(************** Content-type: application/mathematica **************
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(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[     15926,        306]*)
(*NotebookOutlinePosition[     16973,        339]*)
(*  CellTagsIndexPosition[     16929,        335]*)
(*WindowFrame->Normal*)



Notebook[{
Cell["\<\
The following  modules compute the stiffness matrix, consistent \
node body forces,
and corner stresses of the 6-node quadratic iso-P triangle in plane stress.\
\
\>", "Text",
  CellFrame->True,
  CellMargins->{{10, 25}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 1, 0]],

Cell[TextData[{
  "Trig6IsoPMembraneStiffness[ncoor_,Emat_,th_,options_]:= \n \
Module[{i,k,p=3,numer=False,h=th,tcoor,w,c,\n  \
Nf,dNx,dNy,Jdet,Be,Ke=Table[0,{12},{12}]},\n  If [Length[options]>=1, \
numer=options[[1]]];\n  If [Length[options]>=2, p=    options[[2]]];\n  If \
[!MemberQ[{1,-3,3,6,7},p], Print[\"Illegal p\"]; Return[Null]];\n  For [k=1, \
k<=Abs[p], k++,  \n      {tcoor,w}= TrigGaussRuleInfo[{p,numer},k];\n      \
{Nf,dNx,dNy,Jdet}= Trig6IsoPShapeFunDer[ncoor,tcoor];\n      If [numer, \
{Nf,dNx,dNy,Jdet}=N[{Nf,dNx,dNy,Jdet}]];\n      If [Length[th]==6, h=th.Nf]; \
c=w*Jdet*h/2;\n      Be= {Flatten[Table[{dNx[[i]],0       },{i,6}]],\n        \
   Flatten[Table[{0,       dNy[[i]]},{i,6}]],\n           \
Flatten[Table[{dNy[[i]],dNx[[i]]},{i,6}]]};\n      \
Ke+=c*Transpose[Be].(Emat.Be); \n      ]; If[!numer,Ke=Simplify[Ke]]; \
Return[Ke]\n ];\n\n\
Trig6IsoPMembraneBodyForces[ncoor_,rho_,th_,options_,bfor_]:= \n  \
Module[{i,k,p=2,numer=False,h=1,\n  \
bx,by,bx1,by1,bx2,by2,bx3,by3,bx4,by4,bxc,byc,qcoor,\n  \
c,w,Nf,dNx,dNy,Jdet,qctab,fe=Table[0,{12}]}, \n  If [Length[options]==2, \
{numer,p}=options, {numer}=options]; \n  If [Length[fprop]>0, th=fprop[[1]]]; \
\n  If [Length[bfor]==2,{bx,by}=bfor;bx1=bx2=bx3=bx4=bx;by1=by2=by3=by4=by];\n\
  If [Length[bfor]==4,{{bx1,by1},{bx2,by2},{bx3,by3},{bx4,by4}}=bfor];\n  If \
[p!=1&&p!=-3&&p!=3&&p!=6&&p!=7, Print[\"Illegal p\"];Return[Null]];\n  \
bxc={bx1,bx2,bx3,bx4}; byc={by1,by2,by3,by4}; \n  For [k=1, k<=p*p, k++,  \n  \
    {qcoor,w}= QuadGaussRuleInfo[{p,numer},k];\n      \
{Nf,dNx,dNy,Jdet}=Trig6IsoPShapeFunDer[ncoor,qcoor];\n      bx=Nf.bxc; \
by=Nf.byc; If [Length[th]==0, h=th, h=th.Nf]; c=w*Jdet*h; \n      \
bk=Flatten[Table[{Nf[[i]]*bx,Nf[[i]]*by},{i,6}]];\n      fe+=c*bk; \n      ]; \
Return[fe]\n ];\n  \n\
Trig6IsoPMembraneStresses[ncoor_,Emat_,th_,options_,udis_]:= \n  \
Module[{i,k,numer=False,h=th,qcoor,Nf,\n  \
dNx,dNy,Jdet,B,qctab,ue=udis,sige=Table[0,{6},{3}]},  \n  \
qctab={{1,0,0},{0,1,0},{0,0,1},{1/2,1/2,0},{0,1/2,1/2},{1/2,0,1/2}}; \n  \
numer=options[[1]]; If [Length[udis]==6, ue=Flatten[udis]];\n  For [k=1, \
k<=Length[sige], k++, \n      qcoor=qctab[[k]]; If [numer, qcoor=N[qcoor]]; \n\
      {Nf,dNx,dNy,Jdet}=Trig6IsoPShapeFunDer[ncoor,qcoor];\n      B={ \
Flatten[Table[{dNx[[i]], 0      },{i,6}]],\n          Flatten[Table[{0,       \
dNy[[i]]},{i,6}]],\n          Flatten[Table[{dNy[[i]],dNx[[i]]},{i,6}]]}; \n  \
    sige[[k]]=Emat.(B.ue); \n      ]; Return[sige]\n ];\n \n\n\
Trig6IsoPShapeFunDer[ncoor_,tcoor_]:= Module[\n  \
{\[Zeta]1,\[Zeta]2,\[Zeta]3,x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6,\n  \
dx4,dx5,dx6,dy4,dy5,dy6,Jx21,Jx32,Jx13,Jy12,Jy23,Jy31,\n  Nf,dNx,dNy,Jdet}, {\
\[Zeta]1,\[Zeta]2,\[Zeta]3}=tcoor;  \n  \
{{x1,y1},{x2,y2},{x3,y3},{x4,y4},{x5,y5},{x6,y6}}=ncoor;\n  dx4=x4-(x1+x2)/2; \
dx5=x5-(x2+x3)/2; dx6=x6-(x3+x1)/2; \n  dy4=y4-(y1+y2)/2; dy5=y5-(y2+y3)/2; \
dy6=y6-(y3+y1)/2;\n  Nf={\[Zeta]1*(2*\[Zeta]1-1),\[Zeta]2*(2*\[Zeta]2-1),\
\[Zeta]3*(2*\[Zeta]3-1),4*\[Zeta]1*\[Zeta]2,4*\[Zeta]2*\[Zeta]3,4*\[Zeta]3*\
\[Zeta]1};\n  Jx21= x2-x1+4*(dx4*(\[Zeta]1-\[Zeta]2)+(dx5-dx6)*\[Zeta]3);\n  \
Jx32= x3-x2+4*(dx5*(\[Zeta]2-\[Zeta]3)+(dx6-dx4)*\[Zeta]1);\n  Jx13= \
x1-x3+4*(dx6*(\[Zeta]3-\[Zeta]1)+(dx4-dx5)*\[Zeta]2);\n  Jy12= y1-y2+4*(dy4*(\
\[Zeta]2-\[Zeta]1)+(dy6-dy5)*\[Zeta]3);\n  Jy23= y2-y3+4*(dy5*(\[Zeta]3-\
\[Zeta]2)+(dy4-dy6)*\[Zeta]1);\n  Jy31= \
y3-y1+4*(dy6*(\[Zeta]1-\[Zeta]3)+(dy5-dy4)*\[Zeta]2);\n  Jdet = \
Jx21*Jy31-Jy12*Jx13;\n  dNx= {(4*\[Zeta]1-1)*Jy23,(4*\[Zeta]2-1)*Jy31,(4*\
\[Zeta]3-1)*Jy12,4*(\[Zeta]2*Jy23+\[Zeta]1*Jy31),\n         4*(\[Zeta]3*Jy31+\
\[Zeta]2*Jy12),4*(\[Zeta]1*Jy12+\[Zeta]3*Jy23)}/Jdet;\n  dNy= \
{(4*\[Zeta]1-1)*Jx32,(4*\[Zeta]2-1)*Jx13,(4*\[Zeta]3-1)*Jx21,4*(\[Zeta]2*Jx32+\
\[Zeta]1*Jx13),\n         4*(\[Zeta]3*Jx13+\[Zeta]2*Jx21),4*(\[Zeta]1*Jx21+\
\[Zeta]3*Jx32)}/Jdet;\n  Return[Simplify[{Nf,dNx,dNy,Jdet}]]\n];\n\n\n\
TrigGaussRuleInfo[{rule_,numer_},point_]:= Module[\n \
{zeta,p=rule,i=point,g1,g2,info={{Null,Null,Null},0}  },\n  If [p== 1, \
info={{1/3,1/3,1/3},1}];\n  If [p== 3, info={{1,1,1}/6,1/3}; \
info[[1,i]]=2/3];\n  If [p==-3, info={{1,1,1}/2,1/3}; info[[1,i]]=0  ];\n  If \
[p== 6,  g1=(8-Sqrt[10]+Sqrt[38-44*Sqrt[2/5]])/18;\n              \
g2=(8-Sqrt[10]-Sqrt[38-44*Sqrt[2/5]])/18;\n     If [i<4, \
info={{g1,g1,g1},(620+Sqrt[213125-\n              53320*Sqrt[10]])/3720}; \
info[[1,i]]=1-2*g1];\n     If [i>3, info={{g2,g2,g2},(620-Sqrt[213125-\n      \
        53320*Sqrt[10]])/3720}; info[[1,i-3]]=1-2*g2]];\n  If [p== -6, \n     \
If [i<4, info={{1,1,1}/6,3/10}; info[[1,i]]=2/3];  \n     If [i>3, \
info={{1,1,1}/2,1/30}; info[[1,i-3]]=0]]; \n  If [p== 7,  g1=(6-Sqrt[15])/21; \
g2=(6+Sqrt[15])/21;\n     If [i<4,      \
info={{g1,g1,g1},(155-Sqrt[15])/1200};\n                   info[[1,i]]=  \
1-2*g1];\n     If [i>3&&i<7, info={{g2,g2,g2},(155+Sqrt[15])/1200};\n         \
          info[[1,i-3]]=1-2*g2];\n     If [i==7,     \
info={{1/3,1/3,1/3},9/40} ]];    \n  If [numer, Return[N[info]], \
Return[Simplify[info]]];\n];\n\nPlotTrig6Shape[xytrig_,Nsub_,ratio_]:=Module[\
\n  {Ne,Nev,Ni,line2D={},nodes={},xy1,xy2,xy3,i,j,iz1,iz2,iz3,z1,z2,z3,\n\t \
x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6,xc,yc},\n  \
{{x1,y1},{x2,y2},{x3,y3},{x4,y4},{x5,y5},{x6,y6}}=xytrig;\n\t \
xc={x1,x2,x3,x4,x5,x6}; yc={y1,y2,y3,y4,y5,y6};\n \
Ne[z1_,z2_,z3_]:=N[{z1*(2*z1-1),z2*(2*z2-1),z3*(2*z3-1),\n                    \
 4*z1*z2,4*z2*z3,4*z3*z1}];\n Ni=Nsub*3;\n Do [ Do [iz3=Ni-iz1-iz2; If \
[iz3<=0, Continue[]]; d=0;\n\t      If [Mod[iz1+2,3]==0&&Mod[iz2-1,3]==0, d=  \
1];\n          If [Mod[iz1-2,3]==0&&Mod[iz2+1,3]==0, d= -1];\n          If \
[d==0, Continue[]];\n          \
{z1,z2,z3}=N[{iz1+d+d,iz2-d,iz3-d}/Ni];zc1=Ne[z1,z2,z3];\n\t\t  \
{z1,z2,z3}=N[{iz1-d,iz2+d+d,iz3-d}/Ni];zc2=Ne[z1,z2,z3];\n\t      \
{z1,z2,z3}=N[{iz1-d,iz2-d,iz3+d+d}/Ni];zc3=Ne[z1,z2,z3];\n          \
xy1={xc.zc1,yc.zc1};\n          xy2={xc.zc2,yc.zc2};\n          \
xy3={xc.zc3,yc.zc3};\n\t\t AppendTo[line2D,Line[{xy1,xy2,xy3,xy1}]],\n \t \
{iz2,1,Ni-iz1}],{iz1,1,Ni}];\n \t \
Do[AppendTo[nodes,Circle[xytrig[[i]],0.04]],{i,1,6}];\n        \
Show[Graphics[RGBColor[1,0,0]],\n\t    \
Graphics[Thickness[.002]],Graphics[line2D],\n\t    \
Graphics[RGBColor[0,0,0]],Graphics[nodes],\n\t    \
PlotRange->All,AspectRatio->ratio];\n\t];\n\t\n\t\n\
PlotTrig4Shape[xytrig_,Nsub_,ratio_]:=Module[\n  \
{Ne,Nev,Ni,line2D={},nodes={},xy1,xy2,xy3,i,j,iz1,iz2,iz3,z1,z2,z3,\n\t \
x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6,xc,yc},\n  \
{{x1,y1},{x2,y2},{x3,y3},{x4,y4}}=xytrig;\n  x5=(x2+x3)/2; y5=(y2+y3)/2; \
x6=(x1+x3)/2; y6=(y1+y3)/2;\n  xc={x1,x2,x3,x4,x5,x6}; \
yc={y1,y2,y3,y4,y5,y6};\n  Ne[z1_,z2_,z3_]:=N[{\[Zeta]1-2*\[Zeta]1*\[Zeta]2,\
\[Zeta]2-2*\[Zeta]1*\[Zeta]2,\[Zeta]3,4*\[Zeta]1*\[Zeta]2}];\n                \
    \n Ni=Nsub*3;\n Do [ Do [iz3=Ni-iz1-iz2; If [iz3<=0, Continue[]]; d=0;\n\t\
      If [Mod[iz1+2,3]==0&&Mod[iz2-1,3]==0, d=  1];\n          If \
[Mod[iz1-2,3]==0&&Mod[iz2+1,3]==0, d= -1];\n          If [d==0, Continue[]];\n\
          {z1,z2,z3}=N[{iz1+d+d,iz2-d,iz3-d}/Ni];zc1=Ne[z1,z2,z3];\n\t\t  \
{z1,z2,z3}=N[{iz1-d,iz2+d+d,iz3-d}/Ni];zc2=Ne[z1,z2,z3];\n\t      \
{z1,z2,z3}=N[{iz1-d,iz2-d,iz3+d+d}/Ni];zc3=Ne[z1,z2,z3];\n          \
xy1={xc.zc1,yc.zc1};\n          xy2={xc.zc2,yc.zc2};\n          \
xy3={xc.zc3,yc.zc3};\n\t\t AppendTo[line2D,Line[{xy1,xy2,xy3,xy1}]],\n \t \
{iz2,1,Ni-iz1}],{iz1,1,Ni}];\n \t \
Do[AppendTo[nodes,Circle[xytrig[[i]],0.04]],{i,1,4}];\n        \
Show[Graphics[RGBColor[1,0,0]],\n\t    \
Graphics[Thickness[.002]],Graphics[line2D],\n\t    \
Graphics[RGBColor[0,0,0]],Graphics[nodes],\n\t    \
PlotRange->All,AspectRatio->ratio];\n\t];\n\n",
  StyleBox["ClearAll[x1,y1,x2,y2,x3,y3];\n\
{{x1,y1},{x2,y2},{x3,y3}}={{0,0},{6,2},{4,4}};\nx4=(x1+x2)/2; x5=(x2+x3)/2; \
x6=(x3+x1)/2;\ny4=(y1+y2)/2; y5=(y2+y3)/2; y6=(y3+y1)/2;\n e=Table[1,{6}]; \
x={x1,x2,x3,x4,x5,x6}; y={y1,y2,y3,y4,y5,y6};\nncoor= \
{{x1,y1},{x2,y2},{x3,y3},{x4,y4},{x5,y5},{x6,y6}};\n\
{Nf,dNx,dNy,Jdet}=Trig6IsoPShapeFunDer[ncoor,{z1,z2,1-z1-z2}];\n\
Print[\"Nf=\",Nf]; Print[\"dNx=\",dNx]; Print[\"dNy=\",dNy]; \
Print[\"Jdet=\",Jdet];\nPrint[\"dNx.1=\",Simplify[dNx.e]]; \
Print[\"dNy.1=\",Simplify[dNy.e]];\nPrint[\"dNx.x=\",Simplify[dNx.x]]; Print[\
\"dNx.y=\",Simplify[dNx.y]];\nPrint[\"dNy.x=\",Simplify[dNy.x]]; \
Print[\"dNy.y=\",Simplify[dNy.y]];",
    FontColor->RGBColor[0, 0, 1]]
}], "Input",
  CellFrame->True,
  CellMargins->{{10, 25}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]],

Cell["Triangle with straight sides for Ch 24 ", "Text",
  CellFrame->True,
  CellMargins->{{10, 25}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 1, 0]],

Cell["\<\
ClearAll[Em,\[Nu],a,b,e,h]; h=1; Em=288; \[Nu]=1/3;
ncoor={{0,0},{6,2},{4,4},{3,1},{5,3},{2,2}};
PlotTrig6Shape[ncoor,8,4/6];
Emat=Em/(1-\[Nu]^2)*{{1,\[Nu],0},{\[Nu],1,0},{0,0,(1-\[Nu])/2}};
Print[\"Emat=\",Emat//MatrixForm]
Ke=Trig6IsoPMembraneStiffness[ncoor,Emat,h,{False,3}];  
Ke=Simplify[Ke];  Print[Chop[Ke]//MatrixForm];
Print[\"eigs of Ke=\",Chop[Eigenvalues[N[Ke]]]];
Ke=Trig6IsoPMembraneStiffness[ncoor,Emat,h,{False,-3}];  
Ke=FullSimplify[Ke];  Print[Chop[Ke]//MatrixForm];
Print[\"eigs of Ke=\",Chop[Eigenvalues[N[Ke]]]];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{10, 25}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]],

Cell["Triangle with nodes 1-6 lying on a circle for Ch 24", "Text",
  CellFrame->True,
  CellMargins->{{10, 25}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 1, 0]],

Cell["\<\
ClearAll[Em,nu,h]; h=1; Em=7*72; \[Nu]=0; h=1;
{x1,y1}={-1,0}/2; {x2,y2}={1,0}/2; {x3,y3}={0,Sqrt[3]}/2;
{x4,y4}={0,-1/Sqrt[3]}/2; {x5,y5}={1/2,1/Sqrt[3]}; {x6,y6}={-1/2,1/Sqrt[3]};
ncoor= {{x1,y1},{x2,y2},{x3,y3},{x4,y4},{x5,y5},{x6,y6}};
Emat=Em/(1-\[Nu]^2)*{{1,\[Nu],0},{\[Nu],1,0},{0,0,(1-\[Nu])/2}};
PlotTrig6Shape[ncoor,32,1]; 
For [i=2,i<=5,i++, p={1,-3,3,6,7}[[i]];
     Ke=Trig6IsoPMembraneStiffness[ncoor,{Emat,0,0},{h},{True,p}];
     Ke=Chop[Simplify[Ke]];
     Print[\"Ke=\",SetPrecision[Ke,4]//MatrixForm];
     Print[\"Eigenvalues of Ke=\",Chop[Eigenvalues[N[Ke]],.0000001]] 
  ];  \
\>", "Input",
  CellFrame->True,
  CellMargins->{{10, 25}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]],

Cell["\<\
Check of shape function derivatives and jacobians to built shape \
function module\
\>", "Text",
  CellFrame->True,
  CellMargins->{{10, 25}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 1, 0]],

Cell[TextData[StyleBox["ClearAll[x1,x2,x3,x4,x5,x6,y1,y2,y3,y4,y5,y6,\[Zeta]1,\
\[Zeta]2,\[Zeta]3];  \n  x4=(x1+x2)/2+dx4; x5=(x2+x3)/2+dx5; \
x6=(x3+x1)/2+dx6; \n  y4=(y1+y2)/2+dy4; y5=(y2+y3)/2+dy5; y6=(y3+y1)/2+dy6; \n\
  x={x1,x2,x3,x4,x5,x6}; xv={x1,x2,x3,dx4,dx5,dx6};\n  y={y1,y2,y3,y4,y5,y6}; \
yv={y1,y2,y3,dy4,dy5,dy6};\n  r={\[Zeta]1+\[Zeta]2+\[Zeta]3->1,-\[Zeta]1-\
\[Zeta]2-\[Zeta]3->-1,2*\[Zeta]1+2*\[Zeta]2+2*\[Zeta]2->2,-2*\[Zeta]1-2*\
\[Zeta]2-2*\[Zeta]2->-2,\n      \
3*\[Zeta]1+3*\[Zeta]2+3*\[Zeta]3->3,-3*\[Zeta]1-3*\[Zeta]2-3*\[Zeta]3->-3,\n  \
    (1-2*\[Zeta]1-2*\[Zeta]2-2*\[Zeta]3)->-1,(-1+2*\[Zeta]1+2*\[Zeta]2+2*\
\[Zeta]3)->1};\n  \
Nf={\[Zeta]1*(2*\[Zeta]1-1),\[Zeta]2*(2*\[Zeta]2-1),\[Zeta]3*(2*\[Zeta]3-1),4*\
\[Zeta]1*\[Zeta]2,4*\[Zeta]2*\[Zeta]3,4*\[Zeta]3*\[Zeta]1};\n  Nf1=D[Nf,\
\[Zeta]1]; Nf2=D[Nf,\[Zeta]2]; Nf3=D[Nf,\[Zeta]3];\n  \
{Nf1,Nf2,Nf3}=Simplify[{Nf1,Nf2,Nf3}];\n  \
Print[\"Nf1=\",Nf1//InputForm,\";\\nNf2=\",Nf2//InputForm,\n       \";\\nNf3=\
\",Nf3//InputForm,\";\"];\n  \
JJx21=x1-x2+4*((x4-x1)*\[Zeta]1+(x2-x4)*\[Zeta]2+(x5-x6)*\[Zeta]3);\n  \
JJx32=x2-x3+4*((x5-x2)*\[Zeta]2+(x3-x5)*\[Zeta]3+(x6-x4)*\[Zeta]1);\n  \
JJx13=x3-x1+4*((x6-x3)*\[Zeta]3+(x1-x6)*\[Zeta]1+(x4-x5)*\[Zeta]2);\n  \
JJy12=y2-y1+4*((y1-y4)*\[Zeta]1+(y4-y2)*\[Zeta]2+(y6-y5)*\[Zeta]3);\n  \
JJy23=y3-y2+4*((y2-y5)*\[Zeta]2+(y5-y3)*\[Zeta]3+(y4-y6)*\[Zeta]1);\n  \
JJy31=y1-y3+4*((y3-y6)*\[Zeta]3+(y6-y1)*\[Zeta]1+(y5-y4)*\[Zeta]2);\n  Jdet = \
Jx21*Jy31-Jy12*Jx13;\n  dNx= {(4*\[Zeta]1-1)*Jy23,(4*\[Zeta]2-1)*Jy31,(4*\
\[Zeta]3-1)*Jy12,4*(\[Zeta]2*Jy23+\[Zeta]1*Jy31),\n         4*(\[Zeta]3*Jy31+\
\[Zeta]2*Jy12),4*(\[Zeta]1*Jy12+\[Zeta]3*Jy23)}/Jdet;\n  dNy= \
{(4*\[Zeta]1-1)*Jx32,(4*\[Zeta]2-1)*Jx13,(4*\[Zeta]3-1)*Jx21,4*(\[Zeta]2*Jx32+\
\[Zeta]1*Jx13),\n         4*(\[Zeta]3*Jx13+\[Zeta]2*Jx21),4*(\[Zeta]1*Jx21+\
\[Zeta]3*Jx32)}/Jdet;\n      \n  \
{Nf1x,Nf2x,Nf3x}=Simplify[{Nf1.x,Nf2.x,Nf3.x}];\n  \
{Nf1y,Nf2y,Nf3y}=Simplify[{Nf1.y,Nf2.y,Nf3.y}];\n  Jx21= \
x2-x1+4*(dx4*(\[Zeta]1-\[Zeta]2)+(dx5-dx6)*\[Zeta]3);\n  Jx32= x3-x2+4*(dx5*(\
\[Zeta]2-\[Zeta]3)+(dx6-dx4)*\[Zeta]1);\n  Jx13= x1-x3+4*(dx6*(\[Zeta]3-\
\[Zeta]1)+(dx4-dx5)*\[Zeta]2);\n  Print[\"check \
=\",Simplify[{Jx21,Jx32,Jx13}-{JJx21,JJx32,JJx13}]/.r];\n  Jy12= \
y1-y2+4*(dy4*(\[Zeta]2-\[Zeta]1)+(dy6-dy5)*\[Zeta]3);\n  Jy23= y2-y3+4*(dy5*(\
\[Zeta]3-\[Zeta]2)+(dy4-dy6)*\[Zeta]1);\n  Jy31= y3-y1+4*(dy6*(\[Zeta]1-\
\[Zeta]3)+(dy5-dy4)*\[Zeta]2);\n  Print[\"check \
=\",Simplify[{Jy12,Jy23,Jy31}-{JJy12,JJy23,JJy31}]/.r];",
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