Tugas Besar Metode Numerik - Josiah Enrico S

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

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

Data Proceessing

Elasticity Constraint

Area Constraint

Result and Analysis

Modelica Result

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;