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218 lines
8.1 KiB
218 lines
8.1 KiB
% This file's purpose is to compare quadtree, gauss-green, and szego-green...
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% quadrature methods using a small suite of test cases defined by two
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% domains and various integrands:
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% Domains:
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shape=0;
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testIntegrands=1;
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% 1. A circular region whose boundary is defined by four rational curves
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lroffset=[.3957106819596820 -1.5728593603867];
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Circletemp=load("PolyCircle.mat"); Circle= Circletemp.PolyCircle;
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Circle(1:2:end,:)=Circle(1:2:end,:)+lroffset(1);
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Circle(2:2:end,:)=Circle(2:2:end,:)+lroffset(2);
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% 2. A region defined by the intersection of two circular regions.
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offset=.1364728595817;
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C1=Circle; C1(2:2:end,:)=C1(2:2:end,:)+offset;
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C2=Circle; C2(2:2:end,:)=C2(2:2:end,:)-offset;
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InterCircles=boolElsv2(C1,C2,true); InterCircles=InterCircles{1};
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% plot_rat_bern_poly(InterCircles{1},2,.001,'k')
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% Integrands:
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% 1. Monomials up to 6th degree: (1, x, y, ...)
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monfuncts={@(x,y) ones(size(x));
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@(x,y) x;
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@(x,y) y;
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@(x,y) x.^2;
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@(x,y) x.*y;
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@(x,y) y.^2;
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@(x,y) x.^3;
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@(x,y) x.^2.*y;
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@(x,y) x.*y.^2;
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@(x,y) y.^3;
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@(x,y) x.^4;
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@(x,y) x.^3.*y;
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@(x,y) x.^2.*y.^2;
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@(x,y) x.*y.^3;
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@(x,y) y.^4;
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@(x,y) x.^5;
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@(x,y) x.^4.*y;
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@(x,y) x.^3.*y.^2;
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@(x,y) x.^2.*y.^3;
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@(x,y) x.*y.^4;
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@(x,y) y.^5;
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@(x,y) x.^6;
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@(x,y) x.^5.*y;
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@(x,y) x.^4.*y.^2;
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@(x,y) x.^3.*y.^3;
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@(x,y) x.^2.*y.^4;
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@(x,y) x.*y.^5;
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@(x,y) y.^6;}
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% 2. Three polynomials of degree 2 (bilinear), 4 (biquadratic), and 6
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% (bicubic)
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polyfuncts={@(x,y) (2*x.^2 +x.*y - y +2);
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@(x,y) (2*x.^2.*y.^2 +.3*x.^2.*y - y.^4 + 3*x +2);
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@(x,y) (x.^5 - 5*y.^3.*x.^3 + .2*x.^2 + 2*y.*x.^2 +3);}
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% 3. A rational function of degree 4 and an exponential function.
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otherfuncts={@(x,y) (y.^3 - (x.^3.*y.^2) - (x.*y) -3)./((x.^2).*(y.^2) +10);
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@(x,y) 10*(exp( - x.^2 ) + 2*y);
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@(x,y) sqrt((x+10).^2+(x+10).*(y+10).^2 +x)}
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addpath("../Rational_Quadrature/Matlab/Src",...
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"../Rational_Quadrature/Matlab/Tests",...
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"../Rational_Quadrature/Matlab/ThirdPartySupportingCode")
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d=2;
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gaussOrders=[2:15]
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% Test cases: There are 3 functions that we will consider, two of which have
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% known antiderivatives. These functions were taken from
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if shape==0
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shapeObject=Circle;
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elseif shape==1
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shapeObject=InterCircles;
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end
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if testIntegrands==0
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integrands=monfuncts;
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elseif testIntegrands==1
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integrands=polyfuncts;
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elseif testIntegrands==2
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integrands=otherfuncts;
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end
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% figure(7)
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% plot_bern_poly([C1;C2],2,.001,{},{},false)
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% hold on
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% plot_bern_poly(shapeObject,2,.001,{},{},false)
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figure(8)
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% plot_bern_poly([C1;C2],2,.001,{},{},false)
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% hold on
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plot_bern_poly(shapeObject,2,.001,{},{},false)
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RationalOn=0;
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numIntegrands=length(integrands);
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elemSize=size(shapeObject,1)/3;
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int2evals=zeros(length(numIntegrands),1);
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int2errs=zeros(length(numIntegrands),1);
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ggevals=int2evals;
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sgevals=int2evals;
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global evalCounter;
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global scatterEvals;
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scatterEvals=0;
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for i=1:numIntegrands
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figure(1)
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field = @(x,y) field2(x,y,integrands{i});
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% if shape==0
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% truev = integral2(field, -1+lroffset(1),1+lroffset(1), @(x)-sqrt(1-(x-lroffset(1)).^2)+lroffset(2), @(x)sqrt(1-(x-lroffset(1)).^2)+lroffset(2),'AbsTol',1e-17,'RelTol',1e-18);
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% elseif shape==1
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% truev = integral2(field, -sqrt(1-offset^2)+lroffset(1), sqrt(1-offset^2)+lroffset(1), @(x)-sqrt(1-(x-lroffset(1)).^2)+offset+lroffset(2),@(x)sqrt(1-(x-lroffset(1)).^2)-offset+lroffset(2),'AbsTol',1e-17,'RelTol',1e-18);
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% end
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% % fanti=@(a,b) gauss1D(@(x)field(x,b),0,a,15);
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% mfanti=@(a,b) arrayfun(fanti,a,b);
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% SO{1}=shapeObject; truev=RatPolygonIntegrate(SO,mfanti,24,16);
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% for jj=1:14
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% evalCounter=0;
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% if jj==7
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% scatterEvals=1;
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% else
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% scatterEvals=0;
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% end
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% if shape==0
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% int2errs(jj) = truev-integral2(field, -1+lroffset(1),1+lroffset(1), @(x)-sqrt(1-(x-lroffset(1)).^2)+lroffset(2), @(x)sqrt(1-(x-lroffset(1)).^2)+lroffset(2),'RelTol',10^(-jj));
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% int2evals(jj)=evalCounter;
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% elseif shape==1
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% int2errs(jj) = truev-integral2(field, -sqrt(1-offset^2)+lroffset(1), sqrt(1-offset^2)+lroffset(1), @(x)-sqrt(1-(x-lroffset(1)).^2)+offset+lroffset(2),@(x)sqrt(1-(x-lroffset(1)).^2)-offset+lroffset(2),'RelTol',10^(-jj+3));
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% int2evals(jj)=evalCounter;
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% end
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% end
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% int2errs(int2errs==0)=1e-17;
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% figure(1)
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% semilogy(int2evals,abs(int2errs),'k.','MarkerSize',36)
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hold on
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% plot_rat_bern_poly(shapeObject,2,.001,'k');
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% Intersect each element, store moment of each material
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% fanti=@(a,b) gauss1D(@(x)field(x,b),0,a,7);
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% mfanti=@(a,b) arrayfun(fanti,a,b);
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SO{1}=shapeObject;
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% truev=PolygonIntegrate(SO,mfanti,7);
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% IntersectionI1=zeros(nElemMesh1,1);
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% s=plot_bern_poly(Intersection{1},2,.001,{},{},true);
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% plot_rat_bern_poly(Intersection{1},2,.1,'r');
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RationalOn=1;
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ggevals=zeros(length(gaussOrders),1);
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ggerrs=zeros(length(gaussOrders),1);
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sgevals=zeros(length(gaussOrders),1);
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sgerrs=zeros(length(gaussOrders),1);
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for jj=2:length(gaussOrders)
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j=gaussOrders(jj);
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if testIntegrands==0
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kk=2*invTri(i)+6;
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kg=max(ceil(invTri(i)+1),2);
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elseif testIntegrands==1
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kk=4*i+6;
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kg=max(i,2);
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else
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kg=4;
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kk=j;
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end
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evalCounter=0;
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fanti=@(a,b) gauss1D(@(x)field(x,b),0,a,kg);
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% intxw= @(a,b) gaussXW(@(x)field(x,b),0,a,j);
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mfanti=@(a,b) arrayfun(fanti,a,b);
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% mintxw= @(a,b) arrayfun(intxw,a,b);
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if jj==2
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scatterEvals=1;
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else
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scatterEvals=0;
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end
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ggerrs(jj)=PolygonIntegrate(SO,mfanti,j);
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ggevals(jj)=evalCounter;
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% evalCounter=0;
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% sgerrs(jj)=sgPolygonIntegrate(SO,mfanti,j-1,kk)-truev;
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% sgevals(jj)=evalCounter;
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end
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figure(1)
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figure;
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semilogy(ggevals,abs(ggerrs-ggerrs(end)),'b.','MarkerSize',36);
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end
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% semilogy(sgevals,abs(sgerrs),'g.','MarkerSize',36)
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% xlim([0,max(max(ggevals)*3,3*min(int2evals))])
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% end
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% Error = (sum(IntersectionI)-integral2(field,-2*.26180283,2*.26180283,-2*.26180283,2*.26180283))./integral2(field,-2*.26180283,2*.26180283,-2*.26180283,2*.26180283,'AbsTol',0);
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% Error=abs(sum(IntersectionI1(:,gaussOrders),1)-truev);
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% figure
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% nplot=1000;
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% epts=1.5;
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% [x,y]= meshgrid([-epts:(epts/nplot):epts]+3,[(-epts:(epts/nplot):epts)']+3); x=x(:); y=y(:); xp=x; yp=y;
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% xp(x.^2+y.^2>1)=nan; yp(x.^2+y.^2>1)=nan;
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% surf(reshape(xp,2*nplot+1,2*nplot+1),reshape(yp,2*nplot+1,2*nplot+1),zeros(2*nplot+1,2*nplot+1),field(reshape(xp,2*nplot+1,2*nplot+1),reshape(yp,2*nplot+1,2*nplot+1)),'edgecolor','none');
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% view([0 90])
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% hold on
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% for i=1:nElemMesh1
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% % Mesh1{i}=Mesh1{i};
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% plot_bern_poly(Mesh1{i},2,.001,{},{'k'},false)
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% end
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% plot_bern_poly(shapeObject,2,.001,{},{'k'},false)
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% for i=1:nElemMesh1
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% % Mesh1{i}=Mesh1{i};
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% plot_rat_bern_poly(Mesh2{i},2,.001,'k')
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% end
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% plot_rat_bern_poly(shapeObject,2,.001,'b')
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printError= floor(log(Error)/log(10));
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title({sprintf('Background function: $5y^3 + x^2 + 2y +3$, Error $\\approx 10^{%d}$',printError),sprintf('Quadrature points per side of intersection: $%d^2$',gaussOrders(end-1)-1)},'interpreter','latex','FontSize',16)
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% title({sprintf('Background function: $\\frac{y^3 - x^3 y^2 - xy -3}{x^2y^2 + 100}$, Error $\\approx 10^{%d}$',printError),sprintf('Quadrature points per side of intersection: $%d^2$',gaussOrders(end-1)-1)},'interpreter','latex','FontSize',16)
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axis off
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colorbar
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title({sprintf('Background function: $\\frac{y^3 - x^3 y^2 - xy -3}{x^2y^2 + 100}$')},'interpreter','latex','FontSize',16)
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function xw = gaussXW(bound1,bound2,pts)
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% 15 point gauss quadrature weights and nodes on interval [-1,1]
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gaussQuad=load("gaussQuad");
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w=gaussQuad.wv{pts-1};
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x=gaussQuad.abc{pts-1};
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scale=bound2-bound1;
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w=w*scale/2;
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x=(scale/2)*(x+1)+bound1;
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xw=[x w]
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end
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function kk = invTri(i)
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kk=floor(real((sqrt(1/4+2*(i-1))-1/2)));
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end
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