bisection.html

C++编译指导

            <span class="comment">% A new node is added to the mesh. Numerical solution at this</span>
            <span class="comment">% new added node is approximated by linear interpolation.</span>
            mesh.node(newNode,:) = ( mesh.node(mesh.elem(ct,2),:) + <span class="keyword">...</span>
                                     mesh.node(mesh.elem(ct,3),:) )/2;
            mesh.type(newNode) = 2; <span class="comment">% newNode is added by the refinement</span>
            mesh.solu(newNode) = ( mesh.solu(mesh.elem(ct,2)) + <span class="keyword">...</span>
                                   mesh.solu(mesh.elem(ct,3)) )/2;
            <span class="comment">% Find the element which shares the base edge of the current</span>
            <span class="comment">% element. If it is 0, it means the base of the current element</span>
            <span class="comment">% is on the boundary.</span>
            ct = dualEdge(mesh.elem(ct,3),mesh.elem(ct,2));
            <span class="keyword">if</span> (ct==0), index=0; <span class="keyword">end</span> <span class="comment">% base is on the boundary</span>
            <span class="comment">% the while will ended if</span>
            <span class="comment">% 1.  ct==0 means we are on the boundary</span>
            <span class="comment">% 2.  base(ct) is already marked</span>
        <span class="keyword">end</span>
    <span class="keyword">end</span> <span class="comment">% end while for recursive marking</span>
<span class="keyword">end</span> <span class="comment">% end of for loop on all elements</span>
<span class="comment">% Detailed explanation of the algorithm can be found at</span>
<span class="comment">%   Manual --&gt; Algorithms --&gt; Bisection</span>

<span class="comment">%--------------------------------------------------------------------------</span>
<span class="comment">% Refine marked edges for each triangle</span>
<span class="comment">%--------------------------------------------------------------------------</span>
<span class="keyword">for</span> t = 1:NT
    base = d2p(mesh.elem(t,2),mesh.elem(t,3));
    <span class="keyword">if</span> (marker(base)&gt;0)
        p = [mesh.elem(t,:), marker(base)];
        [mesh,eta] = <a href="#_sub1" class="code" title="subfunction [mesh,eta] = divide(mesh,eta,t,p)">divide</a>(mesh,eta,t,p);
        left = d2p(p(1),p(2)); right = d2p(p(3),p(1));
        <span class="keyword">if</span> (marker(right)&gt;0)
            [mesh,eta] = <a href="#_sub1" class="code" title="subfunction [mesh,eta] = divide(mesh,eta,t,p)">divide</a>(mesh,eta,size(mesh.elem,1), <span class="keyword">...</span>
                                [p(4),p(3),p(1),marker(right)]);
        <span class="keyword">end</span>
        <span class="keyword">if</span> (marker(left)&gt;0)
            [mesh,eta] = <a href="#_sub1" class="code" title="subfunction [mesh,eta] = divide(mesh,eta,t,p)">divide</a>(mesh,eta,t,[p(4),p(1),p(2),marker(left)]);
        <span class="keyword">end</span>
    <span class="keyword">end</span> <span class="comment">% end of refinement of one element</span>
<span class="keyword">end</span> <span class="comment">% end of for loop on all elements</span>

<span class="comment">%--------------------------------------------------------------------------</span>
<span class="comment">% Update boundary edges</span>
<span class="comment">%--------------------------------------------------------------------------</span>
mesh.Dirichlet = <a href="#_sub2" class="code" title="subfunction bdEdge = updatebd(bdEdge,marker,d2p)">updatebd</a>(mesh.Dirichlet,marker,d2p);
mesh.Neumann = <a href="#_sub2" class="code" title="subfunction bdEdge = updatebd(bdEdge,marker,d2p)">updatebd</a>(mesh.Neumann,marker,d2p);

<span class="comment">%--------------------------------------------------------------------------</span>
<span class="comment">% Plot refined mesh</span>
<span class="comment">%--------------------------------------------------------------------------</span>
subplot(1,2,2); hold off; 
trisurf(mesh.elem,mesh.node(:,1),mesh.node(:,2),zeros(size(mesh.node,1),1));
view(2), axis equal, axis off;
title(<span class="string">'Mesh after refinement'</span>, <span class="string">'FontSize'</span>, 14)

<span class="comment">%--------------------------------------------------------------------------</span>
<span class="comment">% End of function BISECTION</span>
<span class="comment">%--------------------------------------------------------------------------</span>

<span class="comment">%--------------------------------------------------------------------------</span>
<span class="comment">% Sub functions called by BISECTION</span>
<span class="comment">%--------------------------------------------------------------------------</span>

<a name="_sub1" href="#_subfunctions" class="code">function [mesh,eta] = divide(mesh,eta,t,p)</a>
<span class="comment">% DIVIDE divide triangle t into two triangels and update the error</span>
<span class="comment">% indicator at the same time.</span>
<span class="comment">%</span>
<span class="comment">% USAGE</span>
<span class="comment">%   [mesh,eta] = divide(mesh,eta,t,p)</span>
<span class="comment">%</span>
<span class="comment">% INPUT</span>
<span class="comment">%   mesh:  current mesh</span>
<span class="comment">%    eta:  error indicator for each triangle</span>
<span class="comment">%      t:  triangle to be divided</span>
<span class="comment">%      p:  vertices of triangle t and new vertex</span>
<span class="comment">%            p(1),p(2),p(3) are vertices of t</span>
<span class="comment">%            p(4) new vertex which is opposite to p(1)</span>
<span class="comment">%</span>
<span class="comment">% OUTPUT</span>
<span class="comment">%   mesh:  new mesh</span>
<span class="comment">%    eta:  error indicator for each triangle</span>
<span class="comment">%</span>

<span class="comment">%--------------------------------------------------------------------------</span>
<span class="comment">% Add one element</span>
<span class="comment">%--------------------------------------------------------------------------</span>
new = size(mesh.elem,1) + 1; <span class="comment">% append new element to the end of elem array</span>
mesh.elem(t,:)   = [p(4),p(1),p(2)]; <span class="comment">% t is always a left child</span>
mesh.elem(new,:) = [p(4),p(3),p(1)]; <span class="comment">% new is a right child</span>
<span class="comment">% the newest vertex p(4) is the first vertex in each triangle</span>
eta(new) = eta(t)/2; eta(t) = eta(t)/2; <span class="comment">% update error indicators</span>

<span class="comment">%--------------------------------------------------------------------------</span>
<span class="comment">% End of function DIVIDE</span>
<span class="comment">%--------------------------------------------------------------------------</span>

<a name="_sub2" href="#_subfunctions" class="code">function bdEdge = updatebd(bdEdge,marker,d2p)</a>
<span class="comment">% UPDATEDBD refine the boundary edges</span>
<span class="comment">%</span>
<span class="comment">% USAGE</span>
<span class="comment">%    bdEdge = updatebd(bdEdge,marker,d2p)</span>
<span class="comment">%</span>
<span class="comment">% INPUT</span>
<span class="comment">%   bdEdge:  set of boundary edges</span>
<span class="comment">%   marker:  new node index for marked edge</span>
<span class="comment">%      d2p:  index mapping from dual edge to primary edge</span>
<span class="comment">%</span>
<span class="comment">% OUTPUT</span>
<span class="comment">%   bdEdge:  set of refined boundary edges</span>
<span class="comment">%</span>

NB = size(bdEdge,1);
<span class="keyword">for</span> k = 1:NB
     i = bdEdge(k,1); j = bdEdge(k,2);
     <span class="keyword">if</span> marker(d2p(i,j)) &gt;0
         bdEdge(k,:) = [i,marker(d2p(i,j))];
         bdEdge(size(bdEdge,1)+1,:) = [marker(d2p(i,j)),j];
     <span class="keyword">end</span>
<span class="keyword">end</span>
<span class="comment">%--------------------------------------------------------------------------</span>
<span class="comment">% End of function UPDATEDBD</span>
<span class="comment">%--------------------------------------------------------------------------</span></pre></div>
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