requirements are fulfilled.

2.1.3 Constitutive Relation Discretization

Discretizing constitutive relations determines the link between various cochains which represents the field quantities approximated by the local constitutive equations. This step of discretization offers a great number of different choices [35].

2.1.3.1 Field Function Reconstruction and Projection

For most of the finite volume and finite element schemes, a field projection is used by:

$$\ensuremath{\mathbf{B}} = f_\mu \ensuremath{\mathbf{H}}$$ (2.5)

In the discrete setting the field functions $\ensuremath{\mathbf{B}}$ and $\ensuremath{\mathbf{H}}$ do not belong to the problem's variable. Instead the magnetic flux cochain $\Phi^2$ and the magnetic field cochain $\tilde \Psi^2$ are linked by the relation $\Phi^2 = F_\mu (\tilde \Psi^2)$ .

From the cochain $\tilde \Psi^2$ a field function is derived by a reconstruction operator:

$$\ensuremath{\mathbf{H}} = R_2(\tilde \Psi^2)$$ (2.6)

Next, the local constitutive link, Equation 2.5, is used to derive $\ensuremath{\mathbf{B}}$ . Then, the cochain $\Phi^2$ has to be obtained by means of a projection operator $P^2$ , which produces a cochain for each field function $\ensuremath{\mathbf{B}}$ :

$$\Phi^2 = P^2(\ensuremath{\mathbf{B}})$$ (2.7)

The discrete constitutive link $F_\mu$ is then finally given by:

$$\Phi^2 = F_\mu (\tilde \Psi^2) = P^2 (f_\mu(R_2(\tilde \Psi^2)))$$ (2.8)

A natural requirement for the reconstruction and projection operators is that for each cochain $c^p$ the following relation holds true [35]:

$$P^*(R_*(c^p))=c^p$$ (2.9)

To obtain a sparse matrix representation the reconstruction process is usually performed locally, so the value of the reconstructed field function in a particular point depends only on the values of the original cochain on the cells in a sufficiently small neighborhood of the point.

During the last decade a comprehensive framework for the reconstruction and projection operators, called mimetic discretization [63,64,65,66,37,25], was derived. This framework analyzes in detail how these two operators can be used to mimic the analytical and continuous nature of partial differential equation.