diff --git a/m1_report_electro/M1_report_final.pdf b/m1_report_electro/M1_report_final.pdf new file mode 100644 index 0000000..9813e4b Binary files /dev/null and b/m1_report_electro/M1_report_final.pdf differ diff --git a/m1_report_electro/conclusion.tex b/m1_report_electro/conclusion.tex new file mode 100644 index 0000000..dfcd750 --- /dev/null +++ b/m1_report_electro/conclusion.tex @@ -0,0 +1,39 @@ +\section{Заключение} + +В работе проведено численное решение основной задачи электростатики --- нахождение распределения электрического поля в системе двух параллельных плоских электродов методом моментов (двумерная постановка). + +\textbf{Основные достижения:} + +\begin{enumerate} + \item \textbf{Реализация метода моментов} + \begin{itemize} + \item Реализована дискретизация контуров плоских электродов на равные сегменты + \item Составлена и решена система линейных уравнений для нахождения линейных плотностей зарядов сегментов + \item Корректно учтён собственный потенциал сегментов (диагональные элементы матрицы) + \end{itemize} + + \item \textbf{Расчёт электрического поля} + \begin{itemize} + \item Вычислена напряжённость электрического поля в пространстве между электродами + \item Построены картины силовых линий методом численного интегрирования + \item Построены эквипотенциальные линии (контуры постоянного потенциала) + \end{itemize} + + \item \textbf{Вычисление ёмкости} + \begin{itemize} + \item Рассчитана ёмкость системы двух параллельных пластин + \item Проведено сравнение с аналитической формулой для бесконечного плоского конденсатора + \item Исследована сходимость результатов при увеличении числа сегментов + \end{itemize} +\end{enumerate} + +\subsection{Физическая интерпретация} + +\textbf{Распределение заряда:} + +Линейная плотность заряда на пластинах неоднородна: заряд концентрируется вблизи краёв пластин --- это проявление \textit{краевого эффекта}. Метод моментов позволяет точно учесть эту неоднородность. + +\textbf{Краевые эффекты:} + +В плоском конденсаторе с пластинами конечной длины поле неоднородно вблизи краёв: силовые линии искривляются, выходя за пределы межэлектродного промежутка. Это приводит к увеличению ёмкости по сравнению с формулой для бесконечных пластин $C_\infty = \varepsilon_0 / d$. + diff --git a/m1_report_electro/intro.tex b/m1_report_electro/intro.tex new file mode 100644 index 0000000..adbf031 --- /dev/null +++ b/m1_report_electro/intro.tex @@ -0,0 +1,60 @@ +\newpage + +\section{Введение} + +\Task Два металлических электрода заданной формы поддерживаются при постоянной разности потенциалов $\Delta\varphi = V$. Полные заряды электродов противоположны и одинаковы по модулю. Необходимо найти распределение электрических полей в пространстве между электродами, построить картину силовых линий поля и рассчитать ёмкость системы. + +\Goal +\begin{enumerate} + \item Разработать метод триангуляции поверхности проводников для задания геометрии задачи + \item Реализовать метод моментов для нахождения распределения зарядов на поверхности проводников + \item Рассчитать распределение электрического поля в пространстве между электродами + \item Построить картину силовых линий и эквипотенциальных поверхностей + \item Вычислить ёмкость системы электродов и сравнить с аналитическими формулами +\end{enumerate} +\newpage +\section{Физическая постановка задачи} + + +Рассматривается \textbf{двумерная} система: два параллельных плоских электрода длиной $a$, расположенных горизонтально на расстоянии $d$ друг от друга. Задача однородна вдоль оси $z$, поэтому всё поле определяется поперечным сечением в плоскости $xy$. Проводники поддерживаются при постоянных потенциалах $\varphi_1 = +V/2$ и $\varphi_2 = -V/2$ (разность потенциалов $\Delta\varphi = V$). Требуется определить распределение линейной плотности зарядов на проводниках и электрическое поле в окружающем пространстве. + +\textbf{Геометрия задачи:} +\[ + L_1 = \{(x,\; +d/2) \mid x \in [-a/2,\; a/2]\}, \quad \varphi_1 = +V/2 +\] +\[ + L_2 = \{(x,\; -d/2) \mid x \in [-a/2,\; a/2]\}, \quad \varphi_2 = -V/2 +\] + +\textbf{Основные уравнения электростатики:} + +Потенциал электрического поля в вакууме при отсутствии свободных зарядов удовлетворяет \textbf{уравнению Лапласа}: +\[ +\Delta\varphi = 0 +\] + +с граничными условиями на поверхностях проводников: +\[ +\varphi\big|_{S_1} = \varphi_1 = +\dfrac{V}{2}, \quad \varphi\big|_{S_2} = \varphi_2 = -\dfrac{V}{2} +\] + +и условием на бесконечности: +\[ +\varphi(\vec{r}) \to 0 \quad \text{при} \quad |\vec{r}| \to \infty +\] + +\textbf{Допущения модели:} +\begin{itemize} + \item Проводники идеальные: в двумерной постановке заряды сосредоточены на граничных контурах + \item Среда между проводниками -- вакуум ($\varepsilon = 1$) + \item Задача стационарная (токи отсутствуют) + \item Электроды изолированы друг от друга, полные заряды равны $+Q$ и $-Q$ +\end{itemize} + +\textbf{Физические константы:} +\begin{itemize} + \item $\varepsilon_0 = 8.854 \times 10^{-12}$ Ф/м -- электрическая постоянная + \item $k = \dfrac{1}{4\pi\varepsilon_0} \approx 8.988 \times 10^9$ Н$\cdot$м$^2$/Кл$^2$ -- коэффициент Кулона +\end{itemize} + +\newpage diff --git a/m1_report_electro/main.tex b/m1_report_electro/main.tex new file mode 100644 index 0000000..d71d0cf --- /dev/null +++ b/m1_report_electro/main.tex @@ -0,0 +1,16 @@ +\input{preamble} + +\begin{document} + +\input{title} +\pagestyle{main} +{ +\centering +\tableofcontents +} +\input{intro} +\input{models/models} +\input{numerical_methods} +\input{conclusion} + +\end{document} diff --git a/m1_report_electro/models/capacitance.tex b/m1_report_electro/models/capacitance.tex new file mode 100644 index 0000000..ef18d2c --- /dev/null +++ b/m1_report_electro/models/capacitance.tex @@ -0,0 +1,44 @@ +\subsection{Ёмкость системы проводников} + +Электрическая ёмкость характеризует способность системы проводников накапливать электрический заряд при заданной разности потенциалов. + +\textbf{Определение ёмкости} + +Для системы из двух проводников с зарядами $+Q$ и $-Q$ при разности потенциалов $V$: +\[ +\boxed{C = \dfrac{Q}{V}} +\] + +Полный заряд первого проводника вычисляется как сумма зарядов всех его сегментов: +\[ +Q = \sum_{j \in L_1} q_j +\] + +\textbf{Энергия электростатического поля} + +Энергия системы зарядов выражается через потенциалы и заряды элементов: +\[ +W = \dfrac{1}{2} \sum_{i=1}^{N} q_i \varphi_i +\] + +Для двух проводников: +\[ +W = \dfrac{1}{2} Q V = \dfrac{CV^2}{2} = \dfrac{Q^2}{2C} +\] + +Энергию можно также вычислить через напряжённость поля: +\[ +W = \dfrac{\varepsilon_0}{2} \int \vec{E}^2 \, dV +\] + +Совпадение двух способов вычисления энергии служит проверкой корректности решения. + +\textbf{Аналитическая формула для плоского конденсатора} + +\textit{Плоский конденсатор} --- бесконечные пластины на расстоянии $d$ (на единицу длины): +\[ +\boxed{C_{\infty} = \dfrac{\varepsilon_0}{d}} +\] + +Для пластин конечной длины $a$ необходимо учитывать \textbf{краевые эффекты} --- искривление поля вблизи торцов пластин, которое увеличивает ёмкость по сравнению с формулой для бесконечных пластин. Численный метод моментов автоматически учитывает эти эффекты. + diff --git a/m1_report_electro/models/method_of_moments.tex b/m1_report_electro/models/method_of_moments.tex new file mode 100644 index 0000000..80af4af --- /dev/null +++ b/m1_report_electro/models/method_of_moments.tex @@ -0,0 +1,74 @@ +\subsection{Метод моментов} + +Метод моментов --- численный метод решения интегральных уравнений электростатики, основанный на дискретизации контура проводников. + +\textbf{Интегральное уравнение задачи} + +Граничное условие постоянства потенциала в центре $i$-го сегмента: +\[ +-\dfrac{1}{2\pi\varepsilon_0} \int_{L} \lambda(s') \ln|\vec{r}_i - \vec{r}'| \, ds' = \varphi_i, \quad \vec{r}_i \in L_i +\] + +Это \textbf{интегральное уравнение Фредгольма первого рода} относительно неизвестной линейной плотности заряда $\lambda(s')$. + +\textbf{Дискретизация контура} + +Контур каждого электрода разбивается на $N_{\text{пл}}$ сегментов равной длины $\Delta l$. Каждый сегмент $j$ характеризуется: +\begin{itemize} + \item Центром $\vec{r}_j$ (середина отрезка) + \item Длиной $\Delta l_j$ + \item Зарядом $q_j = \lambda_j \cdot \Delta l_j$ (линейная плотность заряда постоянна в пределах сегмента) +\end{itemize} + +Общее число сегментов: $N = 2 N_{\text{пл}}$. + +\textbf{Дискретная форма интегрального уравнения} + +Потенциал в центре $i$-го сегмента от всех остальных ($j \neq i$): +\[ +\varphi_i = -\dfrac{1}{2\pi\varepsilon_0} \sum_{\substack{j=1 \\ j \neq i}}^{N} q_j \ln|\vec{r}_i - \vec{r}_j| + \varphi_i^{\mathrm{self}} +\] + +\textbf{Собственный потенциал сегмента} + +Равномерно заряженный отрезок длиной $\Delta l_i$ создаёт в своём центре потенциал: +\[ +\varphi_i^{\mathrm{self}} = \dfrac{q_i}{2\pi\varepsilon_0}\!\left(1 - \ln\dfrac{\Delta l_i}{2}\right) +\] + +\textbf{Система линейных уравнений} + +Записывая условие постоянства потенциала для каждого из $N$ сегментов, получаем СЛАУ: +\[ +\boxed{\sum_{j=1}^{N} A_{ij} q_j = \varphi_i, \quad i = 1, 2, \ldots, N} +\] + +Элементы матрицы взаимодействия $\mathbf{A}$: +\[ +A_{ij} = \begin{cases} +-\dfrac{1}{2\pi\varepsilon_0} \ln|\vec{r}_i - \vec{r}_j|, & i \neq j \\[1em] +\dfrac{1}{2\pi\varepsilon_0}\!\left(1 - \ln\dfrac{\Delta l_i}{2}\right), & i = j +\end{cases} +\] + +Правая часть: +\[ +\varphi_i = \begin{cases} ++V/2, & \text{элемент } i \in L_1 \\ +-V/2, & \text{элемент } i \in L_2 +\end{cases} +\] + +В матричной форме: $\mathbf{A}\vec{q} = \vec{\varphi}$, где $\mathbf{A} \in \mathbb{R}^{N \times N}$ --- симметричная матрица. + +\textbf{Восстановление поля после решения} + +После нахождения зарядов $q_j$ электрическое поле в произвольной точке $\vec{r}$ вне проводников: +\[ +\vec{E}(\vec{r}) = \dfrac{1}{2\pi\varepsilon_0} \sum_{j=1}^{N} q_j \dfrac{\vec{r} - \vec{r}_j}{|\vec{r} - \vec{r}_j|^2} +\] + +Потенциал: +\[ +\varphi(\vec{r}) = -\dfrac{1}{2\pi\varepsilon_0} \sum_{j=1}^{N} q_j \ln|\vec{r} - \vec{r}_j| +\] diff --git a/m1_report_electro/models/models.tex b/m1_report_electro/models/models.tex new file mode 100644 index 0000000..482c3f6 --- /dev/null +++ b/m1_report_electro/models/models.tex @@ -0,0 +1,8 @@ +\section{Физические модели} + +\input{models/potential_theory} +\newpage +\input{models/method_of_moments} +\newpage +\input{models/capacitance} +\newpage diff --git a/m1_report_electro/models/potential_theory.tex b/m1_report_electro/models/potential_theory.tex new file mode 100644 index 0000000..cf4509a --- /dev/null +++ b/m1_report_electro/models/potential_theory.tex @@ -0,0 +1,71 @@ +\subsection{Теория потенциала и электрическое поле} + +\textbf{Потенциал системы зарядов} + +В двумерной задаче электрический потенциал, создаваемый распределением линейной плотности заряда $\lambda(s')$ вдоль контура $L$, в точке $\vec{r}$ определяется: +\[ +\varphi(\vec{r}) = -\dfrac{1}{2\pi\varepsilon_0} \int_L \lambda(s') \ln|\vec{r} - \vec{r}'| \, ds' +\] + +Для системы из двух электродов с контурами $L_1$ и $L_2$: +\[ +\varphi(\vec{r}) = -\dfrac{1}{2\pi\varepsilon_0} \left[ \int_{L_1} \lambda_1(s') \ln|\vec{r} - \vec{r}'| \, ds' + \int_{L_2} \lambda_2(s') \ln|\vec{r} - \vec{r}'| \, ds' \right] +\] + +\textbf{Граничные условия на поверхности проводника} + +На поверхности идеального проводника выполняются условия: +\begin{enumerate} + \item Потенциал постоянен: $\varphi\big|_S = \varphi_0$ + \item Электрическое поле перпендикулярно поверхности: $\vec{E}_\tau\big|_S = 0$ + \item Нормальная компонента поля связана с плотностью заряда: + \[ + E_n\big|_S = \dfrac{\sigma}{\varepsilon_0} + \] +\end{enumerate} + +\textbf{Электрическое поле} + +Напряжённость электрического поля выражается через потенциал: +\[ +\vec{E}(\vec{r}) = -\nabla\varphi(\vec{r}) +\] + +Для дискретного набора заряженных сегментов с зарядами $q_j$ в центрах $\vec{r}_j$ (двумерный случай): +\[ +\vec{E}(\vec{r}) = \dfrac{1}{2\pi\varepsilon_0} \sum_{j=1}^{N} q_j \dfrac{\vec{r} - \vec{r}_j}{|\vec{r} - \vec{r}_j|^2} +\] + +\textbf{Силовые линии электрического поля} + +Силовые линии -- кривые, касательные к которым в каждой точке совпадают с направлением вектора $\vec{E}$. Уравнение силовых линий: +\[ +\dfrac{dx}{E_x} = \dfrac{dy}{E_y} = \dfrac{dz}{E_z} +\] + +Для построения силовых линий численно интегрируется система ОДУ: +\[ +\dfrac{d\vec{r}}{ds} = \dfrac{\vec{E}(\vec{r})}{|\vec{E}(\vec{r})|} +\] + +где $s$ -- параметр вдоль линии (длина дуги). + +Силовые линии начинаются на положительных зарядах и заканчиваются на отрицательных. Плотность силовых линий пропорциональна напряжённости поля. + +\textbf{Эквипотенциальные поверхности} + +Эквипотенциальные поверхности -- множества точек с одинаковым потенциалом: +\[ +\varphi(\vec{r}) = C = const +\] + +Они всюду перпендикулярны силовым линиям. Густота эквипотенциальных поверхностей пропорциональна напряжённости поля. + +\textbf{Аналитические решения для простых конфигураций} + +\textit{Плоской конденсатор} (бесконечные пластины на расстоянии $d$): +\[ +\varphi(y) = V\dfrac{y}{d}, \quad E_y = \dfrac{V}{d} = \text{const} +\] + +Это решение используется для верификации численного метода (в пределе $a \gg d$). diff --git a/m1_report_electro/numerical_methods.tex b/m1_report_electro/numerical_methods.tex new file mode 100644 index 0000000..0b303a0 --- /dev/null +++ b/m1_report_electro/numerical_methods.tex @@ -0,0 +1,93 @@ +\section{Численные методы решения} + +Численная реализация метода моментов включает два основных этапа: дискретизацию контуров электродов и сборку и решение СЛАУ, после чего восстанавливается поле и строится визуализация. + +\subsection{Дискретизация контуров электродов} + +В двумерной постановке каждый электрод представлен горизонтальным отрезком длиной $a$. Для численного решения каждый контур разбивается на $N_{\text{сег}}$ сегментов равной длины: +\[ +\Delta l = \dfrac{a}{N_{\text{сег}}} +\] + +Для каждого сегмента $j$ определяются: +\begin{itemize} + \item Центр сегмента $\vec{r}_j = (x_j,\, y_j)$ + \item Длина $\Delta l_j = \Delta l$ + \item Заряд $q_j = \lambda_j \cdot \Delta l_j$, где $\lambda_j$ --- линейная плотность заряда на сегменте +\end{itemize} + +Общее число сегментов: $N = 2 N_{\text{сег}}$ (по $N_{\text{сег}}$ на каждый электрод). + +\subsection{Сборка и решение системы уравнений} + +\textbf{Формирование матрицы} + +Матрица взаимодействия $\mathbf{A}$ размера $N \times N$ заполняется по формулам (из двумерного выражения для потенциала): +\[ +A_{ij} = \begin{cases} +-\dfrac{1}{2\pi\varepsilon_0} \ln|\vec{r}_i - \vec{r}_j|, & i \neq j \\[1em] +\dfrac{1}{2\pi\varepsilon_0}\!\left(1 - \ln\dfrac{\Delta l_i}{2}\right), & i = j +\end{cases} +\] + +Правая часть: +\[ +\varphi_i = \begin{cases} ++V/2, & \text{сегмент } i \in L_1 \\ +-V/2, & \text{сегмент } i \in L_2 +\end{cases} +\] + +Вычислительная сложность формирования матрицы: $O(N^2)$. + +\textbf{Решение СЛАУ} + +Система $\mathbf{A}\vec{q} = \vec{\varphi}$ решается с помощью \texttt{scipy.linalg.solve}). + +Сложность решения: $O(N^3)$, что приемлемо при $N \le 10^3$ сегментов. + +\subsection{Построение силовых линий} + +\textbf{Алгоритм трассировки силовых линий} + +Силовые линии строятся интегрированием системы ОДУ: +\[ +\dfrac{d\vec{r}}{ds} = \pm \dfrac{\vec{E}(\vec{r})}{|\vec{E}(\vec{r})|} +\] + +где знак $+$ для линий, начинающихся на положительном проводнике, $-$ -- на отрицательном. + +Алгоритм: +\begin{enumerate} + \item Выбираем начальные точки на поверхности положительного проводника (равномерно по площади или по плотности заряда) + \item Для каждой начальной точки интегрируем уравнение силовой линии методом Рунге--Кутты 4-го порядка: + \begin{align*} + \vec{k}_1 &= h \cdot \hat{E}(\vec{r}_n) \\ + \vec{k}_2 &= h \cdot \hat{E}(\vec{r}_n + \vec{k}_1/2) \\ + \vec{k}_3 &= h \cdot \hat{E}(\vec{r}_n + \vec{k}_2/2) \\ + \vec{k}_4 &= h \cdot \hat{E}(\vec{r}_n + \vec{k}_3) + \end{align*} + \[ + \vec{r}_{n+1} = \vec{r}_n + \dfrac{1}{6}(\vec{k}_1 + 2\vec{k}_2 + 2\vec{k}_3 + \vec{k}_4) + \] + где $\hat{E} = \vec{E}/|\vec{E}|$ -- единичный вектор поля + + \item Интегрирование прекращается при: + \begin{itemize} + \item достижении отрицательного проводника ($|\vec{r} - \vec{r}_{S_2}| < \delta$) + \item удалении на большое расстояние от системы ($|\vec{r}| > L_{max}$) + \item превышении максимального числа шагов + \end{itemize} +\end{enumerate} + +На каждом шаге интегрирования поле $\vec{E}(\vec{r})$ вычисляется суммированием вкладов всех $N$ зарядов, что требует $O(N)$ операций на шаг. + +\textbf{Построение эквипотенциальных линий} + +Для двумерного сечения ($z = 0$) эквипотенциальные линии строятся: +\begin{enumerate} + \item Вычисляем потенциал на равномерной сетке $(x_i, y_j)$ + \item Применяем алгоритм контурных линий (marching squares / matplotlib \texttt{contour}) + \item Выбираем набор значений потенциала: $\varphi_k = -V/2 + kV/(M+1)$, $k = 1, \ldots, M$ +\end{enumerate} + diff --git a/m1_report_electro/preamble.tex b/m1_report_electro/preamble.tex new file mode 100644 index 0000000..6a8b988 --- /dev/null +++ b/m1_report_electro/preamble.tex @@ -0,0 +1,100 @@ +\documentclass[a4paper,12pt]{article} + +\usepackage{geometry} +% задание полей текста +\geometry{left=10mm,right=10mm,top=20mm,bottom=20mm} +\usepackage{listings} +\usepackage{cmap} % поиск в PDF +\usepackage[T2A]{fontenc} % кодировка +\usepackage[utf8]{inputenc} +\usepackage[english, russian]{babel} % локализация и переносы +\usepackage{natbib} + +\usepackage{graphicx} +\graphicspath{{pix/}} +\usepackage{pgfplots} +\usetikzlibrary{arrows} +\pgfplotsset{width=10cm,compat=1.18} +\pgfkeys{/pgf/trig format=rad} + + +\usepackage{xcolor} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{bm} +\usepackage{enumerate} +\usepackage[normalem]{ulem} +\usepackage{titlesec} + +% Настройка форматирования section для центрирования без растягивания +\titleformat{\section} + {\normalfont\Large\bfseries\centering}{\thesection}{1em}{} +\titleformat{\subsection} + {\normalfont\large\bfseries\centering}{\thesubsection}{1em}{} +\titleformat{\subsubsection} + {\normalfont\normalsize\bfseries\centering}{\thesubsubsection}{1em}{} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + citecolor=blue +} + + +\setlength\parindent{0pt} + + +\sloppy % строго соблюдать границы текста +\linespread{1.2} % коэффициент межстрочного интервала +\setlength{\parskip}{0.5em} % вертик. интервал между абзацами + +\setcounter{secnumdepth}{0} % отключение нумерации разделов +\binoppenalty=1000 % уменьшение переносов в формулах + +% объявление новых макрокоманд + +\newcommand{\Def}{\textbf{Def.} } +\newcommand{\Th}{\textbf{Теорема.} } +\newcommand{\Thbd}{\textbf{Теорема (б/д).} } +\newcommand{\Theor}[1]{\textbf{Теорема ({#1}).} } +\newcommand{\Theorbd}[1]{\textbf{Теорема ({#1}) (б/д).} } +\newcommand{\Consequence}{\textbf{Следствие.} } +\newcommand{\Remind}{\textbf{Remind.} } +\newcommand{\Note}{\textbf{Note.} } +\newcommand{\Statement}{\textbf{Утверждение.} } +\newcommand{\Prop}{\textbf{Свойство:} } +\newcommand{\Props}{\textbf{Свойства:} } +\newcommand{\Proof}{\textbf{Доказательство:} } +\newcommand{\Prooff}{\textbf{Доказать:} } +\newcommand{\Solution}{\textbf{Решение:} } +\newcommand{\Alg}{\textbf{Algorithm.} } +\newcommand{\Lemma}{\textbf{Лемма.} } +\newcommand{\Example}{\textbf{Пример:} } +\newcommand{\Task}{\textbf{Задача.} } +\newcommand{\Goal}{\textbf{Цель работы:} } +\newcommand{\Solve}{\textbf{Решение:} } +\newcommand{\Examples}{\textbf{Примеры.} } + +\allowdisplaybreaks[4] + +%\renewcommand\thesection{\arabic{section}} + +\newcommand{\Endproof}{$\blacksquare$ } + +\newcommand{\tr}{\text{tr}} +\newcommand{\Le}{\leqslant} % русский стиль нестрогих неравенств +\newcommand{\Ge}{\geqslant} +% облегчение математических обозначений +\newcommand{\A}{\mathcal{A}} +\newcommand{\M}{\mathcal{M}} +\newcommand{\F}{\mathcal{F}} +\newcommand{\Gs}{\mathcal{G}} +\newcommand{\R}{\mathbb{R}} +\newcommand{\N}{\mathbb{N}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\Norm}{\mathcal{N}} +%\newcommand{\C}{\mathbb{C}} +\newcommand{\ind}{\perp \!\!\! \perp} diff --git a/m1_report_electro/title.tex b/m1_report_electro/title.tex new file mode 100644 index 0000000..2f2670c --- /dev/null +++ b/m1_report_electro/title.tex @@ -0,0 +1,18 @@ +\begin{titlepage} + \centering + {\scshape\Large Московский физико-технический институт \par} + \vspace{1cm} + {\scshape\Large Высшая школа программной инженерии \par} + \vspace{2cm} + {\huge \textbf{М1. Основная задача электростатики} \par} + \vspace{1.5cm} + {\Large \textit{Численное решение задачи электростатики \\ + методом моментов для системы проводящих тел} \par} + \vspace{2cm} + {\Large Выполнили студенты Б13-402: \par} + {\Large Жердев Егор \par} + {\Large Савельев Данил \par} + \vfill + {\large Долгопрудный \par} + {\large \today \par} +\end{titlepage} diff --git a/pyproject.toml b/pyproject.toml index 1e55ecc..0186d1b 100644 --- a/pyproject.toml +++ b/pyproject.toml @@ -8,6 +8,7 @@ dependencies = [ "altair>=5.5.0", "numpy>=2.3.3", "pandas>=2.3.2", + "pandas-stubs~=2.3.2", "pyarrow>=21.0.0, <22.0.0", "python-dateutil>=2.9.0.post0", "scipy>=1.16.2", diff --git a/src/main.py b/src/main.py index d0cfb45..56f3489 100644 --- a/src/main.py +++ b/src/main.py @@ -1,6 +1,7 @@ import streamlit from models.billiard import page as billiard_page +from models.electrostatics import page as electric_page from models.ideal_gas import page as ideal_gas_page from models.pendulum import page as pendulum_page from models.rolling_ball import page as rolling_ball_page @@ -24,6 +25,11 @@ def main() -> None: title="M7Б Статистика идеального газа со столкновениями частиц", url_path="ideal_gas", ), + streamlit.Page( + electric_page, + title="М1 Электростатика - Метод моментов", + url_path="electrostatics", + ), ]) pages.run() diff --git a/src/models/electrostatics/__init__.py b/src/models/electrostatics/__init__.py new file mode 100644 index 0000000..54018ae --- /dev/null +++ b/src/models/electrostatics/__init__.py @@ -0,0 +1,108 @@ +import numpy as np +import streamlit as st + +from models.electrostatics.charts import ( + create_charge_distribution_chart, + create_field_lines_chart, + create_field_magnitude_chart, + create_potential_contour_chart, +) +from models.electrostatics.objects import ( + MethodOfMomentsModel, + calculate_capacitance, +) +from models.electrostatics.sidebar import create_electrostatics_sidebar +from models.electrostatics.utils import ( + create_charge_distribution_dataframe, + create_field_dataframe, +) + + +def page() -> None: # noqa: PLR0914 + st.set_page_config( + page_title="М1. Электростатика - Метод моментов", layout="wide" + ) + st.title("М1. Электростатика - Метод моментов") + st.write( + "Численное решение основной задачи электростатики для двух " + "параллельных пластин методом моментов" + ) + + model = MethodOfMomentsModel() + params = create_electrostatics_sidebar() + + with st.spinner("Решение системы уравнений методом моментов..."): + r1, r2, q1, q2 = model.solve_charge_distribution(params) + + capacitance = calculate_capacitance(q1, params.voltage) + + st.header("Результаты расчёта") + + col1, col2, col3 = st.columns(3) + with col1: + st.metric("Ёмкость системы", f"{capacitance:.6e} Ф/м") + with col2: + analytical_c = model.epsilon_0 / params.plate_distance + st.metric( + "Аналитическая ёмкость (∞ пластины)", + f"{analytical_c:.6e} Ф/м", + ) + with col3: + ratio = capacitance / analytical_c + st.metric("Отношение C/C_∞", f"{ratio:.4f}") + + st.header("Распределение зарядов на электродах") + charge_chart = create_charge_distribution_chart(r1, r2, q1, q2) + st.altair_chart(charge_chart, use_container_width=True) + + with st.expander("Таблица распределения зарядов"): + charge_df = create_charge_distribution_dataframe(r1, r2, q1, q2) + st.dataframe(charge_df, use_container_width=True) + + with st.spinner("Вычисление электрического поля..."): + a = params.plate_length + d = params.plate_distance + n_grid = params.field_resolution + + x_range = np.linspace(-a, a, n_grid) + y_range = np.linspace(-d * 1.5, d * 1.5, n_grid) + x_grid, y_grid = np.meshgrid(x_range, y_range) + + ex, ey = model.compute_electric_field(r1, r2, q1, q2, x_grid, y_grid) + phi = model.compute_potential(r1, r2, q1, q2, x_grid, y_grid) + + st.header("Визуализация электрического поля") + + field_magnitude_chart = create_field_magnitude_chart( + x_grid, + y_grid, + ex, + ey, + {"plate_length": a, "plate_distance": d}, + ) + st.altair_chart(field_magnitude_chart, use_container_width=True) + + potential_chart = create_potential_contour_chart( + x_grid, + y_grid, + phi, + r1, + r2, + {"plate_length": a, "plate_distance": d}, + ) + st.altair_chart(potential_chart, use_container_width=True) + + with st.expander("Таблица значений поля"): + field_df = create_field_dataframe(x_grid, y_grid, ex, ey, phi) + st.dataframe(field_df, use_container_width=True) + + st.header("Силовые линии электрического поля") + with st.spinner("Построение силовых линий..."): + field_lines_chart = create_field_lines_chart( + x_grid, + y_grid, + ex, + ey, + {"plate_length": a, "plate_distance": d}, + ) + st.altair_chart(field_lines_chart, use_container_width=True) diff --git a/src/models/electrostatics/charts.py b/src/models/electrostatics/charts.py new file mode 100644 index 0000000..478e441 --- /dev/null +++ b/src/models/electrostatics/charts.py @@ -0,0 +1,368 @@ +import altair as alt +import numpy as np +import pandas as pd +from scipy.interpolate import RegularGridInterpolator + + +def create_charge_distribution_chart( + r1: np.ndarray, + r2: np.ndarray, + q1: np.ndarray, + q2: np.ndarray, +) -> alt.Chart: + data = [] + + for x, q in zip(r1[:, 0], q1, strict=False): + data.append({"X (m)": x, "Charge (C)": q, "Plate": "Upper (+V/2)"}) + + for x, q in zip(r2[:, 0], q2, strict=False): + data.append({"X (m)": x, "Charge (C)": q, "Plate": "Lower (-V/2)"}) + + df = pd.DataFrame(data) + + return ( + alt.Chart(df) + .mark_line(point=True) + .encode( + x=alt.X("X (m):Q", title="Позиция X (м)"), + y=alt.Y("Charge (C):Q", title="Заряд сегмента (Кл)"), + color=alt.Color( + "Plate:N", + title="Пластина", + scale=alt.Scale(scheme="category10"), + ), + tooltip=["X (m)", "Charge (C)", "Plate"], + ) + .properties( + title="Распределение зарядов на электродах", + width=800, + height=400, + ) + ) + + +MAX_TOTAL_LINE_POINTS = 4500 +MIN_LINE_LENGTH = 8 +MIN_FIELD_MAGNITUDE = 1e-15 +DENSITY = 1.7 +MAX_STEPS_PER_DIR = 500 +MIN_STRIDE_TARGET = 2 + + +def _trace_streamlines_density( # noqa: PLR0914,PLR0915 + x_1d: np.ndarray, + y_1d: np.ndarray, + ex: np.ndarray, + ey: np.ndarray, + plate_a: float, + plate_d: float, + density: float = DENSITY, +) -> list[list[tuple[float, float]]]: + n_y, n_x = ex.shape + mask_ny = max(2, int(n_y * density)) + mask_nx = max(2, int(n_x * density)) + mask = np.zeros((mask_ny, mask_nx), dtype=bool) + + x_min, x_max = float(x_1d[0]), float(x_1d[-1]) + y_min, y_max = float(y_1d[0]), float(y_1d[-1]) + ds = 0.5 * float(min(x_1d[1] - x_1d[0], abs(y_1d[1] - y_1d[0]))) + plate_eps = max(ds, 5e-3) + + interp_ex = RegularGridInterpolator( + (y_1d, x_1d), ex, bounds_error=False, fill_value=0.0 + ) + interp_ey = RegularGridInterpolator( + (y_1d, x_1d), ey, bounds_error=False, fill_value=0.0 + ) + + def to_mask(x: float, y: float) -> tuple[int, int]: + fx = (x - x_min) / (x_max - x_min) + fy = (y - y_min) / (y_max - y_min) + return int(fy * mask_ny), int(fx * mask_nx) + + def in_bounds(x: float, y: float) -> bool: + return x_min <= x <= x_max and y_min <= y <= y_max + + def near_plate(x: float, y: float) -> bool: + return ( + abs(x) <= plate_a / 2 + plate_eps + and min(abs(y - plate_d / 2), abs(y + plate_d / 2)) < plate_eps + ) + + def trace_dir( + x0: float, y0: float, sign: float, own_cells: set[tuple[int, int]] + ) -> tuple[list[tuple[float, float]], set[tuple[int, int]]]: + pts: list[tuple[float, float]] = [] + new_cells: set[tuple[int, int]] = set() + x, y = x0, y0 + for _ in range(MAX_STEPS_PER_DIR): + ex_v = float(interp_ex([[y, x]])[0]) + ey_v = float(interp_ey([[y, x]])[0]) + mag = (ex_v**2 + ey_v**2) ** 0.5 + if mag < MIN_FIELD_MAGNITUDE: + break + x += sign * ds * ex_v / mag + y += sign * ds * ey_v / mag + if not in_bounds(x, y): + break + if near_plate(x, y): + pts.append((x, y)) + break + iy, ix = to_mask(x, y) + if 0 <= iy < mask_ny and 0 <= ix < mask_nx: + cell = (iy, ix) + if cell not in own_cells and mask[iy, ix]: + break + own_cells.add(cell) + new_cells.add(cell) + pts.append((x, y)) + return pts, new_cells + + cy, cx = mask_ny / 2, mask_nx / 2 + candidates = [(iy, ix) for iy in range(mask_ny) for ix in range(mask_nx)] + candidates.sort(key=lambda c: (c[0] - cy) ** 2 + (c[1] - cx) ** 2) + + streamlines: list[list[tuple[float, float]]] = [] + for iy_c, ix_c in candidates: + if mask[iy_c, ix_c]: + continue + x_seed = x_min + (ix_c + 0.5) / mask_nx * (x_max - x_min) + y_seed = y_min + (iy_c + 0.5) / mask_ny * (y_max - y_min) + if near_plate(x_seed, y_seed): + mask[iy_c, ix_c] = True + continue + + ex_v = float(interp_ex([[y_seed, x_seed]])[0]) + ey_v = float(interp_ey([[y_seed, x_seed]])[0]) + if (ex_v**2 + ey_v**2) ** 0.5 < MIN_FIELD_MAGNITUDE: + mask[iy_c, ix_c] = True + continue + + own: set[tuple[int, int]] = {(iy_c, ix_c)} + fwd_pts, fwd_cells = trace_dir(x_seed, y_seed, +1, own) + bwd_pts, bwd_cells = trace_dir(x_seed, y_seed, -1, own) + + full_line = [*reversed(bwd_pts), (x_seed, y_seed), *fwd_pts] + if len(full_line) < MIN_LINE_LENGTH: + mask[iy_c, ix_c] = True + continue + + for cy_, cx_ in fwd_cells | bwd_cells | {(iy_c, ix_c)}: + mask[cy_, cx_] = True + streamlines.append(full_line) + + return streamlines + + +def create_field_lines_chart( # noqa: PLR0914 + x_grid: np.ndarray, + y_grid: np.ndarray, + ex: np.ndarray, + ey: np.ndarray, + params_dict: dict, +) -> alt.Chart: + a = params_dict["plate_length"] + d = params_dict["plate_distance"] + + x_1d = x_grid[0, :] + y_1d = y_grid[:, 0] + + streamlines = _trace_streamlines_density(x_1d, y_1d, ex, ey, a, d) + + n_lines = max(1, len(streamlines)) + target_per_line = max(5, MAX_TOTAL_LINE_POINTS // n_lines) + + rows: list[dict] = [] + while target_per_line >= MIN_STRIDE_TARGET: + rows = [] + for idx, line in enumerate(streamlines): + n = len(line) + if n == 0: + continue + stride = max(1, n // target_per_line) + sub = line[::stride] + if sub[-1] != line[-1]: + sub = [*sub, line[-1]] + for i, (x, y) in enumerate(sub): + rows.append({ + "X": float(x), + "Y": float(y), + "Line": idx, + "Order": i, + }) + if len(rows) <= MAX_TOTAL_LINE_POINTS: + break + target_per_line = max(MIN_STRIDE_TARGET, target_per_line - 2) + df_lines = pd.DataFrame(rows) + + plates_data = [ + {"X": -a / 2, "Y": d / 2, "Plate": "+V/2", "Order": 0}, + {"X": a / 2, "Y": d / 2, "Plate": "+V/2", "Order": 1}, + {"X": -a / 2, "Y": -d / 2, "Plate": "-V/2", "Order": 0}, + {"X": a / 2, "Y": -d / 2, "Plate": "-V/2", "Order": 1}, + ] + df_plates = pd.DataFrame(plates_data) + + if not df_lines.empty: + x_pad = 0.05 * a + y_pad = 0.1 * d + x_min = min(float(df_lines["X"].min()) - x_pad, -a) + x_max = max(float(df_lines["X"].max()) + x_pad, a) + y_min = min(float(df_lines["Y"].min()) - y_pad, -d * 1.2) + y_max = max(float(df_lines["Y"].max()) + y_pad, d * 1.2) + else: + x_min, x_max = -a, a + y_min, y_max = -d * 1.5, d * 1.5 + + lines_chart = ( + alt.Chart(df_lines) + .mark_line(strokeWidth=1.4, color="#3B5BA5", opacity=0.85) + .encode( + x=alt.X( + "X:Q", + title="X (м)", + scale=alt.Scale(domain=[x_min, x_max]), + ), + y=alt.Y( + "Y:Q", + title="Y (м)", + scale=alt.Scale(domain=[y_min, y_max]), + ), + order="Order:Q", + detail="Line:N", + ) + ) + + plates_chart = ( + alt.Chart(df_plates) + .mark_line(strokeWidth=5) + .encode( + x="X:Q", + y="Y:Q", + order="Order:Q", + detail="Plate:N", + color=alt.Color( + "Plate:N", + title="Пластины", + scale=alt.Scale( + domain=["+V/2", "-V/2"], + range=["#d62728", "#1f77b4"], + ), + ), + ) + ) + + return (lines_chart + plates_chart).properties( + title="Силовые линии электрического поля", + width=800, + height=500, + ) + + +def create_potential_contour_chart( + x_grid: np.ndarray, + y_grid: np.ndarray, + phi: np.ndarray, + r1: np.ndarray, + r2: np.ndarray, + params_dict: dict, +) -> alt.Chart: + data = [] + for i in range(x_grid.shape[0]): + data.extend( + { + "X": x_grid[i, j], + "Y": y_grid[i, j], + "Potential": phi[i, j], + } + for j in range(x_grid.shape[1]) + ) + + df = pd.DataFrame(data) + + a = params_dict["plate_length"] + d = params_dict["plate_distance"] + + return ( + alt.Chart(df) + .mark_rect() + .encode( + x=alt.X( + "X:Q", + title="X (м)", + bin=alt.Bin(maxbins=50), + scale=alt.Scale(domain=[-a, a]), + ), + y=alt.Y( + "Y:Q", + title="Y (м)", + bin=alt.Bin(maxbins=50), + scale=alt.Scale(domain=[-d * 1.5, d * 1.5]), + ), + color=alt.Color( + "mean(Potential):Q", + title="Потенциал (В)", + scale=alt.Scale(scheme="redblue", reverse=True), + ), + ) + .properties( + title="Эквипотенциальные линии (контуры потенциала)", + width=800, + height=500, + ) + ) + + +def create_field_magnitude_chart( + x_grid: np.ndarray, + y_grid: np.ndarray, + ex: np.ndarray, + ey: np.ndarray, + params_dict: dict, +) -> alt.Chart: + e_mag = np.sqrt(ex**2 + ey**2) + + data = [] + for i in range(x_grid.shape[0]): + data.extend( + { + "X": x_grid[i, j], + "Y": y_grid[i, j], + "E_magnitude": e_mag[i, j], + } + for j in range(x_grid.shape[1]) + ) + + df = pd.DataFrame(data) + + a = params_dict["plate_length"] + d = params_dict["plate_distance"] + + return ( + alt.Chart(df) + .mark_rect() + .encode( + x=alt.X( + "X:Q", + title="X (м)", + bin=alt.Bin(maxbins=50), + scale=alt.Scale(domain=[-a, a]), + ), + y=alt.Y( + "Y:Q", + title="Y (м)", + bin=alt.Bin(maxbins=50), + scale=alt.Scale(domain=[-d * 1.5, d * 1.5]), + ), + color=alt.Color( + "mean(E_magnitude):Q", + title="Напряжённость (В/м)", + scale=alt.Scale(scheme="viridis"), + ), + ) + .properties( + title="Напряжённость электрического поля", + width=800, + height=500, + ) + ) diff --git a/src/models/electrostatics/config.py b/src/models/electrostatics/config.py new file mode 100644 index 0000000..faef112 --- /dev/null +++ b/src/models/electrostatics/config.py @@ -0,0 +1,4 @@ +PHYSICAL_CONSTANTS = { + "epsilon_0": 8.854e-12, + "k": 8.988e9, +} diff --git a/src/models/electrostatics/objects.py b/src/models/electrostatics/objects.py new file mode 100644 index 0000000..c51411b --- /dev/null +++ b/src/models/electrostatics/objects.py @@ -0,0 +1,204 @@ +from dataclasses import dataclass + +import numpy as np +import streamlit as st +from scipy import linalg + +from models.electrostatics.config import PHYSICAL_CONSTANTS + + +@dataclass +class ElectrostaticsParameters: + plate_length: float = 1.0 + plate_distance: float = 0.2 + voltage: float = 100.0 + n_segments: int = 50 + field_resolution: int = 50 + n_field_lines: int = 20 + + +def calculate_capacitance( + q1: np.ndarray, + voltage: float, +) -> float: + q_total = np.sum(q1) + return q_total / voltage + + +class MethodOfMomentsModel: + def __init__(self) -> None: + self.epsilon_0 = PHYSICAL_CONSTANTS["epsilon_0"] + + @st.cache_data(ttl=300) + def solve_charge_distribution( # noqa: PLR0914 + _self, # noqa: N805 + params: ElectrostaticsParameters, + ) -> tuple[np.ndarray, np.ndarray, np.ndarray, np.ndarray]: + a = params.plate_length + d = params.plate_distance + n = params.n_segments + + delta_l = a / n + + x1 = np.linspace(-a / 2 + delta_l / 2, a / 2 - delta_l / 2, n) + y1 = np.full(n, d / 2) + + x2 = np.linspace(-a / 2 + delta_l / 2, a / 2 - delta_l / 2, n) + y2 = np.full(n, -d / 2) + + r1 = np.stack([x1, y1], axis=1) + r2 = np.stack([x2, y2], axis=1) + + r_all = np.vstack([r1, r2]) + + N = 2 * n # noqa: N806 + A = np.zeros((N, N)) # noqa: N806 + + for i in range(N): + for j in range(N): + if i == j: + A[i, j] = ( + 1 + / (2 * np.pi * _self.epsilon_0) + * (1 - np.log(delta_l / 2)) + ) + else: + dist = np.linalg.norm(r_all[i] - r_all[j]) + A[i, j] = -1 / (2 * np.pi * _self.epsilon_0) * np.log(dist) + + phi = np.concatenate([ + np.full(n, params.voltage / 2), + np.full(n, -params.voltage / 2), + ]) + + q = linalg.solve(A, phi) + + q1 = q[:n] + q2 = q[n:] + + return r1, r2, q1, q2 + + def compute_electric_field( + self, + r1: np.ndarray, + r2: np.ndarray, + q1: np.ndarray, + q2: np.ndarray, + x_grid: np.ndarray, + y_grid: np.ndarray, + ) -> tuple[np.ndarray, np.ndarray]: + ex = np.zeros_like(x_grid) + ey = np.zeros_like(y_grid) + + r_all = np.vstack([r1, r2]) + q_all = np.concatenate([q1, q2]) + + for i in range(len(x_grid.flat)): + x_flat = x_grid.flat[i] + y_flat = y_grid.flat[i] + + for j, (rx, ry) in enumerate(r_all): + dx = x_flat - rx + dy = y_flat - ry + r_sq = dx**2 + dy**2 + + if r_sq < 1e-6: # noqa: PLR2004 + continue + + factor = q_all[j] / (2 * np.pi * self.epsilon_0 * r_sq) + ex.flat[i] += factor * dx + ey.flat[i] += factor * dy + + return ex, ey + + def compute_potential( + self, + r1: np.ndarray, + r2: np.ndarray, + q1: np.ndarray, + q2: np.ndarray, + x_grid: np.ndarray, + y_grid: np.ndarray, + ) -> np.ndarray: + phi = np.zeros_like(x_grid) + + r_all = np.vstack([r1, r2]) + q_all = np.concatenate([q1, q2]) + + for i in range(len(x_grid.flat)): + x_flat = x_grid.flat[i] + y_flat = y_grid.flat[i] + + for j, (rx, ry) in enumerate(r_all): + dx = x_flat - rx + dy = y_flat - ry + r = np.sqrt(dx**2 + dy**2) + + if r < 1e-6: # noqa: PLR2004 + continue + + phi.flat[i] -= ( + q_all[j] / (2 * np.pi * self.epsilon_0) * np.log(r) + ) + + return phi + + def trace_field_line( # noqa: PLR0914 + self, + r1: np.ndarray, + r2: np.ndarray, + q1: np.ndarray, + q2: np.ndarray, + start_point: tuple[float, float], + max_steps: int = 3000, + step_size: float = 0.003, + bounds: tuple[float, float, float, float] = (-2, 2, -1, 1), + ) -> tuple[np.ndarray, np.ndarray]: + x_min, x_max, y_min, y_max = bounds + + x_line = [start_point[0]] + y_line = [start_point[1]] + + r_all = np.vstack([r1, r2]) + q_all = np.concatenate([q1, q2]) + + for _ in range(max_steps): + x_curr = x_line[-1] + y_curr = y_line[-1] + + if ( + x_curr < x_min + or x_curr > x_max + or y_curr < y_min + or y_curr > y_max + ): + break + + ex, ey = 0.0, 0.0 + near_conductor = False + + for j, (rx, ry) in enumerate(r_all): + dx = x_curr - rx + dy = y_curr - ry + r_sq = dx**2 + dy**2 + + if r_sq < 1e-5: # noqa: PLR2004 + near_conductor = True + break + + factor = q_all[j] / (2 * np.pi * self.epsilon_0 * r_sq) + ex += factor * dx + ey += factor * dy + + if near_conductor: + break + + e_mag = np.sqrt(ex**2 + ey**2) + + if e_mag < 1e-10: # noqa: PLR2004 + break + + x_line.append(x_curr + step_size * ex / e_mag) + y_line.append(y_curr + step_size * ey / e_mag) + + return np.array(x_line), np.array(y_line) diff --git a/src/models/electrostatics/sidebar.py b/src/models/electrostatics/sidebar.py new file mode 100644 index 0000000..8716072 --- /dev/null +++ b/src/models/electrostatics/sidebar.py @@ -0,0 +1,79 @@ +import streamlit as st + +from models.electrostatics.objects import ElectrostaticsParameters + + +def create_electrostatics_sidebar() -> ElectrostaticsParameters: + with st.sidebar: + st.header("Параметры системы") + + plate_length = st.slider( + "Длина пластин a (м)", + min_value=0.2, + max_value=2.0, + value=1.0, + step=0.1, + help="Длина каждой пластины", + key="slider_plate_length", + ) + + plate_distance = st.slider( + "Расстояние между пластинами d (м)", + min_value=0.05, + max_value=1.0, + value=0.2, + step=0.05, + help="Вертикальное расстояние между пластинами", + key="slider_plate_distance", + ) + + voltage = st.slider( + "Разность потенциалов V (В)", + min_value=10.0, + max_value=500.0, + value=100.0, + step=10.0, + help="Напряжение между пластинами", + key="slider_voltage", + ) + + st.header("Параметры расчета") + + n_segments = st.slider( + "Число сегментов на пластину", + min_value=10, + max_value=100, + value=50, + step=10, + help="Дискретизация контура электродов", + key="slider_n_segments", + ) + + field_resolution = st.slider( + "Разрешение сетки поля", + min_value=20, + max_value=100, + value=50, + step=10, + help="Плотность точек для визуализации", + key="slider_field_resolution", + ) + + n_field_lines = st.slider( + "Число силовых линий", + min_value=5, + max_value=50, + value=20, + step=5, + help="Количество силовых линий для визуализации", + key="slider_n_field_lines", + ) + + return ElectrostaticsParameters( + plate_length=plate_length, + plate_distance=plate_distance, + voltage=voltage, + n_segments=n_segments, + field_resolution=field_resolution, + n_field_lines=n_field_lines, + ) diff --git a/src/models/electrostatics/utils.py b/src/models/electrostatics/utils.py new file mode 100644 index 0000000..ebaa2a5 --- /dev/null +++ b/src/models/electrostatics/utils.py @@ -0,0 +1,57 @@ +import numpy as np +import pandas as pd + + +def create_charge_distribution_dataframe( + r1: np.ndarray, + r2: np.ndarray, + q1: np.ndarray, + q2: np.ndarray, +) -> pd.DataFrame: + data = [] + + for i, ((x, y), q) in enumerate(zip(r1, q1, strict=False)): + data.append({ + "Segment": i + 1, + "Plate": "Upper (+V/2)", + "X (m)": x, + "Y (m)": y, + "Charge (C)": q, + }) + + for i, ((x, y), q) in enumerate(zip(r2, q2, strict=False)): + data.append({ + "Segment": i + 1, + "Plate": "Lower (-V/2)", + "X (m)": x, + "Y (m)": y, + "Charge (C)": q, + }) + + return pd.DataFrame(data) + + +def create_field_dataframe( + x_grid: np.ndarray, + y_grid: np.ndarray, + ex: np.ndarray, + ey: np.ndarray, + phi: np.ndarray, +) -> pd.DataFrame: + e_magnitude = np.sqrt(ex**2 + ey**2) + + data = [] + for i in range(x_grid.shape[0]): + data.extend( + { + "X": x_grid[i, j], + "Y": y_grid[i, j], + "Ex": ex[i, j], + "Ey": ey[i, j], + "E_magnitude": e_magnitude[i, j], + "Potential": phi[i, j], + } + for j in range(x_grid.shape[1]) + ) + + return pd.DataFrame(data) diff --git a/tests/m1_electrostatics/__init__.py b/tests/m1_electrostatics/__init__.py new file mode 100644 index 0000000..e69de29 diff --git a/tests/m1_electrostatics/test_method_of_moments.py b/tests/m1_electrostatics/test_method_of_moments.py new file mode 100644 index 0000000..cd32904 --- /dev/null +++ b/tests/m1_electrostatics/test_method_of_moments.py @@ -0,0 +1,165 @@ +import numpy as np +import pytest + +from models.electrostatics.objects import ( + ElectrostaticsParameters, + MethodOfMomentsModel, + calculate_capacitance, +) + + +@pytest.fixture +def model() -> MethodOfMomentsModel: + return MethodOfMomentsModel() + + +@pytest.fixture +def basic_params() -> ElectrostaticsParameters: + return ElectrostaticsParameters( + plate_length=1.0, + plate_distance=0.2, + voltage=100.0, + n_segments=20, + field_resolution=30, + n_field_lines=10, + ) + + +def test_charge_distribution_shape( + model: MethodOfMomentsModel, basic_params: ElectrostaticsParameters +) -> None: + r1, r2, q1, q2 = model.solve_charge_distribution(basic_params) + + assert len(r1) == basic_params.n_segments + assert len(r2) == basic_params.n_segments + assert len(q1) == basic_params.n_segments + assert len(q2) == basic_params.n_segments + + assert r1.shape == (basic_params.n_segments, 2) + assert r2.shape == (basic_params.n_segments, 2) + + +def test_charge_conservation( + model: MethodOfMomentsModel, basic_params: ElectrostaticsParameters +) -> None: + _, _, q1, q2 = model.solve_charge_distribution(basic_params) + + q1_total = np.sum(q1) + q2_total = np.sum(q2) + + assert np.isclose(q1_total, -q2_total, rtol=1e-6) + + +def test_plate_geometry( + model: MethodOfMomentsModel, basic_params: ElectrostaticsParameters +) -> None: + r1, r2, _, _ = model.solve_charge_distribution(basic_params) + + assert np.allclose(r1[:, 1], basic_params.plate_distance / 2) + assert np.allclose(r2[:, 1], -basic_params.plate_distance / 2) + + assert np.min(r1[:, 0]) >= -basic_params.plate_length / 2 + assert np.max(r1[:, 0]) <= basic_params.plate_length / 2 + assert np.min(r2[:, 0]) >= -basic_params.plate_length / 2 + assert np.max(r2[:, 0]) <= basic_params.plate_length / 2 + + +def test_capacitance_positive( + model: MethodOfMomentsModel, basic_params: ElectrostaticsParameters +) -> None: + _, _, q1, _ = model.solve_charge_distribution(basic_params) + + capacitance = calculate_capacitance(q1, basic_params.voltage) + + assert capacitance > 0 + + +def test_capacitance_vs_analytical( + model: MethodOfMomentsModel, basic_params: ElectrostaticsParameters +) -> None: + _, _, q1, _ = model.solve_charge_distribution(basic_params) + + capacitance = calculate_capacitance(q1, basic_params.voltage) + + analytical_c = model.epsilon_0 / basic_params.plate_distance + + ratio = capacitance / analytical_c + assert ratio > 1.0 + assert ratio < 2.0 + + +def test_electric_field_computation( + model: MethodOfMomentsModel, basic_params: ElectrostaticsParameters +) -> None: + r1, r2, q1, q2 = model.solve_charge_distribution(basic_params) + + x_grid = np.array([[0.0, 0.1], [0.2, 0.3]]) + y_grid = np.array([[0.0, 0.0], [0.05, 0.05]]) + + ex, ey = model.compute_electric_field(r1, r2, q1, q2, x_grid, y_grid) + + assert ex.shape == x_grid.shape + assert ey.shape == y_grid.shape + assert not np.any(np.isnan(ex)) + assert not np.any(np.isnan(ey)) + + +def test_potential_computation( + model: MethodOfMomentsModel, basic_params: ElectrostaticsParameters +) -> None: + r1, r2, q1, q2 = model.solve_charge_distribution(basic_params) + + x_grid = np.array([[0.0, 0.1], [0.2, 0.3]]) + y_grid = np.array([[0.0, 0.0], [0.05, 0.05]]) + + phi = model.compute_potential(r1, r2, q1, q2, x_grid, y_grid) + + assert phi.shape == x_grid.shape + assert not np.any(np.isnan(phi)) + + +def test_field_line_tracing( + model: MethodOfMomentsModel, basic_params: ElectrostaticsParameters +) -> None: + r1, r2, q1, q2 = model.solve_charge_distribution(basic_params) + + start_point = (0.0, basic_params.plate_distance / 2 + 0.01) + + x_line, y_line = model.trace_field_line( + r1, + r2, + q1, + q2, + start_point, + max_steps=500, + step_size=0.01, + ) + + assert len(x_line) > 0 + assert len(y_line) > 0 + assert len(x_line) == len(y_line) + + +def test_convergence_with_segments(model: MethodOfMomentsModel) -> None: + params_coarse = ElectrostaticsParameters( + plate_length=1.0, + plate_distance=0.2, + voltage=100.0, + n_segments=20, + ) + + params_fine = ElectrostaticsParameters( + plate_length=1.0, + plate_distance=0.2, + voltage=100.0, + n_segments=50, + ) + + _, _, q1_coarse, _ = model.solve_charge_distribution(params_coarse) + _, _, q1_fine, _ = model.solve_charge_distribution(params_fine) + + c_coarse = calculate_capacitance(q1_coarse, params_coarse.voltage) + c_fine = calculate_capacitance(q1_fine, params_fine.voltage) + + 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