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Optimization of Particle in Cell (PIC) Method

Jayabalan, Varun LU (2024) FYSM34 20242
Department of Physics
Particle and nuclear physics
Abstract
Simulating the behavior of charged particles in linear accelerators requires calculating the effects of electromagnetic fields on the particles. One source of these fields is the particles themselves, through the so-called space charge self-field. To compute this field, the Poisson equation must be solved using the charge distribution of the particle bunch. In most cases, this cannot be done analytically, necessitating the use of numerical solvers. To address these issues, the Particle-In-Cell (PIC) method is employed to simulate charged particle dynamics adopting various numerical techniques. This thesis focuses on solving the Poisson equation using the Finite Difference Method (FDM) and the B-spline Differential Quadrature Method(DQM).... (More)
Simulating the behavior of charged particles in linear accelerators requires calculating the effects of electromagnetic fields on the particles. One source of these fields is the particles themselves, through the so-called space charge self-field. To compute this field, the Poisson equation must be solved using the charge distribution of the particle bunch. In most cases, this cannot be done analytically, necessitating the use of numerical solvers. To address these issues, the Particle-In-Cell (PIC) method is employed to simulate charged particle dynamics adopting various numerical techniques. This thesis focuses on solving the Poisson equation using the Finite Difference Method (FDM) and the B-spline Differential Quadrature Method(DQM). The FDM discretizes the computational domain into small intervals and computes the electric potential at each point based on the values at neighboring points. In contrast, the DQM employs B-spline functions to approximate derivatives by determining weighting coefficients. The B-spline approach integrates Shu’s method for efficient computation of these coefficients. Both Dirichlet and mixed boundary conditions are implemented and analyzed using these methods. Error analysis is performed to evaluate the accuracy of the two methods by varying the degrees of B-splines and grid sizes. A comparative analysis reveals that while FDM requires a higher number of grid points to achieve greater accuracy, it also demands more computational time. The B-spline DQM, the other hand, proves to be highly effective for complex geometries, offering superior accuracy with fewer grid points and reduced computation time. The results demonstrate that the differential quadrature method based on B-splines is an efficient and accurate approach for solving the 2D Poisson equation. (Less)
Popular Abstract
In particle accelerators, a large number of particles with the same charges are accelerated. This results in the spreading and divergence of the beams, a phenomenon known as the space charge effect. To address this issue, particles can be transported using the Particle-in-Cell (PIC) method. The PIC algorithm operates iteratively, with each cycle comprising four essential steps.
The first step involves generating particles and binning them within the computational domain. The second step calculates the weight of each particle within this domain. In the third step, the Poisson equation is solved to determine the potential—a critical aspect of the PIC algorithm. Numerous methods are available for solving partial differential equations... (More)
In particle accelerators, a large number of particles with the same charges are accelerated. This results in the spreading and divergence of the beams, a phenomenon known as the space charge effect. To address this issue, particles can be transported using the Particle-in-Cell (PIC) method. The PIC algorithm operates iteratively, with each cycle comprising four essential steps.
The first step involves generating particles and binning them within the computational domain. The second step calculates the weight of each particle within this domain. In the third step, the Poisson equation is solved to determine the potential—a critical aspect of the PIC algorithm. Numerous methods are available for solving partial differential equations (PDEs). This thesis employs two numerical techniques: the Finite Difference Method (FDM) and the Differential Quadrature Method (DQM).
Various methods can also be utilized for particle weighting. In this study, the B-spline method is applied in conjunction with the DQM approach. Subsequently, the error accuracy is calculated for both methods, and the results are compared. The findings demonstrate that the DQM method combined with the B-spline approach is an efficient and accurate technique for solving two-dimensional partial differential equations (Less)
Please use this url to cite or link to this publication:
author
Jayabalan, Varun LU
supervisor
organization
course
FYSM34 20242
year
type
H2 - Master's Degree (Two Years)
subject
keywords
ESS, PIC method, FDM, DQM, B-spline, DQM-B-spline, Poisson equation, Degree, space charge effect
language
English
id
9178903
date added to LUP
2024-12-19 08:00:14
date last changed
2024-12-19 08:00:14
@misc{9178903,
  abstract     = {{Simulating the behavior of charged particles in linear accelerators requires calculating the effects of electromagnetic fields on the particles. One source of these fields is the particles themselves, through the so-called space charge self-field. To compute this field, the Poisson equation must be solved using the charge distribution of the particle bunch. In most cases, this cannot be done analytically, necessitating the use of numerical solvers. To address these issues, the Particle-In-Cell (PIC) method is employed to simulate charged particle dynamics adopting various numerical techniques. This thesis focuses on solving the Poisson equation using the Finite Difference Method (FDM) and the B-spline Differential Quadrature Method(DQM). The FDM discretizes the computational domain into small intervals and computes the electric potential at each point based on the values at neighboring points. In contrast, the DQM employs B-spline functions to approximate derivatives by determining weighting coefficients. The B-spline approach integrates Shu’s method for efficient computation of these coefficients. Both Dirichlet and mixed boundary conditions are implemented and analyzed using these methods. Error analysis is performed to evaluate the accuracy of the two methods by varying the degrees of B-splines and grid sizes. A comparative analysis reveals that while FDM requires a higher number of grid points to achieve greater accuracy, it also demands more computational time. The B-spline DQM, the other hand, proves to be highly effective for complex geometries, offering superior accuracy with fewer grid points and reduced computation time. The results demonstrate that the differential quadrature method based on B-splines is an efficient and accurate approach for solving the 2D Poisson equation.}},
  author       = {{Jayabalan, Varun}},
  language     = {{eng}},
  note         = {{Student Paper}},
  title        = {{Optimization of Particle in Cell (PIC) Method}},
  year         = {{2024}},
}