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# Simulation of floating potentials in industrial applications by boundary element methods

- Dominic Amann
^{1}, - Andreas Blaszczyk
^{2}, - Günther Of
^{1}Email author and - Olaf Steinbach
^{1}

**4**:13

https://doi.org/10.1186/2190-5983-4-13

© Amann et al.; licensee Springer. 2014

**Received:**17 April 2014**Accepted:**12 October 2014**Published:**30 October 2014

## Abstract

We consider the electrostatic field computations with floating potentials in a multi-dielectric setting. A floating potential is an unknown equipotential value associated with an isolated perfect electric conductor, where the flux through the surface is zero. The floating potentials can be integrated into the formulations directly or can be approximated by a dielectric medium with high permittivity. We apply boundary integral equations for the solution of the electrostatic field problem. In particular, an indirect single layer potential ansatz and a direct formulation based on the Steklov-Poincaré interface equation are considered. All these approaches are discussed in detail and compared for several examples including some industrial applications. In particular, we will demonstrate that the formulations involving constraints are vastly superior to the penalized formulations with high permittivity, which are widely used in practice.

**MSC:** 65N38, 78A30.

## Keywords

- electrostatic field computations
- boundary element methods
- industrial applications

## 1 Introduction

For the solution of 3D electrostatic field problems, boundary element methods are widely used and are, in particular, advantageous in the presence of an unbounded domain. In addition to Dirichlet and Neumann boundary conditions, so-called floating potentials may occur. Isolated perfect electric conductors result in equipotential surfaces. The equipotential value of the surface is unknown, but, in addition, the flux through the closed surface must be zero in the absence of sources. Such floating electrodes are found in, e.g., some lightning protection systems and can modify the breakdown probability of air gaps [1].

While there are numerous papers on the solution of electrostatic field problems with floating potentials by boundary element methods and related methods, like the charge simulation method [2, 3], in the engineering literature, a detailed view based on a mathematically profound basis seems to be missing. In this paper, we try to close this gap and, in addition, compare several formulations in practical examples. In particular, we consider boundary element methods, see, e.g., [4–6], for the solution of electrostatic field problems with floating potentials in a multi-dielectric setting. We apply an indirect approach based on the single layer potential and a domain decomposition method based on symmetric approximations of the local Dirichlet to Neumann maps, the so-called Steklov-Poincaré operators, see e.g. [7–10]. These two methods have been compared for magnetostatic problems in [11, 12]. Here, we apply these formulations for electrodes at floating potentials.

As electrodes at floating potentials can be considered equivalent to dielectric bodies with infinite permittivity, a common strategy is to substitute electrodes at floating potential by a dielectric media with high permittivity, see e.g. [13]. This approximation can be interpreted as a penalty approach. We compare this strategy to the direct incorporation of the constant but unknown potential and of the zero flux constraint into the formulations. While the penalty approach of a dielectric media with high permittivity needs no additional implementation work in a code which can cope with jumping permittivities, the approach with constraints is highly preferable from a mathematical point of view. Our numerical examples will demonstrate that the results of the approach with constraints are superior to the ones of the penalty approach. In addition, we will show that the formulations based on the Steklov-Poincaré interface equation have advantages over the single layer potential ansatz in case of corners and edges.

The paper is organized as follows: A model problem of the electrostatic field computation with floating potential is introduced in Section 2. In Section 3, the formulations based on the Steklov-Poincaré interface equation and the single layer potential ansatz are presented, and the unique solvability of the variational formulations is proven. The boundary element discretization of both formulations is described in Section 4, and first academic examples in Section 5 show the advantages and disadvantages of the considered approaches. Finally we discuss several extensions of the model problem and apply the methods to examples of industrial applications like an arrester, a bushing, and an insulator with partial wetting in Section 6.

## 2 Floating potentials in electrostatic field problems

*φ*such that ${\phi}_{D}={\phi}_{|{\mathrm{\Omega}}_{D}}$, ${\phi}_{0}={\phi}_{|{\mathrm{\Omega}}_{0}^{c}}$, and a constant $\alpha ={\phi}_{|{\mathrm{\Omega}}_{F}}$ are the solution of

Here ${n}_{i}$ denotes the exterior unit normal vector on ${\mathrm{\Gamma}}_{i}:=\partial {\mathrm{\Omega}}_{i}$, $i\in \{0,D,E,F\}$, and is defined almost everywhere. On the surface ${\mathrm{\Gamma}}_{E}$ of the electrode a constant potential *g* is given in (3), while we enforce continuity of the potential as well as of the flux for the dielectrics by (4) and (5). In addition, we introduce Γ as the union of all boundaries. For our simple model problem $\mathrm{\Gamma}={\mathrm{\Gamma}}_{0}$. Note that we distinguish ${\mathrm{\Gamma}}_{0}$ and Γ from the beginning, such that the developed boundary integral equations can be applied to more general settings which are discussed in Section 6. The dielectric domain is characterized by its relative permittivity ${\epsilon}_{D}$ and the exterior domain ${\mathrm{\Omega}}_{0}^{c}$ by ${\epsilon}_{0}$. For the floating potential, we assume a constant but unknown potential *α* on the boundary ${\mathrm{\Gamma}}_{F}$ in (7), but the total flux through this surface is zero, see (8).

*α*and the constraint (8) directly. The second approach, which is widely used in practice due to its simple implementation, is to approximate the floating potential by considering ${\mathrm{\Omega}}_{F}$ to be a dielectric medium with high relative permittivity ${\epsilon}_{F}$, i.e., to determine a potential ${\phi}_{F}$ instead of

*α*. In this case we end up with a system consisting of (1)-(6) with additional transmission conditions on ${\mathrm{\Gamma}}_{F}$:

We will demonstrate by some numerical examples in Section 5 that the penalty approach of an additional dielectric medium with a high relative permittivity gives bad approximations of the floating potential in general.

## 3 Boundary integral equations

If we use ${\mathrm{\Gamma}}_{C}=\mathrm{\Gamma}\setminus {\mathrm{\Gamma}}_{E}={\mathrm{\Gamma}}_{D}\cup {\mathrm{\Gamma}}_{F}$ instead of ${\mathrm{\Gamma}}_{D}$ in (4) and (5) and set the permittivities *ε* correctly, the model with a high relative permittivity ${\epsilon}_{F}$ is the special case (1)-(6) of the full model (1)-(8) with a floating potential. Thus we will derive the boundary integral equations for the full model only. For the model with a high relative permittivity we just need to drop the boundary integral equations related to ${\mathrm{\Gamma}}_{F}$ and take into account the ones of ${\mathrm{\Gamma}}_{D}$ for ${\mathrm{\Gamma}}_{F}$ in addition.

We consider an approach which is based on the Steklov-Poincaré interface equation known from domain decomposition methods, see e.g. [14, 15], and an indirect ansatz leading to a single layer boundary integral equation. While the latter approach is popular due to the ease of implementation, the domain decomposition approach will result in better approximations for the examples in Sections 5 and 6.

### 3.1 Steklov-Poincaré interface equation

Thus we need to determine the unknown parts of the Cauchy data $[{t}_{i},{\phi}_{i}]$, $i\in \{0,D\}$.

### 3.2 Single layer boundary integral operator formulation

*φ*of the boundary value problem (1)-(8). With this choice the local partial differential equations (1) and (2), the continuity condition (4) as well as the radiation condition (6) are satisfied. The remaining Dirichlet boundary condition (3), the floating potential condition (7), the flux transmission condition (5), and the scaling condition (8) provide the equations to determine the unknown density $w\in {H}^{-1/2}(\mathrm{\Gamma})$:

*V*denotes the global single layer boundary integral operator and ${K}^{\prime}$ is the global adjoint double layer boundary integral operator for $x\in \mathrm{\Gamma}$:

### 3.3 Unique solvability

**Lemma 3.1** *There exists a unique solution* $({\phi}_{D},\alpha )\in {H}^{1/2}({\mathrm{\Gamma}}_{D})\times \mathbb{R}$ *satisfying* (11)-(12).

*Proof*Using the splitting $\phi ={\phi}_{D}+\alpha {1}_{F}+g{1}_{E}$, we can reformulate (11)-(12) as: Find $\phi \in X:=\{\psi \in {H}^{1/2}(\mathrm{\Gamma}):{\psi}_{|{\mathrm{\Gamma}}_{F}}=\alpha ,\alpha \in \mathbb{R},{\psi}_{|{\mathrm{\Gamma}}_{E}}=0\}$:

This variational formulation admits a unique solution, as $X\subset {H}^{1/2}(\mathrm{\Gamma})$, the exterior Steklov-Poincaré operator ${S}^{0}$ is ${H}^{1/2}(\mathrm{\Gamma})$-elliptic, and the interior Steklov-Poincaré operator ${S}^{D}$ is ${H}^{1/2}({\mathrm{\Gamma}}_{D})$-semi-elliptic, see, e.g., [[14], Lemma 1.83, p. 49f] and [[6], Section 6.6.3, p.149]. □

**Lemma 3.2**

*Let*$({\phi}_{D},\alpha )\in {H}^{1/2}({\mathrm{\Gamma}}_{D})\times \mathbb{R}$

*be a solution of the Steklov*-

*Poincaré interface equations*(11)-(12),

*and let*$w\in {H}^{-1/2}(\mathrm{\Gamma})$

*be a solution of the indirect single layer approach*(13)-(16).

*Then there holds the relation*

*Proof*Obviously, the statement holds true for all $x\in {\mathrm{\Gamma}}_{E}$, as the condition (13) for the single layer potential approach coincide with the choice of

*φ*for the Steklov-Poincaré interface equation. On ${\mathrm{\Gamma}}_{F}$ the assertion holds true due to (14). On ${\mathrm{\Gamma}}_{D}$ we start from the continuity (15) of the flux for the single layer potential approach, use $w={V}^{-1}Vw$ and the symmetry relation ${K}^{\prime}{V}^{-1}={V}^{-1}K$, see e.g. [[6], Corollary 6.19, p.138],

Taking into account the floating potential (14), these two equations coincide with the formulation (11)-(12) of the Steklov-Poincaré interface equation, and hence we conclude $\phi =Vw$ on Γ. □

Due to equivalence of the two formulations we conclude the unique solvability of the indirect approach (13)-(16) from Lemma 3.1.

## 4 Boundary element methods

For the discretization of the considered boundary integral formulations, we assume a quasi-uniform mesh of the surface Γ with *N* plane triangles and *M* nodes. The considered trial and ansatz spaces are the space ${S}_{h}^{0}(\mathrm{\Gamma})=span{\{{\psi}_{\ell}^{0}\}}_{\ell =1}^{N}$ of piecewise constant functions and the space ${S}_{h}^{1}(\mathrm{\Gamma})=span{\{{\psi}_{\ell}^{1}\}}_{\ell =1}^{M}$ of piecewise linear and continuous functions. We use Galerkin variational formulations for the discretization of the domain decomposition method (11)-(12) and of the single layer boundary integral equations (13)-(16).

### 4.1 Steklov-Poincaré interface equation

We transfer the splitting $\phi ={\phi}_{D}+\alpha {1}_{F}+g{1}_{E}$ of the solution of (11)-(12) to the Steklov-Poincaré operators such that ${S}_{ij}^{0}$ indicates that the operator ${S}^{0}$ is applied to a function defined on ${\mathrm{\Gamma}}_{j}$ only and evaluated on ${\mathrm{\Gamma}}_{i}$ for $i,j\in \{D,E,F\}$.

Due to the positive semi-definiteness of ${S}_{h}^{D}$ and the positive definiteness of ${S}_{h}^{0}$ the linear system (17) is uniquely solvable, see Lemma 3.1.

The unique solvability of both discrete formulations (17) and (18) is a consequence of the positive definiteness of the approximation ${S}_{h}^{0}$ of the exterior Steklov-Poincaré operator ${S}^{0}$ and the positive semi-definiteness of the approximation ${S}_{h}^{i}$ of the other Steklov-Poincaré operator ${S}^{i}$ ($i\in \{D,F\}$), see, e.g., [[14], Lemma 1.93, p.58f] and [[6], Lemma 12.11, p.285].

### 4.2 Single layer boundary integral operator formulation

for $i,j\in \{D,E,F\}$, $m,n=1,\dots ,{N}_{i}$ or ${N}_{j}$.

To our best knowledge, the stability of these indirect boundary element formulations is still an open problem for general Lipschitz surfaces due to the inconsistent, but widely used discretization of the adjoint double layer potential in ${L}_{2}(\mathrm{\Gamma})$.

## 5 Numerical examples

In this section, we consider a few rather academic examples to compare the introduced approaches to solve the electrostatic potential problem (1)-(8). We compare four formulations in total. We apply the Steklov-Poincaré (SP) operator formulation (18) and the indirect single layer potential (SL) ansatz (21) for the full dielectric approach (full dielectric) with a high relative permittivity ${\epsilon}_{F}=10\text{,}000$ to approximate the floating potential. For the direct incorporation (floating) of the floating potential we solve the Steklov-Poincaré (SP) system (17) and the indirect single layer potential (SL) ansatz (20), respectively.

For the computations, we used an implementation [17] of the proposed boundary element methods which is based on the Fast Multipole Method [18] for fast and data-sparse realizations of the involved boundary integral operators. The Steklov-Poincaré operator formulation is implemented by means of MPI and we used one process per active subdomain, i.e. two processes for (17) and three processes for (18). The implementation of the Fast Multipole Method utilizes OpenMP and we used two threads for each instance of the program. The computations were done on a Workstation with 2 Intel Xeon E5620 processors and 24 GB RAM.

We use the concept of operators of opposite order [19] for the preconditioning of the Steklov-Poincaré operator formulations (17) and (18). We apply the artificial multilevel preconditioning [20, 21] for the inner inversion of the Galerkin matrix of the single layer boundary integral operator in the Steklov-Poincaré operator formulations. For the systems (20) and (21) of the indirect single layer potential ansatz, we use the artificial multilevel preconditioning for the block of the single layer boundary integral operator and a diagonal scaling for the block of the adjoint double layer potential operator.

### 5.1 Two spheres

The two spheres of our first example [22] have the same diameter which is twice the distance of the two spheres. The first sphere ${\mathrm{\Omega}}_{E}$ is an electrode with a given potential of $\phi =g=100$ on its surface. The second sphere ${\mathrm{\Omega}}_{F}$ is either a floating potential or a dielectric with relative permittivity ${\epsilon}_{F}=10\text{,}000$, depending on the considered approach. The surrounding air has the relative permittivity ${\epsilon}_{0}=1$.

*α*and the computational times of the four formulations for several refinement levels. For this setting an approximate solution of an axial symmetric charge simulation solver (ELFI, [23]) is used for comparison. One purpose of the new solvers is to overcome the restrictions of ELFI to axial symmetric geometries. For the full dielectric approach, we do not determine

*α*directly but provide the mean value of the potential on ${\mathrm{\Gamma}}_{F}$. Even on the finest refinement level the potential ${\phi}_{F}$ is not constant, it has a range of 0.1983 for the indirect approach and 0.0209 for the Steklov-Poincaré operator formulation.

**Approximate values of the floating potential**
α
**on**
${\mathbf{\Gamma}}_{\mathit{F}}$
**and computational times for the example of two spheres**

Number of elements | 256 | 1,040 | 4,160 | |||
---|---|---|---|---|---|---|

SP floating | 32.41 | 1 s | 33.63 | 8 s | 33.86 | 35 s |

SL floating | 32.41 | 1 s | 33.63 | 4 s | 33.86 | 20 s |

SP full dielectric | 32.40 | 2 s | 33.62 | 12 s | 33.85 | 50 s |

SL full dielectric | 32.39 | 1 s | 33.62 | 5 s | 33.86 | 28 s |

2D ELFI | 33.9 |

We notice that all four formulations result in good and similar approximate solutions. Only the indirect single layer formulation (21) for the full dielectric model gives a potential which is not quite constant although we consider approximations of smooth objects. We will encounter this behavior to a greater extend in the next example.

### 5.2 Sphere and bicone

*α*and the computational times. Again an approximate solution of the potential on the surface ${\mathrm{\Gamma}}_{F}$ of the cone by an axial symmetric FEM solver is used for comparison.

**Approximate values of the floating potential**
α
**on**
${\mathbf{\Gamma}}_{\mathit{F}}$
**and computational times for the example of a sphere and a bicone**

Number of elements | 384 | 1,536 | 6,144 | 24,576 | ||||
---|---|---|---|---|---|---|---|---|

SP floating | 44.512 | 2 s | 45.339 | 15 s | 45.572 | 79 s | 45.637 | 329 s |

SL floating | 44.512 | 2 s | 45.341 | 7 s | 45.573 | 28 s | 45.637 | 124 s |

SP full dielectric | 44.512 | 3 s | 45.340 | 27 s | 45.573 | 106 s | 45.636 | 569 s |

SL full dielectric | 44.433 | 2 s | 45.355 | 11 s | 45.553 | 34 s | 45.632 | 168 s |

2D ELFI | 45.7 |

**Range of the floating potential**
${\mathit{\phi}}_{\mathit{F}}$
**for the sphere and the bicone**

Number of elements | 384 | 1,536 | 6,144 | 24,576 |
---|---|---|---|---|

SP full dielectric | 44.51-44.53 | 45.33-45.36 | 45.55-45.59 | 45.62-45.65 |

SL full dielectric | 40.38-61.91 | 42.34-58.26 | 43.34-54.93 | 44.02-52.41 |

## 6 Extensions and applications

We observed poor approximations by the single layer potential ansatz for large jumps in the permittivities *ε* in the example of the sphere and the bicone due to artificial singularities in the discrete solution, see the formulation ‘SL full dielectric’ in Table 3. For such simple examples the single layer potential ansatz with direct realization of the floating potential (‘SL floating’) gives still good results. But for examples with large jumps of the coefficients ${\epsilon}_{D}$ and ${\epsilon}_{0}$ this is not the case anymore. As the third row in (20) is similar to the second and third row in (21), we are facing the same problem as for the formulation ‘SL full dielectric’ in Table 3. We observed these problems with artificial singularities in the discrete solution in the presence of dielectric media already for relative permittivity ${\epsilon}_{D}$ of 800 and higher, see e.g. [11, 12].

But for more general examples we have to cope with such jumps in the relative permittivities. In such cases, the approximation error of the single layer approach is more than one order of magnitude larger than the one of the Steklov-Poincaré operator formulations for the same mesh, see [11, 12]. Thus one needs significantly finer meshes for the single layer potential ansatz to come up with the same accuracy for this class of problems. This results in larger computational times than for the Steklov-Poincaré operator formulations. Due to these significant drawbacks of the single layer potential ansatz, we will consider the Steklov-Poincaré operator formulations only.

For real world examples, we need to consider more general settings. For the ease of presentation we have restricted the description of the formulations to one representative of each kind of subdomains and to well separated subdomains. We will now comment on some extensions.

have to be incorporated.

If two subdomains are in contact, we have to make some additional modifications. If a dielectric subdomain is in contact with an electrode, we use a discrete extension $\tilde{g}$ of the given potential *g* to the surface ${\mathrm{\Gamma}}_{D}$ of the dielectric and determine the unknown remainder ${\phi}_{D}-\tilde{g}$ of ${\phi}_{D}$. If the floating potential is surrounded by a dielectric medium instead of the exterior air domain the vector $\underline{a}$ and the coefficient *λ* in (17) involve ${\epsilon}_{D}{S}^{D}$ instead of ${\epsilon}_{0}{S}^{0}$. In (18), ${\epsilon}_{D}{S}^{D}$ and ${\epsilon}_{0}{S}^{0}$ are interchanged.

This extended constraint can be transferred straightforward to the approach of the Steklov-Poincaré interface equation by the means of the related Steklov-Poincaré operators. For the indirect single layer potential ansatz, the simplification of the related constraint (19) seems not to be possible in general.

### 6.1 IEC arrester

**Approximate values of the floating potentials and computational times for the IEC arrester**

${\mathit{\alpha}}_{\mathbf{1}}$ | ${\mathit{\alpha}}_{\mathbf{2}}$ | Time | |
---|---|---|---|

SP floating | 57.44 | 24.28 | 4,559 s |

SP full dielectric | 57.41 | 24.23 | 9,659 s |

2D ELFI | 57.62 | 24.42 |

### 6.2 Bushing

^{−3}. The computational mesh consists of 7,936 global nodes. Consequently the distance between elements created on the parallel foil surfaces is approximately 100 times smaller than the size of the elements. In spite of these extreme geometrical relations the floating potentials calculated for all foils with the Steklov-Poincaré operator approach show a good agreement with the 2D solution as presented in Table 5. The system (17) of linear equations is solved in 31 steps of a preconditioned CG method to a relative accuracy of 10

^{−6}.

**Approximate values of the floating potentials for the bushing**

${\mathit{\alpha}}_{\mathbf{1}}$ | ${\mathit{\alpha}}_{\mathbf{2}}$ | ${\mathit{\alpha}}_{\mathbf{3}}$ | ${\mathit{\alpha}}_{\mathbf{4}}$ | |
---|---|---|---|---|

SP floating | 70.7 | 51.4 | 35.1 | 19.0 |

2D ELFI | 70.8 | 51.4 | 35.0 | 18.9 |

### 6.3 Insulator with partial wetting

**Geometrical dimensions of the insulator with partial wetting**

Quantity | In mm |
---|---|

Electrodes distance | 14 |

Insulation thickness between electrode and air | 5 |

Shed diameter | 160 |

Shed thickness | 6 |

Water layer thickness | 1 |

^{−6}.

**Approximate values of the floating potentials for the insulator with partial wetting**

${\mathit{\alpha}}_{\mathbf{1}}$ | ${\mathit{\alpha}}_{\mathbf{2}}$ | |
---|---|---|

SP floating | 80.93 | 51.71 |

2D ELFI | 81.49 | 52.74 |

## 7 Conclusions

We insistently recommend the formulations which integrate the floating potential directly and the zero flux condition by a constraint. These formulations give better results and are faster than the approximation obtained by a dielectric media with high permittivity because of a smaller number of degrees of freedom and a smaller number of steps of the iterative solver. Thus the additional effort for the implementation of the modified system pays off.

In case of no or small jumps of the permittivity, the indirect approach by the single layer potential and the direct formulation based on the Steklov-Poincaré interface equation give good results. Here the computational times of the single layer approach are smaller.

In case of larger jumps of the permittivity, the direct formulation based on the Steklov-Poincaré interface equation turned out to be superior to the indirect single layer potential ansatz, as the results are of much better quality. In particular, the indirect approach shows unphysical singularities close to edges and corners in the case of large jumps of the permittivity. Comparing the computational times for a desired accuracy and not for a fixed mesh the Steklov-Poincaré formulations turns out to be faster.

## Declarations

### Acknowledgements

This work was supported by the FP7 Marie Curie IAPP Project CASOPT (Controlled Component and Assembly Level Optimisation of Industrial Devices, http://www.casopt.com).

## Authors’ Affiliations

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