之前转载了一篇博客http://blog.sina.com.cn/s/blog_6163bdeb0102dvay.html，讲Matlab网格划分程序Distmesh，看了看程序，感觉程序写得有很多值得学的地方，所以又自己重新看了一看，加了一些注释，最后再总结一下学到的东西吧。

源代码的地址已经改变，如下http://people.sc.fsu.edu/~jburkardt/m_src/distmesh/distmesh.html。程序给出了20个划分的例子，文件名为p1_demo.m~p20_demo.m，直接运行程序可以看到划分的动画效果，每个例子基本都是设置一些参数，然后调用distmesh_2d函数进行网格划分，最后得到划分的节点p和各三角形t，最后将划分的三角形绘制出来。划分结果如下

p1_demo p14_demo

p5_demo p18_demo

进一步加注释的程序(需要学习的地方用颜色标记)：

function [ p, t ] = distmesh_2d ( fd, fh, h0, box,

iteration_max, pfix, varargin )

%% DISTMESH_2D is a 2D mesh generator using distance

functions.

% Example:

% Uniform

Mesh on Unit Circle:

% fd = inline('sqrt(sum(p.^2,2))-1','p');

% [p,t] = distmesh_2d(fd,@huniform,0.2,[-1,-1;1,1],100,[]);

% Rectangle

with circular hole, refined at circle boundary:

% fd =

inline('ddiff(drectangle(p,-1,1,-1,1),dcircle(p,0,0,0.5))','p');

% fh = inline('min(4*sqrt(sum(p.^2,2))-1,2)','p');

% [p,t] =

distmesh_2d(fd,fh,0.05,[-1,-1;1,1],500,[-1,-1;-1,1;1,-1;1,1]);

% Parameters:

% Input,

function FD, signed distance function d(x,y).

% fd:d=fd(p)，p=[x y],fd为给定任一点到边界的有符号距离函数，负号表示在区域内，正号为在区域外

% Input,

function FH, scaled edge length function h(x,y).

% fh:就是网格大小的函数

% Input, real

H0, the initial edge length.

% h0:也就是h,

网格的初始大小

% Input, real

BOX(2,2), the bounding box [xmin,ymin; xmax,ymax].

% box:最大外围矩形范围

% Input,

integer ITERATION_MAX, the maximum number of iterations.

% The

iteration might terminate sooner than this limit, if the program

decides

% that the

mesh has converged.

% iteration_max:允许的最大迭代次数

% Input, real

PFIX(NFIX,2), the fixed node positions.

% pfix：网格中需要固定的点坐标，也就是一定需要出现在网格中的点

% Input,

VARARGIN, aditional parameters passed to FD and FH.%

% Output,

real P(N,2), the node positions.

% p:网格点的x,y坐标

% Output,

integer T(NT,3), the triangle indices.

% t:输出网格任意一个三角形的三个顶点

% Local parameters:

% Local, real

GEPS, a tolerance for determining whether a point is "almost"

inside

% the

region. Setting GEPS = 0 makes this an exact

test. The program currently

% sets it to

0.001 * H0, that is, a very small multiple of the desired side

length

% of a

triangle. GEPS is also used to determine whether

a triangle falls inside

% or outside

the region. In this case, the test is a little

tighter. The centroid

% PMID is

required to satisfy FD ( PMID ) <= -GEPS.

% 局部变量geps：容忍度，一个点是否属于区域，也在判断三角形是否属于区域内时使用

dptol =

0.001; % 收敛精度

ttol =

0.1; % 三角形划分精度(百分比)

Fscale =

1.2; % 放大比例

deltat =

0.2; % 相当于柔度

geps = 0.001 *

h0; % 容忍度

deps = sqrt ( eps ) * h0; %

微小变化dx

iteration =

0; % 迭代次数

triangulation_count =

0; %

三角形划分次数

% 1. Create the initial point distribution by

generating a rectangular mesh

% in the bounding box.

% 根据初始网格的大小h0，先把能涵盖欲划分区域的最大矩形划分为结构网格。

[ x, y ] = meshgrid ( box(1,1) :

h0 : box(2,1), ...

box(1,2) :

h0*sqrt(3)/2 : box(2,2) );

% Shift the even rows of the mesh to create a

"perfect" mesh of equilateral triangles.

% Store the X and Y coordinates together as our

first estimate of "P", the mesh points

% we want.

% 然后把偶数行的点整体向右平移半格，得到正三角形划分

x(2:2:end,:) = x(2:2:end,:) + h0 / 2;

p = [ x(:), y(:) ];

% 2. Remove mesh points that are outside the

region,

% then satisfy the density constraint.

% Keep only points inside (or almost inside) the

region, that is, FD(P) < GEPS.

% 根据fd的函数定义，移除边界外的点

p = p( feval_r( fd, p, varargin{:} )

<= geps, : ); % 1 % Set R0, the relative probability to keep a point, based on the mesh

density function.

r0 = 1 ./ feval_r( fh, p, varargin{:} ).^2;

% Apply the rejection method to thin out points

according to the density.

% 根据网格密度函数fh，每个点上产生一个0-1随机数，判断是否小于r0/max(r0)大于的话，该点被删除

p = [ pfix; p(rand(size(p,1),1) < r0 ./ max ( r0 ),:

) ];

[ nfix, dummy ] = size ( pfix );

% Especially when the user has included fixed

points, we may have a few

% duplicates. Get rid of any

you spot.

% 当指定了某些点要保留的时候，把保留的点加入，删除重复的点。

p = unique ( p, 'rows' ); % 2

N = size ( p, 1 );

% If ITERATION_MAX is 0, we're almost

done.

% For just this case, do the triangulation, then

exit.

% Setting ITERATION_MAX to 0 means that we can

see the initial mesh

% before any of the improvements have been

made.

% 如果最大迭代次数为负，则直接结束

if ( iteration_max <= 0 )

t =

delaunayn ( p ); % 3

triangulation_count = triangulation_count + 1;

return

end

pold = inf; % 第一次迭代前设置旧的点的坐标为无穷

while ( iteration < iteration_max )

iteration =

iteration + 1;

if ( mod (

iteration, 10 ) == 0 )

fprintf ( 1, ' %d iterations,', iteration

);

fprintf ( 1, ' %d triangulations.\n',

triangulation_count );

end

% 3. Retriangulation by the Delaunay

algorithm.

% Was there large enough movement to

retriangulate?

% If so, save the current positions, get the list

of

% Delaunay triangles, compute the centroids, and

keep

% the interior triangles (whose centroids are

within the region).

%

%

先判断上次移动后的点和旧的点之间的移动距离，如果小于某个阀值，停止迭代

if ( ttol

< max ( sqrt ( sum ( ( p - pold ).^2, 2 ) ) / h0 )

)

pold =

p; % 如果还可以移动，保存当前的节点

t = delaunayn ( p

); %

利用delauny算法，生成三角形网格

triangulation_count = triangulation_count +

1; % 划分次数加1

pmid = ( p(t(:,1),:) + p(t(:,2),:) + p(t(:,3),:) ) / 3; %

计算三角形的重心

t = t( feval_r( fd, pmid, varargin{:} ) <= -geps, :

); % 移除重心在区域外的三角形

% 4. Describe each bar by a unique pair of

nodes.

% 生成网格的边的集合，也就是相邻点之间连接的线段

bars = [ t(:,[1,2]); t(:,[1,3]); t(:,[2,3]) ];

bars = unique ( sort ( bars, 2 ), 'rows' );

% 5. Graphical output of the current mesh

trimesh ( t, p(:,1), p(:,2),

zeros(N,1)

) % 绘制划分的三角形% 3

view(2), axis equal, axis off, drawnow

end

% 6. Move mesh points based on bar lengths L and

forces F

% Make a list of the bar vectors and

lengths.

% Set L0 to the desired lengths, F to the scalar

bar forces,

% and FVEC to the x, y components of the bar

forces.

% At the fixed positions, reset the force to

0.

barvec =

p(bars(:,1),:) -

p(bars(:,2),:); %

生成bar的矢量

L = sqrt (

sum ( barvec.^2, 2 )

);