 Solve_problem_method.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 程序编程中的解决方法"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " **简化问题**\n",
    "* 将问题清楚重述, 思考如何将问题化简为多步解决, 搞清楚每一步需要做什么, 如何做做到解决每一小步的问题, 最后如何串联起每一步, 将整个问题整体解决."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**寻求帮助**\n",
    "* 在网上寻求 $\\text{Google}$ 的帮助, 在简化问题的基础上, 去查询自己需要解决的问题是什么, 目的明确, 在 $\\text{Google}$ 中查找问题,需要掌握, 用英语怎么提问, 怎么去查找, $\\text{Google}$ 中的资料更为完善, 除此之外, 还可以采用 **百度**,诸如这样的提问网站 `https://meta.stackoverflow.com/questions` 等. \n",
    "\n",
    "$\\textbf{例如:}$用英语提问:\n",
    "\n",
    "`How to rotate a square numpy array with different times efficiently by `np.rot90`?`\n",
    "\n",
    "![scale=0.1](./figures/solve-problem.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 对于网格生成的简单举例"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "####  以三角形网格为例\n",
    "\n",
    " **网格的数据结构**\n",
    "\n",
    "* 网格节点坐标数组 `point`:\n",
    "    + $N\\times 2$ 的二维数组\n",
    "    + `point[i, 0]` 和 `point[i, 1]` 分别存储第 $i$ 个网格节点的 $x$ 和 $y$ 坐标\n",
    "* 单元顶点编号数组 `cell`:\n",
    "    + $NC\\times 3$ 的二维数组\n",
    "    + `cell[i, 0]`, `cell[i, 1]` 和 `cell[i, 2]` 分别存储第 $i$ 个单元三个顶点的全局编号(即 `point` 中的行号)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "point = np.array(\n",
    "    [(0.0, 0.0),\n",
    "     (1.0, 0.0),\n",
    "     (1.0, 1.0),\n",
    "     (0.0, 1.0)], dtype=np.float)\n",
    "cell = np.array([\n",
    "        (1, 2, 0), \n",
    "        (3, 0, 2)], dtype=np.int)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 调用三角形网格生成算法 $\\text{fealpy}$ 算法包\n",
    "\n",
    "* 导入三角形网格生成算法."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from fealpy.mesh.TriangleMesh import TriangleMesh"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "调用 $\\text{TriangleMesh}$ 算法生成三角形网格"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "tmesh = TriangleMesh(point, cell)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "网格生成可以获得的信息, 可以采用加密网格生成算法."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "tmesh.point  #网格的顶点\n",
    "tmesh.ds.edge #网格的边\n",
    "tmesh.ds.cell #网格的单元\n",
    "tmesh.ds.edge2cell\n",
    "tmesh.ds.cell_to_edge()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "如何将网格生成, 呈现出来."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![scale=0.1](./figures/solve-problem.png)导入画图模块"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "fig = plt.figure()# 生成画布\n",
    "axes = fig.gca()# 生成坐标系\n",
    "tmesh.add_plot(axes)#画图生成网格\n",
    "tmesh.find_point(axes, showindex=True) #显示网格顶点编号\n",
    "tmesh.find_edge(axes, showindex=True)# 显示网格边编号\n",
    "tmesh.find_cell(axes, showindex=True)# 显示网格单元编号\n",
    "plt.savefig(\"test.png\")# 保存图\n",
    "plt.show() # 显示网格"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![scale=0.1](./figures/test.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "### 对于一个具体的有限元程序的编程举例\n",
    "\n",
    "* 有限元问题系统的理论及编程过程"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " **问题**： \n",
    "\n",
    "\\begin{equation}\n",
    "-\\Delta u =f\\quad{(1)}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "有限元的基本问题可以归纳为:\n",
    "\n",
    "* 把问题转化成变分形式\n",
    "* 选定单元的形状, 对求解域做剖分. 一维情形的单元是小区间, 二维情形的重要单元有两种, 三角形和四边形(矩形和任意凸四边形). 三维情形的单元就更复杂, 四面体,六面体等.\n",
    "* 构造基函数或单元形状函数, 行成有限元空间.\n",
    "* 形成有限元方程, $Ax=b$.\n",
    "* 提供有限元方程的有效解法.\n",
    "* 收敛性及误差估计."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**解决问题:**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 问题的变分\n",
    "\n",
    "引入函数空间 $H^1(\\Omega)$, 对于任意 $v \\in H^1(\\Omega)$, $v$ 和它的一阶导数都在 $\\Omega$ 上 $L^2$ 可积. 这里的 $H^1(\\Omega)$ 是一个无限维的空间.\n",
    "\n",
    "另外, 引入空间 $H_0^1(\\Omega) := \\{v\\in H^1(\\Omega), v|_{\\partial\\Omega} = 0\\}$.\n",
    "\n",
    "对于任意的 $v\\in H^1_0(\\Omega)$, 同乘以方程 (1) 的两端, 然后做分部积分可得: \n",
    "\n",
    "\\begin{equation}\n",
    "\\int_{\\Omega}\\nabla u\\cdot\\nabla v\\mathrm{d}x = \\int_{\\Omega}fv\\mathrm{d}x,\\quad\\forall v \\in H^1_0(\\Omega).\\quad{(2)}\n",
    "\\end{equation}\n",
    "\n",
    "引入双线性泛函\n",
    "\n",
    "\\begin{equation}\n",
    "a(u,v) = <f,v> \\text{ for all }v\\in H_0^1(\\Omega).\\quad{(3)}\n",
    "\\end{equation}\n",
    "\n",
    "$$\n",
    "a(u,v) = \\int_{\\Omega}\\nabla u\\cdot\\nabla v\\mathrm{d}x,\\quad <f,v> =  \\int_{\\Omega}fv\\mathrm{d}x.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "1. 变分形式\n",
    "$$\n",
    "(\\nabla u,\\nabla v)=(f,v)\n",
    "$$\n",
    "\n",
    "2. 离散形式\n",
    "\n",
    "$$\n",
    "(\\nabla u_h,\\nabla v_h)=(f,v_h)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 网格的选取(三角形网格)\n",
    "\n",
    "**二维情形**\n",
    "\n",
    "* 选择均匀网格加密.\n",
    "\n",
    "**准备初始网格**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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