Difference between revisions of "Faizal Rikaz Al Muntaqo"

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===Ujian Akhir Semester/ UAS (13/01/21)===
 
===Ujian Akhir Semester/ UAS (13/01/21)===
 +
Pada pertemuan kali ini diadakan Ujian Akhir Semester yang diadakan secara Daring/ Online. Berikut merupakan hasil pekerjaan/ jawaban dari UAS Metode Numerik saya.
 +
 +
'''Jawaban UAS No. 1'''
 +
 +
[[File:UAS_METNUM_Faizal Rikaz A_No. 1.jpg|500px|center]]
 +
 +
'''Jawaban UAS No. 2 & 3'''
 +
 +
[[File:UAS_METNUM_Faizal Rikaz A_No. 2 & 3.jpg|500px|center]]
 +
 +
'''Jawaban UAS No. 4 & 5'''
 +
 +
[[File:UAS_METNUM_Faizal Rikaz A_No. 4 & 5.jpg|500px|center]]
 +
 +
'''Jawaban UAS No. 6'''
 +
 +
[[File:UAS_METNUM_Faizal Rikaz A_No. 6.jpg|500px|center]]
 +
 +
'''Jawaban UAS No. 7'''
 +
 +
Berikut merupakan code untuk menjawab soal no. 7
 +
 +
model Metnum02_UAS_Faizal_Rikaz
 +
  //define initial variable
 +
  parameter Integer Points = size(P, 1);
 +
    //Number of Points
 +
  parameter Integer Trusses = size(C, 1);
 +
    //Number of Trusses
 +
  parameter Real Yield = 290e6;
 +
    //Yield Strength (Pa)
 +
  parameter Real Area = 4350;
 +
    //Area L Profile
 +
  parameter Real Elas = 193e9;
 +
    //Elasticity Stainless Steel 201  (Pa)
 +
    //define connection
 +
  parameter Integer C[:, 2] = [1, 2; 1, 3; 1, 4];
 +
    //define coordinates (please put orderly)
 +
  parameter Real P[:, 6] = [0, 0, 0, 1, 1, 1;          //1
 +
                            6.5, 6.5, 36.5, 1, 1, 1;  //2
 +
                            6.5, 6.5, 36.5, 1, 1, 1;  //3
 +
                            6.5, 6.5, 36.5, 1, 1, 1];  //4
 +
    //define external force (please put orderly)
 +
  parameter Real F[Points * 3] = {0,0,0,0,3708180,0,0,3708180,0,0,3708180,0};
 +
    //solution
 +
  Real displacement[N], reaction[N];
 +
  Real check[3];
 +
  Real stress1[Trusses];
 +
  Real safety[Trusses];
 +
  Real dis[3];
 +
  Real Str[3];
 +
protected
 +
  parameter Integer N = 3 * Points;
 +
  Real q1[3], q2[3], g[N, N], G[N, N], G_star[N, N], id[N, N] = identity(N), cx, cy, cz, L, X[3, 3];
 +
  Real err = 10e-10, ers = 10e-4;
 +
algorithm
 +
    //Creating Global Matrix
 +
  G := id;
 +
  for i in 1:Trusses loop
 +
    for j in 1:3 loop
 +
      q1[j] := P[C[i, 1], j];
 +
      q2[j] := P[C[i, 2], j];
 +
    end for;
 +
    L := Modelica.Math.Vectors.length(q2 - q1);
 +
    cx := (q2[1] - q1[1]) / L;
 +
    cy := (q2[2] - q1[2]) / L;
 +
    cz := (q2[3] - q1[3]) / L;
 +
    X := Area * Elas / L * [cx ^ 2, cx * cy, cx * cz; cy * cx, cy ^ 2, cy * cz; cz * cx, cz * cy, cz ^ 2];
 +
    g := zeros(N, N);
 +
    for m, n in 1:3 loop
 +
      g[3 * (C[i, 1] - 1) + m, 3 * (C[i, 1] - 1) + n] := X[m, n];
 +
      g[3 * (C[i, 2] - 1) + m, 3 * (C[i, 2] - 1) + n] := X[m, n];
 +
      g[3 * (C[i, 2] - 1) + m, 3 * (C[i, 1] - 1) + n] := -X[m, n];
 +
      g[3 * (C[i, 1] - 1) + m, 3 * (C[i, 2] - 1) + n] := -X[m, n];
 +
    end for;
 +
    G_star := G + g;
 +
    G := G_star;
 +
  end for;
 +
//Solving Matrix
 +
//Transforming to global matrix
 +
//Implementing boundary
 +
  for x in 1:Points loop
 +
    if P[x, 4] <> 0 then
 +
      for a in 1:Points * 3 loop
 +
        G[x * 3 - 2, a] := 0;
 +
        G[x * 3 - 2, x * 3 - 2] := 1;
 +
      end for;
 +
    end if;
 +
    if P[x, 5] <> 0 then
 +
      for a in 1:Points * 3 loop
 +
        G[x * 3 - 1, a] := 0;
 +
        G[x * 3 - 1, x * 3 - 1] := 1;
 +
      end for;
 +
    end if;
 +
    if P[x, 6] <> 0 then
 +
      for a in 1:Points * 3 loop
 +
        G[x * 3, a] := 0;
 +
        G[x * 3, x * 3] := 1;
 +
      end for;
 +
    end if;
 +
  end for;
 +
//Solving displacement
 +
  displacement := Modelica.Math.Matrices.solve(G, F);
 +
//Solving reaction
 +
  reaction := G_star * displacement - F;
 +
//Eliminating float error
 +
  for i in 1:N loop
 +
    reaction[i] := if abs(reaction[i]) <= err then 0 else reaction[i];
 +
    displacement[i] := if abs(displacement[i]) <= err then 0 else displacement[i];
 +
  end for;
 +
//Checking Force
 +
  check[1] := sum({reaction[i] for i in 1:3:N - 2}) + sum({F[i] for i in 1:3:N - 2});
 +
  check[2] := sum({reaction[i] for i in 2:3:N - 1}) + sum({F[i] for i in 2:3:N - 1});
 +
  check[3] := sum({reaction[i] for i in 3:3:N}) + sum({F[i] for i in 3:3:N});
 +
  for i in 1:3 loop
 +
    check[i] := if abs(check[i]) <= ers then 0 else check[i];
 +
  end for;
 +
//Calculating stress in each truss
 +
  for i in 1:Trusses loop
 +
    for j in 1:3 loop
 +
      q1[j] := P[C[i, 1], j];
 +
      q2[j] := P[C[i, 2], j];
 +
      dis[j] := abs(displacement[3 * (C[i, 1] - 1) + j] - displacement[3 * (C[i, 2] - 1) + j]);
 +
    end for;
 +
    L := Modelica.Math.Vectors.length(q2 - q1);
 +
    cx := (q2[1] - q1[1]) / L;
 +
    cy := (q2[2] - q1[2]) / L;
 +
    cz := (q2[3] - q1[3]) / L;
 +
    X := Elas / L * [cx ^ 2, cx * cy, cx * cz; cy * cx, cy ^ 2, cy * cz; cz * cx, cz * cy, cz ^ 2];
 +
    Str := X * dis;
 +
    stress1[i] := Modelica.Math.Vectors.length(Str);
 +
  end for;
 +
//Solving Matrix
 +
//Safety factor
 +
  for i in 1:Trusses loop
 +
    if stress1[i] > 0 then
 +
      safety[i] := Yield / stress1[i];
 +
    else
 +
      safety[i] := 0;
 +
    end if;
 +
  end for;
 +
 +
end Metnum02_UAS_Faizal_Rikaz;
 +
 +
Berikut merupakan hasil coding yang telah disimulasikan pada aplikasi OpenModelica
 +
 +
[[File:Hasil Coding 1.jpg|750px|center]]
 +
 +
[[File:Hasil Coding 2.jpg|750px|center]]
 +
 +
[[File:Hasil Coding 3.jpg|750px|center]]
 +
 +
[[File:Hasil Coding 4.jpg|750px|center]]
 +
 +
Berikut merupakan hasil dari simulasi pada aplikasi OpenModelica
 +
 +
[[File:Hasil_Simulasi_1.jpg|300px|center]] [[File:Hasil Simulasi 2.jpg|300px|center]]
 +
 +
[[File:Hasil Simulasi 3.jpg|300px|center]] [[File:Hasil Simulasi 4.jpg|300px|center]]
 +
 +
[[File:Hasil Simulasi 5.jpg|300px|center]] [[File:Hasil Simulasi 6.jpg|300px|center]]

Latest revision as of 23:00, 14 January 2021

Selamat Datang di laman Wiki saya!

Biodata

Faizal Rikaz Al Muntaqo

Nama  : Faizal Rikaz Al Muntaqo

NPM  : 1806201245

TTL  : Jakarta, 15 Desember 1999

Hobi  : Futsal

Saya adalah seorang mahasiswa Teknik Mesin UI angkatan 2018. Ketertarikan saya kepada Jurusan Teknik Mesin salah satunya didasari oleh rasa penasaran dan kesukaan saya terhadap teknologi yang terus berkembang tiap harinya dan dan juga tidak sedikit saya mendapatkan pengaruh secara tidak langsung dari ayah saya yang juga merupakan seorang Engineer. Saya sangat berharap nantinya ilmu ilmu yang telah saya dapatkan dari perkuliahan dapat diaplikasikan dikehidupan saya kedepannya dan semoga dapat berguna bagi keluarga, agama dan bangsa Indonesia.

Metode Numerik

Tujuan Pembelajaran

Tujuan pembelajaran dari mata kuliah Metode Numerik ini adalah:

  • 1.Memahami konsep dan prinsip dasar dalam metode numerik (contoh:Persamaan aljabar, algorithma, dll)
  • 2. Mengerti dan dapat mengaplikasikan aplikasi metode numerik
  • 3. Mampu menerapkan ilmu-ilmu metode numerik dalam persoalan teknik
  • 4. Mendapat nilai tambah adab sehingga dapat menjadi orang yang lebih beradab

Review Materi(Sebelum UTS)

Beberapa topik yang dibahas pada saat sebelum UTS meliputi:

  • 1. Deret Taylor dan McLaurin
  • 2. Regresi Linear
  • 3. Interpolasi
  • 4. Pseudocode

Pertemuan 1 (11/11/20)

Tugas Pertemuan 1 adalah mempelajari Modelica melalui situs Youtube lalu membuat video pembelajaran tentang Modelica kemudian di-upload di channel Youtube. Saya mempelajari Modelica melalui youtube pada situs berikut:

https://www.youtube.com/watch?v=m0Ahs8fEN28

Dan berikut merupakan Tugas video pembelajaran saya pada aplikasi modelica.

Pertemuan 2 (18/11/20)

Pada pertemuan kali ini kami sharing tugas yang telah diberikan diminggu lalu. Perbedaan Modelica dan Bahasa Coding lainnya (seperti: Phyton, dll) adalah modelica merupakan bahasa permodelan dan berbeda dengan bahasa coding sperti phyton dll. Pada proses simulate Open Modellica cukup lama karena proses dari bahasa modelica ke C lalu baru ke bahasa mesin. Namun ketika kita hanya merubah parameternya saja dengan cara me-resimulate maka proses akan cepat, karena tidak melewati proses perubahan bahasa lagi.

Mengapa menggunakan open modelica?

  • - Cocok dengan engineer
  • - Sistem perhitungan cepat
  • - Banyak penggunanya
  • - Free/ open technology

Tugas Video: https://www.youtube.com/watch?v=nVaCw_QBNaY

Pertemuan 3 (25/11/20)

Pada pertemuan ke 3 ini mempelajari tentang pseoudocode gauss elemination, lalu Pseudocode gauss dijelaskan oleh teman sekelas kami yaitu Christo. Lalu kami juga ditugaskan untuk membuat program sederhana menggunakan Open Modelica untuk menyelesaikan sebuah permasalahan yaitu mengenai Truss. Berikut Tugas Truss yang telah saya kerjakan.

TUGAS TRUSSES 

Faizal Rikaz A Trusses.jpg
class Tugas_Trusses

parameter Integer N=8; //Global Matrice
parameter Real A=0.001; //Area m2
parameter Real E=200e9; //Pa
Real KG[N,N]; //global
Real KGinitial[N,N]; //global
Real Sol[N]; //global dispplacement
Real X[N]={0,0,-1035.2762,-3863.7033,0,0,-1035.2762,-3863.7033};
Real R[N]; 
Real SolMat[N,1];
Real XMat[N,1];
Real L1=1;
Real L2=1;
Real L3=1.6;
Real L4=1.25;
Real L5=1.6;
Real Tetha1=degtorad(0);
Real Tetha2=degtorad(0);
Real Tetha3=degtorad(231.34);
Real Tetha4=degtorad(270);
Real Tetha5=degtorad(308.66);

//Boundary condition
 Integer b1=1;
 Integer b2=3;

//Truss 1
 Real X1=Tetha1;
 Real k1=A*E/L1;
 Real K1[4,4]; //stiffness matrice
 Real KG1[N,N];
 Integer p1o=1;
 Integer p1i=2;

//Truss 2
 Real X2=Tetha2;
 Real k2=A*E/L2;
 Real K2[4,4]; //stiffness matrice
 Real KG2[N,N];
 Integer p2o=2;
 Integer p2i=3;

//Truss 3
 Real X3=Tetha1;
 Real k3=A*E/L3;
 Real K3[4,4]; //stiffness matrice
 Real KG3[N,N];
 Integer p3o=2;
 Integer p3i=4;

//Truss 4
 Real X4=Tetha4;
 Real k4=A*E/L4;
 Real K4[4,4]; //stiffness matrice
 Real KG4[N,N];
 Integer p4o=1;
 Integer p4i=4;

//Truss 5
 Real X5=Tetha5;
 Real k5=A*E/L5;
 Real K5[4,4]; //stiffness matrice
 Real KG5[N,N];
 Integer p5o=3;
 Integer p5i=4;

algorithm

//Pembuatan Matrice Global
 K1:=Stiffness_Matrices(X1);
 KG1:=k1*Local_Global(K1,N,p1o,p1i);

 K2:=Stiffness_Matrices(X2);
 KG2:=k2*Local_Global(K2,N,p2o,p2i);

 K3:=Stiffness_Matrices(X3);
 KG3:=k3*Local_Global(K3,N,p3o,p3i);

 K4:=Stiffness_Matrices(X4);
 KG4:=k4*Local_Global(K4,N,p4o,p4i);

 K5:=Stiffness_Matrices(X5);
 KG5:=k5*Local_Global(K5,N,p5o,p5i);

 KG:=KG1+KG2+KG3+KG4+KG5;
 KGinitial:=KG;

//Implementing Boundary Condition
 for i in 1:N loop
  KG[2*b1-1,i]:=0;
  KG[2*b1,i]:=0;
  KG[2*b2-1,i]:=0;
  KG[2*b2,i]:=0;
 end for;

 KG[2*b1-1,2*b1-1]:=1;
 KG[2*b1,2*b1]:=1;
 KG[2*b2-1,2*b2-1]:=1;
 KG[2*b2,2*b2]:=1;

//Solving Displacement
 Sol:=Gauss_Jordan(N,KG,X);

//Solving Reaction Force
 SolMat:=matrix(Sol);
 XMat:=matrix(X);
 R:=Reaction_Trusses(N,KGinitial,SolMat,XMat);


end Tugas_Trusses;

Fungsi Degtorad 

function Degtorad
 input Real deg;
 output Real rad;
 protected
 constant Real pi = 3.14;
algorithm

rad:=deg*pi/180;

end Degtorad;

Fungsi Stiffness Matrices 

function Stiffness_Matrices
 input Real A;
 output Real X[4,4];

protected
 Real Y;
 Real float_error = 10e-10;
 final constant Real pi=2*Modelica.Math.asin(1.0);

algorithm

 Y:=A/180*pi;
    
X:=[(Modelica.Math.cos(Y))^2,Modelica.Math.cos(Y)*Modelica.Math.sin(Y),-(Modelica.Math.cos(Y))^2,- 
Modelica.Math.cos(Y)*Modelica.Math.sin(Y);

Modelica.Math.cos(Y)*Modelica.Math.sin(Y),(Modelica.Math.sin(Y))^2,-Modelica.Math.cos(Y)*Modelica.Math.sin(Y),- 
(Modelica.Math.sin(Y))^2;

-(Modelica.Math.cos(Y))^2,-Modelica.Math.cos(Y)*Modelica.Math.sin(Y), 
(Modelica.Math.cos(Y))^2,Modelica.Math.cos(Y)*Modelica.Math.sin(Y);

-Modelica.Math.cos(Y)*Modelica.Math.sin(Y),-(Modelica.Math.sin(Y))^2,Modelica.Math.cos(Y)*Modelica.Math.sin(Y), 
(Modelica.Math.sin(Y))^2];

 for i in 1:4 loop
 for j in 1:4 loop
   if abs(X[i,j]) <= float_error then
     X[i,j] := 0;
   end if;
 end for;
 end for;

end Stiffness_Matrices;

Fungsi Local Global 

function Local_Global
 input Real Y[4,4];
 input Integer B;
 input Integer p1;
 input Integer p2;
 output Real G[B,B];

algorithm

 for i in 1:B loop
 for j in 1:B loop
   G[i,j]:=0;
 end for;
 end for;

G[2*p1,2*p1]:=Y[2,2];
G[2*p1-1,2*p1-1]:=Y[1,1];
G[2*p1,2*p1-1]:=Y[2,1];
G[2*p1-1,2*p1]:=Y[1,2];

G[2*p2,2*p2]:=Y[4,4];
G[2*p2-1,2*p2-1]:=Y[3,3];
G[2*p2,2*p2-1]:=Y[4,3];
G[2*p2-1,2*p2]:=Y[3,4]; 

G[2*p2,2*p1]:=Y[4,2];
G[2*p2-1,2*p1-1]:=Y[3,1];
G[2*p2,2*p1-1]:=Y[4,1];
G[2*p2-1,2*p1]:=Y[3,2];

G[2*p1,2*p2]:=Y[2,4];
G[2*p1-1,2*p2-1]:=Y[1,3];
G[2*p1,2*p2-1]:=Y[2,3];
G[2*p1-1,2*p2]:=Y[1,4];

end Local_Global;

Fungsi Reaction Trusses 

function Reaction_Trusses
 input Integer N;
 input Real A[N,N];
 input Real B[N,1];
 input Real C[N,1];
 output Real Sol[N];
protected
 Real X[N,1];
 Real float_error = 10e-10;

algorithm
 X:=A*B-C;

 for i in 1:N loop
 if abs(X[i,1]) <= float_error then
   X[i,1] := 0;
 end if;
 end for;

 for i in 1:N loop
  Sol[i]:=X[i,1];
 end for;

end Reaction_Trusses;

TUGAS GAUSS JORDAN 

function Tugas_GaussJordan

input Real [:,:] A;
output Real [:,:] B;

protected // local variable
Integer h = 1;    //pivot row
Integer k = 1;    //pivot coloumn
Integer m = size(A,1); //Number of row
Integer n = size(A,2); //Number of column
Integer c = 0;
Integer max_row; // Row index of max number in pivot column

Real [:] pivot_column;
Real [:] pivot_row;
Real [:,:] temp_array;
Real r;

Real float_error = 10e-10;


algorithm

//function input A and output B 
B := A;
  
while h <= m and k <= n loop

  for i in 1 : m loop
    for j in 1 : n loop
     if abs(B[i,j]) <= float_error then
       B[i,j] := 0;
      end if;
    end for;
  end for;

//Finding pivot 
  pivot_column:= {B[i,h] for i in h:m};
  
    //Search for the bottom row that has the highest pivot value
    c:=h-1;
    for element in pivot_column loop
      c:= c+1;
      if abs(element)== max(abs(pivot_column)) then
        max_row :=c;
      end if;
    end for;
    
  //If there are no pivots in this column, move to the next column
  if B[max_row,k] == 0 then
    k:=k+1;
   
  else 
    // switch row h - max_row
    temp_array := B;
    temp_array[h] := B[max_row];
    temp_array[max_row] := B[h];
    B:= temp_array;
    
    //devide pivot row by pivot number
     B[h] := B[h]/B[h,k];
     
     for i in (h+1) :m loop
       r := B[i,k]/B[h,k];
      
      B[i,k]:=0;
      
      for j in (k+1) : n loop
        B[i,j] := B[i,j]-B[h,j] * r;
      end for;
    end for;
    
    //move to pivot column and the next row
    h := h+1;
    k := k+1;
    
  end if;
  
end while;

// from the top right
h :=m;
k :=n;

while h >=1 and k>=1 loop
  
  //dealing with error
  for i in 1:m loop
    for j in 1:n loop
      if abs(B[i,j]) <=float_error then
        B[i,j]:=0;
      end if;
    end for;
  end for; 

//finding pivot 
    pivot_row := {B[h,i] for i in 1:k};
    
    //Get position index k of pivot 
    c := 0;
    for element in pivot_row loop
      c := c+1;
      if element <> 0 then
        break;
      end if;
    end for;
    k:= c;
    
  // no pivot in this row, move to next row
  if B[h,k] == 0 then 
    h:= h-1;
    
  else
    //perform row operatation
    for i in 1:(h-1) loop
      r := B[i,k];
      B[i] := B[i] - B[h] *r;
    end for;
    
    //move to next pivot row dan column
    h:=h+1;
    k:=k+1;
    
  end if;
  
end while;
    
     
end Tugas_GaussJordan;

Pertemuan 4 (02/12/20)

Membuat diagram class dan flowchart

QUIZ 1 

Berikut jawaban beserta Flowchart dari QUIZ 1 saya.
Faizal Rikaz A Quiz 1 Page 1.jpg
Faizal Rikaz A Quiz 1 Page 2.jpg

TUGAS PERTEMUAN 4 

Berikut merupakan jawaban dari Tugas Pertemuan 4 saya
Faizal Rikaz A Soal 8.jpg
Faizal Rikaz A Jawaban Soal 8 Slide 1.jpg
Faizal Rikaz A Jawaban Soal 8 Slide 2.jpg
Faizal Rikaz A Jawaban Soal 8 Slide 3.jpg
Faizal Rikaz A Jawaban Soal 8 Slide 4.jpg
Faizal Rikaz A Jawaban Soal 8 Grafik 1.jpg
Faizal Rikaz A Jawaban Soal 8 Grafik 2.jpg


Pertemuan 5 (16/12/20)

Pertemuan kali ini kami belajar melalui video yang diberikan Bu Chandra tentang aplikasi metode numerik dalam optimasi sebuah sistem menggunakan OpenModelica. Dibahas pula contoh dari optimasi menggunakan metode Bracket. Berikut pseudocode yang digunakan pada OpenModelica : 

FungsiObjek.mo
function FungsiObjek
 
input Real x;
output Real y;

algorithm

y:= 2*Modelica.Math.sin(x)-x^2/10;

end FungsiObjek;

Fungsi panggil 

BracketOptimal.mo
model BracketOptimal

parameter Integer n = 8;
Real x1[n];
Real x2[n];
Real xup;
Real xlow;
Real f1[n];
Real f2[n];
Real xopt;
Real yopt;
Real d;

algorithm
xup := 4;
xlow := 0;

for i in 1:n loop
  d:=((5^(1/2)-1)/2) * (xup-xlow);
  x1[i] := xlow+d;
  x2[i] := xup-d;
  f1[i] := FungsiObjek(x1[i]);
  f2[i] := FungsiObjek(x2[i]);
  
  if f1[i]>f2[i] then
    xup := xup;
    xlow := x2[i];
    xopt := xup;
    yopt := f1[i];
    else
      xlow :=xlow;
      xup := x1[i];
      xopt := xup;
  end if;
end for;
 

end BracketOptimal;

Pertemuan 6 (23/12/20)

TUGAS BESAR

Pada pertemuan ke 6 ini kami diberi sebuah tugas besar yang dimana mahasiswa diharuskan mendesign suatu rangka dengan hasil yang terbaik namun juga memerhatikan cost agar mndapat harga yang relatif murah pula. Selain mempertimbangkan biaya serta material, area cross section truss pun juga ikut dipertimbangkan. 

Soal Tubes Metnum Faizal Rikaz A 1.jpg  Soal Tubes Metnum Faizal Rikaz A 2.jpg

Pendefinisian

Berikut merupakan pendefinisian node dan elemen pada struktur batang. 

Soal Tubes Metnum Faizal Rikaz A 3.jpg

Asumsi dan Constraint pada problem

Asumsi 

*Material
*Luas permukaan truss
*Tidak adanya bending
*Beban terdistribusi di node - node
*Batas displacement 0,001m sebelum buckling
*Nilai safety factor 2

Constraint: 

*Node 1, 2, 3, dan 4 fixed.
*Beban F1 dan F2 terdistribusi ke node sekitarnya
*Node 13 & 16 = 1000N
*Node 14 & 15 = 500N

Data yang digunakan

Berikut merupakan dua metode pendekatan yang digunakan dalam pencarian dimensi area dan material teroptimal dalam pembentukan rangka. 

Tubes Metnum Faizal Rikaz A 1.jpg

Permodelan Numerik

Berikut merupakan permodelan numerik dan komputasi pada Open Modelica 

//define initial variable
parameter Integer Points=size(P,1); //Number of Points
parameter Integer Trusses=size(C,1); //Number of Trusses
parameter Real Yield=2.0e8; //Yield Strength (Pa)
parameter Real Area=0.0001727;   //Area L Profile (Dimension=0.03, Thickness=0,004) (m2)
parameter Real Elas=192e11;     //Elasticity SS 316  (Pa)

model TugasBesar_FaizalRikaz

//define connection
parameter Integer C[:,2]=[1,5; 
                          2,6;
                          3,7;
                          4,8;
                          5,6;  //1st floor
                          6,7;  //1st floor
                          7,8;  //1st floor
                          5,8;  //1st floor
                          5,9;
                         6,10;
                         7,11;
                         8,12;
                         9,10; //2nd floor
                         10,11;//2nd floor 
                         11,12;//2nd floor
                          9,12; //2nd floor
                          9,13;
                         10,14;
                         11,15;
                         12,16;
                         13,14;//3rd floor
                         14,15;//3rd floor
                         15,16;//3rd floor
                        13,16];//3rd floor
                                                              
//define coordinates (please put orderly)
parameter Real P[:,6]=[0.3,-0.375,0,1,1,1;     //1
                       -0.3,-0.375,0,1,1,1;    //2
                       -0.3,0.375,0,1,1,1;     //3
                       0.3,0.375,0,1,1,1;      //4
                           
                       0.3,-0.375,0.6,0,0,0;   //5
                       -0.3,-0.375,0.6,0,0,0;  //6
                       -0.3,0.375,0.6,0,0,0;   //7
                       0.3,0.375,0.6,0,0,0;    //8
                            
                       0.3,-0.375,1.2,0,0,0;   //9
                       -0.3,-0.375,1.2,0,0,0;  //10  
                       -0.3,0.375,1.2,0,0,0;   //11
                       0.3,0.375,1.2,0,0,0;    //12
                            
                       0.3,-0.375,1.8,0,0,0;   //13
                       -0.3,-0.375,1.8,0,0,0;  //14
                       -0.3,0.375,1.8,0,0,0;   //15
                       0.3,0.375,1.8,0,0,0];   //16
                          
//define external force (please put orderly)
parameter Real F[Points*3]={0,0,0,
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,0, 
                            0,0,-500, 
                            0,0,-1000, 
                            0,0,-1000, 
                            0,0,-500}; 

//solution
Real displacement[N], reaction[N];
Real check[3];

Real stress1[Trusses];
Real safety[Trusses];
Real dis[3];
Real Str[3];
 
protected
parameter Integer N=3*Points;
Real q1[3], q2[3], g[N,N], G[N,N], G_star[N,N], id[N,N]=identity(N), cx, cy, cz, L, X[3,3];
Real err=10e-10, ers=10e-4;

algorithm
//Creating Global Matrix
G:=id;
for i in 1:Trusses loop
 for j in 1:3 loop
  q1[j]:=P[C[i,1],j];
  q2[j]:=P[C[i,2],j];
 end for;
      
   //Solving Matrix
   L:=Modelica.Math.Vectors.length(q2-q1);
   cx:=(q2[1]-q1[1])/L;
   cy:=(q2[2]-q1[2])/L;
   cz:=(q2[3]-q1[3])/L; 
   X:=(Area*Elas/L)*[cx^2,cx*cy,cx*cz;
                     cy*cx,cy^2,cy*cz;
                     cz*cx,cz*cy,cz^2];

   //Transforming to global matrix
   g:=zeros(N,N); 
   for m,n in 1:3 loop
     g[3*(C[i,1]-1)+m,3*(C[i,1]-1)+n]:=X[m,n];
     g[3*(C[i,2]-1)+m,3*(C[i,2]-1)+n]:=X[m,n];
     g[3*(C[i,2]-1)+m,3*(C[i,1]-1)+n]:=-X[m,n];
     g[3*(C[i,1]-1)+m,3*(C[i,2]-1)+n]:=-X[m,n];
   end for;  

 G_star:=G+g;
 G:=G_star;
end for;

//Implementing boundary
for x in 1:Points loop
 if P[x,4] <> 0 then
   for a in 1:Points*3 loop
     G[(x*3)-2,a]:=0;
     G[(x*3)-2,(x*3)-2]:=1;
   end for;
 end if;
 if P[x,5] <> 0 then
   for a in 1:Points*3 loop
     G[(x*3)-1,a]:=0;
     G[(x*3)-1,(x*3)-1]:=1;
   end for;
 end if;
 if P[x,6] <> 0 then
   for a in 1:Points*3 loop
     G[x*3,a]:=0;
     G[x*3,x*3]:=1;
   end for;
 end if;
end for;

//Solving displacement
displacement:=Modelica.Math.Matrices.solve(G,F);

//Solving reaction
reaction:=(G_star*displacement)-F;

//Eliminating float error
for i in 1:N loop
 reaction[i]:=if abs(reaction[i])<=err then 0 else reaction[i];
 displacement[i]:=if abs(displacement[i])<=err then 0 else displacement[i];
end for;

//Checking Force
check[1]:=sum({reaction[i] for i in (1:3:(N-2))})+sum({F[i] for i in (1:3:(N-2))});
check[2]:=sum({reaction[i] for i in (2:3:(N-1))})+sum({F[i] for i in (2:3:(N-1))});
check[3]:=sum({reaction[i] for i in (3:3:N)})+sum({F[i] for i in (3:3:N)});
  
for i in 1:3 loop
 check[i] := if abs(check[i])<=ers then 0 else check[i];
end for;

//Calculating stress in each truss
for i in 1:Trusses loop
for j in 1:3 loop
  q1[j]:=P[C[i,1],j];
  q2[j]:=P[C[i,2],j];
  dis[j]:=abs(displacement[3*(C[i,1]-1)+j]-displacement[3*(C[i,2]-1)+j]);
end for;
      
   //Solving Matrix
   L:=Modelica.Math.Vectors.length(q2-q1);
   cx:=(q2[1]-q1[1])/L;
   cy:=(q2[2]-q1[2])/L;
   cz:=(q2[3]-q1[3])/L; 
   X:=(Elas/L)*[cx^2,cx*cy,cx*cz;
                cy*cx,cy^2,cy*cz;
                cz*cx,cz*cy,cz^2];
   
   Str:=(X*dis);
   stress1[i]:=Modelica.Math.Vectors.length(Str);
end for;

//Safety factor
for i in 1:Trusses loop
 if stress1[i]>0 then
   safety[i]:=Yield/stress1[i];
 else
   safety[i]:=0;
 end if; 
end for;

end TugasBesar_FaizalRikaz;

Kurva Curve Fitting 

function Curve_Fitting

input Real X[:];
input Real Y[size(X,1)];
input Integer order=2;
output Real Coe[order+1];

protected
Real Z[size(X,1),order+1];
Real ZTr[order+1,size(X,1)];
Real A[order+1,order+1];
Real B[order+1];

algorithm

for i in 1:size(X,1) loop
 for j in 1:(order+1) loop
 Z[i,j]:=X[i]^(order+1-j);
 end for;
end for;
ZTr:=transpose(Z);

A:=ZTr*Z;
B:=ZTr*Y;
Coe:=Modelica.Math.Matrices.solve(A,B);

end Curve_Fitting;

Golden Section 

model Opt_Gold

parameter Real xd[:];
parameter Real yd[size(xd,1)];
parameter Real xlo=87e-6;
parameter Real xhi=504e-6; 
parameter Integer N=10; // maximum iteration
parameter Real es=0.0001; // maximum error

Real f1[N], f2[N], x1[N], x2[N], ea[N], y[3];
Real xopt,  fx;
protected
Real d, xl, xu, xint, R=(5^(1/2)-1)/2;

algorithm
xl := xlo; 
xu := xhi;
y  := Curve_Fitting(xd,yd);
 
for i in 1:N loop
 d:= R*(xu-xl);
 x1[i]:=xl+d;
 x2[i]:=xu-d;
 f1[i]:=y[1]*x1[i]^2+y[2]*x1[i]+y[3];
 f2[i]:=y[1]*x2[i]^2+y[2]*x2[i]+y[3];
 xint:=xu-xl;
 
 if f1[i]>f2[i] then
   xl:=x2[i];
   xopt:=x1[i];
   fx:=f1[i];
   else
     xu:=x1[i];
     xopt:=x2[i];
     fx:=f2[i];
 end if;
 
 ea[i]:=(1-R)*abs((xint)/xopt);
 if ea[i]<es then
   break;
 end if;
end for;

end Opt_Gold;


Hasil Perhitungan

  • Fixed Elasticity

Berikut merupakan hasil perhitungan dengan fixed elasticity

Dan berikut merupakan grafik yang diperoleh dengan menggunakan metode Curve Fitting 

Tubes Metnum Faizal Rikaz A 3 Fixed Elasticity.jpg

Tubes Metnum Faizal Rikaz A 5 Fixed Elasticity.jpg

Dari hasil diatas dapat dilihat bahwa hasil yang paling optimum adalah SS316 dengan dimensi 30x30x3mm


  • Fixed Area

Berikut merupakan hasil perhitungan dengan fixed area

Dan berikut merupakan grafik yang diperoleh dengan menggunakan metode Curve Fitting 

Tubes Metnum Faizal Rikaz A 4 Fixed Area.jpg

Tubes Metnum Faizal Rikaz A 6 Fixed Area.jpg

Dari hasil diatas dapat dilihat bahwa hasil yang paling optimum adalah SS316 dengan dimensi 30x30x3mm


Ujian Akhir Semester/ UAS (13/01/21)

Pada pertemuan kali ini diadakan Ujian Akhir Semester yang diadakan secara Daring/ Online. Berikut merupakan hasil pekerjaan/ jawaban dari UAS Metode Numerik saya.

Jawaban UAS No. 1

UAS METNUM Faizal Rikaz A No. 1.jpg

Jawaban UAS No. 2 & 3

UAS METNUM Faizal Rikaz A No. 2 & 3.jpg

Jawaban UAS No. 4 & 5

UAS METNUM Faizal Rikaz A No. 4 & 5.jpg

Jawaban UAS No. 6

UAS METNUM Faizal Rikaz A No. 6.jpg

Jawaban UAS No. 7

Berikut merupakan code untuk menjawab soal no. 7 

model Metnum02_UAS_Faizal_Rikaz
 //define initial variable
  parameter Integer Points = size(P, 1);
    //Number of Points
  parameter Integer Trusses = size(C, 1);
    //Number of Trusses
  parameter Real Yield = 290e6;
    //Yield Strength (Pa)
  parameter Real Area = 4350;
    //Area L Profile
  parameter Real Elas = 193e9;
    //Elasticity Stainless Steel 201  (Pa)
    //define connection
  parameter Integer C[:, 2] = [1, 2; 1, 3; 1, 4];
    //define coordinates (please put orderly)
  parameter Real P[:, 6] = [0, 0, 0, 1, 1, 1;          //1
                            6.5, 6.5, 36.5, 1, 1, 1;   //2
                            6.5, 6.5, 36.5, 1, 1, 1;   //3
                            6.5, 6.5, 36.5, 1, 1, 1];  //4
    //define external force (please put orderly)
  parameter Real F[Points * 3] = {0,0,0,0,3708180,0,0,3708180,0,0,3708180,0};
    //solution
  Real displacement[N], reaction[N];
  Real check[3];
  Real stress1[Trusses];
  Real safety[Trusses];
  Real dis[3];
  Real Str[3];
protected
  parameter Integer N = 3 * Points;
  Real q1[3], q2[3], g[N, N], G[N, N], G_star[N, N], id[N, N] = identity(N), cx, cy, cz, L, X[3, 3];
  Real err = 10e-10, ers = 10e-4;
algorithm
    //Creating Global Matrix
  G := id;
  for i in 1:Trusses loop
    for j in 1:3 loop
      q1[j] := P[C[i, 1], j];
      q2[j] := P[C[i, 2], j];
    end for;
    L := Modelica.Math.Vectors.length(q2 - q1);
    cx := (q2[1] - q1[1]) / L;
    cy := (q2[2] - q1[2]) / L;
    cz := (q2[3] - q1[3]) / L;
    X := Area * Elas / L * [cx ^ 2, cx * cy, cx * cz; cy * cx, cy ^ 2, cy * cz; cz * cx, cz * cy, cz ^ 2];
    g := zeros(N, N);
    for m, n in 1:3 loop
      g[3 * (C[i, 1] - 1) + m, 3 * (C[i, 1] - 1) + n] := X[m, n];
      g[3 * (C[i, 2] - 1) + m, 3 * (C[i, 2] - 1) + n] := X[m, n];
      g[3 * (C[i, 2] - 1) + m, 3 * (C[i, 1] - 1) + n] := -X[m, n];
      g[3 * (C[i, 1] - 1) + m, 3 * (C[i, 2] - 1) + n] := -X[m, n];
    end for;
    G_star := G + g;
    G := G_star;
  end for;
//Solving Matrix
//Transforming to global matrix
//Implementing boundary
  for x in 1:Points loop
    if P[x, 4] <> 0 then
      for a in 1:Points * 3 loop
        G[x * 3 - 2, a] := 0;
        G[x * 3 - 2, x * 3 - 2] := 1;
      end for;
    end if;
    if P[x, 5] <> 0 then
      for a in 1:Points * 3 loop
        G[x * 3 - 1, a] := 0;
        G[x * 3 - 1, x * 3 - 1] := 1;
      end for;
    end if;
    if P[x, 6] <> 0 then
      for a in 1:Points * 3 loop
        G[x * 3, a] := 0;
        G[x * 3, x * 3] := 1;
      end for;
    end if;
  end for;
//Solving displacement
  displacement := Modelica.Math.Matrices.solve(G, F);
//Solving reaction
  reaction := G_star * displacement - F;
//Eliminating float error
  for i in 1:N loop
    reaction[i] := if abs(reaction[i]) <= err then 0 else reaction[i];
    displacement[i] := if abs(displacement[i]) <= err then 0 else displacement[i];
  end for;
//Checking Force
  check[1] := sum({reaction[i] for i in 1:3:N - 2}) + sum({F[i] for i in 1:3:N - 2});
  check[2] := sum({reaction[i] for i in 2:3:N - 1}) + sum({F[i] for i in 2:3:N - 1});
  check[3] := sum({reaction[i] for i in 3:3:N}) + sum({F[i] for i in 3:3:N});
  for i in 1:3 loop
    check[i] := if abs(check[i]) <= ers then 0 else check[i];
  end for;
//Calculating stress in each truss
  for i in 1:Trusses loop
    for j in 1:3 loop
      q1[j] := P[C[i, 1], j];
      q2[j] := P[C[i, 2], j];
      dis[j] := abs(displacement[3 * (C[i, 1] - 1) + j] - displacement[3 * (C[i, 2] - 1) + j]);
    end for;
    L := Modelica.Math.Vectors.length(q2 - q1);
    cx := (q2[1] - q1[1]) / L;
    cy := (q2[2] - q1[2]) / L;
    cz := (q2[3] - q1[3]) / L;
    X := Elas / L * [cx ^ 2, cx * cy, cx * cz; cy * cx, cy ^ 2, cy * cz; cz * cx, cz * cy, cz ^ 2];
    Str := X * dis;
    stress1[i] := Modelica.Math.Vectors.length(Str);
  end for;
//Solving Matrix
//Safety factor
  for i in 1:Trusses loop
    if stress1[i] > 0 then
      safety[i] := Yield / stress1[i];
    else
      safety[i] := 0;
    end if;
  end for;

end Metnum02_UAS_Faizal_Rikaz;

Berikut merupakan hasil coding yang telah disimulasikan pada aplikasi OpenModelica

Hasil Coding 1.jpg
Hasil Coding 2.jpg
Hasil Coding 3.jpg
Hasil Coding 4.jpg

Berikut merupakan hasil dari simulasi pada aplikasi OpenModelica

Hasil Simulasi 1.jpg
Hasil Simulasi 2.jpg
Hasil Simulasi 3.jpg
Hasil Simulasi 4.jpg
Hasil Simulasi 5.jpg
Hasil Simulasi 6.jpg
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