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

Force

F1	2000
F2	1000

• 

  Figure 1 Truss geometry and 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

• 

  Table 1 Angle bar standard dimension and area

• 

  Table 2 Cost of material in market

Data Proceessing

• 

  Figure 2 Modelica Simulation Result of point displacement, reaction force, each truss stress, each truss safety factor

model Truss_3D_Tugas_Besar_Safety // Base Code by Josiah Enrico S. 1906356286 // Editing and Testing by Ahmad M. Fahmi 1806181836 // Commenting and Cleaning by Christopher S. Erwin 1806201056 // === DEFINITION PHASE === // Define the properties of the truss arrangement parameter Integer Nodes = size(node_points,1); // Number of Nodes parameter Integer Members = size(node_pairs,1); // Number of Members // Define the properties of the truss material parameter Real Yield = 215e6; // Yield Strength (Pa) parameter Real Area = 0.000224; // Area L Profile [Dimension=0.03, Thickness=0,004] (m^2) parameter Real Elas = 193e9; // Elasticity of material [SS 304] (Pa) // Define connections; list pair of nodes joined by a member parameter Integer node_pairs[:,2]=[1,5; 2,6; 3,7; 4,8; // Lowest vertical members 5,6; 6,7; 7,8; 5,8; // 1st shelf 5,9; 6,10; 7,11; 8,12; // Middle vertical members 9,10; 10,11; 11,12; 9,12; // 2nd shelf 9,13; 10,14; 11,15; 12,16; // Top vertical members 13,14; 14,15; 15,16; 13,16];// 3rd shelf // Define coordinates and boundaries of each node // [x, y, z, boundary_x, boundary_y, boundary_z] // If the node is bound (stationary) along that axis, then boundary_axis = 1 // otherwise the node is free and boundary_axis = 0 parameter Real node_points[:,6]=[0.3,-0.375,0,1,1,1; // Node 1 -0.3,-0.375,0,1,1,1; // Node 2 -0.3,0.375,0,1,1,1; // Node 3 0.3,0.375,0,1,1,1; // Node 4 0.3,-0.375,0.6,0,0,0; // Node 5 -0.3,-0.375,0.6,0,0,0; // Node 6 -0.3,0.375,0.6,0,0,0; // Node 7 0.3,0.375,0.6,0,0,0; // Node 8 0.3,-0.375,1.2,0,0,0; // Node 9 -0.3,-0.375,1.2,0,0,0; // Node 10 -0.3,0.375,1.2,0,0,0; // Node 11 0.3,0.375,1.2,0,0,0; // Node 12 0.3,-0.375,1.8,0,0,0; // Node 13 -0.3,-0.375,1.8,0,0,0; // Node 14 -0.3,0.375,1.8,0,0,0; // Node 15 0.3,0.375,1.8,0,0,0]; // Node 16 // Define external forces in Newtons on each node // [Fx, Fy, Fz] parameter Real Forces[nodes_xyz]={0,0,0, // Node 1 0,0,0, // Node 2 0,0,0, // Node 3 0,0,0, // Node 4 0,0,0, // Node 5 0,0,0, // Node 6 0,0,0, // Node 7 0,0,0, // Node 8 0,0,0, // Node 9 0,0,0, // Node 10 0,0,0, // Node 11 0,0,0, // Node 12 0,0,-500, // Node 13 0,0,-1000, // Node 14 0,0,-1000, // Node 15 0,0,-500}; // Node 16 // === SOLUTION PHASE === Real displacement[nodes_xyz], reaction[nodes_xyz]; Real check[3]; // To check if forces in xyz directions sum to zero (structure is static) Real safety[Members]; // Safety Factor of each member (based on stress experienced) Real mem_dis[3]; // Displacement in xyz directions experienced by member Real stress_vector[3]; // Stress experienced by member in xyz directions Real stress_mag[Members]; // Magnitude of stress experienced by member protected parameter Integer nodes_xyz = 3 * Nodes; // Size of vector of {node_#_x, node_#_y, node_#_z} containing all nodes Real node_i[3], node_j[3]; // Coordinates in xyz for member's node pairs // Global Stiffness Matrix construction Real G_element[nodes_xyz, nodes_xyz], G_total[nodes_xyz, nodes_xyz], G_total_new[nodes_xyz, nodes_xyz]; // Transformation Matrix construction Real cos_x, cos_y, cos_z, L, T[3,3]; // Limits to eliminate Float Errors Real err_solve = 10e-10, err_check = 10e-4; algorithm //Creating Global Matrix G_total := identity(nodes_xyz); for mem in 1:Members loop // Get coordinates of member's nodes for i in 1:3 loop node_i[i] := node_points[node_pairs[mem,1],i]; node_j[i] := node_points[node_pairs[mem,2],i]; end for; // Finding the Stiffness Matrix [Ke] // {F} = [Ke]{U} // [Ke] = k[T] L := Modelica.Math.Vectors.length(node_j - node_i); cos_x := (node_j[1]-node_i[1])/L; cos_y := (node_j[2]-node_i[2])/L; cos_z := (node_j[3]-node_i[3])/L; T := (Area * Elas / L)*[cos_x^2,cos_x*cos_y,cos_x*cos_z; cos_y*cos_x,cos_y^2,cos_y*cos_z; cos_z*cos_x,cos_z*cos_y,cos_z^2]; // Transforming to Global Matrix G_element := zeros(nodes_xyz,nodes_xyz); for m,n in 1:3 loop G_element[3*(node_pairs[mem,1]-1)+m, 3*(node_pairs[mem,1]-1)+n] := T[m,n]; G_element[3*(node_pairs[mem,2]-1)+m, 3*(node_pairs[mem,2]-1)+n] := T[m,n]; G_element[3*(node_pairs[mem,2]-1)+m, 3*(node_pairs[mem,1]-1)+n] := -T[m,n]; G_element[3*(node_pairs[mem,1]-1)+m, 3*(node_pairs[mem,2]-1)+n] := -T[m,n]; end for; // Adding to total Global Matrix G_total_new := G_total + G_element; G_total := G_total_new; end for; //Implementing boundary conditions to Global Matrix for nod in 1:Nodes loop // Boundary x-axis if node_points[nod,4] <> 0 then for i in 1:Nodes*3 loop G_total[(nod*3)-2,i] := 0; G_total[(nod*3)-2,(nod*3)-2] := 1; end for; end if; // Boundary y-axis if node_points[nod,5] <> 0 then for i in 1:Nodes*3 loop G_total[(nod*3)-1,i] := 0; G_total[(nod*3)-1,(nod*3)-1] := 1; end for; end if; // Boundary z-axis if node_points[nod,6] <> 0 then for i in 1:Nodes*3 loop G_total[nod*3,i] := 0; G_total[nod*3,nod*3] := 1; end for; end if; end for; // Solving displacement (U) // [G_total]{U} = {F} displacement := Modelica.Math.Matrices.solve(G_total, Forces); // Solving reaction (R) // {R} = [G_total]{U} - {F} reaction := (G_total_new * displacement) - Forces; // Eliminating float errors for i in 1:nodes_xyz loop reaction[i] := if abs(reaction[i])<= err_solve then 0 else reaction[i]; displacement[i] := if abs(displacement[i])<= err_solve then 0 else displacement[i]; end for; // Checking Sum of Forces // For static structures, should add up to 0 check[1]:=sum({reaction[i] for i in (1:3:(nodes_xyz-2))}) + sum({Forces[i] for i in (1:3:(nodes_xyz-2))}); check[2]:=sum({reaction[i] for i in (2:3:(nodes_xyz-1))}) + sum({Forces[i] for i in (2:3:(nodes_xyz-1))}); check[3]:=sum({reaction[i] for i in (3:3:(nodes_xyz))}) + sum({Forces[i] for i in (3:3:(nodes_xyz))}); // Eliminating float errors for i in 1:3 loop check[i] := if abs(check[i])<= err_check then 0 else check[i]; end for; // === POSTPROCESSING PHASE === //Calculating stress in each truss for mem in 1:Members loop // Getting node coordinate data and displacement for current member for i in 1:3 loop node_i[i] := node_points[node_pairs[mem,1],i]; node_j[i] := node_points[node_pairs[mem,2],i]; mem_dis[i] := abs(displacement[3*(node_pairs[mem,1]-1)+i] - displacement[3*(node_pairs[mem,2]-1)+i]); end for; // Solving the Displacement Matrix for current member // Stress = F/A = (E/L) * [dL] L:=Modelica.Math.Vectors.length(node_j-node_i); cos_x:=(node_j[1]- node_i[1])/L; cos_y:=(node_j[2]- node_i[2])/L; cos_z:=(node_j[3]- node_i[3])/L; T := (Elas/L)*[cos_x^2,cos_x*cos_y,cos_x*cos_z; cos_y*cos_x,cos_y^2,cos_y*cos_z; cos_z*cos_x,cos_z*cos_y,cos_z^2]; stress_vector:=(T*mem_dis); // The stress in vector components {stress_x, stress_y, stress_z} stress_mag[mem]:=Modelica.Math.Vectors.length(stress_vector); // Magnitude/length of the stress vectors for each member end for; // Safety Factor // Factor of Safety = (Yield Strength of material) / (Stress experienced by Member) for mem in 1:Members loop if stress_mag[mem]>0 then safety[mem] := Yield / stress_mag[mem]; else safety[mem] := 0; end if; end for; end Truss_3D_Tugas_Besar_Safety;

Elasticity Constraint

Area Constraint

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;