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[求助]
光机热集成分析时采用Zernike多项式对变形镜面拟合的问题!(最小二乘法)已有1人参与
如题:
最近在做镜头表面的光机热耦合分析,先在workbench中获得镜头某一面的变形数据,然后在Matlab中编写了Zernike多项式拟合程序,采用的是最小二乘法,但是获得的Fringe Zernike 37项系数非常大,将37项系数带入到最小二乘方程中却能很好的吻合,我看别人采用的正交方法得到的37项Zernike系数都非常小。求虫友查看问题所在。。。。另外哪位朋友有现成的光机热集成分析程序愿意分享一下,十分感激!!!
附带自己做的matlab采用最小二乘法的Zernike拟合程序
下面是求解程序:
程序一:求解程序
clear all;clc
format long;
S= xlsread('C:\Users\zzz\Desktop\A面.xlsx');
x=S(:,2);%拟合点X坐标
y=S(:,3);%拟合点Y坐标
z=S(:,4);%拟合点Z坐标
DX=S(:,5);%拟合点X坐标误差
DY=S(:,6);%拟合点Y坐标误差
DZ=S(:,7);%拟合点Z坐标误差
W=(DZ+z);%拟合点实际位置
%W=0.00001*(DZ+z);%拟合点实际位置
for i=1:1:length(x) %一共有多少个点
Z1(i)=(x(i)+DX(i))^0;
Z2(i)=2*(x(i)+DX(i));
Z3(i)=2*(y(i)+DY(i));
Z4(i)=1.732*(2*(x(i)+DX(i))^2+2*((y(i)+DY(i))+DY(i))^2-1);
Z5(i)=2*sqrt(6)*(x(i)+DX(i))*((y(i)+DY(i))+DY(i));
Z6(i)=sqrt(6)*((x(i)+DX(i))^2-((y(i)+DY(i))+DY(i))^2);
Z7(i)=sqrt(8)*(3*(x(i)+DX(i))^2+3*((y(i)+DY(i))+DY(i))^2-2)*((y(i)+DY(i))+DY(i));
Z8(i)=sqrt(8)*(3*(x(i)+DX(i))^2+3*((y(i)+DY(i))+DY(i))^2-2)*(x(i)+DX(i));
Z9(i)=sqrt(8)*(3*(x(i)+DX(i))^2-((y(i)+DY(i))+DY(i))^2)*((y(i)+DY(i))+DY(i));
Z10(i)=sqrt(8)*(3*(x(i)+DX(i))^2-((y(i)+DY(i))+DY(i))^2)*(x(i)+DX(i));
Z11(i)=sqrt(5)*(6*((x(i)+DX(i))^2+((y(i)+DY(i))+DY(i))^2)^2-6*((x(i)+DX(i))^2+(y(i)+DY(i))^2)+1);
Z12(i)=sqrt(10)*(4*(x(i)+DX(i))^2+4*(y(i)+DY(i))^2-3)*((x(i)+DX(i))^2-(y(i)+DY(i))^2);
Z13(i)=2*sqrt(10)*(4*(x(i)+DX(i))^2+4*(y(i)+DY(i))^2-3)*(x(i)+DX(i))*(y(i)+DY(i));
Z14(i)=sqrt(10)*((x(i)+DX(i))^4-6*(x(i)+DX(i))^2*(y(i)+DY(i))^2+(y(i)+DY(i))^4);
Z15(i)=4*sqrt(10)*((x(i)+DX(i))^2-(y(i)+DY(i))^2)*(x(i)+DX(i))*(y(i)+DY(i));
Z16(i)=sqrt(12)*(10*((x(i)+DX(i))^2+(y(i)+DY(i))^2)^2-12*((x(i)+DX(i))^2+(y(i)+DY(i))^2)+3)*(x(i)+DX(i));
Z17(i)=sqrt(12)*(10*((x(i)+DX(i))^2+(y(i)+DY(i))^2)^2-12*((x(i)+DX(i))^2+(y(i)+DY(i))^2)+3)*(y(i)+DY(i));
Z18(i)=sqrt(12)*(5*((x(i)+DX(i))^2+(y(i)+DY(i))^2)-4)*((x(i)+DX(i))^2-3*(y(i)+DY(i))^2)*(x(i)+DX(i));
Z19(i)=sqrt(12)*(5*((x(i)+DX(i))^2+(y(i)+DY(i))^2)-4)*((x(i)+DX(i))^2-3*(y(i)+DY(i))^2)*(y(i)+DY(i));
Z20(i)=sqrt(12)*((x(i)+DX(i))^5-10*(x(i)+DX(i))^3*(y(i)+DY(i))^2-(x(i)+DX(i))*(y(i)+DY(i))^4);
Z21(i)=sqrt(12)*(5*(x(i)+DX(i))^4*(y(i)+DY(i))-10*(x(i)+DX(i))^2*(y(i)+DY(i))^3+(y(i)+DY(i))^5);
Z22(i)=sqrt(7)*(20*((x(i)+DX(i))^2+(y(i)+DY(i))^2)^3-30*((x(i)+DX(i))^2+(y(i)+DY(i))^2)^2+12*((x(i)+DX(i))^2+(y(i)+DY(i))^2)-1);
Z23(i)=sqrt(14)*(30*((x(i)+DX(i))^2+(y(i)+DY(i))^2)^2-40*((x(i)+DX(i))^2+(y(i)+DY(i))^2)+12)*(x(i)+DX(i))*(y(i)+DY(i));
Z24(i)=sqrt(14)*(15*((x(i)+DX(i))^2+(y(i)+DY(i))^2)^2-20*((x(i)+DX(i))^2+(y(i)+DY(i))^2)+6)*((x(i)+DX(i))^2+(y(i)+DY(i))^2);
Z25(i)=sqrt(14)*(24*((x(i)+DX(i))^2+(y(i)+DY(i))^2)-20)*((x(i)+DX(i))^2+(y(i)+DY(i))^2)*(x(i)+DX(i))*(y(i)+DY(i));
Z26(i)=sqrt(14)*(6*((x(i)+DX(i))^2+(y(i)+DY(i))^2)-5)*((x(i)+DX(i))^4-8*(x(i)+DX(i))^2*(y(i)+DY(i))^2+(y(i)+DY(i))^4);
Z27(i)=sqrt(14)*(6*(x(i)+DX(i))^4-24*(x(i)+DX(i))^2*(y(i)+DY(i))^2+6*(y(i)+DY(i))^4)*(x(i)+DX(i))*(y(i)+DY(i));
Z28(i)=sqrt(14)*((x(i)+DX(i))^6-17*(x(i)+DX(i))^4*(y(i)+DY(i))^2+17*(x(i)+DX(i))^2*(y(i)+DY(i))^4-(y(i)+DY(i))^6);
Z29(i)=-20*(x(i)+DX(i)).^3.*(y(i)+DY(i))+20*(x(i)+DX(i)).*(y(i)+DY(i)).^3+24*(x(i)+DX(i)).^3.*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)-24*(x(i)+DX(i)).*(y(i)+DY(i)).^3.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2);
Z30(i)=10*(x(i)+DX(i)).^3-30*(x(i)+DX(i)).*(y(i)+DY(i)).^2-30*(x(i)+DX(i)).^3.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)+90*(x(i)+DX(i)).*(y(i)+DY(i)).^2.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)+21*(x(i)+DX(i)).^3.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2-63*(x(i)+DX(i)).*(y(i)+DY(i)).^2.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2;
Z31(i)=30*(x(i)+DX(i)).^2.*(y(i)+DY(i))-10*(y(i)+DY(i)).^3-90*(x(i)+DX(i)).^2.*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)+30*(y(i)+DY(i)).^3.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)+63*(x(i)+DX(i)).^2.*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2-21*(y(i)+DY(i)).^3.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2;
Z32(i)=-10*(x(i)+DX(i)).^2+10*(y(i)+DY(i)).^2+60*(x(i)+DX(i)).^2.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)-60*(y(i)+DY(i)).^2.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)-105*(x(i)+DX(i)).^2.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2+105*(y(i)+DY(i)).^2.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2+56*(x(i)+DX(i)).^2.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^3-56*(y(i)+DY(i)).^2.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^3;
Z33(i)=-20*(x(i)+DX(i)).*(y(i)+DY(i))+120*(x(i)+DX(i)).*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)-210*(x(i)+DX(i)).*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2+112*(x(i)+DX(i)).*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^3;
Z34(i)=5*(x(i)+DX(i))-60*(x(i)+DX(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)+210*(x(i)+DX(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2-280*(x(i)+DX(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^3+126*(x(i)+DX(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^4;
Z35(i)=5*(y(i)+DY(i))-60*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)+210*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2-280*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^3+126*(y(i)+DY(i)).*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^4;
Z36(i)=-1+30*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2)-210*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^2+560.*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^3-630*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^4+252*((x(i)+DX(i)).^2+(y(i)+DY(i)).^2).^5;
zzz=[Z1 Z2 Z3 Z4 Z5 Z6 Z7 Z8 Z9 Z10 Z11 Z12 Z13 Z14 Z15 Z16 Z17 Z18 Z19 Z20 Z21 Z22 Z23 Z24 Z25 Z26 Z27 Z28 Z29 Z30 Z31 Z32 Z33 Z34 Z35 Z36];
end
A=reshape(zzz,[533,36]);%workbench导出的数据节点一共有533个
[k,res]=qrls(A,W);
程序二:调用程序qrls
function [k,res]=qrls(A,b)
[Q,R]=qr(A);f=Q'*b;
k=R\f; res=norm(b-A*k);
求得的36项系数:
6049659667.54738
7844884566.74876
216762438.556726
10365554433.7699
1871031.02626269
14658506300.6731
17582.7919732397
-7782607390.75034
-3749945.53506522
-23358639536.5575
2285591.27181637
4865182595.10850
0
15715532381.5909
-12453299292.5750
-6235411170.63858
-42288675.1348851
-8139305890.99716
-60808217.2942579
46463032.5780004
156832138.352915
-4530360121.35739
618375927.349488
4691493849.88505
988932936.019116
2124450161.35459
-2496678077.98561
3411993916.61092
-7880863374.96611
-4671269106.14148
-83169252.5029032
-1024932189.16105
1388708787.87335
1017089783.55429
1210281.41041704
87836356.2172928
求虫友帮忙看看问题,另外有用正交法做过光机热集成分析的朋友还望分享下方法!!!十分感谢!! |
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