Difference between revisions of "Metnum03-Bimo Adinugroho"

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== '''Pertemuan Kelima''' ==
 
== '''Pertemuan Kelima''' ==
 +
 +
Pada pertemuan kali ini, melanjutkan bahasan mengenai tugas yang diberikan minggu lalu. Tugas tersebut membahas mengenai gaya reaksi pada elemen beam penyusun yang sudah pernah dipelajari saat mata kuliah statika struktur. Kemudian dilanjutkan dengan pembahasan tugas dari beberapa rekan saya. Pertemuan ini menjelaskan hasil dari tugas rekan kami, Ahmad Mohammad Fahmi, mensenai coding yang ia buat pada tugas minggu lalu tentang Plane Truss dan Space Truss.
 +
 +
Diskusi berlangsung cutup interaktif dengan penjelasan dari Fahmi sendiri mengenai hasil tugas ia. Namun, masih banyak hal yang belum saya sepenuhnya mengerti, walau begitu tetap meneruskan progress diskusi pada platform ini. Soal baru diberikan kepada kami dari Pak Dai sebagai berikut
 +
 +
[[File:Soal3.jpeg|300px|center]]
 +
 +
Pada algoritma Ahmad Muhammad Fahmi, metode yang digunakan untuk penyelesaian sama dengan ''flowchart'' yang sudah saya buat untuk Quiz. Penyelesaian kedua soal ini memiliki ''parameter input'':
 +
 +
A (Luas Permukaan), E (Modulus Elastisitas), node, load
 +
 +
Pada algoritma ini dibuat terlebih dahulu beberapa ''function'' untuk mempermudah pemanggilan pada ''class''. Disini terdapat ''function'' yaitu Stiffness Matrix Global, Sum Stiffness Matrix Global, Boundary Stiffness Matrix Global, Gauss Jordan, Reaction Force, dan Check Force.
 +
 +
''' Pengaplikasian Coding Terhadap Soal '''
 +
 +
Berikut
 +
 +
{| class="wikitable"
 +
|-
 +
| style='border-style: none  none  solid  solid;' |
 +
'''Stiffness Matrix Element Function'''
 +
 +
  function StiffnessMatrixElement
 +
    input Real [:,6] inisiasi_mat;
 +
    output Real [size(inisiasi_mat,1),6,6] Ke_mat;
 +
  protected
 +
    Real cos_x;
 +
    Real cos_y;
 +
    Real cos_z;
 +
    Real [6] StiffTrig;
 +
    Real [6,6] StiffTrans;
 +
    Real [size(inisiasi_mat,1)] L;
 +
    Real [size(inisiasi_mat,1)] k_vec;
 +
  algorithm
 +
  L := {(sqrt(inisiasi_mat[i,2]^2 + inisiasi_mat[i,3]^2 + inisiasi_mat[i,4]^2)) for i in 1:size(inisiasi_mat,1)};
 +
  k_vec := {(inisiasi_mat[i,5] * inisiasi_mat[i,6] / L[i]) for i in 1:size(inisiasi_mat,1)};
 +
  // Finding stiffness matrix of each element member
 +
    for i in 1:size(inisiasi_mat,1) loop
 +
  // Clearing the matrices
 +
    StiffTrig := zeros(6);
 +
    StiffTrans := zeros(6,6);
 +
  // Converting degrees to radians
 +
    cos_x := inisiasi_mat[i,2]/L[i];
 +
    cos_y := inisiasi_mat[i,3]/L[i];
 +
    cos_z := inisiasi_mat[i,4]/L[i];
 +
  // {cos^2, sin^2, sincos}
 +
    StiffTrig := {(cos_x)^2,
 +
                  (cos_y)^2,
 +
                  (cos_z)^2,
 +
                  (cos_x*cos_y),
 +
                  (cos_x*cos_z),
 +
                  (cos_y*cos_z)};
 +
  // Construct stiffness transformation matrix
 +
  StiffTrans := [  StiffTrig[1],    StiffTrig[4],    StiffTrig[5], -1*StiffTrig[1], -1*StiffTrig[4], -1*StiffTrig[5];
 +
                  StiffTrig[4],    StiffTrig[2],    StiffTrig[6], -1*StiffTrig[4], -1*StiffTrig[2], -1*StiffTrig[6];
 +
                  StiffTrig[5],    StiffTrig[6],    StiffTrig[3], -1*StiffTrig[5], -1*StiffTrig[6], -1*StiffTrig[3];
 +
                -1*StiffTrig[1], -1*StiffTrig[4], -1*StiffTrig[5],    StiffTrig[1],    StiffTrig[4],    StiffTrig[5];
 +
                -1*StiffTrig[4], -1*StiffTrig[2], -1*StiffTrig[6],    StiffTrig[4],    StiffTrig[2],    StiffTrig[6];
 +
                -1*StiffTrig[5], -1*StiffTrig[6], -1*StiffTrig[3],    StiffTrig[5],    StiffTrig[6],    StiffTrig[3]];           
 +
  // Multiply in stiffness constant of element, add final stiffness matrix to Ke_mat
 +
  for m in 1:6 loop
 +
    for n in 1:6 loop
 +
      Ke_mat[i,m,n] := k_vec[i] * StiffTrans[m,n];
 +
    end for;
 +
  end for;
 +
  end for;
 +
  end StiffnessMatrixElement;
 +
 +
|}
 +
 +
{| class="wikitable"
 +
|-
 +
| style='border-style: none  none  solid  solid;' |
 +
'''Stiffness Matrix Global Function'''
 +
 +
function StiffnessMatrixGlobal
 +
  input Integer x;
 +
  input Integer [:,2] n;
 +
  input Real [:,6,6] Ke_mat;
 +
  output Real [size(Ke_mat,1),3*x,3*x] Kg_mat;
 +
algorithm
 +
  Kg_mat := zeros(size(Ke_mat,1),3*x,3*x);
 +
  for i in 1:size(Ke_mat,1) loop
 +
    Kg_mat[i,3*n[i,1],3*n[i,1]]:=Ke_mat[i,3,3];
 +
    Kg_mat[i,3*n[i,1],3*n[i,1]-1]:=Ke_mat[i,3,2];
 +
    Kg_mat[i,3*n[i,1],3*n[i,1]-2]:=Ke_mat[i,3,1];
 +
    Kg_mat[i,3*n[i,1]-1,3*n[i,1]]:=Ke_mat[i,2,3];
 +
    Kg_mat[i,3*n[i,1]-1,3*n[i,1]-1]:=Ke_mat[i,2,2];
 +
    Kg_mat[i,3*n[i,1]-1,3*n[i,1]-2]:=Ke_mat[i,2,1];
 +
    Kg_mat[i,3*n[i,1]-2,3*n[i,1]]:=Ke_mat[i,1,3];
 +
    Kg_mat[i,3*n[i,1]-2,3*n[i,1]-1]:=Ke_mat[i,1,2];
 +
    Kg_mat[i,3*n[i,1]-2,3*n[i,1]-2]:=Ke_mat[i,1,1];
 +
 +
    Kg_mat[i,3*n[i,2],3*n[i,2]]:=Ke_mat[i,6,6];
 +
    Kg_mat[i,3*n[i,2],3*n[i,2]-1]:=Ke_mat[i,6,5];
 +
    Kg_mat[i,3*n[i,2],3*n[i,2]-2]:=Ke_mat[i,6,4];
 +
    Kg_mat[i,3*n[i,2]-1,3*n[i,2]]:=Ke_mat[i,5,6];
 +
    Kg_mat[i,3*n[i,2]-1,3*n[i,2]-1]:=Ke_mat[i,5,5];
 +
    Kg_mat[i,3*n[i,2]-1,3*n[i,2]-2]:=Ke_mat[i,5,4];
 +
    Kg_mat[i,3*n[i,2]-2,3*n[i,2]]:=Ke_mat[i,4,6];
 +
    Kg_mat[i,3*n[i,2]-2,3*n[i,2]-1]:=Ke_mat[i,4,5];
 +
    Kg_mat[i,3*n[i,2]-2,3*n[i,2]-2]:=Ke_mat[i,4,4];
 +
 +
    Kg_mat[i,3*n[i,2],3*n[i,1]]:=Ke_mat[i,6,3];
 +
    Kg_mat[i,3*n[i,2],3*n[i,1]-1]:=Ke_mat[i,6,2];
 +
    Kg_mat[i,3*n[i,2],3*n[i,1]-2]:=Ke_mat[i,6,1];
 +
    Kg_mat[i,3*n[i,2]-1,3*n[i,1]]:=Ke_mat[i,5,3];
 +
    Kg_mat[i,3*n[i,2]-1,3*n[i,1]-1]:=Ke_mat[i,5,2];
 +
    Kg_mat[i,3*n[i,2]-1,3*n[i,1]-2]:=Ke_mat[i,5,1];
 +
    Kg_mat[i,3*n[i,2]-2,3*n[i,1]]:=Ke_mat[i,4,3];
 +
    Kg_mat[i,3*n[i,2]-2,3*n[i,1]-1]:=Ke_mat[i,4,2];
 +
    Kg_mat[i,3*n[i,2]-2,3*n[i,1]-2]:=Ke_mat[i,4,1];
 +
 +
    Kg_mat[i,3*n[i,1],3*n[i,2]]:=Ke_mat[i,3,6];
 +
    Kg_mat[i,3*n[i,1],3*n[i,2]-1]:=Ke_mat[i,3,5];
 +
    Kg_mat[i,3*n[i,1],3*n[i,2]-2]:=Ke_mat[i,3,4];
 +
    Kg_mat[i,3*n[i,1]-1,3*n[i,2]]:=Ke_mat[i,2,6];
 +
    Kg_mat[i,3*n[i,1]-1,3*n[i,2]-1]:=Ke_mat[i,2,5];
 +
    Kg_mat[i,3*n[i,1]-1,3*n[i,2]-2]:=Ke_mat[i,2,4];
 +
    Kg_mat[i,3*n[i,1]-2,3*n[i,2]]:=Ke_mat[i,1,6];
 +
    Kg_mat[i,3*n[i,1]-2,3*n[i,2]-1]:=Ke_mat[i,1,5];
 +
    Kg_mat[i,3*n[i,1]-2,3*n[i,2]-2]:=Ke_mat[i,1,4];
 +
  end for;
 +
end StiffnessMatrixGlobal;
 +
 +
|}
 +
 +
{| class="wikitable"
 +
|-
 +
| style='border-style: none  none  solid  solid;' |
 +
''' Sum Stiffness Matrix Element Function'''
 +
 +
function SumStiffnessMatrixGlobal
 +
  input Real [:,:,:] Kg_mat;
 +
  output Real [size(Kg_mat,2),size(Kg_mat,2)] KgTot_mat;
 +
algorithm
 +
  for a in 1:size(Kg_mat,2) loop
 +
    for b in 1:size(Kg_mat,2) loop
 +
      KgTot_mat[a,b] := sum(Kg_mat [:,a,b]);
 +
    end for;
 +
    end for;
 +
end SumStiffnessMatrixGlobal;
 +
 +
|}
 +
 +
{| class="wikitable"
 +
|-
 +
| style='border-style: none  none  solid  solid;' |
 +
''' Boundary Stiffness Matrix Global Function'''
 +
 +
function BoundaryStiffnessMatrixGlobal
 +
  input Real [:,:] KgTot_met;
 +
  input Integer[:] Boundary_xyz;
 +
  input Integer[:] Boundary_xy;
 +
  input Integer[:] Boundary_xz;
 +
  input Integer[:] Boundary_yz;
 +
  input Integer[:] Boundary_x;
 +
  input Integer[:] Boundary_y;
 +
  input Integer[:] Boundary_z;
 +
  output Real [size(KgTot_met,1),size(KgTot_met,1)] KgB_met;
 +
 
 +
algorithm
 +
  for a in 1:size(KgTot_met,1) loop
 +
    for b in 1:size(KgTot_met,1) loop
 +
    KgB_met[a,b] := KgTot_met [a,b];
 +
    end for;
 +
  end for;
 +
 
 +
  if Boundary_xyz[1] <> 0 then
 +
  for i in 1:size(KgTot_met,1) loop
 +
  for a in 1:size(Boundary_xyz,1) loop
 +
    for b in 0:2 loop
 +
      KgB_met[3*(Boundary_xyz[a])-b,i]:=0;
 +
      KgB_met[3*Boundary_xyz[a]-b,3*Boundary_xyz[a]-b]:=1;
 +
    end for;
 +
  end for;
 +
  end for;
 +
  end if;
 +
 +
  if Boundary_xy[1] <> 0 then
 +
  for i in 1:size(KgTot_met,1) loop
 +
  for a in 1:size(Boundary_xy,1) loop
 +
    for b in 1:2 loop
 +
      KgB_met[3*(Boundary_xy[a])-b,i]:=0;
 +
      KgB_met[3*Boundary_xy[a]-b,3*Boundary_xy[a]-b]:=1;
 +
    end for;
 +
  end for;
 +
  end for;
 +
  end if;
 +
 
 +
  if Boundary_xz[1] <> 0 then
 +
  for i in 1:size(KgTot_met,1) loop
 +
  for a in 1:size(Boundary_xz,1) loop
 +
    for b in 0:2:2 loop
 +
      KgB_met[3*(Boundary_xz[a])-b,i]:=0;
 +
      KgB_met[3*Boundary_xz[a]-b,3*Boundary_xz[a]-b]:=1;
 +
    end for;
 +
  end for;
 +
  end for;
 +
  end if;
 +
 
 +
  if Boundary_yz[1] <> 0 then
 +
  for i in 1:size(KgTot_met,1) loop
 +
  for a in 1:size(Boundary_yz,1) loop
 +
    for b in 0:1 loop
 +
      KgB_met[3*(Boundary_yz[a])-b,i]:=0;
 +
      KgB_met[3*Boundary_yz[a]-b,3*Boundary_yz[a]-b]:=1;
 +
    end for;
 +
  end for;
 +
  end for;
 +
  end if;
 +
 
 +
  if Boundary_x[1] <> 0 then
 +
  for i in 1:size(KgTot_met,1) loop
 +
  for a in 1:size(Boundary_x,1) loop
 +
    KgB_met[3*(Boundary_x[a])-2,i]:=0;
 +
    KgB_met[3*Boundary_x[a]-2,3*Boundary_x[a]-2]:=1;
 +
  end for;
 +
  end for;
 +
  end if;
 +
 
 +
  if Boundary_y[1] <> 0 then
 +
  for i in 1:size(KgTot_met,1) loop
 +
  for a in 1:size(Boundary_y,1) loop
 +
    KgB_met[3*(Boundary_y[a])-1,i]:=0;
 +
    KgB_met[3*Boundary_y[a]-1,3*Boundary_y[a]-1]:=1;
 +
  end for;
 +
  end for;
 +
  end if;
 +
 
 +
  if Boundary_z[1] <> 0 then
 +
  for i in 1:size(KgTot_met,1) loop
 +
  for a in 1:size(Boundary_z,1) loop
 +
      KgB_met[3*Boundary_z[a],i]:=0;
 +
      KgB_met[3*Boundary_z[a],3*Boundary_z[a]]:=1;
 +
  end for;
 +
  end for;
 +
  end if;
 +
end BoundaryStiffnessMatrixGlobal;
 +
 +
|}
 +
 +
{| class="wikitable"
 +
|-
 +
| style='border-style: none  none  solid  solid;' |
 +
'''Gauss Jordan Function'''
 +
 +
function GaussJordan
 +
  input Real [:,:] KgB_met;
 +
  input Real [size(KgB_met,1)] load_met;
 +
  output Real [size(KgB_met,1)] U_met;
 +
 
 +
  protected
 +
  Real float_error = 10e-10;
 +
 +
algorithm
 +
  U_met:=Modelica.Math.Matrices.solve(KgB_met,load_met);
 +
 +
  for i in 1:size(KgB_met,1) loop
 +
    if abs(U_met[i]) <= float_error then
 +
    U_met[i] := 0;
 +
    end if;
 +
  end for;
 +
 +
end GaussJordan;
 +
 +
|}
 +
 +
{| class="wikitable"
 +
|-
 +
| style='border-style: none  none  solid  solid;' |
 +
'''Reaction Force Function'''
 +
 +
function ReactionForce
 +
  input Real [:,:] KgTot_met;
 +
  input Real [size(KgTot_met,1)] U_met;
 +
  input Real [size(KgTot_met,1)] load_met;
 +
  output Real [size(KgTot_met,1)] R_met;
 +
  protected Real float_error = 10e-10;
 +
 +
algorithm
 +
  R_met := KgTot_met*U_met-load_met;
 +
 
 +
  for t in 1:size(KgTot_met,1) loop
 +
    if abs(R_met[t]) <= float_error then
 +
      R_met[t] := 0;
 +
    end if;
 +
  end for;
 +
 
 +
end ReactionForce;
 +
 +
|}
 +
 +
{| class="wikitable"
 +
|-
 +
| style='border-style: none  none  solid  solid;' |
 +
'''Check Force Function'''
 +
 +
function CheckForce
 +
  input Real [:] load;
 +
  input Real [size(load,1)] R;
 +
  output Real [3] F;
 +
  protected Real float_error = 10e-10;
 +
 
 +
  protected
 +
    Real load_x;
 +
    Real load_y;
 +
    Real load_z;
 +
    Real R_x;
 +
    Real R_y;
 +
    Real R_z;
 +
   
 +
algorithm
 +
  load_x := sum({load[i] for i in 1:3:(size(load,1)-2)});
 +
  load_y := sum({load[i] for i in 2:3:(size(load,1)-1)});
 +
  load_z := sum({load[i] for i in 3:3:size(load,1)});
 +
  R_x := sum({R[i] for i in 1:3:(size(load,1)-2)});
 +
  R_y := sum({R[i] for i in 2:3:(size(load,1)-1)});
 +
  R_z := sum({R[i] for i in 3:3:size(load,1)});
 +
 
 +
  F[1] := load_x + R_x;
 +
  F[2] := load_y + R_y;
 +
  F[3] := load_z + R_z;
 +
 
 +
  for i in 1:3 loop
 +
    if abs(F[i]) <= float_error then
 +
    F[i] := 0;
 +
    end if;
 +
  end for;
 +
 +
end CheckForce;
 +
 +
|}
 +
 +
Berikut merupakan ''class'' yang digunakan untuk memanggil ''function'' diatas dan menyelesaikan ''Example'' 3.3
 +
 +
{| class="wikitable"
 +
|-
 +
| style='border-style: none  none  solid  solid;' |
 +
'''Class'''
 +
 +
class QuizSoal1
 +
  //inisiasi = [ elemen#, dX, dY, dZ, A, E]
 +
  parameter Real [:,6] inisiasi = [1,  6,  0, -3, 1.56, 10.6e6; //isi sesuai data
 +
                                  2,  0,  0, -6, 1.56, 10.6e6;
 +
                                  3,  0,  6, -3, 1.56, 10.6e6;
 +
                                  4, -6,  0, -3, 1.56, 10.6e6;
 +
                                  5, -6,  6,  0, 1.56, 10.6e6;
 +
                                  6,  0,  6,  3, 1.56, 10.6e6];
 +
 +
  //node = [ i, j]                               
 +
  parameter Integer [size(inisiasi,1),2] node = [1, 2; //isi sesuai data
 +
                                                1, 3;
 +
                                                1, 4;
 +
                                                2, 3;
 +
                                                2, 4;
 +
                                                3, 4];
 +
 +
  //jumlah node
 +
  parameter Integer n = 4; //isi sesuai data
 +
 +
  //titik node boundary xyz
 +
  parameter Integer [:] Boundary_xyz = {1}; //isi sesuai data
 +
 +
  //titik node boundary xy
 +
  parameter Integer [:] Boundary_xy = {4}; //isi sesuai data
 +
 +
  //titik node boundary xz
 +
  parameter Integer [:] Boundary_xz = {0}; //isi sesuai data
 +
 +
  //titik node boundary yz
 +
  parameter Integer [:] Boundary_yz = {0}; //isi sesuai data
 +
 +
  //titik node boundary x
 +
  parameter Integer [:] Boundary_x = {3}; //isi sesuai data
 +
 +
  //titik node boundary y
 +
  parameter Integer [:] Boundary_y = {0}; //isi sesuai data
 +
 +
  //titik node boundary z
 +
  parameter Integer [:] Boundary_z = {0}; //isi sesuai data
 +
                           
 +
  //load = [ F1x, F1y, F1z,..., Fnx, Fny, Fnz]
 +
  parameter Real [3*n] load = {0,    0, 0,  //isi sesuai data
 +
                              0, -200, 0,
 +
                              0,    0, 0,
 +
                              0,    0, 0};
 +
 +
  Real [size(inisiasi,1)] L;
 +
  Real [size(inisiasi,1)] k;
 +
  Real [size(inisiasi,1),6,6] Ke;
 +
  Real [size(inisiasi,1),3*n,3*n] Kg;
 +
  Real [3*n,3*n] KgTot;
 +
  Real [3*n,3*n] KgB;
 +
  Real [3*n] U;
 +
  Real [3*n] R;
 +
 +
  //check force
 +
  Real [3] F;
 +
 +
equation
 +
L = {(sqrt(inisiasi[i,2]^2 + inisiasi[i,3]^2 + inisiasi[i,4]^2)) for i in 1:size(inisiasi,1)};
 +
 +
k = {(inisiasi[i,5] * inisiasi[i,6] / L[i]) for i in 1:size(inisiasi,1)};
 +
 +
Ke = StiffnessMatrixElement(inisiasi);
 +
 +
Kg = StiffnessMatrixGlobal(n, node, Ke);
 +
 +
KgTot = SumStiffnessMatrixGlobal(Kg);
 +
 +
KgB = BoundaryStiffnessMatrixGlobal(KgTot, Boundary_xyz, Boundary_xy, Boundary_xz, Boundary_yz, Boundary_x, Boundary_y, Boundary_z);
 +
 +
U = GaussJordan(KgB, load);
 +
 +
R = ReactionForce(KgTot, U, load);
 +
 +
F = CheckForce(load,R);
 +
 +
end QuizSoal1;
 +
 +
|}
 +
 +
Berikut merupakan hasil dan grafik yang didapat
 +
 +
[[File:1607924165277.jpg|200px]]
 +
[[File:1607924118583.jpg|200px]]
 +
[[File:1607924101463.jpg|200px]]
 +
 +
[[File:1607924228684.jpg|600px|center]]
  
 
== '''Pertemuan Keenam''' ==
 
== '''Pertemuan Keenam''' ==
  
 
Kelas pada hari Senin 21 Desember2020 diisi oleh Ibu Chandra. Beliau memaparkan materi pengaplikasian metode numerik pada OpenModelica.
 
Kelas pada hari Senin 21 Desember2020 diisi oleh Ibu Chandra. Beliau memaparkan materi pengaplikasian metode numerik pada OpenModelica.
 +
 +
Pada pembelajaran metode numerik ini, saya selalu berusaha mengikuti pembelajaran di kelas pada setiap pertemuan. Tentang konsep statika struktur saya sudah cukup paham, karena saya juga sudah pernah mempelajari statika struktur pada semester semester sebelumnya.
 +
 +
Untuk pembelajaran programming menggunakan OpenModelica, saya merasa memang cukup sulit, karena saya belum pernah mempelajari basic programming, saya mengalami kendala dalam pemahaman programmingnya(syntax). Saya selalu mengikuti tutorial dari teman sekelas ataupun platform Youtube tapi hasilnya sering kali "syntax error"
 +
 +
== '''Tugas Besar Metode Numerik''' ==
 +
 +
'''Tujuan :'''
 +
 +
Mencari harga pembuatan rangka truss sederhana dengan memvariasi dimensi dan elastisitas material secara optimal!
 +
 +
'''Berikut kasus yang diberikan :'''
 +
 +
[[File:Tugas Besar Metnum Geometri Jos.jpg]]
 +
 +
 +
'''Urutan penyelesaian :'''
 +
 +
'''1. Mendefinisikan karakteristik dari Truss'''
 +
 +
Dari soal diminta untuk mencari material pada struktur tersebut yang paling optimal. Besi siku(angle bar akan menjadi strukturnya. Membandingkan optimasi material dan optimasi luas penampang. Struktur pada kasus ini memiliki dimensi 0.75m x 0.6m x 1.8m. Pada bagian atas diberikan dua gaya yang memengaruhi struktur
 +
 +
F1 : 2000 N
 +
 +
F2 : 1000 N
 +
 +
'''2. Menentukan asusmsi dan node-node pada pengoperasian sistem'''
 +
 +
Gaya F1 dan F2 akan terdistribusi di point penghubung pada struktur. Struktur pada kasus ini memiliki sifat truss
 +
 +
'''3. Mencari data profil besi siku yang digunakan'''
 +
 +
[[File:Hargamaterialbm.jpeg]]
 +
 +
'''4. Melakukan simulasi pada OpenModelica'''
 +
 +
//define initial variable
 +
parameter Integer Points=size(P,1); //Number of Points
 +
parameter Integer Trusses=size(C,1); //Number of Trusses
 +
parameter Real Yield=215e6; //Yield Strength (Pa)
 +
parameter Real Area=0.000224;  //Area L Profile (Dimension=0.03, Thickness=0,004) (m2)
 +
parameter Real Elas=193e9;    //Elasticity SS 304  (Pa)
 +
 +
//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-6;
 +
 +
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 Trusses_3D_Tugas_Besar_Safety;
 +
 +
'''5. Optimasi 1 dan Optimasi 2'''
 +
 +
'''6. Mendapatkan kurva hasil optimasi yang telah dilakukan dan menganalisanya'''
 +
 +
 +
 +
== Ujian Akhir Semester ==
 +
 +
[[File:UASbm1.jpeg]]
 +
 +
[[File:UASbm2.jpeg]]
 +
 +
[[File:UASbm3.jpeg]]
 +
 +
[[File:UASbm4.jpeg]]

Latest revision as of 17:42, 13 January 2021

بِسْمِ اللّهِ الرَّحْمَنِ الرَّحِيْمِ

Kelas Metode Numerik 03

Nama : Bimo Adinugroho

NPM  : 1706025586

Pertemuan Pertama

Pertemuan pertama pada kelas metode numerik dengan Pak Dai saya diminta untuk menginstall aplikasi OpenModelica. Fungsi dari aplikasi ini untuk mensimulaikan suatu sistem yang didistribusikan dalam bentuk kode dan biner untuk penelitian. Pada pertemuan tersebut Pak Dai belum menjelaskan menggunakan aplikasi tersebut secara dalam, melainkan Pak Dai meminta mahasiswa untuk mereview kembali materi yang sudah diajarkan sebelum UTS, dan juga mencari fenomena apa saja yang pernah dilakukan dan dapat diaplikasikan dengan metode numerik.

Pertemuan Kedua

Om1bm.jpeg

Om2bm.jpeg

Om3bm.jpeg


Pertemuan Ketiga

Pada hari Senin, 23 November 2020 kelas Metnum diawali dengan membaca buku mengenai spring-mass systems

Soalpr.jpeg


Menghitung defleksi yang terjadi di setiap elemen dan gaya reaksinya!

Syntaxtugas3metnum.jpeg

Functionrumusbm.jpeg

Simulasitugas3metnum.jpeg


Quiz 1

Soal

Soal quiz.jpg

Jawabsoalkuisbm.jpeg

Gambar 2020-12-14 032203.png

Pertemuan Keempat

Pertemuan Kelima

Pada pertemuan kali ini, melanjutkan bahasan mengenai tugas yang diberikan minggu lalu. Tugas tersebut membahas mengenai gaya reaksi pada elemen beam penyusun yang sudah pernah dipelajari saat mata kuliah statika struktur. Kemudian dilanjutkan dengan pembahasan tugas dari beberapa rekan saya. Pertemuan ini menjelaskan hasil dari tugas rekan kami, Ahmad Mohammad Fahmi, mensenai coding yang ia buat pada tugas minggu lalu tentang Plane Truss dan Space Truss.

Diskusi berlangsung cutup interaktif dengan penjelasan dari Fahmi sendiri mengenai hasil tugas ia. Namun, masih banyak hal yang belum saya sepenuhnya mengerti, walau begitu tetap meneruskan progress diskusi pada platform ini. Soal baru diberikan kepada kami dari Pak Dai sebagai berikut

Soal3.jpeg

Pada algoritma Ahmad Muhammad Fahmi, metode yang digunakan untuk penyelesaian sama dengan flowchart yang sudah saya buat untuk Quiz. Penyelesaian kedua soal ini memiliki parameter input:

A (Luas Permukaan), E (Modulus Elastisitas), node, load

Pada algoritma ini dibuat terlebih dahulu beberapa function untuk mempermudah pemanggilan pada class. Disini terdapat function yaitu Stiffness Matrix Global, Sum Stiffness Matrix Global, Boundary Stiffness Matrix Global, Gauss Jordan, Reaction Force, dan Check Force.

Pengaplikasian Coding Terhadap Soal

Berikut

Stiffness Matrix Element Function 

 function StiffnessMatrixElement
   input Real [:,6] inisiasi_mat;
   output Real [size(inisiasi_mat,1),6,6] Ke_mat;
 protected
   Real cos_x;
   Real cos_y;
   Real cos_z;
   Real [6] StiffTrig;
   Real [6,6] StiffTrans;
   Real [size(inisiasi_mat,1)] L;
   Real [size(inisiasi_mat,1)] k_vec;
 algorithm
 L := {(sqrt(inisiasi_mat[i,2]^2 + inisiasi_mat[i,3]^2 + inisiasi_mat[i,4]^2)) for i in 1:size(inisiasi_mat,1)};
 k_vec := {(inisiasi_mat[i,5] * inisiasi_mat[i,6] / L[i]) for i in 1:size(inisiasi_mat,1)};
 // Finding stiffness matrix of each element member
    for i in 1:size(inisiasi_mat,1) loop
 // Clearing the matrices
    StiffTrig := zeros(6);
    StiffTrans := zeros(6,6);
 // Converting degrees to radians
    cos_x := inisiasi_mat[i,2]/L[i];
    cos_y := inisiasi_mat[i,3]/L[i];
    cos_z := inisiasi_mat[i,4]/L[i];
 // {cos^2, sin^2, sincos}
    StiffTrig := {(cos_x)^2,
                 (cos_y)^2,
                 (cos_z)^2,
                 (cos_x*cos_y),
                 (cos_x*cos_z),
                 (cos_y*cos_z)};
 // Construct stiffness transformation matrix
 StiffTrans := [  StiffTrig[1],    StiffTrig[4],    StiffTrig[5], -1*StiffTrig[1], -1*StiffTrig[4], -1*StiffTrig[5];
                  StiffTrig[4],    StiffTrig[2],    StiffTrig[6], -1*StiffTrig[4], -1*StiffTrig[2], -1*StiffTrig[6];
                  StiffTrig[5],    StiffTrig[6],    StiffTrig[3], -1*StiffTrig[5], -1*StiffTrig[6], -1*StiffTrig[3];
               -1*StiffTrig[1], -1*StiffTrig[4], -1*StiffTrig[5],    StiffTrig[1],    StiffTrig[4],    StiffTrig[5];
               -1*StiffTrig[4], -1*StiffTrig[2], -1*StiffTrig[6],    StiffTrig[4],    StiffTrig[2],    StiffTrig[6];
               -1*StiffTrig[5], -1*StiffTrig[6], -1*StiffTrig[3],    StiffTrig[5],    StiffTrig[6],    StiffTrig[3]];             
 // Multiply in stiffness constant of element, add final stiffness matrix to Ke_mat
 for m in 1:6 loop
   for n in 1:6 loop
     Ke_mat[i,m,n] := k_vec[i] * StiffTrans[m,n];
   end for;
 end for;
 end for;
 end StiffnessMatrixElement;

Stiffness Matrix Global Function 

function StiffnessMatrixGlobal
 input Integer x;
 input Integer [:,2] n;
 input Real [:,6,6] Ke_mat; 
 output Real [size(Ke_mat,1),3*x,3*x] Kg_mat;
algorithm
 Kg_mat := zeros(size(Ke_mat,1),3*x,3*x);
 for i in 1:size(Ke_mat,1) loop
   Kg_mat[i,3*n[i,1],3*n[i,1]]:=Ke_mat[i,3,3];
   Kg_mat[i,3*n[i,1],3*n[i,1]-1]:=Ke_mat[i,3,2];
   Kg_mat[i,3*n[i,1],3*n[i,1]-2]:=Ke_mat[i,3,1];
   Kg_mat[i,3*n[i,1]-1,3*n[i,1]]:=Ke_mat[i,2,3];
   Kg_mat[i,3*n[i,1]-1,3*n[i,1]-1]:=Ke_mat[i,2,2];
   Kg_mat[i,3*n[i,1]-1,3*n[i,1]-2]:=Ke_mat[i,2,1];
   Kg_mat[i,3*n[i,1]-2,3*n[i,1]]:=Ke_mat[i,1,3];
   Kg_mat[i,3*n[i,1]-2,3*n[i,1]-1]:=Ke_mat[i,1,2];
   Kg_mat[i,3*n[i,1]-2,3*n[i,1]-2]:=Ke_mat[i,1,1];

   Kg_mat[i,3*n[i,2],3*n[i,2]]:=Ke_mat[i,6,6];
   Kg_mat[i,3*n[i,2],3*n[i,2]-1]:=Ke_mat[i,6,5];
   Kg_mat[i,3*n[i,2],3*n[i,2]-2]:=Ke_mat[i,6,4];
   Kg_mat[i,3*n[i,2]-1,3*n[i,2]]:=Ke_mat[i,5,6];
   Kg_mat[i,3*n[i,2]-1,3*n[i,2]-1]:=Ke_mat[i,5,5];
   Kg_mat[i,3*n[i,2]-1,3*n[i,2]-2]:=Ke_mat[i,5,4];
   Kg_mat[i,3*n[i,2]-2,3*n[i,2]]:=Ke_mat[i,4,6];
   Kg_mat[i,3*n[i,2]-2,3*n[i,2]-1]:=Ke_mat[i,4,5];
   Kg_mat[i,3*n[i,2]-2,3*n[i,2]-2]:=Ke_mat[i,4,4];

   Kg_mat[i,3*n[i,2],3*n[i,1]]:=Ke_mat[i,6,3];
   Kg_mat[i,3*n[i,2],3*n[i,1]-1]:=Ke_mat[i,6,2];
   Kg_mat[i,3*n[i,2],3*n[i,1]-2]:=Ke_mat[i,6,1];
   Kg_mat[i,3*n[i,2]-1,3*n[i,1]]:=Ke_mat[i,5,3];
   Kg_mat[i,3*n[i,2]-1,3*n[i,1]-1]:=Ke_mat[i,5,2];
   Kg_mat[i,3*n[i,2]-1,3*n[i,1]-2]:=Ke_mat[i,5,1];
   Kg_mat[i,3*n[i,2]-2,3*n[i,1]]:=Ke_mat[i,4,3];
   Kg_mat[i,3*n[i,2]-2,3*n[i,1]-1]:=Ke_mat[i,4,2];
   Kg_mat[i,3*n[i,2]-2,3*n[i,1]-2]:=Ke_mat[i,4,1];

   Kg_mat[i,3*n[i,1],3*n[i,2]]:=Ke_mat[i,3,6];
   Kg_mat[i,3*n[i,1],3*n[i,2]-1]:=Ke_mat[i,3,5];
   Kg_mat[i,3*n[i,1],3*n[i,2]-2]:=Ke_mat[i,3,4];
   Kg_mat[i,3*n[i,1]-1,3*n[i,2]]:=Ke_mat[i,2,6];
   Kg_mat[i,3*n[i,1]-1,3*n[i,2]-1]:=Ke_mat[i,2,5];
   Kg_mat[i,3*n[i,1]-1,3*n[i,2]-2]:=Ke_mat[i,2,4];
   Kg_mat[i,3*n[i,1]-2,3*n[i,2]]:=Ke_mat[i,1,6];
   Kg_mat[i,3*n[i,1]-2,3*n[i,2]-1]:=Ke_mat[i,1,5];
   Kg_mat[i,3*n[i,1]-2,3*n[i,2]-2]:=Ke_mat[i,1,4];
 end for;
end StiffnessMatrixGlobal;

Sum Stiffness Matrix Element Function 

function SumStiffnessMatrixGlobal
 input Real [:,:,:] Kg_mat;
 output Real [size(Kg_mat,2),size(Kg_mat,2)] KgTot_mat;
algorithm
  for a in 1:size(Kg_mat,2) loop
   for b in 1:size(Kg_mat,2) loop
     KgTot_mat[a,b] := sum(Kg_mat [:,a,b]);
    end for;
   end for;
end SumStiffnessMatrixGlobal;

Boundary Stiffness Matrix Global Function 

function BoundaryStiffnessMatrixGlobal
 input Real [:,:] KgTot_met;
 input Integer[:] Boundary_xyz;
 input Integer[:] Boundary_xy;
 input Integer[:] Boundary_xz;
 input Integer[:] Boundary_yz;
 input Integer[:] Boundary_x;
 input Integer[:] Boundary_y;
 input Integer[:] Boundary_z;
 output Real [size(KgTot_met,1),size(KgTot_met,1)] KgB_met;
 
algorithm
 for a in 1:size(KgTot_met,1) loop
   for b in 1:size(KgTot_met,1) loop
    KgB_met[a,b] := KgTot_met [a,b];
   end for;
  end for; 
 
 if Boundary_xyz[1] <> 0 then
 for i in 1:size(KgTot_met,1) loop
  for a in 1:size(Boundary_xyz,1) loop
   for b in 0:2 loop
     KgB_met[3*(Boundary_xyz[a])-b,i]:=0;
     KgB_met[3*Boundary_xyz[a]-b,3*Boundary_xyz[a]-b]:=1;
   end for;
  end for;
 end for;
 end if;

 if Boundary_xy[1] <> 0 then
 for i in 1:size(KgTot_met,1) loop
  for a in 1:size(Boundary_xy,1) loop
   for b in 1:2 loop
     KgB_met[3*(Boundary_xy[a])-b,i]:=0;
     KgB_met[3*Boundary_xy[a]-b,3*Boundary_xy[a]-b]:=1;
   end for;
  end for;
 end for;
 end if;
 
 if Boundary_xz[1] <> 0 then
 for i in 1:size(KgTot_met,1) loop
  for a in 1:size(Boundary_xz,1) loop
   for b in 0:2:2 loop
     KgB_met[3*(Boundary_xz[a])-b,i]:=0;
     KgB_met[3*Boundary_xz[a]-b,3*Boundary_xz[a]-b]:=1;
   end for;
  end for;
 end for;
 end if;
 
 if Boundary_yz[1] <> 0 then
 for i in 1:size(KgTot_met,1) loop
  for a in 1:size(Boundary_yz,1) loop
   for b in 0:1 loop
     KgB_met[3*(Boundary_yz[a])-b,i]:=0;
     KgB_met[3*Boundary_yz[a]-b,3*Boundary_yz[a]-b]:=1;
   end for;
  end for;
 end for;
 end if;
 
 if Boundary_x[1] <> 0 then
 for i in 1:size(KgTot_met,1) loop
  for a in 1:size(Boundary_x,1) loop
   KgB_met[3*(Boundary_x[a])-2,i]:=0;
   KgB_met[3*Boundary_x[a]-2,3*Boundary_x[a]-2]:=1;
  end for;
 end for;
 end if;
 
 if Boundary_y[1] <> 0 then
 for i in 1:size(KgTot_met,1) loop
  for a in 1:size(Boundary_y,1) loop
   KgB_met[3*(Boundary_y[a])-1,i]:=0;
   KgB_met[3*Boundary_y[a]-1,3*Boundary_y[a]-1]:=1;
  end for;
 end for;
 end if;
 
 if Boundary_z[1] <> 0 then
 for i in 1:size(KgTot_met,1) loop
  for a in 1:size(Boundary_z,1) loop
     KgB_met[3*Boundary_z[a],i]:=0;
     KgB_met[3*Boundary_z[a],3*Boundary_z[a]]:=1;
  end for;
 end for;
 end if;
end BoundaryStiffnessMatrixGlobal;

Gauss Jordan Function 

function GaussJordan
 input Real [:,:] KgB_met;
 input Real [size(KgB_met,1)] load_met;
 output Real [size(KgB_met,1)] U_met;
 
 protected
 Real float_error = 10e-10;

algorithm
 U_met:=Modelica.Math.Matrices.solve(KgB_met,load_met);

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

end GaussJordan;

Reaction Force Function 

function ReactionForce
 input Real [:,:] KgTot_met;
 input Real [size(KgTot_met,1)] U_met;
 input Real [size(KgTot_met,1)] load_met;
 output Real [size(KgTot_met,1)] R_met;
 protected Real float_error = 10e-10;

algorithm
 R_met := KgTot_met*U_met-load_met;
 
 for t in 1:size(KgTot_met,1) loop
   if abs(R_met[t]) <= float_error then
     R_met[t] := 0;
   end if;
 end for;
 
end ReactionForce;

Check Force Function 

function CheckForce
 input Real [:] load;
 input Real [size(load,1)] R;
 output Real [3] F;
 protected Real float_error = 10e-10;
 
 protected
   Real load_x;
   Real load_y;
   Real load_z;
   Real R_x;
   Real R_y;
   Real R_z;
   
algorithm
 load_x := sum({load[i] for i in 1:3:(size(load,1)-2)});
 load_y := sum({load[i] for i in 2:3:(size(load,1)-1)});
 load_z := sum({load[i] for i in 3:3:size(load,1)});
 R_x := sum({R[i] for i in 1:3:(size(load,1)-2)});
 R_y := sum({R[i] for i in 2:3:(size(load,1)-1)});
 R_z := sum({R[i] for i in 3:3:size(load,1)});
 
 F[1] := load_x + R_x;
 F[2] := load_y + R_y;
 F[3] := load_z + R_z;
 
 for i in 1:3 loop
   if abs(F[i]) <= float_error then
    F[i] := 0;
   end if;
 end for;

end CheckForce;

Berikut merupakan class yang digunakan untuk memanggil function diatas dan menyelesaikan Example 3.3

Class 

class QuizSoal1
 //inisiasi = [ elemen#, dX, dY, dZ, A, E]
 parameter Real [:,6] inisiasi = [1,  6,  0, -3, 1.56, 10.6e6; //isi sesuai data
                                  2,  0,  0, -6, 1.56, 10.6e6;
                                  3,  0,  6, -3, 1.56, 10.6e6;
                                  4, -6,  0, -3, 1.56, 10.6e6;
                                  5, -6,  6,  0, 1.56, 10.6e6;
                                  6,  0,  6,  3, 1.56, 10.6e6];

 //node = [ i, j]                                 
 parameter Integer [size(inisiasi,1),2] node = [1, 2; //isi sesuai data
                                                1, 3;
                                                1, 4;
                                                2, 3;
                                                2, 4;
                                                3, 4];

 //jumlah node
 parameter Integer n = 4; //isi sesuai data

 //titik node boundary xyz
 parameter Integer [:] Boundary_xyz = {1}; //isi sesuai data

 //titik node boundary xy
 parameter Integer [:] Boundary_xy = {4}; //isi sesuai data

 //titik node boundary xz
 parameter Integer [:] Boundary_xz = {0}; //isi sesuai data

 //titik node boundary yz
 parameter Integer [:] Boundary_yz = {0}; //isi sesuai data

 //titik node boundary x
 parameter Integer [:] Boundary_x = {3}; //isi sesuai data

 //titik node boundary y
 parameter Integer [:] Boundary_y = {0}; //isi sesuai data

 //titik node boundary z
 parameter Integer [:] Boundary_z = {0}; //isi sesuai data
                            
 //load = [ F1x, F1y, F1z,..., Fnx, Fny, Fnz]
 parameter Real [3*n] load = {0,    0, 0,  //isi sesuai data
                              0, -200, 0, 
                              0,    0, 0, 
                              0,    0, 0}; 

 Real [size(inisiasi,1)] L;
 Real [size(inisiasi,1)] k;
 Real [size(inisiasi,1),6,6] Ke;
 Real [size(inisiasi,1),3*n,3*n] Kg;
 Real [3*n,3*n] KgTot;
 Real [3*n,3*n] KgB;
 Real [3*n] U;
 Real [3*n] R;

 //check force
 Real [3] F;

equation
L = {(sqrt(inisiasi[i,2]^2 + inisiasi[i,3]^2 + inisiasi[i,4]^2)) for i in 1:size(inisiasi,1)}; 

k = {(inisiasi[i,5] * inisiasi[i,6] / L[i]) for i in 1:size(inisiasi,1)};

Ke = StiffnessMatrixElement(inisiasi);

Kg = StiffnessMatrixGlobal(n, node, Ke);

KgTot = SumStiffnessMatrixGlobal(Kg);

KgB = BoundaryStiffnessMatrixGlobal(KgTot, Boundary_xyz, Boundary_xy, Boundary_xz, Boundary_yz, Boundary_x, Boundary_y, Boundary_z);

U = GaussJordan(KgB, load);

R = ReactionForce(KgTot, U, load);

F = CheckForce(load,R);

end QuizSoal1;

Berikut merupakan hasil dan grafik yang didapat

1607924165277.jpg 1607924118583.jpg 1607924101463.jpg

1607924228684.jpg

Pertemuan Keenam

Kelas pada hari Senin 21 Desember2020 diisi oleh Ibu Chandra. Beliau memaparkan materi pengaplikasian metode numerik pada OpenModelica.

Pada pembelajaran metode numerik ini, saya selalu berusaha mengikuti pembelajaran di kelas pada setiap pertemuan. Tentang konsep statika struktur saya sudah cukup paham, karena saya juga sudah pernah mempelajari statika struktur pada semester semester sebelumnya.

Untuk pembelajaran programming menggunakan OpenModelica, saya merasa memang cukup sulit, karena saya belum pernah mempelajari basic programming, saya mengalami kendala dalam pemahaman programmingnya(syntax). Saya selalu mengikuti tutorial dari teman sekelas ataupun platform Youtube tapi hasilnya sering kali "syntax error"

Tugas Besar Metode Numerik

Tujuan :

Mencari harga pembuatan rangka truss sederhana dengan memvariasi dimensi dan elastisitas material secara optimal!

Berikut kasus yang diberikan :

Tugas Besar Metnum Geometri Jos.jpg


Urutan penyelesaian :

1. Mendefinisikan karakteristik dari Truss

Dari soal diminta untuk mencari material pada struktur tersebut yang paling optimal. Besi siku(angle bar akan menjadi strukturnya. Membandingkan optimasi material dan optimasi luas penampang. Struktur pada kasus ini memiliki dimensi 0.75m x 0.6m x 1.8m. Pada bagian atas diberikan dua gaya yang memengaruhi struktur

F1 : 2000 N

F2 : 1000 N

2. Menentukan asusmsi dan node-node pada pengoperasian sistem

Gaya F1 dan F2 akan terdistribusi di point penghubung pada struktur. Struktur pada kasus ini memiliki sifat truss

3. Mencari data profil besi siku yang digunakan

Hargamaterialbm.jpeg

4. Melakukan simulasi pada OpenModelica

//define initial variable parameter Integer Points=size(P,1); //Number of Points parameter Integer Trusses=size(C,1); //Number of Trusses parameter Real Yield=215e6; //Yield Strength (Pa) parameter Real Area=0.000224; //Area L Profile (Dimension=0.03, Thickness=0,004) (m2) parameter Real Elas=193e9; //Elasticity SS 304 (Pa)

//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-6;

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 Trusses_3D_Tugas_Besar_Safety;

5. Optimasi 1 dan Optimasi 2

6. Mendapatkan kurva hasil optimasi yang telah dilakukan dan menganalisanya


Ujian Akhir Semester

UASbm1.jpeg

UASbm2.jpeg

UASbm3.jpeg

UASbm4.jpeg

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