High-order integral equations for electromagnetic problems in layered media with applications in biology and solar cells

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Zinser, B. (2016). High-order integral equations for electromagnetic problems in layered media with applications in biology and solar cells. Unc Charlotte Electronic Theses And Dissertations.
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Abstract

We present two distinct mathematical models where high-order integral equations are applied to electromagnetic problems. The first problem is to find the electric potential in and around ion channels and Janus particles. The second problem is to find the electromagnetic scattering caused by a set of simple geometric objects.In biology, we consider two types of inhomogeneities: the first one is a simple model of an ion channel which consists of a finite height cylindrical cavity embedded in a layered electrolytes/membrane environment, and the second one is a Janus particle made of two different semi-spherical dielectric materials. A boundary element method (BEM) for the Poisson-Boltzmann equation based on Muller's hyper-singular second kind integral equation formulation is used to accurately compute electrostatic potentials. The proposed BEM gives O(1) condition numbers and we show that the second order basis converges faster and is more accurate than the first order basis.For solar cells, we develop a Nystrom volume integral equation (VIE) method for calculating the electromagnetic scattering according to the Maxwell equations. The Cauchy principal values (CPVs) that arise from the VIE are computed using a finite size exclusion volume with explicit correction integrals. Outside the exclusion, the hyper-singular integrals are computed using an interpolated quadrature formulae with tensor-product quadrature nodes. We considered cubes, rectangles, cylinders, spheres, and ellipsoids. As the new quadrature weights are pre-calculated and tabulated, the integrals are calculated efficiently at runtime. Simulations with many scatterers demonstrate the efficiency of the interpolated quadrature formulae. We also demonstrate that the resulting VIE has high accuracy and p-convergence.

Details
Author

Zinser, Brian

Title

High-order integral equations for electromagnetic problems in layered media with applications in biology and solar cells

Physical Description

1 online resource (139 pages) : PDF

Date

2016

Degree Granting Institution

University of North Carolina at Charlotte

Abstract

We present two distinct mathematical models where high-order integral equations are applied to electromagnetic problems. The first problem is to find the electric potential in and around ion channels and Janus particles. The second problem is to find the electromagnetic scattering caused by a set of simple geometric objects.In biology, we consider two types of inhomogeneities: the first one is a simple model of an ion channel which consists of a finite height cylindrical cavity embedded in a layered electrolytes/membrane environment, and the second one is a Janus particle made of two different semi-spherical dielectric materials. A boundary element method (BEM) for the Poisson-Boltzmann equation based on Muller's hyper-singular second kind integral equation formulation is used to accurately compute electrostatic potentials. The proposed BEM gives O(1) condition numbers and we show that the second order basis converges faster and is more accurate than the first order basis.For solar cells, we develop a Nystrom volume integral equation (VIE) method for calculating the electromagnetic scattering according to the Maxwell equations. The Cauchy principal values (CPVs) that arise from the VIE are computed using a finite size exclusion volume with explicit correction integrals. Outside the exclusion, the hyper-singular integrals are computed using an interpolated quadrature formulae with tensor-product quadrature nodes. We considered cubes, rectangles, cylinders, spheres, and ellipsoids. As the new quadrature weights are pre-calculated and tabulated, the integrals are calculated efficiently at runtime. Simulations with many scatterers demonstrate the efficiency of the interpolated quadrature formulae. We also demonstrate that the resulting VIE has high accuracy and p-convergence.

Genre

doctoral dissertations

Subjects--Topics

Mathematics
Cytology
Electromagnetism

Degree

Ph.D.

Keywords

Dyadic Green's Function
Electromagnetic Scattering
Ion Channel
Janus Particles
Nystrom Method
Poisson Boltzmann Equation

Subject Area

Mathematics

Advisor(s)

Cai, Wei

Committee Members

Chen, Duan
Deng, Shaozhong
Gbur, Greg

Degree Note

Thesis (Ph.D.)--University of North Carolina at Charlotte, 2016.

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Rights Holder Information

Copyright is held by the author unless otherwise indicated.

Identifier

Zinser_uncc_0694D_11032

Permalink

http://hdl.handle.net/20.500.13093/etd:1326