Hominttriangle

Contents

• Triangular area for global cartesian coordinates.
• Definite multi parameter parametric matrix [Nparametric]
• Definite multi parameter homogen matrix [Nhomogen]
• Homogen or parametric function integrate on triangular area
• Homogen integrate(1): Parametric function
• Parametric integrate(2):Homogen function

%__________________________________________________________________________
%|HOMOGEN INTEGRATE ON TRIANGULAR AREAS                    (A.Ö)27.01.2007 |
%|_________________________________________________________________________|
%|Integrate technique :Gamma function format   |   PARAMETRIC APPLICATION  |
%|                    :Beta function format    |     Symbolic toolbox      |
%|_____________________________________________|___________________________|
%|FUNCTION                                                                 |
%|function [Integrate]=TRhomogenint(Cor,mainfunction,flag)                 |
%|                                                                         |
%|Integrate: Tetrahedral volume homogen integrate value                    |
%|Cor: Triangular element global cartesian node coordinates                |
%|mainfunction:                                                            |
%|             1-Only multi parametric parameter polinom or function       |
%|  FLAG=1     2-Multi parameter parametric matrix                         |
%|  FLAG=2     3-Homogen multi parameter function or matrix                |
%|                                                                         |
%|_____________________________________________________Matlab ver(7.1)_____|

clc
clear

syms X Y Z real

Triangular area for global cartesian coordinates.

     %[X]  [Y]
Cor=[1.00  1.00       %Node(1)
     3.00  2.00       %Node(2)
     2.00  3.00]      %Node(3)

Cor =

     1     1
     3     2
     2     3

Definite multi parameter parametric matrix [Nparametric]

                         %#7# Point-Gaussian integrate results
Nparametric = [1         %3/2     (Exact!)
               X         %3       (Exact!)
               X^2       %25/4    (Exact!)
               X*Y       %49/8    (Exact!)
               X^3       %27/2    (Exact!)
               X^2*Y ]'  %259/20  (Exact!)

 
Nparametric =
 
[     1,     X,   X^2,   X*Y,   X^3, X^2*Y]
 
 

Definite multi parameter homogen matrix [Nhomogen]

syms  e1 e2 e3 real
Nhomogen= [   e1*(-1+2*e1)
              e2*(-1+2*e2)
              e3*(-1+2*e3)
              (2*e1-1+2*e2+2*e3)*(e1-1+e2+e3)
              4*e1*e2
              4*e2*e3
              4*e3*e1
             -4*e1*(e1-1+e2+e3)
             -4*e2*(e1-1+e2+e3)
             -4*e3*(e1-1+e2+e3) ]

 
Nhomogen =
 
                     e1*(-1+2*e1)
                     e2*(-1+2*e2)
                     e3*(-1+2*e3)
 (-1+2*e1+2*e2+2*e3)*(e1-1+e2+e3)
                          4*e1*e2
                          4*e2*e3
                          4*e3*e1
               -4*e1*(e1-1+e2+e3)
               -4*e2*(e1-1+e2+e3)
               -4*e3*(e1-1+e2+e3)
 
 

Homogen or parametric function integrate on triangular area

syms t1 real

px=Cor(:,1) ;
py=Cor(:,2) ;


%Triangular area
%Area=1/2*abs(det([ 1   1   1
%                   X1  X2  X3
%                   Y1  Y2  Y3 ]));

A=1/2*abs(det([ 1      1      1
               px(1)  px(2)  px(3)
               py(1)  py(2)  py(3)]));


%Cartesian axises are connecting homogen axises
%X = e1*px(1) + e2*px(2) + e3*px(3)
%Y = e1*py(1) + e2*py(2) + e3*py(3)
%Z = e1*pz(1) + e2*pz(2) + e3*pz(3)


%Homogen axises depends
%e2=(1-e1)*t1

%e3=(1-e1-e2) =>
%e3=(1-e1-(1-e1)*t1) =>
%e3=(1-e1)*(1-t1)


% homogen axis(1)                       e1 = e1
% homogen axis(2)                       e2 = (1-e1)*t1
% homogen axis(3)                       e3 = (1-e1)*(1-t1)


Acon=subs(Nparametric ,{X,Y}, ...
                       {e1*px(1) + e2*px(2) + e3*px(3), ...
                        e1*py(1) + e2*py(2) + e3*py(3) });

%Parametric function transform
Ahparametric = subs(Acon,{e2,e3},...
                         {(1-e1)*t1,...
                          (1-e1)*(1-t1) });



%Homogen function transform
Ahhomogen = subs(Nhomogen , {e2,e3},...
                            {(1-e1)*t1,...
                             (1-e1)*(1-t1) });

Homogen integrate(1): Parametric function

%Chain-Rules on integration
%Integra1=int(6*V*Ah*(1-e1)*(1-t1),e1,0,1);
%Integra2=int(Integra1,t1,0,1);


Integra1=int(int(2*A*Ahparametric*(1-e1),e1,0,1),t1,0,1);
Integra1

 
Integra1 =
 
[    3/2,      3,   25/4,   49/8,   27/2, 259/20]
 
 

Parametric integrate(2):Homogen function

Integra2=int(int(2*A*Ahhomogen*(1-e1),e1,0,1),t1,0,1)'

 
Integra2 =
 
[   0,   0,   0,   0, 1/2, 1/2, 1/2,   0,   0,   0]
 
 

Published with MATLAB® 7.1