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Numerical Calculation: Triangular Integral
The book continues from the previous chapter "Gauss-Lelande Integral Formula"
**Requirements:** In the given space triangle ΔABCΔABC\Delta ABC, A(x1,y1,z1),B(x2,y2,z2),C(x3,y3,z3)A(x1,y1,z1 ),B(x2,y2,z2),C(x3,y3,z3)A(x_1,y_1,z_1),B(x_2,y_2,z_2),C(x_3,y_3,z_3), known function f (x,y,z)f(x,y,z)f(x,y,z), use numerical method to solve the integral: ∬ΔABCf(x,y,z)dS∬ΔABCf(x,y,z) dS\iint_{\Delta ABC}f(x,y,z)\text dS.
**Solution:** Refer to the integration method given by triangle_lyness_rule. I don’t understand the specific details, but the idea is similar to Gaussian integration. Calculate the integration points and coefficient weights on the plane for integration .
Triangular integration points and integral weight calculation
Triangle Llyness Rule
triangle_lyness_rule gives the coefficient point positions and weight coefficients in standard triangles at different orders. For example, the figure below shows the integration point position and weight coefficient when Rule=10Rule=10Rule=10. The following table shows the integration accuracy PrecisionPrecisionPrecision number of integration points orderorderorder and whether the integration point includes the triangle centercentercentercenter under different RuleRuleRule.
| Rule | Order | Precision | Center |
|---|---|---|---|
| 0 | 1 | 1 | YES |
| 1 | 3 | 2 | NO |
| 2 | 4 | 2 | YES |
| 3 | 4 | 3 | YES |
| 4 | 7 | 3 | YES |
| 5 | 6 | 4 | NO |
| 6 | 10 | 4 | YES |
| 7 | 9 | 4 | NO |
| 8 | 7 | 5 | YES |
| 9 | 10 | 5 | YES |
| 10 | 12 | 6 | NO |
| 11 | 16 | 6 | YES |
| 12 | 13 | 6 | YES |
| 13 | 13 | 7 | YES |
| 14 | 16 | 7 | YES |
| 15 | 16 | 8 | YES |
| 16 | 21 | 8 | NO |
| 17 | 16 | 8 | YES |
| 18 | 19 | 9 | YES |
| 19 | 22 | 9 | YES |
| 20 | 27 | 11 | NO |
| 21 | 28 | 11 | YES |
Coordinate transformation
The coordinate transformation relationship is calculated using the shape function method of the "Gauss-Lelande Integral Formula" in the previous article . The three-node shape function is obtained, and the remaining steps are similar to those in the "Gauss-Lelande Integral Formula" .
⎧⎩⎨N1(s,t)=1−s−tN2(s,t)=sN3(s,t)=t{N1(s,t)=1−s−tN2(s,t)=sN3(s,t)=t\begin{cases}
N_1(s,t)=1-s-t\
N_2(s,t)=s\
N_3(s,t)=t\
\end{cases}\
Improve
Under the guidance of Brother KY, the above steps can be further simplified. The reason is that the shape function of triangle coordinate transformation is simple and coordinate operations can be performed directly. The Jacobi coefficient is directly equal to the area of the triangle. See the code for details.
points test
The following is a test integration function. The data stored in LYNESSRULE.txtLYNESSRULE.txtLYNESS_RULE.txt is too long, so it is placed in the Gitee: link to be updated warehouse.
%% 测试三角形积分
clc;clear;
global TriCoeff
% 导入积分系数
TriCoeff=loadLynessFromTxT("LYNESS_RULE.txt");
P1=[0,0,0];
P2=[2,0,0];
P3=[0,3,0];
% 积分函数
func=@(x,y,z) (x^6+y^3+1);
count=1;
for rule=0:1:21
[P_W] = getTrianglePoints([P1;P2;P3],rule);
[N,~]=size(P_W);
res=0;
for i=1:1:N
res=res+func(P\_W(i,1),P\_W(i,2),P\_W(i,3))*P\_W(i,4);
end
resA(count,1)=res;
resA(count,2)=rule;
resA(count,3)=N;
count=count+1;
end
%% matlab 自带积分函数
pfun = @(x,y) (x.^6+y.^3+1);
xmin = 0;
xmax = 2;
ymin = 0;
ymax = @(x) -3/2*x+3;
r = integral2(pfun,xmin,xmax,ymin,ymax);
%% plot
figure(22)
plot(resA(:,2),resA(:,1),'r-o');hold on;
plot(resA(:,2),r*ones(22,1),'b-');grid on;
xticks([0:2:22]);
xlim([0,22]);
legend("TRIANGLE LYNESS RULE 积分","Matlab integral2积分");
text(10,14,"积分函数：(x^6+y^3+1)")
text(10,12,"积分区域：(0,0,0),(2,0,0),(0,3,0)");
xlabel("Lyness Rule");
ylabel("积分数值");
Comparison of points results
code
getTrianglePoints.m
function [P_W] = getTrianglePoints(Triangle,Rule)
% getTrianglePoints 三角形面元积分
% https://people.sc.fsu.edu/~jburkardt/cpp\_src/triangle\_lyness\_rule/triangle\_lyness\_rule.html
% 输入：
% Triangle(3,3)：三角形面元三个点
% Rule：triangle\_lyness\_rule
% 输出：
% P\_W(:,4)：P\_W(:,1:3)积分点、P\_W(:,4)权重系数
%% 任意空间三角形 =》平面直角三角形 坐标转换
% 形函数
N1=@(s,t) -s-t+1;
N2=@(s,t) s;
N3=@(s,t) t;
N1_s=@(s,t) -1;
N2_s=@(s,t) 1;
N3_s=@(s,t) 0;
N1_t=@(s,t) -1;
N2_t=@(s,t) 0;
N3_t=@(s,t) 1;
P1=Triangle(1,:);
P2=Triangle(2,:);
P3=Triangle(3,:);
global TriCoeff;
data=TriCoeff{Rule+1,1};
[order,~]=size(data);
P_W=zeros(order,4);
for i=1:1:order
P\_W(i,1:3)=Loc2Glo(data(i,1:2));
P\_W(i,4)=data(i,3)*Jacobi(data(i,1:2));
end
function Pglobal=Loc2Glo(loc)
%loc(1,2)
Pglobal=N1(loc(1),loc(2))*P1+...
N2(loc(1),loc(2))*P2+...
N3(loc(1),loc(2))*P3;
end
function J=Jacobi(Loc)
s=N1\_s(Loc(1),Loc(2))*P1+...
N2\_s(Loc(1),Loc(2))*P2+...
N3\_s(Loc(1),Loc(2))*P3;
t=N1\_t(Loc(1),Loc(2))*P1+...
N2\_t(Loc(1),Loc(2))*P2+...
N3\_t(Loc(1),Loc(2))*P3;
%三角形，这里多除了一个2
J=norm(cross(s,t))/2;
end
end
getTrianglePointsSimplified.m
function [P_W] = getTrianglePointsSimplified(Triangle,Rule)
global TriCoeff;
points = TriCoeff{Rule+1};
weights = points(:,3)';
points(:,3) = 1-points(:,1)-points(:,2);
P_W = zeros(size(points,1),4);
P\_W(:,1:3)=points*Triangle;
area = 0.5*norm(cross(Triangle(1,:)-Triangle(2,:),Triangle(1,:)-Triangle(3,:)));
P\_W(:,4)=weights*area;
end
loadLynessFromTxT.m
function TriCoeff = loadLynessFromTxT(filename)
%LOADLYNESSFROMTXT 加载系数
TriCoeff=cell(22,1);
fp=fopen(filename,'r');
data=textscan(fp,"%f,%f,%f");
fclose(fp);
ALL=[data{1,1},data{1,2},data{1,3}];
[N,~]=size(ALL);
i=1;
while i<N
rule=ALL(i,1);
order=ALL(i,2);
TriCoeff{rule+1,1}=ALL(i+1:i+order,:);
i=i+order+1;
end
end