Theses and Dissertations--Electrical and Computer Engineering. Electrical and Computer Engineering

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University of Kentucky University of Kentucky

UKnowledge

Theses and Dissertations--Electrical and

Computer Engineering Electrical and Computer Engineering

2021

Boundary Integral Equation Method for Electrostatic Field

Prediction in Piecewise-Homogeneous Electrolytes

Christopher Keith Pratt

University of Kentucky, chrispratt400@gmail.com

Digital Object Identifier: https://doi.org/10.13023/etd.2021.136

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Recommended Citation Recommended Citation

Pratt, Christopher Keith, "Boundary Integral Equation Method for Electrostatic Field Prediction in Piecewise-Homogeneous Electrolytes" (2021). Theses and Dissertations--Electrical and Computer Engineering. 164.

https://uknowledge.uky.edu/ece_etds/164

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STUDENT AGREEMENT: STUDENT AGREEMENT:

I represent that my thesis or dissertation and abstract are my original work. Proper attribution has been given to all outside sources. I understand that I am solely responsible for obtaining any needed copyright permissions. I have obtained needed written permission statement(s) from the owner(s) of each third-party copyrighted matter to be included in my work, allowing electronic distribution (if such use is not permitted by the fair use doctrine) which will be submitted to UKnowledge as Additional File.

I hereby grant to The University of Kentucky and its agents the irrevocable, non-exclusive, and royalty-free license to archive and make accessible my work in whole or in part in all forms of media, now or hereafter known. I agree that the document mentioned above may be made available immediately for worldwide access unless an embargo applies.

I retain all other ownership rights to the copyright of my work. I also retain the right to use in future works (such as articles or books) all or part of my work. I understand that I am free to register the copyright to my work.

REVIEW, APPROVAL AND ACCEPTANCE REVIEW, APPROVAL AND ACCEPTANCE

The document mentioned above has been reviewed and accepted by the student’s advisor, on behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of the program; we verify that this is the final, approved version of the student’s thesis including all changes required by the advisory committee. The undersigned agree to abide by the statements above.

Christopher Keith Pratt, Student Dr. John C. Young, Major Professor Dr. Daniel Lau, Director of Graduate Studies

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BOUNDARY INTEGRAL EQUATION METHOD FOR ELECTROSTATIC FIELD PREDICTION IN PIECEWISE-HOMOGENEOUS ELECTROLYTES

________________________________________ THESIS

________________________________________ A thesis submitted in partial fulfillment of the

requirements for the degree of Master of Science in Electrical Engineering

in the College of Engineering at the University of Kentucky

By

Christopher Pratt Lexington, Kentucky

Director: Dr. John Young, Professor of Electrical Engineering Lexington, Kentucky

2021

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ABSTRACT OF THESIS

This thesis presents a method to predict electrostatic fields, potentials, and currents in regions containing piecewise-homogeneous electrolytes. Additionally, an efficient electric field calculation is presented. A boundary integral equation is formulated for the boundary potentials and currents and is discretized using the Locally Corrected Nyström method. Solution convergence with respect to the mesh discretization and basis order is investigated. The techniques are validated through analysis of problems with either analytic solutions, with published data, or with other solution methods.

KEYWORDS: Corrosion-related field, Boundary integral equation, Locally Corrected Nyström method

Christopher K. Pratt May 14, 2021 Date

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By

Christopher Pratt

Dr. John Young Director of Thesis

Dr. Daniel Lau Director of Graduate Studies

May 14, 2021 Date

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iii

ACKNOWLEDGMENTS

I am very grateful to have been given this opportunity to work with my advisor Dr. John C. Young. The advice and instruction provided by Dr. Young during my research gave me a great learning experience in this complex and interesting field. The courses in theoretical and computational electromagnetics instructed by Dr. Young inspired me to further my education and purse graduate research in this field. I am thankful for Dr. Young’s time and expertise throughout my studies.

I would also like to thank my committee members Dr. Robert J. Adams and Dr. Cai-Cheng Lu for reviewing my thesis and for their excellent courses that assisted in my understanding in electromagnetics and computational methods. I am grateful for Dr. Johné Parker’s inspiration to seek higher education during my time assisting a part of her research as an undergraduate.

I am appreciative of the opportunities and the support of this research provided by the Office of Naval Research under grant N00014-16-1-3066.

Lastly, I would like to thank my family for supporting me during my education. I would not have been able to have this achievement without their love and support.

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iv

TABLE OF CONTENTS

ACKNOWLEDGMENTS ... iii

LIST OF FIGURES ... v

CHAPTER 1. Introduction ... 1

CHAPTER 2. Piecewise-Homogeneous Electrolyte Method ... 3

2.1 Boundary Integral Equation Method ... 3

2.2 Locally Corrected Nyström Discretization ... 8

CHAPTER 3. Electric Field Calculation ... 12

CHAPTER 4. Results and Validation... 15

4.1 Insulated box filled with two electrolytes. ... 15

4.2 Insulated Box with Split Electrolyte Interface ... 21

4.3 Insulated Box with Multiple Electrolyte Interfaces ... 25

4.4 Insulated Box containing Sphere Electrolyte Interface ... 31

4.5 Box with Linearly Varying Conductivity ... 35

4.6 Box with Aperture ... 43

4.7 Insulated cylindrical tank filled with two electrolytes. ... 48

4.8 Sphere encapsulating an electrode ... 51

4.9 Sphere encapsulating a Brick... 54

CHAPTER 5. Conclusion ... 57

References ... 58

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v

LISTOFFIGURES

Figure 2.1: Conductivity models with I homogeneous regions ...3 Figure 4.1: A two-meter brick with two homogeneous electrolytes. ...16 Figure 4.2: (a) Electric potential vs. position and (b) electric field vs. position for various

basis orders p containing homogeneous electrolyte with _σ1₌_σ2 ₌_{4 S/m}_{and meshed}

using quadrilateral cells of edge length 0.25 m. ...17 Figure 4.3: (a) Relative error vs. maximum edge length of the electric potential and (b) electric field for various basis orders p containing homogeneous electrolyte with

1 2 _{4 S/m}

σ =σ = . ...18 Figure 4.4: (a) Electric potential vs. position and (b) electric field vs. position for various

basis orders p containing piecewise-homogeneous electrolytes with _σ1 ₌_{6 S/m}_and

2 _{3 S/m}

σ = and meshed using quadrilateral cells of edge length 0.25 m. ...19

Figure 4.5: (a) Relative error vs. maximum edge length of the electric potential and (b) electric field for various basis orders p containing piecewise-homogeneous electrolytes with _σ1₌_{6 S/m}_and_σ2 ₌_{3 S/m}_{. ...20}

Figure 4.6: A two-meter brick with two homogeneous electrolytes and a split electrolyte interface. The arrows indicate the direction of the surface normals. ...22 Figure 4.7: (a) Electric potential vs. position and (b) electric field vs. position for various

basis orders p containing homogeneous electrolytes with _σ1₌_σ2 ₌_{4 S/m}_{and meshed}

using quadrilateral cells of edge length 0.25 m. ...23 Figure 4.8 (a) Electric potential vs. position and (b) electric field vs. position for various

basis orders p containing piecewise-homogeneous electrolytes with _σ1 ₌_{3 S/m}_and

2 _{5 S/m}

σ = and meshed using quadrilateral cells of edge length 0.25 m. ...24

Figure 4.9: A three-meter brick with three homogeneous electrolytes. ...26 Figure 4.10 (a) Electric potential vs. position and (b) electric field vs. position for various

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vi

Figure 4.11: (a) Relative error vs. maximum edge length of the electric potential and (b) electric field for various basis orders p containing homogeneous electrolyte with

1 2 _{4 S/m}

σ =σ = . ...28 Figure 4.12 (a) Electric potential vs. position and (b) electric field vs. position for various

basis orders p containing piecewise-homogeneous electrolytes with _σ1 ₌_{6 S/m}_,

2 _{4 S/m}

σ = , and _σ3 ₌_{2 S/m}_{and meshed using quadrilateral cells of edge length 0.25 m.}

...29 Figure 4.13: (a) Relative error vs. maximum edge length of the electric potential and (b) electric field for various basis orders p containing piecewise-homogeneous electrolytes with _σ1₌_{6 S/m}_,_σ2 ₌_{4 S/m}_{, and}_σ3 ₌_{2 S/m}_{. ...30}

Figure 4.14: A two-meter long brick with two homogeneous electrolytes and a spherical electrolyte interface. ...32 Figure 4.15: (a) Electric potential vs. position and (b) electric field vs. position for various

using quadrilateral cells of edge length 0.1 m. ...33 Figure 4.16: (a) Electric potential vs. position and (b) electric field vs. position for various

basis orders p and piecewise-homogeneous electrolytes with _σ1 ₌_{2 S/m}_{, and}_σ2 ₌_{3 S/m}

and meshed using quadrilateral cells of edge length 0.1 m. ...34

Figure 4.17: (a-e) One-meter box with n =1,...,5 homogeneous electrolyte(s). ...37

Figure 4.18: Electric potential vs. position and (b) electric field vs. position for various

basis orders p containing electrolyte varying linearly from _{z =}0 _{0 m}_with_σ0 ₌_{1 S/m}_and

1 _{1 m}

z = with _σ1₌_{2 S/m}_with_{n = regions and meshed using quadrilateral cells of edge}₁

length 0.25 m. ...38 Figure 4.19: Electric potential vs. position and (b) electric field vs. position for various

2 _{1 m}

z = with _σ1 ₌_{2 S/m}_with_{n = regions and meshed using quadrilateral cells of edge}₂

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vii

Figure 4.20: Electric potential vs. position and (b) electric field vs. position for various

3 _{1 m}

z = with _σ1 ₌_{2 S/m}_with _{n = regions and meshed using quadrilateral cells of}₃

maximum edge length 0.25 m. ...40 Figure 4.21: Electric potential vs. position and (b) electric field vs. position for various

4 _{1 m}

z = with _σ1 ₌_{2 S/m}_with _{n = regions and meshed using quadrilateral cells of}₄

maximum edge length 0.25 m. ...41 Figure 4.22: Electric potential vs. position and (b) electric field vs. position for various

5 _{1 m}

z = with _σ1 ₌_{2 S/m}_with _{n = regions and meshed using quadrilateral cells of}₅

maximum edge length 0.25 m. ...42 Figure 4.23: A box with aperture inside a larger insulating box. ...44 Figure 4.24: Electrical potential vs. position within the interior box (a) without using zoning and (b) using zoning for various basis orders p and both boxes are meshed using quadrilateral cells of maximum edge length 5 cm. ...45 Figure 4.25: Electric potential vs. position outside the interior box (a) without using zoning and (b) using zoning for various basis orders p and both boxes are meshed using quadrilateral cells of maximum edge length 5 cm. ...46 Figure 4.26: (a) Electrical potential vs. position within the interior box and (b) electric potential vs. position outside interior box (b) for various basis orders p calculated with zoning without the exterior box and the interior box is meshed using quadrilateral cells of maximum edge length 5 cm. ...47 Figure 4.27: A cylindrical tank with a false surface placed along the y-axis. ...48

Figure 4.28: E_x vs. position (a) without using zoning and (b) using zoning for various basis

orders p and the tank is meshed using quadrilateral cells of maximum edge length 20 cm with a depth of 2.625 m. ...49

Figure 4.29: e vs. position (a) without using zoning and (b) using zoning for various basis _z

orders p and the tank is meshed using quadrilateral cells of maximum edge length 20 cm with a depth of 2.625 m. ...50

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viii

Figure 4.30: A pair of electrodes with a fictious sphere encompassing an electrode. ...51 Figure 4.31: Electric potential vs. position (a) without using zoning and (b) using zoning for various basis orders p and meshed using quadrilateral cells of maximum edge length 0.25 m. ...52 Figure 4.32: Electric field vs. position (a) without using zoning and (b) using zoning for various basis orders p and meshed using quadrilateral cells of maximum edge length 0.25 m. ...53 Figure 4.33: A brick encapsulated by a fictious sphere. ...54 Figure 4.34: Electric potential vs. position (a) without using zoning and (b) using zoning for various basis orders p and meshed using quadrilateral cells of maximum edge length 0.25 m. ...55 Figure 4.35: Electric field vs. position (a) without using zoning and (b) using zoning for various basis orders p and meshed using quadrilateral cells of maximum edge length 0.25 m. ...56

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1

CHAPTER 1. INTRODUCTION

Modern engineering involving conductors surrounded by or immersed in electrolyte, such as the hull of a ship in seawater, is faced with the challenge of mitigating corrosion [1-4]. These conducting structures may be difficult to protect from corrosion over time due to the conditions the structure is exposed to. While the use of insulating paint can be used to prevent corrosion of conducting surfaces, this coating may be damaged during regular operation of the system. In other cases, insulating paint may not be an option for some surfaces.

One preventative method for slowing corrosion is an impressed-current corrosion protection system. This system slows the corrosion process by injecting electric current into the electrolyte to adjust the structure’s surface potential. Impressed-current corrosion protection systems are complex to design and are often dependent on numerical modeling of the system to predict the corrosion-related currents and electromagnetic fields. Numerical modelling techniques include finite-difference methods [5], finite-element methods [6, 7], and boundary element methods (BEM) [8]. The BEM is a convenient method since the electric currents and fields can be solved using a surface integral equation without the need to mesh the surrounding electrolytic medium.

In the analysis of a marine structure and buried pipes, the prediction of corrosion currents and fields in an electrolyte may assume the surrounding electrolyte to be homogeneous. In reality, the conductivity of electrolytes, e.g., seawater or soil, typically varies with location. This implies that the assumption of a constant conductivity may be a poor approximation in some cases. For a better approximation of inhomogeneous conductivity, a piecewise-constant conductivity model, referred to as zoning in this work, is discussed. Herein, each region with a homogeneous electrolyte is called a zone. The application of zoning will be applied with marine structures in mind, but these methods may be applicable to other systems with a surrounding inhomogeneous conductivity.

In this thesis, a boundary integral equation is formulated for the analysis of piecewise-homogeneous electrolytes. The integral equation is discretized using the Locally Corrected Nyström method [9-11], and the solution convergence with respect to mesh discretization and basis order is investigated. An efficient electric field calculation

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2

is presented as well and is verified through comparison to literature data [12, 13], analytic solutions, and data for single electrolyte problems.

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3

CHAPTER 2. PIECEWISE-HOMOGENEOUS ELECTROLYTE METHOD

2.1 Boundary Integral Equation Method

Consider a piecewise-homogeneous electrolyte model for I homogeneous regions

i

V where region _Vi_{has homogeneous conductivity}_σi_{for i = 1… I. Let the region}_V0_be

the region outside the problem domain. Figure 2.1 illustrates an electrolyte with I

homogeneous regions. The shared surface between regions _Vi_and_Vj_{is denoted}_{Γ = Γ}ij ji

with the unit surface normal ˆn pointing from region ij Vi into region Vj. The exterior

boundary of the total region is _Γ0_{, which may recede to infinity for unbounded problems,}

and the total boundary of _Vi_is_Γi_.

Figure 2.1: I regions with piecewise-homogeneous conductivity _σi_.

The potential _Φi_within_Vi_{is given by [12]}

( )

ˆ

(

,

)

(

,

)

ˆ

( )

i i i i _G _d _G i _d Γ Γ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ Φ r + Φ

_

_∫

r n ⋅∇ r r Γ =

_

_∫

r r n ⋅ Φ∇ r Γ (2.1)

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4

where ˆ′n points out of region _Vi_{. The electrostatic Green’s function}

(

,

)

1/ 4

G r r′ = π r r− ′ is assumed unless noted otherwise. For simplicity, it will be

assumed all surfaces _Γi_{are bounded except possibly for}_Γ0_.

To obtain a set of coupled integral equations with a unique solution, the shared surfaces between each region must have both the electric potential be continuous

( )

i j

Φ r = Φ r (2.2)

and the normal electric current density be continuous

( )

ˆ i ˆ j ij⋅ = ij⋅ n J r n J r (2.3) for _r_∈Γij_where

( )

, i i i i V σ = − ∇Φ ∈ J r r r . (2.4)

The boundary condition on insulating surfaces _Γi,insulating _{⊂ Γ}i_{is that the normal current}

density is zero

( )

ˆ_⋅ i ₌0

n J r (2.5)

for _r_∈Γi,insulating_{. The excitations in the model are the boundary conditions over the} electrodes _Γi,electrode_{⊂ Γ}i_{and are either a fixed potential}

( )

,electrode

( )

i i

Φ r = Φ r (2.6)

or a fixed normal current density

( )

,electrode

( )

ˆ i i

i J

− ⋅n J r = r (2.7)

for _r_∈Γi,electrode_{based on whether the boundary condition is Dirichlet or Neumann,}

respectively.

For interior, linear Neumann problems, a one-dimensional null space corresponding to a constant potential offset is addressed by enforcing the average potential over some

surface _Γavg_{to a constant C}

( )

avg . d C Γ ′ ′ Φ Γ =

∫

r (2.8)

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5 ,electrode ˆ 0 i i d Γ ′ ⋅ Γ =

∫

n J  (2.9)

is satisfied over all electrode surfaces _Γi,electrode_{. Additionally, (2.9) must be a priori satisfied}

for linear Neumann problems but is not required to be enforced explicitly for linear, Dirichlet or linear, mixed boundary condition problems since these problems already have the potential fixed somewhere in the problem. Enforcing the conditions (2.2) - (2.9) produces a uniquely solvable system.

The reference direction of the unit surface normal ˆn on a shared interface ij Γij is

always directed from i into j. On a shared interface _Γij_{, let}_Φij

( )

_r _{= Φ}i

( )

_r _{= Φ}j

( )

_r _and

( )

ˆ ij ˆ i ˆ j

ij⋅ = ij⋅ = ij⋅

n J r n J r n J r where i < j to fix the reference direction for the

normal current density. In the following, i j< is assumed unless otherwise noted.

At _r_∈Γij_{, if we approach along a path from outside the respective region, then}

( )

lim

( )

0 and

( )

lim

( )

0,

j i

i i j j

A r = −_r_→_r+Φ r = A r = −_{r r}_→ +Φ r = (2.10)

where _r+_{indicates the limit is taken along a path outside the respective region. The}

limiting operation for a small surface∆S [14] is

( )

(

)

( )

(

)

( )

0 0 1 ˆ ˆ lim , P.V. , 2 S S G ds G ds ± → _∆

∫

Φ ′ ′ ′ ⋅ ′ ′= _∆

∫

Φ ′ ′ ′ ⋅ ′ ′± Φ r r r ∇ r r n r ∇ r r n r (2.11) where

(

( )

)

0 ˆ lim h + h ± → = ±

r r n r and the surface normal is ˆn . P.V. denotes a principal value

integral and is suppressed throughout.

To enforce (2.2) and (2.3) on a shared interface _Γij_{, a linear combination of}_Ai_and

j

A

( )

0

i i_A j_Aj

α r +α r = (2.12)

is taken for _r_∈Γij_{. Splitting the integrals into ones over}_Γij_{and not over}_Γij_{and applying}

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6

( )

(

)

( )

(

)

(

)

( )

(

)

( )

in, in, 1 0 ( ) 2 ˆ , ˆ , 1 _, _ˆ 1 _, _ˆ i ij ij i ij ij i ij i ij i i ij i i ij i ij i i A G d G d G d G d σ σ Γ −Γ Γ Γ −Γ Γ = = Φ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ − Φ ⋅ Γ + Φ ⋅ Γ ′ ′ ′ ′ ′ ′ ′ ′ − ⋅ Γ + ⋅ Γ

∫

r r r n r r r n r r r r n J r r r n J r ∇ ∇ (2.13) and

( )

(

)

( )

(

)

(

)

( )

(

)

( )

in, in, 1 0 ( ) 2 ˆ , ˆ , 1 _, _ˆ 1 _, _ˆ j ij ij j ij ij j ij j ij j ij j ij j ij j j A G d G d G d G d σ σ Γ −Γ Γ Γ −Γ Γ = = Φ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ − Φ ⋅ Γ − Φ ⋅ Γ ′ ′ ′ ′ ′ ′ ′ ′ − ⋅ Γ − ⋅ Γ

∫

r r r n r r r n r r r r n J r r r n J r ∇ ∇ (2.14)

where nîn,i = −n is into the electrolyte of region î Vi and nîn,j = −n is into the electrolyte ˆj

of region _Vj_{. Here}_αi ₌_αj ₌₁_{is chosen to eliminate the strong singularity in the integral}

over Φ . Applying these constants gives

( )

(

)

( )

(

)

(

)

( )

(

)

( )

(

)

( )

in, in, in, in, ˆ , ˆ , 1 _, _ˆ 1 _, _ˆ 1 1 _, _ˆ 0 i ij j ij i ij j ij ij i i j j ij i i j j i i i j j j ij ij i j A A G d G d G d G d G d α α σ σ σ σ Γ −Γ Γ −Γ Γ −Γ Γ −Γ Γ ′ ′ ′ ′ ′ + = Φ − Φ ⋅ Γ ′ ′ ′ ′ ′ − Φ ⋅ Γ ′ ′ ′ ′ − ⋅ Γ ′ ′ ′ ′ − ⋅ Γ   _{′ ′} _′ _′ + _ − _ ⋅ Γ   =

∫

r r r r n r r r n r r r r n J r r r n J r r r n J r ∇ ∇ (2.15) for _r_∈Γij_.

To enforce (2.2) and (2.3), a linear combination

( )

0 i i_B j_Bj β r +β r = (2.16) is taken where

( )

in, in, ˆ ˆ

lim 0 and lim 0

j i

i i j j

i j

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7

for _r_∈Γij_{and where the limits are taken along paths outside the respective regions.}

Separating these integrals gives

( )

(

)

( )

(

)

( )

(

)

( )

(

)

( )

in, in, in, in, 0 in, in, in, in, ˆ ˆ 0 , ˆ ˆ lim , 2 ˆ , ˆ ˆ , ˆ i ij ij a i ij i i i i i ij i ij i i i a S i i i i i i ij i i i B G d G d a G d G d σ σ σ σ σ σ σ Γ −Γ → Γ − Γ −Γ ′  ′ ′  ′ = = Φ ⋅ _ ⋅ _ Γ  Φ   _′ _ _′ _′ _ _′  + _ Φ ⋅ _ ⋅ _ Γ + _     ′ ′ ′ ′ ′   + _ ⋅ _ ⋅ Γ ′ ′ ′ ′ ′   + _ ⋅ _ ⋅

∫

r r n n r r r r n n r r n r r n J r n r r n J r ∇ ∇ ∇ ∇ ∇ ∇ ˆin,

( )

2 ij ij i i i σ σ Γ ⋅ Γ −

∫

n J r (2.18) and

( )

(

)

( )

(

)

( )

(

)

( )

(

)

( )

in, in, in, in, 0 in, in, in, in, ˆ ˆ 0 , ˆ ˆ lim , 2 ˆ , ˆ ˆ , ˆ j ij ij a j ij j j j j j ij j ij j j j a S j j j j j j ij j j j B G d G d a G d G d σ σ σ σ σ σ σ Γ −Γ → Γ − Γ −Γ ′  ′ ′  ′ = = Φ ⋅ _ ⋅ _ Γ  _Φ   _′ _ _′ _′ _ _′  + _ Φ ⋅ _ ⋅ _ Γ + _     ′ ′ ′ ′ ′   + _ ⋅ _ ⋅ Γ ′ ′ ′ ′ ′   + _ ⋅ _ ⋅

∫

r r n n r r r r n n r r n r r n J r n r r n J r ∇ ∇ ∇ ∇ ∇ ∇ ˆin,

( )

_. 2 ij ij j j j σ σ Γ ⋅ Γ −

∫

n J r (2.19)

The procedure for dealing with the ∇∇ term is taken from [15] where S is a _a

circular disk of radius a centered on the field point _r_∈Γij_{. A linear combination is taken}

using i 1 i β σ = and j 1 j β σ

= − , which eliminates the terms over _Γij_{and the circular disk}

a

S in (2.18) and (2.19) while maintaining the non-integrated _{J r}ij

( )

_{terms. Applying this}

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8

( )

(

)

( )

(

)

(

)

( )

(

)

( )

in, in, in, in, in, in, in, in, 1 1 _ˆ _ˆ _, ˆ ˆ , 1 _ˆ _, _ˆ 1 _ˆ _, _ˆ 1 1 i ij j ij i ij j ij i j i i i i i j i j j i j j i i i i j i j j i j B B G d G d G d G d σ σ σ σ σ σ σ σ σ σ Γ −Γ Γ −Γ Γ −Γ Γ −Γ ′  ′ ′  ′ − = Φ ⋅ _ ⋅ _ Γ ′  ′ ′  ′ + Φ ⋅ _ ⋅ _ Γ ′ ′ ′ ′ ′   + _ ⋅ _ ⋅ Γ ′ ′ ′ ′ ′   + _ ⋅ _ ⋅ Γ  + −

∫

r r r n n r r r n n r r n r r n J r n r r n J r ∇ ∇ ∇ ∇ ∇ ∇

(

)

( )

in, in, in, ˆ , ˆ ˆ 1 1 2 0 ij ij i i ij i i j G d σ σ Γ  _ _⋅ _′ _′ _ _′ _⋅ _′ _Γ_′       ′ ⋅   −_ + _   =

∫

n r r n J r n J r ∇ (2.20) for _r_∈Γij_.

For _r_∈Γi_{, such that}_{r is not on a surface between electrolytes, it is only necessary}

to enforce

( )

0 lim i 0 + → Φ = r r r for 0 0∈V

r , giving the equation

( )

in,

(

)

(

)

in,

( )

1 _ˆ 1 _ˆ 0 , , . 2 i i i i i i G i d _σi G i i d Γ Γ ′ ′ ′ ′ ′ ′ ′ ′ ′ = Φ r − Φ

_∫

r n ⋅∇ r r Γ −

_∫

r r n ⋅J r Γ (2.21)

Finally, note that in (2.15), (2.20), and (2.21), whenever the source point r′ is on a

shared surface _Γik_{, i.e.,}_r_′∈Γik_{, then}_Φi

( )

_r _{= Φ}ik

( )

_r _and

( )

ˆ i ˆ ik , i⋅ = + ik⋅ i k< n J r n J r (2.22) and

( )

ˆ i ˆ ki , i⋅ = − ki⋅ i k> n J r n J r . (2.23)

Similarly, for terms in region _Vj_.

2.2 Locally Corrected Nyström Discretization

The integral equations are discretized using the Locally-Corrected Nyström method as described in [9-11]. To illustrate the method, consider the integral equation

(20)

9

( )

(

,

) ( )

, , h G f ds Ω ′ ′ ′ =

∫

∈Ω r r r r r (2.24)

where h is the excitation, f is the unknown, Ω is the domain, and G is the kernel. The

domain Ω can be divided into N subdomains s Ω such that n

1 s N n n= Ω =



Ω , giving the equation

( )

(

) ( )

1 , s n N n h G f ds = Ω ′ ′ ′ =

∑ ∫

r r r r . (2.25)

In the Nyström method, the integral in each subdomain is approximated using Q -point n

quadrature rules

{

r_q_n,ω_q_n

}

where r are the quadrature points and _q_n ω_q_n are the quadrature

weights. Then, (2.25) is enforced at the field points

{ }

r to obtain _q_m

(

) ( )

1 1 ( )_m s n _m, _n _n _n ( )_n n N Q q q q q q q n q h G f ω g = = ≈

∑∑

r r r r r , (2.26)

where g( )r_q_n is the cell Jacobian at the qth quadrature point on the cell n. The matrix

equation formed from (2.26) is

[ ][ ] [ ]

Z f = h . (2.27)

The interaction between the quadrature point r of the field cell q_m Ω and the quadrature m

point r on the source cell q_n Ω is n ωq qm, nsuch that

[ ]

Zmn q q_m,_n =ωq qm,n , (2.28)

where

[ ]

Z_mn is the sub-matrix of

[ ]

Z for the field cell Ω and source cell m Ω and n

{

ωq qm n,

}

is the corresponding matrix element value.

For a field cell Ω and source cell m Ω separated such that the integrand in (2.26) is n

sufficiently smooth, the matrix entry ωq qm n, is

(

)

, , ( ).

m n m n n n

q q G q q q g q

(21)

10

For field and source cell interactions close enough that standard integration rules begin to lose accuracy, a set of locally corrected quadrature weights may be utilized. The matrix entries

{

ω_{q q}_{m n}_,

}

are locally corrected weights calculated by solving

( )

,

(

)

( )

1 , n n m n m n n Q k q q q q k q f ω G f ds = Ω ′ ′ ′ =

∑

r 

∫

r r r (2.30)

for k =1,...,Kn, where

{

fk

( )

r

}

is an appropriate set of K basis functions defined on the n

source cell Ω . The weights n

{

ωq qm n,

}

can be solved from

[ ]

m,n m

n n q q q

L _ω  _{ }= d _ , (2.31)

where the local correction matrix

[ ]

_,

( )

_n

n n _{k q} k q L = f r and _m

(

_m,

)

( )

n n q _k q k d G f ds Ω   = ′ ′ ′  

∫

r r r .

The basis functions

{

f_k

( )

r are the set

}

{

( ) ( )

1 2

}

m n

P u P u where

m

P and P are Legendre polynomials of degree m and n, respectively, such that n

0<m n p, ≤ for quadrilateral cells with local coordinates

(

_{u u}1_, 2

)

and a representation of

order p. Furthermore, for an order p representation on a quadrilateral cell, K_n =

(

p+1

)

2.

In discretizing (2.15), (2.20), and (2.21) using the Locally-Corrected Nyström

method, the Nyström unknowns _f_q_n_ are the sampled potential i

( )

_n _n

q fqΦ

Φ r = and the

sampled normal current ˆ

( )

n n

i

q fq

⋅ = J

n J r at the quadrature point r subject to and _q_n

continuity constraints and the sign convention given in (2.22) and (2.23) on shared

surfaces. The excitation vector

[ ]

h is determined from the known boundary conditions

over the electrodes and the insulating surfaces. If the normal current ˆ ⋅n J is known

(Neumann boundary type) over an electrode, then the integrals over n Jˆ ⋅ from (2.15),

(2.20), and (2.21) are part of the excitation vector

[ ]

h while the integrals over Φ are part

of the system matrix [ ]Z . If the potential Φ is known (Dirichlet boundary type) over the

electrode, then the integrals over Φ from (2.15), (2.20), and (2.21) are part of

[ ]

h while

(22)

11

( )

ˆ⋅ =0

n J r , all integrals over ˆ ⋅n J are 0 a priori satisfied. On the shared surfaces between

the electrolytes, both Φ and ˆ ⋅n J are unknown.

Interior, linear Neumann problems admit a one-dimensional null space which corresponds to a constant potential offset [16] that may be removed by enforcing the

average potential over some surface _Γavg_{to a constant value C, as in (2.8). To enforce the}

(23)

12

CHAPTER 3. ELECTRIC FIELD CALCULATION

Consider a region _Vi_{of homogenous conductivity}_σi_{. The potential Φ within this}

region is

( )

1

(

,

) ( )

( )

ˆ , i i i i i i V V G ρ dv G G ds V ε ′ ′ ′ _Γ  ′ ′ ′ ′  ′ ′ Φ r =

_∫

r r r −

_

_∫

_Φ r ∇ − ∇Φ r _⋅n r∈ , (3.1)

where ˆn is the unit normal to _Γi_{out of}_Vi_,_ε_{is the permittivity of the material, and}

(

,

)

1/ 4

G r r′ = π r r− ′ is electrostatic Green’s function. If there is no volume charge

density ρ_V in _Vi_{, then}

( )

ˆ , . i i i _{G G} i _ds _Vi Γ ′ ′ ′ ′ ′ ′   Φ _r = −

_

_∫

_Φ _r ∇ − ∇Φ _r _⋅_n _r∈ (3.2)

The electric field E in the region _Vi_is

( )

(

,

)

ˆ

( )

ˆ

(

,

)

i i i i i _G _d i _G _d Γ Γ = − Φ ′ ′ ′ ′ ′  ′ ′ ′ ′ Φ _ ⋅ _ Γ − _ ⋅ __ _ Γ

∫

E r r r r r n J r n r r



∇ = ∇ ∇ ∇ (3.3) where

(

)

(

)

3 2 ˆ 1 1 , , 4 4 G ′ = − ′G ′ = − _π − ′ = − _π ′ − r r u r r r r r r u ∇ ∇ (3.4)

with u=

(

r r− ′

)

. From the identity [17]

(

A B⋅

)

= × × + × × +A B B A B

(

⋅

)

A A+

(

⋅

)

B, ∇ ∇ ∇ ∇ ∇ (3.5) (3.3) becomes

( )

( )(

ˆ

)

(

,

)

( )

ˆ

(

,

)

. i i i i _G _d i _G _d Γ Γ ′ ′ ′ ′ ′  ′ ′ ′ ′ = Φ

_∫

⋅ _ _ Γ −

_∫

_ ⋅ __ _ Γ E r

_

r n ∇ ∇ r r

_

J r n ∇ r r (3.6) Applying (3.4) to (3.6) and expanding the equation gives

( )

[

]

( )

3 2 2 1 _ˆ 3 _ˆ 4 ˆ 1 ˆ . 4 i i i i i d d π π Γ Γ     ′ ′ ′ ′ = Φ _ − ⋅ _ Γ       ′ ′ ′   − _ ⋅ _{ }−  Γ   

∫

E r r n n u u u u u J r n u



(3.7)

The integral over _Γi_{is divided into}_Ni_cells n

(24)

13

( )

i i n n i n n IΦ I Ω ∈Γ Ω ∈Γ =

∑

−

∑

J E r , (3.8) where

( )

3 2

[

]

1 _ˆ 3 _ˆ 4 n i n I d π Φ Ω    _′ _′  _′ = Φ _ − ⋅ _ Γ    

∫

r r n n u u u u (3.9) and

( )

2 ˆ 1 ˆ 4 n i n I _π d Ω   ′ ′   = _ ⋅ _{ }−  Γ   

∫

J _r _{J r n} u u . (3.10)

For field points sufficiently distant from the source cell Ω , the integrals are approximated n

using Q -point quadrature rules _n

{

r_q_n,ω_q_n

}

to obtain

( )

3 2 1 1 _ˆ 3 _ˆ _{( )} 4 n n n n n n n n n n n Q i n q q q q q q q q _q _q I ω g π Φ =    _′ _ _′ _  ≈ Φ _ − _ ⋅ _ _    

∑

r r n n u u r u u (3.11) and I_nJ_becomes

( )

2 1 ˆ 1 ˆ ( ) 4 n n n n n n n n Q q i n q q q q q _q I ω g π =      ′  ≈ _ ⋅ _{ }−   

∑

J _r _{J r} _n u _r u (3.12)

where g( )r_q_n is the Jacobian of the source cell Ω , n nˆqn =n r , and ˆ

( )

qn uqn =

(

r r− qn

)

.

When the field point is close to the source cell Ω , the quadrature rule approximation n

is not sufficiently accurate. Then, the original integrals are used with the potential approximated as

( )

1 n K i k k k f γ = Φ _r ≈

∑

_r (3.13)

and the normal current approximated as

( )

1 ˆ i Kn k k k f γ = ⋅ ≈

∑

n J r r , (3.14)

(25)

14

where

{ }

γ_k is a set of coefficients for the K set of basis functions n

{

fk

( )

rqn

}

. For

quadrilateral cells, the basis functions

{

fk

( )

r are the set of Legendre polynomials

}

products defined in the previous section. The coefficients

{ }

γk are solutions to

( )

1 n n n K i q k k q k f γ = Φ _r ≈

∑

_r (3.15) and

( )

1 ˆ n n n n K i q q k k q k f γ = ⋅ ≈

∑

n J r r (3.16) for qn =1,...,Qn. Here,

( )

n n i q fqΦ Φ r = and ˆ _n i

( )

_n _n q ⋅ q = fqJ

n J r are from the Locally-Corrected Nyström solution to (2.27) and subject to the sign convention given in (2.22) and (2.23) on shared surfaces.

(26)

15

CHAPTER 4. RESULTS AND VALIDATION

The problems presented in this section are for structures filled with piecewise-homogeneous electrolytes to verify the accuracy and correctness of the method.

4.1 Insulated box filled with two electrolytes.

Consider a rectangular box structure centered vertically along the z-axis with a length of 2 m and side lengths of 1 m with a fictitious surface at the center of the structure, as

shown in Figure 4.1. Electrodes are placed at the top of the structure _{z =}2 _{1 m}_{and bottom}

0_{= 1 m}

z − , sourcing and sinking 1 A, respectively. The surfaces on the sides of the structure

are insulating material. The average potential over the bottom electrode is forced to be

zero. The structure was analyzed using basis orders p =0,1,2.

The analytic electric potential and electric field for I regions of homogeneous

conductivity _σi_{with the average potential on the bottom electrode enforce to be zero are}

( )

1

(

1

)

1 1 1 ( _{) ,} T i i i k k i k i k I _z _z z z _V S σ σ − − − =  −  Φ = _ − + _ ∈ 

∑

 r r (4.1) and

( )

T ˆ, i i i I _V Sσ = − ∈ E r z r , (4.2)

respectively, where _IT_{is the electrode current,}_{S is the cross-sectional area of the box,}

and _zi_{is the axial locations of the electrodes and surfaces in between regions. This analytic}

solution is relevant for the box problems presented in Sections 4.1 – 4.4.

In Figure 4.2 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis. The electrolytes filling the boxes have a

conductivity of _σ1₌_σ2 ₌₄_{S/m. The gap seen in the data is a result of calculating the}

potential and electric field within each region of the structure. The results are in good agreement with the analytic solution. In Figure 4.3 (a) and (b), respectively, are the relative error of the computed potential and electric field versus the maximum mesh edge length

for basis orders p =0,1,2,3. The potential and electric field for the relative error was

(27)

16

the potential and electric field begin the stagnate between ₁₀−3_and₁₀−4_{, but the reason for}

this has yet to be determined.

In Figure 4.4 (a) and (b) are plotted the computed and analytic electric potential and electric field calculated similarly but with different electrolytes in each box. The bottom

electrolyte has a conductivity of _σ1₌₆_{S/m and the top electrolyte has a conductivity of}

2

σ = 3 S/m. The results are consistent with the analytic solution provided in (4.1) and (4.2)

. In Figure 4.5 (a) and (b), respectively, are the relative error of the computed potential and

electric field versus the maximum mesh edge length for basis orders p =0,1,2,3. The

potential and electric field for the relative error was calculated no closer than 5 cm to the surfaces between the regions. Both the potential and electric field begin the stagnate

between ₁₀−3_and₁₀−4_{, but the reason for this has yet to be determined.}

(28)

17 (a)

(b)

Figure 4.2: (a) Electric potential vs. position and (b) electric field vs. position for various

basis orders p and conductivity _σ1 ₌_σ2 ₌_{4 S/m}_{and meshed using quadrilateral cells of}

(29)

18 (a)

(b)

Figure 4.3: Relative error vs. maximum mesh edge length for (a) the electric potential and

(30)

19 (a)

(b)

basis orders p and conductivity _σ1₌_{6 S/m}_and_σ2 ₌_{3 S/m}_{and meshed using}

(31)

20 (a)

(b)

Figure 4.5: Relative error vs. maximum edge length for (a) the electric potential and (b)

(32)

21 4.2 Insulated Box with Split Electrolyte Interface

Similar to the previous problem, the interior problem presented here is a rectangular box centered vertically along the z-axis with a length of 2 m and side lengths of 1 m with a fictitious surface at the center of the structure but this fictitious surface, shown in Figure 4.6, is split along the y-axis. The surfaces of the fictitious surface have oppositely directed surface normals. Although this breaks the convention that the reference direction for the

current _Jij_{is defined as i j}_< _{, the equations may be modified to allow scenarios when the}

surface normal are not pointing in the appropriate direction. Electrodes are placed at the

top of the structure _{z =}2 _{1 m}_{and bottom}_{z = −}0 _{1 m}_{, sourcing and sinking 1 A,}

respectively. The surfaces on the sides of the structure are insulating. The structure was

analyzed using basis orders p =0,1,2.

conductivity of _σ1 ₌_σ2 ₌_{4 S/m}_{. The results are in good agreement with the analytic}

solution. A comparison of the data shown in Figure 4.7 to the data in Figure 4.2 provides verification of the robustness of the method.

In Figure 4.8 (a) and (b), respectively, are plotted the potential and electric field along the z-axis as well but with different electrolytes filling the structure. The bottom electrolyte

has a conductivity _σ1₌_{3 S/m}_{and the top electrolyte has a conductivity}_σ2 ₌_{5 S/m}_{. The}

(33)

22

Figure 4.6: A two-meter brick with two homogeneous electrolytes and a split electrolyte interface. The arrows indicate the direction of the surface normals.

(34)

23 (a)

(b)

(35)

24 (a)

(b)

Figure 4.8 (a) Electric potential vs. position and (b) electric field vs. position for various

basis orders p and conductivity _σ1 ₌_{3 S/m}_and_σ2 ₌_{5 S/m}_{and meshed using}

(36)

25

4.3 Insulated Box with Multiple Electrolyte Interfaces

Consider a rectangular box centered vertically along the z-axis with a length of 3 m and side lengths of 1 m with fictitious surfaces placed 0.5 m above and below the z-axis within the structure, as shown in Figure 4.9. Electrodes are placed at the top of the structure

3 _{1.5 m}

z = and bottom of the structure _{z = −}0 _{1.5 m}_{, sourcing and sinking 1 A,}

respectively. The surfaces on the sides of the structure are insulating material. The structure

was analyzed using basis orders p =0,1,2.

conductivity of_σ1₌_σ2 ₌_σ3 ₌_{4 S/m}_{. The results are in good agreement with the analytic}

solution. In Figure 4.11 (a) and (b), respectively, are the relative error of the computed potential and electric field versus the maximum mesh edge length for basis orders

0,1,2,3

p = . The potential and electric field for the relative error was calculated no closer

than 5 cm to the surfaces between the regions. Both the potential and electric field relative

errors begin to stagnate around ₁₀−4_{, but the reason for this has yet to be determined.}

In Figure 4.12 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis when the box is filled with different

electrolytes. The bottom electrolyte has a conductivity _σ1 ₌_{6 S/m, the middle electrolyte}

has a conductivity of _σ2 ₌_{4 S/m, and the top electrolyte has a conductivity of}_σ3 ₌_{2 S/m.}

The results show good agreement with the analytic solution provided in (4.1) and (4.2). In Figure 4.13 (a) and (b), respectively, are the relative error of the computed potential and

electric field versus the maximum mesh edge length for basis orders p =0,1,2,3. The

potential and electric field for the relative error was calculated no closer than 5 cm to the surfaces between the regions. Both the potential and electric field relative errors begin to

(37)

26

(38)

27 (a)

(b)

(39)

28 (a)

(b)

(40)

29 (a)

(b)

basis orders p and conductivity _σ1 ₌_{6 S/m}_,_σ2 ₌_{4 S/m}_{, and}_σ3 ₌_{2 S/m}_{and meshed}

(41)

30 (a)

(b)

the electric field for various basis orders p and conductivity _σ1 ₌_{6 S/m}_,_σ2 ₌_{4 S/m}_,

(42)

31

4.4 Insulated Box containing Sphere Electrolyte Interface

Consider a rectangular box centered vertically along the z-axis with a length of 2 m and side lengths of 1 m with a fictitious surface at the center of the structure, as shown in Figure 4.14. A spherical fictitious surface is centered in the structure with a radius of 0.25

m. Electrodes are placed at the top _{z =}2 _{1 m}_{and bottom of the rectangular box}_{z = −}0 _{1 m}

, sourcing and sinking 1 A, respectively. The surfaces on the sides of the structure are

insulating material. The structure was analyzed using basis orders p =0,1,2.

In Figure 4.15 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis. The electrolyte filling the boxes has a

conductivity of _σ1 ₌_σ2 ₌_{4 S/m. The results match well with the analytic solution but have}

some error in the electric field when observing the fields closer to the fictious sphere. This may be because the field on the curved surfaces of the sphere varies more rapidly which makes the approximation more inaccurate as the field point approaches a surface of the sphere. The error decreases significantly as the basis order is increased.

In Figure 4.16 (a) and (b) are plotted the computed and analytic electric potential and electric field, respectively, along the z-axis. In this case, the bottom box and the bottom

half of the sphere are filled with an electrolyte of conductivity _σ1 ₌_{2 S/m and the top box}

along with the top half of the sphere are filled with an electrolyte of conductivity _σ2 ₌₃

S/m. The results are in good agreement with the analytic solution in (4.1) and (4.2). The results also show the same type of error as discussed in Figure 4.15 (a) and (b) with the electric field approximation losing accuracy as it approaches the curved surfaces of the sphere at a low basis order.

(43)

32

Figure 4.14: A two-meter long brick with two homogeneous electrolytes and a spherical electrolyte interface.

(44)

33 (a)

(b)

Figure 4.15: (a) Electric potential vs. position and (b) electric field vs. position for

various basis orders p and conductivity _σ1 ₌_σ2 ₌_{4 S/m}_{and meshed using quadrilateral}

(45)

34 (a)

(b)

Figure 4.16: (a) Electric potential vs. position and (b) electric field vs. position for

various basis orders p and conductivity _σ1₌_{2 S/m}_{, and}_σ2 ₌_{3 S/m}_{and meshed using}

(46)

35 4.5 Box with Linearly Varying Conductivity

Consider a cubic box aligned vertically with the z-axis and with sides of length 1

m. The bottom of the box is at z =0 m and the top of the box is at z =1 m. Electrodes

are placed at the top of the box and the bottom of the box, sourcing and sinking, respectively, 1 A. The surfaces on the sides are insulating. The electrolyte has a linear

conductivity profile _σ

( )

_r ₌_σb₊

(

_{σ σ}t₋ b

)

_z_where_σb ₌_{1 S/m}_and_σt ₌_{2 S/m}_{. The}

average potential over the bottom electrode is forced to be zero.

The analytic electric potential and electric field inside the box are

( )

₍

_tIT _b

₎

ln ( )_bz S σ σ σ σ Φ = − r (4.3) and

( )

IT

_{( )}

ˆ Sσ z = − E r z , (4.4)

respectively, where _IT_{is the electrode current and}_{S is the cross-sectional area of the}

box.

The cubic box is approximated by being divided evenly along z into N boxes where

n n

z N

= for n=  locates the surfaces bounding the regions in z. The conductivity 0, ,N

of the nth box is _σn ₌_σ

(

_r_centern

)

_{which is the average of the actual conductivity in the nth}

box and where _r_centern _{is the center of the nth box for} _n_{=  . A set of 5 models,}_{1, ,}_N

illustrated in Figure 4.17, was analyzed for N =  . 1, ,5

In Figure 4.18 (a) and (b) are plotted the computed and analytic electric potential and

electric field, respectively, along the z-axis with N = regions. The electrolyte filling the 1

box has a conductivity of _σ1 ₌_{1.5 S/m}_{. The results do not converge well to the analytic}

solution since a single, homogeneous electrolyte is a poor approximation of the linear conductivity profile. The electric field can only be piecewise-constant, thus it does not represent the analytic solution well.

(47)

36

conductivities of _σ1₌_{1.25 S/m}_and_σ2 ₌_{1.75 S/m}_{. The results are converging to the}

analytic solution. The electric field can only be piecewise-constant, thus it does not represent the analytic solution well.

electric field, respectively, along the z-axis with N = regions. The electrolytes filling the 3

boxes have conductivities of _σ1₌_{1.166 S/m}_,_σ2 ₌_{1.5 S/m}_{, and}_σ3 ₌_{1.833 S/m}_{. The}

results are converging nicely, but the electric field can only be piecewise-constant, thus it does not represent the analytic solution well.

electric field, respectively, along the z-axis with N = regions. The electrolytes filling the 4

boxes have conductivities of _σ1₌_{1.125 S/m}_, _σ2 ₌_{1.375 S/m}_, _σ3 ₌_{1.625 S/m}_{, and}

4 _{1.875 S/m}

σ = . The results are converging nicely, but the electric field can only be

piecewise-constant, thus it does not represent the analytic solution well.

Finally, in Figure 4.22 (a) and (b) are plotted the computed and analytic electric

potential and electric field, respectively, along the z-axis withN = regions. The 5

electrolytes filling the boxes with conductivities of _σ1 ₌_{1.1 S/m}_, _σ2 ₌_{1.3 S/m}_,

3 _{1.5 S/m}

σ = , _σ4 ₌_{1.7 S/m}_{, and}_σ5 ₌_{1.9 S/m}_{. The results are converging nicely, but the}

electric field can only be piecewise-constant, thus it does not represent the analytic solution well.

(48)

37

(a) (b)

(c) (d)

(e)

(49)

38 (a)

(b)

Figure 4.18: (a) Electric potential vs. position and (b) electric field vs. position for N = 1

sub-boxes and various basis orders p and meshed using quadrilateral cells of edge length 0.25 m.

(50)

39 (a)

(b)

sub-boxes and various basis orders p and meshed using quadrilateral cells of edge length 0.25 m.

(51)

40 (a)

(b)

sub-boxes and various basis orders p and meshed using quadrilateral cells of maximum edge length 0.25 m.

(52)

41 (a)

(b)

(53)

42 (a)

(b)