(************** Content-type: application/mathematica **************
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Notebook[{
Cell["\<\
This module computes the stiffness matrix of the 4-node transition \
triangle for plane stress.\
\>", "Text",
  CellFrame->True,
  CellMargins->{{13, 48}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
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Cell[CellGroupData[{

Cell[TextData[{
  "Trig4IsoPMembraneStiffness[ncoor_,mprop_,fprop_,opt_]:= \n \
Module[{i,k,l,p=3,numer=False,Emat,th={fprop},h,tcoor,w,c,\n  \
Nf,dNx,dNy,Jdet,Be,Ke=Table[0,{8},{8}]},\n  Emat=mprop[[1]]; If \
[Length[fprop]>0, th=fprop[[1]]]; \n  If [Length[opt]>0,   numer=opt[[1]]];\n \
 If [Length[opt]>1,   p=  opt[[2]]];\n  If [p!=1&&p!=3&&p!=-3&&p!=6&&p!=7, \
Print[\"Illegal p\"];Return[Null]];\n  For [k=1, k<=Abs[p], k++,",
  StyleBox[" ",
    FontColor->RGBColor[1, 0, 0]],
  "  \n      {tcoor,w}=          TrigGaussRuleInfo[{p,numer},k];\n      \
{Nf,dNx,dNy,Jdet}=  Trig4IsoPShapeFunDer[ncoor,tcoor];\n      If \
[Length[th]==0, h=th, h=th.Nf]; c=w*Jdet*h/2;\n      \
Be={Flatten[Table[{dNx[[i]],      0},{i,4}]],\n          Flatten[Table[{0,    \
  dNy[[i]]},{i,4}]],\n          Flatten[Table[{dNy[[i]],dNx[[i]]},{i,4}]]};\n \
     Ke+=c*Transpose[Be].(Emat.Be);     \n      ]; Return[Ke]\n   ]; \n   \n\
Trig4IsoPShapeFunDer[xycoor_,tcoor_]:= Module[\n  \
{\[Zeta]1,\[Zeta]2,\[Zeta]3,x1,x2,x3,x4,y1,y2,y3,y4,\n  \
Jx1,Jx2,Jx3,Jy1,Jy2,Jy3,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}}=xycoor;\n  \
Nf={\[Zeta]1-2*\[Zeta]1*\[Zeta]2,\[Zeta]2-2*\[Zeta]1*\[Zeta]2,\[Zeta]3,4*\
\[Zeta]1*\[Zeta]2};\n  Jx1=x1*(1-2*\[Zeta]2)-2*x2*\[Zeta]2+4*x4*\[Zeta]2;\n  \
Jx2=x2*(1-2*\[Zeta]1)-2*x1*\[Zeta]1+4*x4*\[Zeta]1; Jx3=x3;\n  Jy1=y1*(1-2*\
\[Zeta]2)-2*y2*\[Zeta]2+4*y4*\[Zeta]2;\n  \
Jy2=y2*(1-2*\[Zeta]1)-2*y1*\[Zeta]1+4*y4*\[Zeta]1; Jy3=y3; \n  Jx21=Jx2-Jx1; \
Jx32=Jx3-Jx2; Jx13=Jx1-Jx3;\n  Jy12=Jy1-Jy2; Jy23=Jy2-Jy3; Jy31=Jy3-Jy1;\n  \
Jdet=Jx21*Jy31-Jy12*Jx13;\n  dNx={Jy23-2*Jy31*\[Zeta]1-2*Jy23*\[Zeta]2, \
Jy31-2*Jy31*\[Zeta]1-2*Jy23*\[Zeta]2, \n       Jy12,4*(Jy31*\[Zeta]1+Jy23*\
\[Zeta]2)}/Jdet;\n  dNy={Jx32-2*Jx13*\[Zeta]1-2*Jx32*\[Zeta]2, Jx13-2*Jx13*\
\[Zeta]1-2*Jx32*\[Zeta]2, \n       \
Jx21,4*(Jx13*\[Zeta]1+Jx32*\[Zeta]2)}/Jdet;\n  Return[{Nf,dNx,dNy,Jdet}]\n];\n\
\nTrigGaussRuleInfo[{rule_,numer_},point_]:= Module[\n \
{zeta,p=rule,i=point,g1,g2,g3,g4,w1,w2,w3,eps=10.^(-24),\n \
jkl={{1,2,3},{2,1,3},{1,3,2},{3,1,2},{2,3,1},{3,2,1}},\n \
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\n\n(* Plot module not used *)\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]2,4*\[Zeta]2*\[Zeta]3,\
4*\[Zeta]1*\[Zeta]3}];\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",
  StyleBox["   ",
    FontColor->RGBColor[1, 0, 0]],
  "\nClearAll[Em,nu,a,b,e,h]; h=1; Em=2880; nu=1/3;\n\
ncoor={{0,0},{6,2},{4,4},{3,1}};\n\
Emat=Em/(1-nu^2)*{{1,nu,0},{nu,1,0},{0,0,(1-nu)/2}};\n\
Print[\"Emat=\",Emat//MatrixForm];\n\
Ke=Trig4IsoPMembraneStiffness[ncoor,{Emat,0,0},{h},{False,3}];  \n\
Ke=Simplify[Ke];  Print[Chop[Ke]//MatrixForm];\nPrint[\"eigs of \
Ke=\",Chop[Eigenvalues[N[Ke]]]];"
}], "Input",
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                {"1080", "3240", "0"},
                {"0", "0", "1080"}
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              {"1080", \(-810\), \(-540\), "540", "1620", \(-810\), \(-2160\),
                 "1080"},
              {\(-810\), "1440", "540", \(-180\), \(-810\), "3780", 
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              {\(-2880\), "1440", \(-720\), \(-360\), \(-2160\), "1080", 
                "5760", \(-2160\)},
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              }], "\[NoBreak]", ")"}],
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                }], "\[NoBreak]", ")"}],
          
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      Editable->False]], "Print",
  FontSize->18]
}, Open  ]],

Cell["\<\
This module evaluates the shape functions and their derivatives at \
a point of
triangular coordinates (\[Zeta]1,\[Zeta]2,\[Zeta]3) of a 4-node quadratic \
transition triangle.\
\>", "Text",
  CellFrame->True,
  CellMargins->{{13, 48}, {Inherited, Inherited}},
  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{0, 1}, {0, 1}}],

Cell[TextData[{
  "Trig4IsoPShapeFunDer[xycoor_,tcoor_]:= Module[\n  {\[Zeta]1,\[Zeta]2,\
\[Zeta]3,x1,x2,x3,x4,y1,y2,y3,y4,\n  \
Jx1,Jx2,Jx3,Jy1,Jy2,Jy3,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}}=xycoor;\n  \
Nf={\[Zeta]1-2*\[Zeta]1*\[Zeta]2,\[Zeta]2-2*\[Zeta]1*\[Zeta]2,\[Zeta]3,4*\
\[Zeta]1*\[Zeta]2};\n  Jx1=x1*(1-2*\[Zeta]2)-2*x2*\[Zeta]2+4*x4*\[Zeta]2;\n  \
Jx2=x2*(1-2*\[Zeta]1)-2*x1*\[Zeta]1+4*x4*\[Zeta]1; Jx3=x3;\n  Jy1=y1*(1-2*\
\[Zeta]2)-2*y2*\[Zeta]2+4*y4*\[Zeta]2;\n  \
Jy2=y2*(1-2*\[Zeta]1)-2*y1*\[Zeta]1+4*y4*\[Zeta]1; Jy3=y3; \n  Jx21=Jx2-Jx1; \
Jx32=Jx3-Jx2; Jx13=Jx1-Jx3;\n  Jy12=Jy1-Jy2; Jy23=Jy2-Jy3; Jy31=Jy3-Jy1;\n  \
Jdet=Jx21*Jy31-Jy12*Jx13;\n  dNx={Jy23-2*Jy31*\[Zeta]1-2*Jy23*\[Zeta]2, \n    \
  Jy31-2*Jy31*\[Zeta]1-2*Jy23*\[Zeta]2, Jy12, \n      4*(Jy31*\[Zeta]1+Jy23*\
\[Zeta]2)}/Jdet;\n  dNy={Jx32-2*Jx13*\[Zeta]1-2*Jx32*\[Zeta]2, \n      \
Jx13-2*Jx13*\[Zeta]1-2*Jx32*\[Zeta]2, Jx21, \n      4*(Jx13*\[Zeta]1+Jx32*\
\[Zeta]2)}/Jdet;\n  Return[{Nf,dNx,dNy,Jdet}]\n];\n",
  StyleBox[" \nClearAll[Em,nu,a,b,e,h];   \nxycoor={{0,0},{6,2},{4,4},{3,1}};\
\n{Nf,dNx,dNy,Jdet}=Trig4IsoPShapeFunDer[xycoor,{\[Zeta]1,\[Zeta]2,\[Zeta]3}];\
\nPrint[Simplify[{Nf,dNx,dNy,Jdet}]];\n\n(*\n\
{Nf,dNx,dNy,Jdet}=Trig4IsoPShapeFunDer[xycoor,{1/3,1/3,1/3}];\n\
Print[{Nf,dNx,dNy,Jdet}];\n\
{Nf,dNx,dNy,Jdet}=Trig4IsoPShapeFunDer[xycoor,{2/3,1/6,1/6}];\n\
Print[{Nf,dNx,dNy,Jdet}];\n\
{Nf,dNx,dNy,Jdet}=Trig4IsoPShapeFunDer[xycoor,{1/6,2/3,1/6}];\n\
Print[{Nf,dNx,dNy,Jdet}];\n\
{Nf,dNx,dNy,Jdet}=Trig4IsoPShapeFunDer[xycoor,{1/6,1/6,2/3}];\n\
Print[{Nf,dNx,dNy,Jdet}];*)",
    FontColor->RGBColor[1, 0, 0]]
}], "Input",
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Cell["\<\
J={{1,1,1},{Jx1,Jx2,Jx3},{Jy1,Jy2,Jy3}};
R={{0,0},{1,0},{0,1}};
X=Inverse[J].R;  X=Simplify[X];
Print[X//MatrixForm];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{13, 48}, {Inherited, Inherited}},
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Cell["\<\
Symbolic derivations of shape function stuff - used in shape \
function module after cut & paste \
\>", "Text",
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  CellLabelMargins->{{5, Inherited}, {Inherited, Inherited}},
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  Background->RGBColor[1, 1, 0]],

Cell[CellGroupData[{

Cell["\<\
ClearAll[x1,x2,x3,x4,y1,y2,y3,y4,\[Zeta]1,\[Zeta]2,\[Zeta]3];

Nf={\[Zeta]1-2*\[Zeta]1*\[Zeta]2,\[Zeta]2-2*\[Zeta]1*\[Zeta]2,\[Zeta]3,4*\
\[Zeta]1*\[Zeta]2};
{N1,N2,N3,N4}=Nf;  Print[Simplify[N1+N2+N3+N4]];
Jx1=x1*D[N1,\[Zeta]1]+x2*D[N2,\[Zeta]1]+x3*D[N3,\[Zeta]1]+x4*D[N4,\[Zeta]1];
Jx2=x1*D[N1,\[Zeta]2]+x2*D[N2,\[Zeta]2]+x3*D[N3,\[Zeta]2]+x4*D[N4,\[Zeta]2]; 
Jx3=x1*D[N1,\[Zeta]3]+x2*D[N2,\[Zeta]3]+x3*D[N3,\[Zeta]3]+x4*D[N4,\[Zeta]3]; 
Jy1=y1*D[N1,\[Zeta]1]+y2*D[N2,\[Zeta]1]+y3*D[N3,\[Zeta]1]+y4*D[N4,\[Zeta]1]; 
Jy2=y1*D[N1,\[Zeta]2]+y2*D[N2,\[Zeta]2]+y3*D[N3,\[Zeta]2]+y4*D[N4,\[Zeta]2]; 
Jy3=y1*D[N1,\[Zeta]3]+y2*D[N2,\[Zeta]3]+y3*D[N3,\[Zeta]3]+y4*D[N4,\[Zeta]3]; 
Jdet=1;
ClearAll[Jy23,Jy31,Jy12,Jx32,Jx13,Jx21];
dNx={D[N1,\[Zeta]1]*Jy23+D[N1,\[Zeta]2]*Jy31+D[N1,\[Zeta]3]*Jy12,
    D[N2,\[Zeta]1]*Jy23+D[N2,\[Zeta]2]*Jy31+D[N2,\[Zeta]3]*Jy12,
    D[N3,\[Zeta]1]*Jy23+D[N3,\[Zeta]2]*Jy31+D[N3,\[Zeta]3]*Jy12,
    D[N4,\[Zeta]1]*Jy23+D[N4,\[Zeta]2]*Jy31+D[N4,\[Zeta]3]*Jy12}/Jdet;       
dNy={D[N1,\[Zeta]1]*Jx32+D[N1,\[Zeta]2]*Jx13+D[N1,\[Zeta]3]*Jx21,
    D[N2,\[Zeta]1]*Jx32+D[N2,\[Zeta]2]*Jx13+D[N2,\[Zeta]3]*Jx21,
    D[N3,\[Zeta]1]*Jx32+D[N3,\[Zeta]2]*Jx13+D[N3,\[Zeta]3]*Jx21,
    D[N4,\[Zeta]1]*Jx32+D[N4,\[Zeta]2]*Jx13+D[N4,\[Zeta]3]*Jy21}/Jdet;
Print[\"Jx1=\",Jx1]; Print[\"Jx2=\",Jx2]; Print[\"Jx3=\",Jx3];
Print[\"Jy1=\",Jy1]; Print[\"Jy2=\",Jy2]; Print[\"Jy3=\",Jy3];  
Print[\"dNx=\",Simplify[dNx]//InputForm]; 
Print[\"dNy=\",Simplify[dNy]//InputForm];\
\>", "Input",
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Cell[BoxData[
    \(\[Zeta]1 + \[Zeta]2 + \[Zeta]3\)], "Print"],

Cell[BoxData[
    InterpretationBox[\("Jx1="\[InvisibleSpace]\(x1\ \((1 - 2\ \[Zeta]2)\) - 
          2\ x2\ \[Zeta]2 + 4\ x4\ \[Zeta]2\)\),
      SequenceForm[ "Jx1=", 
        Plus[ 
          Times[ x1, 
            Plus[ 1, 
              Times[ -2, \[Zeta]2]]], 
          Times[ -2, x2, \[Zeta]2], 
          Times[ 4, x4, \[Zeta]2]]],
      Editable->False]], "Print"],

Cell[BoxData[
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          2\ x1\ \[Zeta]1 + 4\ x4\ \[Zeta]1\)\),
      SequenceForm[ "Jx2=", 
        Plus[ 
          Times[ x2, 
            Plus[ 1, 
              Times[ -2, \[Zeta]1]]], 
          Times[ -2, x1, \[Zeta]1], 
          Times[ 4, x4, \[Zeta]1]]],
      Editable->False]], "Print"],

Cell[BoxData[
    InterpretationBox[\("Jx3="\[InvisibleSpace]x3\),
      SequenceForm[ "Jx3=", x3],
      Editable->False]], "Print"],

Cell[BoxData[
    InterpretationBox[\("Jy1="\[InvisibleSpace]\(y1\ \((1 - 2\ \[Zeta]2)\) - 
          2\ y2\ \[Zeta]2 + 4\ y4\ \[Zeta]2\)\),
      SequenceForm[ "Jy1=", 
        Plus[ 
          Times[ y1, 
            Plus[ 1, 
              Times[ -2, \[Zeta]2]]], 
          Times[ -2, y2, \[Zeta]2], 
          Times[ 4, y4, \[Zeta]2]]],
      Editable->False]], "Print"],

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