Difference between revisions of "Tugas Besar Metode Numerik - Josiah Enrico S"

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'''Powell Method'''
 
'''Powell Method'''
  
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model Powell_Method_Opt_Jos_Ratio
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model Powell_Method_Opt_Jos_Ratio
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parameter Integer order=2; //order of regression
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parameter Integer order=2; //order of regression
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parameter Integer GoldN=15; //maximum iteration of linear gold optimization
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parameter Integer GoldN=15; //maximum iteration of linear gold optimization
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parameter Integer N=10; //maximum iteration
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parameter Integer N=10; //maximum iteration
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parameter Real maxerror=1e-50; //maximum error
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parameter Real maxerror=1e-50; //maximum error
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//Assumed to be X function
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//Assumed to be X function
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parameter Real RatioArea[size(Area,1)]={3.6534e-5, 3.75348e-5, 3.79639e-5, 3.79507e-05, 3.74821e-05};
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parameter Real RatioArea[size(Area,1)]={3.6534e-5, 3.75348e-5, 3.79639e-5, 3.79507e-05, 3.74821e-05};
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parameter Real CostArea[size(Area,1)]={829776,1053955,1308215,1722235,2202651};
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parameter Real CostArea[size(Area,1)]={829776,1053955,1308215,1722235,2202651};
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parameter Real Area[:]={141e-6, 184e-6, 231e-6, 304e-6, 384e-6};
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parameter Real Area[:]={141e-6, 184e-6, 231e-6, 304e-6, 384e-6};
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Real CoeArea[order+1];
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Real CoeArea[order+1];
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Real CoeAreaCost[order+1];
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Real CoeAreaCost[order+1];
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//Assumed to be Y function
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//Assumed to be Y function
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parameter Real RatioElas[size(Elas,1)]={2.83875e-5, 8.5637e-5, 12.5153e-5, 5.70447e-5};
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parameter Real RatioElas[size(Elas,1)]={2.83875e-5, 8.5637e-5, 12.5153e-5, 5.70447e-5};
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parameter Real CostElas[size(Elas,1)]={1367994,732878,492691,622939};
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parameter Real CostElas[size(Elas,1)]={1367994,732878,492691,622939};
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parameter Real Elas[:]={192e9, 195e9, 198e9, 201e+9};
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parameter Real Elas[:]={192e9, 195e9, 198e9, 201e+9};
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Real CoeElas[order+1];
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Real CoeElas[order+1];
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Real CoeElasCost[order+1];
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Real CoeElasCost[order+1];
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//Guessing Start (1 point, 2 vector) - the second vector will be auto-generated
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//Guessing Start (1 point, 2 vector) - the second vector will be auto-generated
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parameter Real StartPoint[2]={171e-6,195e9};
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parameter Real StartPoint[2]={171e-6,195e9};
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parameter Real VectorPoint[2]={205e-6,200e9};
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parameter Real VectorPoint[2]={205e-6,200e9};
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//Solution
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//Solution
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Real RatioZ[N]; //Optimum result
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Real RatioZ[N]; //Optimum result
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Real CostZ; //Cost Optimum result
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Real CostZ; //Cost Optimum result
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Real x3[2]; //Optimum point coordinate (x,y)
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Real x3[2]; //Optimum point coordinate (x,y)
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Real error[N]; //Error  
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Real error[N]; //Error  
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protected
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protected
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Real jos;
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Real jos;
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Real RatioZX;
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Real RatioZX;
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Real RatioZY;
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Real RatioZY;
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Real CostZX;
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Real CostZX;
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Real CostZY;
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Real CostZY;
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Real x0tes[2];
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Real x0tes[2];
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Real x0[2];
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Real x0[2];
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Real x1[2];
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Real x1[2];
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Real x2[2];
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Real x2[2];
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Real grad1;
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Real grad1;
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Real grad2;
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Real grad2;
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Real grad3;
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Real grad3;
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//Checking approach  
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//Checking approach  
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Real cek0[N];
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Real cek0[N];
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Real cek1[N];
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Real cek1[N];
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Real cek2[N];
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Real cek2[N];
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Real cek3[N];
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Real cek3[N];
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Real yek0[N];
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Real yek0[N];
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Real yek1[N];
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Real yek1[N];
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Real yek2[N];
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Real yek2[N];
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Real yek3[N];
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Real yek3[N];
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algorithm
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algorithm
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//Creating function at both  
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//Creating function at both  
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CoeArea:=Curve_Fitting(Area,RatioArea,order);
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CoeArea:=Curve_Fitting(Area,RatioArea,order);
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CoeElas:=Curve_Fitting(Elas,RatioElas,order);
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CoeElas:=Curve_Fitting(Elas,RatioElas,order);
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CoeAreaCost:=Curve_Fitting(Area,CostArea,order);
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CoeAreaCost:=Curve_Fitting(Area,CostArea,order);
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CoeElasCost:=Curve_Fitting(Elas,CostElas,order);
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CoeElasCost:=Curve_Fitting(Elas,CostElas,order);
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//Creating 2 vector defined in 2 gradient (grad1,grad2)
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//Creating 2 vector defined in 2 gradient (grad1,grad2)
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x0:=StartPoint;
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x0:=StartPoint;
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x0tes:=VectorPoint;
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x0tes:=VectorPoint;
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grad1:=(x0tes[2]-x0[2])/(x0tes[1]-x0[1]);
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grad1:=(x0tes[2]-x0[2])/(x0tes[1]-x0[1]);
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grad2:=-0.5*grad1;
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grad2:=-0.5*grad1;
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//Provide error variable
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//Provide error variable
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RatioZX:=CoeArea[1]*x0[1]^2+CoeArea[2]*x0[1]+CoeArea[3];
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RatioZX:=CoeArea[1]*x0[1]^2+CoeArea[2]*x0[1]+CoeArea[3];
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RatioZY:=CoeElas[1]*x0[2]^2+CoeElas[2]*x0[2]+CoeElas[3];
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RatioZY:=CoeElas[1]*x0[2]^2+CoeElas[2]*x0[2]+CoeElas[3];
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jos:=(RatioZX+RatioZY)/2;
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jos:=(RatioZX+RatioZY)/2;
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//Computing approach
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//Computing approach
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for i in 1:N loop
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for i in 1:N loop
 
   cek0[i]:=x0[1];
 
   cek0[i]:=x0[1];
 
   yek0[i]:=x0[2];
 
   yek0[i]:=x0[2];
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     break;
 
     break;
 
   end if;
 
   end if;
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end for;
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end for;
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end Powell_Method_Opt_Jos_Ratio;
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end Powell_Method_Opt_Jos_Ratio;
 
|}
 
|}
  

Revision as of 15:58, 6 January 2021

Abstract

Introduction

Objective

  • Membuat program analisis truss dan optimasi sederhana dengan Modelica
  • Mengoptimasi harga pembuatan rangka truss sederhana dengan memvariasi dimensi dan elastisitas material.

Methodology

  • Metode Optimasi Golden Section (univariabel)
  • Metode Optimasi Golden Section (multivariabel)

Procedure

Geometri dan Load

Tugas Besar Metnum Geometri Jos.jpg


Constraint:

- Spesifikasi L (Panjang) dan geometri rangka truss

- Gaya beban terhadap struktur (1000 N dan 2000 N)


Asumsi:

- Beban akan terdistribusi hanya pada point penghubung dan semua gaya yang dipengaruhi moment diamggap tidak ada karena sistem bersifat truss


Koleksi Data

Tugas Besar Metnum Dimension Jos.jpg Tugas Besar Metnum Sample Jos.jpg

Data Proceessing

Elasticity Constraint

Tugas Besar Metnum ExcelElas Jos.jpg
Tugas Besar Metnum GraphElas Jos.jpg

Area Constraint

Tugas Besar Metnum ExcelArea Jos.jpg
Tugas Besar Metnum GraphArea Jos.jpg

Result and Analysis

Modelica Result

Powell Method 

model Powell_Method_Opt_Jos_Ratio

parameter Integer order=2; //order of regression
parameter Integer GoldN=15; //maximum iteration of linear gold optimization
parameter Integer N=10; //maximum iteration
parameter Real maxerror=1e-50; //maximum error

//Assumed to be X function
parameter Real RatioArea[size(Area,1)]={3.6534e-5, 3.75348e-5, 3.79639e-5, 3.79507e-05, 3.74821e-05};
parameter Real CostArea[size(Area,1)]={829776,1053955,1308215,1722235,2202651};
parameter Real Area[:]={141e-6, 184e-6, 231e-6, 304e-6, 384e-6};
Real CoeArea[order+1];
Real CoeAreaCost[order+1];

//Assumed to be Y function
parameter Real RatioElas[size(Elas,1)]={2.83875e-5, 8.5637e-5, 12.5153e-5, 5.70447e-5};
parameter Real CostElas[size(Elas,1)]={1367994,732878,492691,622939};

parameter Real Elas[:]={192e9, 195e9, 198e9, 201e+9};
Real CoeElas[order+1];
Real CoeElasCost[order+1];


//Guessing Start (1 point, 2 vector) - the second vector will be auto-generated
parameter Real StartPoint[2]={171e-6,195e9};
parameter Real VectorPoint[2]={205e-6,200e9};

//Solution
Real RatioZ[N]; //Optimum result
Real CostZ; //Cost Optimum result
Real x3[2]; //Optimum point coordinate (x,y)
Real error[N]; //Error 

protected
Real jos;
Real RatioZX;
Real RatioZY;
Real CostZX;
Real CostZY;
Real x0tes[2];
Real x0[2];
Real x1[2];
Real x2[2];
Real grad1;
Real grad2;
Real grad3;

//Checking approach 
Real cek0[N];
Real cek1[N];
Real cek2[N];
Real cek3[N];
Real yek0[N];
Real yek1[N];
Real yek2[N];
Real yek3[N];

algorithm
//Creating function at both 
CoeArea:=Curve_Fitting(Area,RatioArea,order);
CoeElas:=Curve_Fitting(Elas,RatioElas,order);
CoeAreaCost:=Curve_Fitting(Area,CostArea,order);
CoeElasCost:=Curve_Fitting(Elas,CostElas,order);

//Creating 2 vector defined in 2 gradient (grad1,grad2)
x0:=StartPoint;
x0tes:=VectorPoint;
grad1:=(x0tes[2]-x0[2])/(x0tes[1]-x0[1]);
grad2:=-0.5*grad1;

//Provide error variable
RatioZX:=CoeArea[1]*x0[1]^2+CoeArea[2]*x0[1]+CoeArea[3];
RatioZY:=CoeElas[1]*x0[2]^2+CoeElas[2]*x0[2]+CoeElas[3];
jos:=(RatioZX+RatioZY)/2;

//Computing approach
for i in 1:N loop
 cek0[i]:=x0[1];
 yek0[i]:=x0[2];
 x1:=Gold_Opt_Func_2D_Vector(x0,grad1,GoldN,CoeArea,CoeElas);
 cek1[i]:=x1[1];
 yek1[i]:=x1[2];
 x2:=Gold_Opt_Func_2D_Vector(x1,grad2,GoldN,CoeArea,CoeElas);
 cek2[i]:=x2[1];
 yek2[i]:=x2[2];
 
   if (x0[1]==x2[1]) and (x0[2]<>x2[2]) then
     grad3:=1/0; //divergent gradient, please subtitute vector
     x3:=Gold_Opt_Func_2D_Vector_1(x2,1,GoldN,CoeArea,CoeElas);    
   else  
     grad3:=(x2[2]-x0[2])/(x2[1]-x0[1]);
     x3:=Gold_Opt_Func_2D_Vector(x0,grad3,GoldN,CoeArea,CoeElas);  
   end if;
 
 cek3[i]:=x3[1];
 yek3[i]:=x3[2];
 grad1:=grad2;
 grad2:=grad3; 
   
 //Function Created
 RatioZX:=CoeArea[1]*x3[1]^2+CoeArea[2]*x3[1]+CoeArea[3];
 RatioZY:=CoeElas[1]*x3[2]^2+CoeElas[2]*x3[2]+CoeElas[3];
 RatioZ[i]:=(RatioZX+RatioZY)/2;
 
 CostZX:=CoeAreaCost[1]*x3[1]^2+CoeAreaCost[2]*x3[1]+CoeAreaCost[3];
 CostZY:=CoeElasCost[1]*x3[2]^2+CoeElasCost[2]*x3[2]+CoeElasCost[3];
 CostZ:=(CostZX+CostZY)/2;
 x0:=x3;
 
 error[i]:=abs(1-jos/RatioZ[i]);
 jos:=RatioZ[i];
 if error[i]<maxerror then
   break;
 end if;
end for;

end Powell_Method_Opt_Jos_Ratio;

Acknowledgement

Reference

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