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采用特定离散化方案的二维网格矩阵图

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采用特定离散化方案的二维网格矩阵图

我有一个 2D 网格，当使用有限差分法 (5 点模板) 用二阶中心差分格式进行离散化时，将产生一个稀疏矩阵。有没有办法生成一个填充矩阵，显示网格点各个位置的条目。

（这个问题针对的是具有数学背景的人，特别是那些了解稀疏线性系统和离散化方案的人）

矩阵的示例图像：

在此处输入图片描述

注意：上图不是下面网格离散化后得到的矩阵结构，但我看到的是使用二阶中心差分格式对我的网格进行离散化后得到的类似上图的图像（有关详细信息，可以查看这一页）

我的网格点的代码

\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
[darkstyle/.style={circle,draw,minimum size=25}]
 \foreach \x in {0,...,4}
    \foreach \y in {0,...,4} 
       {\pgfmathtruncatemacro{\label}{\x - 5 *  \y +21}
       \node [darkstyle]  (\x\y) at (2.5*\x,2.5*\y) {\label};} 

  \foreach \x in {0,...,4}
    \foreach \y [count=\yi] in {0,...,3}  
      \draw (\x\y)--(\x\yi) (\y\x)--(\yi\x) ;
\end{tikzpicture}
\end{document}
%resulting grid shown below

上述代码生成的网格

我想到的解决方案没有将网格链接到我的离散化方案并生成矩阵间隙（这似乎是一项有点复杂的任务，需要组装所谓的刚度矩阵，我对 TEX 的要求太高了 :P）尽管如此，我还是设法用一段非常糟糕的代码（我不喜欢）生成了我想要的图形。它在这里：

\begin{tikzpicture}
 %draws the outer box
 \draw (0.15,-0.05) rectangle (-3.8,3.9);

  \foreach \x in  {0,...,25}
    \foreach \y in  {0,...,25}
     \foreach \position in {(0.15*-\x,0.15*\y)}
      {\ifnum \x=\y
            { \draw[fill=black] \position rectangle +(0.1,0.1);}
        \fi}

\foreach \x in  {1,...,25}
    \foreach \y in  {1,...,25}
     \foreach \position in {(0.15 + 0.15*-\x,0.15*\y)}
      {\ifnum \x = \y
            {\draw[fill=black] \position rectangle +(0.1,0.1);}
        \fi}

\foreach \x in  {0,...,25}
    \foreach \y in  {0,...,24}
     \foreach \position in {( -0.15+0.15*-\x,0.15*\y)}
      {\ifnum \x = \y
            { \draw[fill=black] \position rectangle +(0.1,0.1);}
        \fi}

\foreach \x in  {5,...,25}
    \foreach \y in  {0,...,25}
     \foreach \position in {( 0.75+0.15*-\x,0.15*\y)}
      {\ifnum \x = \y
            {\draw[fill=black] \position rectangle +(0.1,0.1);}
        \fi}

\foreach \x in  {0,...,20}
    \foreach \y in  {0,...,25}
     \foreach \position in {( -0.75+0.15*-\x,0.15*\y)}
      {\ifnum \x = \y
            {\draw[fill=black] \position rectangle +(0.1,0.1);}
      \fi}
\end{tikzpicture}

矩阵的图形如下所示：

在此处输入图片描述

但如果有人能给我建议来清理代码，我会很高兴:)

答案1

好的，谢谢你的回答，现在我开始明白你想要什么了。这是一个稍微简单的代码。

\documentclass[tikz,border=7pt]{standalone}
\usetikzlibrary{plotmarks}
\begin{document}
\pgfsetplotmarksize{0.05cm}
\begin{tikzpicture}
 %draws the outer box
 \draw (0.15,-0.15) rectangle (-3.95,3.9);

  \foreach \x in  {0,...,25}
    {\foreach \y in  {0,...,25}
     {\pgfmathtruncatemacro{\Z}{ifthenelse(abs(\x-\y)<2,0,1)}
     \ifnum\Z=0
     \node at (-0.15*\x,0.15*\y) {\pgfuseplotmark{square*}};
     \else
     \pgfmathtruncatemacro{\Z}{ifthenelse(abs(\x-\y)==4,0,1)}
     \ifnum\Z=0
     \node at (-0.15*\x,0.15*\y) {\pgfuseplotmark{square*}};
     \fi
     \fi}
    }
\end{tikzpicture}
\end{document}

在此处输入图片描述

我还想知道您是否想要像问题中那样的块结构，如果是的话，请告诉我。（顺便说一句，我认为最好将答案移到问题上。）以防万一：

\documentclass[tikz,border=7pt]{standalone}
\usetikzlibrary{plotmarks,calc}
\begin{document}
\pgfsetplotmarksize{0.05cm}
\begin{tikzpicture}
 %draws the outer box
 \draw (0.15,-0.15) coordinate (br) rectangle (-3.95,3.9) coordinate (tl);

  \foreach \x in  {0,...,25}
    {\foreach \y in  {0,...,25}
     {\pgfmathtruncatemacro{\Z}{ifthenelse(abs(\x-\y)<2,0,1)}
     \ifnum\Z=0
     \node at (-0.15*\x,0.15*\y) {\pgfuseplotmark{square*}};
     \else
     \pgfmathtruncatemacro{\Z}{ifthenelse(abs(\x-\y)==4,0,1)}
     \ifnum\Z=0
     \node at (-0.15*\x,0.15*\y) {\pgfuseplotmark{square*}};
     \fi
     \fi}
    }
 \fill[white] ([xshift=1mm,yshift=1mm]br -| tl) rectangle ([xshift=0.2mm]$(br)!0.5!(tl)$);  
 \fill[white] ([xshift=-1mm,yshift=-1mm]br |- tl) rectangle ([xshift=0.2mm]$(br)!0.5!(tl)$);    
\end{tikzpicture}
\end{document}

在此处输入图片描述

答案2

在此处输入图片描述五年前我写了这个解决方案。抱歉结果不好，但我删除了输入行以保持在 30000 行以内

% !TEX program = LuaLaTeX

%\documentclass[a4paper]{report}
%\usepackage[margin=2cm]{geometry}

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{calc}

\def\LuaCalc#1{\directlua{tex.print(#1)}}

\def\pixels#1#2#3#4[#5]{
% #1 = pixel list
% #2 = X resolution
% #3 = Y resolution
% #4 = south west vertex
% #5 = grid key-values
\path[step=1,#5] (#4)grid(#2,#3);

\foreach \pixel in {#1}
{
\edef\column{\LuaCalc{math.ceil(\pixel/#3)}}

\ifnum\column>1
      \edef\row{\LuaCalc{#3-\pixel+(\column-1)*#3}}
\else
      \edef\row{\LuaCalc{#3-\pixel}}
\fi

\ifnum#2<\column
\edef\maxPix{\LuaCalc{#2*#3}}
\errmessage{Pixel value is \pixel. The maximum must be \maxPix}
\fi


\fill (\column-1,\row)rectangle++(1,1);
}
}



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\begin{tikzpicture}[x=.2mm,y=.2mm]
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10440,10448,10449,10450,10451,10452,%
10453,10454,10461,10470,10471,10472,%
10473,10479,10480,10481,10484,10487,%
10488,10492,10493,10494,10505,10511,%
10512,10514,10526,10527,10528,10530,%
10532,10552,10556,10557,10567,10568,%
10569,10570,10588,10589,10590,10591,%
10594,10596,10597,10606,10607,10608,%
10621,10622,10623,10624,10625,10626,%
10627,10632,10633,10634,10638,10651,%
10652,10653,10661,10668,10669,10670,%
10672,10673,10679,10683,10684,10687,%
10688,10689,10690,10691,10692,10693,%
10711,10712,10713,10714,10726,10727,%
10728,10730,10732,10750,10751,10752,%
10756,10767,10768,10769,10791,10806,%
10822,10823,10824,10825,10827,10832,%
10833,10834,10849,10850,10851,10860,%
10861,10873,10874,10875,10889,10891,%
10892,10893,10903,10904,10905,10914,%
10928,10950,10951,10952,10956,10966,%
10967,10968,10969,10986,10987,10991,%
10992,10996,10997,11006,11007,11009,%
11024,11027,11032,11034,11046,11074,%
11075,11077,11088,11089,11090,11091,%
11097,11098,11101,11102,11103,11104,%
11105,11127,11128,11154,11156,11160,%
11186,11187,11191,11192,11193,11196,%
11204,11205,11206,11207,11208,11209,%
11223,11224,11231,11232,11234,11246,%
11247,11248,11268,11274,11277,11290,%
11291,11297,11298,11302,11304,11305,%
11327,11332,11354,11355,11356,11357,%
11359,11360,11362,11363,11364,11369,%
11386,11387,11388,11391,11393,11394,%
11395,11396,11409,11429,11430,11431,%
11432,11444,11445,11446,11447,11448,%
11453,11465,11466,11467,11468,11469,%
11474,11475,11477,11496,11497,11498,%
11527,11532,11533,11534,11554,11555,%
11556,11557,11558,11559,11560,11561,%
11562,11563,11564,11569,11570,11572,%
11587,11588,11593,11594,11596,11608,%
11609,11610,11629,11631,11644,11648,%
11653,11658,11659,11663,11664,11665,%
11669,11674,11675,11676,11677,11690,%
11695,11696,11697,11698,11699,11723,%
11724,11725,11734,11754,11755,11756,%
11757,11761,11764,11765,11769,11772,%
11773,11774,11775,11776,11777,11778,%
11779,11780,11787,11788,11796,11809,%
11829,11830,11831,11843,11844,11848,%
11852,11853,11858,11859,11863,11864,%
11865,11867,11868,11869,11874,11888,%
11889,11890,11891,11897,11925,11926,%
11927,11928,11931,11932,11933,11934,%
11935,11960,11961,11963,11964,11965,%
11966,11967,11968,11969,11977,11980,%
11988,11989,12028,12029,12033,12034,%
12035,12036,12037,12038,12039,12042,%
12043,12044,12045,12046,12048,12053,%
12058,12059,12067,12069,12070,12089,%
12091,12124,12125,12127,12131,12132,%
12158,12159,12160,12164,12165,12166,%
12167,12168,12169,12177,12188,12228,%
12229,12239,12242,12244,12245,12251,%
12252,12253,12257,12258,12259,12270,%
118890,118891,118892,118893,118906,118907,%
118908,118912,118930,118940,118941,118942,%
118944,118945,118947,118949,118972,118973,%
118986,118995,118996,119001,119002,119003,%
119013,119014,119018,119035,119036,119038,%
119064,119065,119066,119067,119070,119089,%
119090,119091,119093,119106,119107,119108,%
119109,119110,119111,119112,119130,119131,%
119136,119141,119144,119145,119149,119150,%
119172,119186,119187,119196,119198,119199,%
119202,119203,119204,119214,119233,119234,%
119235,119236,119237,119238,119240,119241,%
119242,119243,119244,119265,119270,119289,%
119310,119330,119331,119336,119337,119338,%
119341,119343,119344,119350,119387,119388,%
119389,119398,119399,119401,119402,119403,%
119404,119405,119414,119436,119437,119443,%
119444,119445,119467,119468,119469,119470,%
119471,119489,119493,119494,119495,119496,%
119497,119531,119536,119537,119538,119543,%
119555,119556,119589,119594,119595,119596,%
119597,119598,119599,119600,119604,119605,%
119614,119615,119616,119617,119637,119642,%
119643,119644,119645,119671,119672,119673,%
119674,119693,119696,119731,119736,119737,%
119740,119742,119743,119751,119752,119753,%
119754,119755,119756,119757,119787,119788,%
119789,119794,119795,119797,119803,119804,%
119814,119815,119836,119837,119839,119840,%
119841,119842,119844,119845,119871,119873,%
119893,119894,119896,119897,119931,119937,%
119940,119941,119942,119943,119955,119956,%
119957,119987,119989,119990,119991,119992,%
119994,119995,119997,119998,119999,120000%
}{600}{200}{0,0}[]

\end{tikzpicture}
\end{document}

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