Contents
- Tetrahedral volumes for global cartesian coordinates.
- Definite multi parameter parametric matrix [Nparametric]
- Definite multi parameter homogen matrix [Nhomogen]
- Homogen or parametric function integrate on tetrahedral volumes
- Homogen integrate(1): Parametric function
- Parametric integrate(2):Homogen function
%__________________________________________________________________________ %|HOMOGEN INTEGRATE ON TETRAHEDRAL VOLUME (A.Ö)27.01.2007 | %|_________________________________________________________________________| %|Integrate technique :Gamma function format | PARAMETRIC APPLICATION | %| :Beta function format | Symbolic toolbox | %|_____________________________________________|___________________________| %|FUNCTION | %|function [Integrate]=THhomogenint(Cor,mainfunction,flag) | %| | %|Integrate: Tetrahedral volume homogen integrate value | %|Cor: Tetrahedral 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
Tetrahedral volumes for global cartesian coordinates.
%[X] [Y] [Z] Cor=[0.00 0.00 0.00 %Node(1) 3.00 0.00 1.00 %Node(2) 1.00 3.00 0.00 %Node(3) 1.00 1.00 4.00]; %Node(4)
Definite multi parameter parametric matrix [Nparametric]
%#5# Point-Gaussian integrate results Nparametric = [1 %17/3 (Exact!) X %85/12 (Exact!) X^2 %51/5 (Exact!) X*Y %34/5 (Exact!) X^3 %493/50 (Exact!) X^2*Y %136/15 (Exact!) X*Y*Z ]' %374/45 (Exact!)
Nparametric = [ 1, X, X^2, X*Y, X^3, X^2*Y, X*Y*Z]
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) ]';
Homogen or parametric function integrate on tetrahedral volumes
syms X Y Z e1 e2 e3 e4 t1 t2 real px=Cor(:,1) ; py=Cor(:,2) ; pz=Cor(:,3) ; %Tetrahedral volume %Volume=1/6*abs(det([ 1 1 1 1 % X1 X2 X3 X4 % Y1 Y2 Y3 Y4 % Z1 Z2 Z3 Z4 ])); V=1/6*abs(det([ 1 1 1 1 px(1) px(2) px(3) px(4) py(1) py(2) py(3) py(4) pz(1) pz(2) pz(3) pz(4) ])); %Cartesian axises are connecting homogen axises %X = e1*px(1) + e2*px(2) + e3*px(3) + e4*px(4) %Y = e1*py(1) + e2*py(2) + e3*py(3) + e4*py(4) %Z = e1*pz(1) + e2*pz(2) + e3*pz(3) + e4*pz(4) %Homogen axises depends %e2=(1-e1)*t2 %e3=(1-e1-e2)*t1 => %e3=(1-e1-(1-e1)*t2)*t1 => %e3=(1-e1)*(1-t2)*t1 %e4=(1-e1-e2-e3)=> %e4=(1-e1-(1-e1)*t2-(1-e1)*(1-t2)*t1) => %e4=(1-e1)*[(1-t2)*(1-t1)] % e1 = e1 % e2 = (1-e1)*t2 % e3 = (1-e1)*(1-t2)*t1 % e4 = (1-e1)*[(1-t2)*(1-t1)] Acon=subs(Nparametric ,{X,Y,Z}, ... {e1*px(1) + e2*px(2) + e3*px(3) + e4*px(4), ... e1*py(1) + e2*py(2) + e3*py(3) + e4*py(4), ... e1*pz(1) + e2*pz(2) + e3*pz(3) + e4*pz(4) }); %Parametric function transform Ahparametric = subs(Acon,{e2,e3,e4},... {(1-e1)*t2,... (1-e1)*(1-t2)*t1,... (1-e1)*(1-t2)*(1-t1)}); %Homogen function transform Ahhomogen = subs(Nhomogen , {e2,e3,e4},... {(1-e1)*t2,... (1-e1)*(1-t2)*t1,... (1-e1)*(1-t2)*(1-t1)});
Homogen integrate(1): Parametric function
%Chain-Rules on integration %Integra1=int(6*V*Ah*(1-e1)*(1-e1)*(1-t2),e1,0,1); %Integra2=int(Integra1,t1,0,1); %Integra3=int(Integra2,t2,0,1); Integra1=int(int(int(6*V*Ahparametric*(1-e1)*(1-e1)*(1-t2),e1,0,1),t1,0,1),t2,0,1); Integra1
Integra1 = [ 17/3, 85/12, 51/5, 34/5, 493/30, 136/15, 374/45]
Parametric integrate(2):Homogen function
Integra2=int(int(int(6*V*Ahhomogen*(1-e1)*(1-e1)*(1-t2),e1,0,1),t1,0,1),t2,0,1); Integra2
Integra2 = [ -17/60, -17/60, -17/60, -17/60, 17/15, 17/15, 17/15, 17/15, 17/15, 17/15]