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Notebook[{

Cell[CellGroupData[{
Cell["\<\
 Incremental Solution Methods For Two-Bar Arch \
Modeled
 by the Total Lagrangian Formulation\
\>", "Section",
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Cell["\<\
This Notebook is a testbed of incremental solution methods, \
converted from ancient
Fortran.  The methods are applied to the 2-bar arch problem of Chapter 8.  
It is used for the  first two Exercises of Chapter 18.  

The present implementation includes three integration methods:
    FE:      Forward Euler
    MR:     Midpoint Rule (a 2-step Runge Kutta)
    RK4:    Classical 4-step Runge Kutta
 with three Increment Control Strategies
    LC:        Lambda Control, also called Load Control
    DC:       Displacement Control
    AC :      ArcLength Control
 The implementations below use a constant stepsize ell and a positive-work \
criterion to
 traverse limit points.
 
 To run the main programs in Cells 12ff, Cells 1 through 11 must be \
initialized.  \
\>", "Text",
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Cell[TextData[{
  "Cell 1.  The modules ",
  StyleBox["FormTanStiffnessOfTwoBarArch",
    FontWeight->"Bold"],
  " and ",
  StyleBox["FormInternalForceOfTwoBarArch",
    FontWeight->"Bold"],
  "\n below form return the internal force p and tangent stiffness  K for the \
two-bar arch structure of Chapter 8,\n which is discretized by two Total \
Lagrangian (TL)  bar elements.\n They are extracted directly from the \
Mathematica modules of that Chapter, with minor edits.\n",
  StyleBox[" DetTanStiffness",
    FontWeight->"Bold"],
  " returns exactly that.\n",
  StyleBox[" FormIncrementalLoadVector",
    FontWeight->"Bold"],
  " simply returns its argument force as q.  This trivial operation\n is \
implemented as a module to have a placeholder for more complicated cases.\n \n\
 The inputs to  ",
  StyleBox["FormTanStiffnessOfTwoBarArch",
    FontWeight->"Bold"],
  " and ",
  StyleBox["FormInternalForceOfTwoBarArch\n ",
    FontWeight->"Bold"],
  "includes  four structural properties collected in sprop:\n       S         \
          arch span between supports\n       H                  arch height \
under zero load\n       Em               bar elastic modulus\n       A0       \
         cross section area in reference state\n  and the displacements \
u={uX,uY} of the crown from the reference state"
}], "Text",
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Cell[TextData[{
  "FormTanStiffnessOfTwoBarArch[{S_,H_,Em_,A0_},{uX_,uY_}]:=\n \
Module[{c,K},c=4*Em*A0/(4*H^2+S^2)^(3/2); \n   \
K=c*{{(S^2+6*uX^2+4*H*uY+2*uY^2), \n          4*uX*(H+uY)},{4*uX*(H+uY), \n   \
       2*(2*H^2+uX^2+6*H*uY+3*uY^2)}};\n   Return[K]];\n   \n\
FormInternalForceOfTwoBarArch[{S_,H_,Em_,A0_},{uX_,uY_}]:=\n  \
Module[{c,p},c=4*Em*A0/(4*H^2+S^2)^(3/2); \n   \
p=c*{uX*(S^2+2*uX^2+4*H*uY+2*uY^2),\n        2*(H+uY)*(uX^2+2*H*uY+uY^2)};\n  \
 Return[p]];\n    \nFormIncrementalLoadVector[lambda_,force_,u_]:=Module[{},\n\
   Return[force]];\n \nDetTanStiffness[sprop_,u_]:=Module[{K11,K12,K21,K22},\n\
  {{K11,K12},{K21,K22}}=FormTanStiffnessOfTwoBarArch[sprop,u];\n   \
Return[K11*K22-K12*K21]];\n    ",
  StyleBox["\nClearAll[uX,uY];\nK= \
FormTanStiffnessOfTwoBarArch[{S,H,Em,A0},{uX,uY}];\np= \
FormInternalForceOfTwoBarArch[{S,H,Em,A0},{uX,uY}];\nKK={D[p,uX],D[p,uY]};\n\
Print[\"Check K=grad(p): \",Simplify[K-KK]];\n\
Print[Simplify[Together[DetTanStiffness[{S,H,Em,A0},{0,uY}]]]];\n",
    FontColor->RGBColor[1, 0, 0]]
}], "Input",
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Cell[TextData[{
  "Cell 2.  ",
  StyleBox["SolveTwoLinearEqs",
    FontWeight->"Bold"],
  "  is an ad-hoc module that solves a system of two linear equations: \n\
Ax=b.   It returns x and a regularity/singularity indicator."
}], "Text",
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Cell[TextData[{
  "SolveTwoLinearEqs[A_,b_]:=Module[{A11,A12,A21,A22,b1,b2,\n  \
dnorm,Adet,x,eps=10.^(-15)},\n   {{A11,A12},{A21,A22}}=A; \
Adet=A11*A22-A12*A21; {b1,b2}=b;\n   dnorm=Max[Abs[A11*A22],Abs[A12*A21]];\n  \
 If [Abs[Adet]<=eps*dnorm, \n       Return[{Null,False}]];\n   \
x={A22*b1-A12*b2,-A21*b1+A11*b2}/Adet; \n   Return[{x,True}]];\n",
  StyleBox["   \nPrint[SolveTwoLinearEqs[{{2,6},{1,3}},{8,4}]];\n\
Print[SolveTwoLinearEqs[{{2,5},{1,3}},{7,4}]];",
    FontColor->RGBColor[1, 0, 0]]
}], "Input",
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Cell[TextData[{
  "Cell  3.   ",
  StyleBox["IncVelocity  ",
    FontWeight->"Bold"],
  "computes the incremental velocity vector v at a given state (lambda,u).  \
On entry\n K and q are evaluated.  If K is numerically singular u is \
perturbed by a tiny random amount and\n the process repeated up to 10 times.  \
If K is not numerically singular, the velocity v=Kinv.q is returned."
}], "Text",
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Cell[TextData[{
  "IncVelocity[sprop_,force_,{lambda_,u_}]:= \n  \
Module[{q,nudge,un=u,v,OK,ueps=10.^(-10)},\n    nudge=0; SeedRandom[7654321];\
\n    While [nudge<=10,\n      K=FormTanStiffnessOfTwoBarArch[sprop,un]; \
K=N[K];\n    ",
  StyleBox["  (*Print[\"IncVelocity K=\",K];*)",
    FontColor->RGBColor[1, 0, 0]],
  "\n      q=FormIncrementalLoadVector[lambda,force,un]; q=N[q];\n      \
{v,OK}=SolveTwoLinearEqs[K,q]; \n      ",
  StyleBox[" (*Print[\"IncVelocity v=\",v,\"  OK= \",OK];*)\n      ",
    FontColor->RGBColor[1, 0, 0]],
  "If [OK, Break[]];\n      nudge++; un+=Table[Random[]*ueps,{2}] ];\n    If \
[OK, Return [{N[v],N[q],\" \"}]];\n    Return [Null,Null,\"IncVelocity: Cant \
escape singularity\"]];\n    ",
  StyleBox["\nClearAll[uX,uY];\n{v,q,status}= \
IncVelocity[{2,3,1,1},{4,0},{1,{1,1}}];\nPrint[\"v=\",v];  Print[\"q=\",q]; \n\
If[status!=\" \", Print[\"status=\",status]];",
    FontColor->RGBColor[1, 0, 0]]
}], "Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  InitializationCell->True,
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Cell[TextData[{
  "Cell 4.  ",
  StyleBox["IncSolution ",
    FontWeight->"Bold"],
  "advances the solution over one increment, without executing\nequilibrium \
corrections.  The module arguments provide the following input data:\n \n     \
 method                 Keywords specifying the solution method:  {integ,ics} \
where \n                                    integ identifies the integrator:  \
FE, MR or RK4\n                                    ics identifies the \
increment control strategy:  LC,  DC  or  AC\n      sprop                     \
Properties of arch structure:  {H,S,Em,A0}.  See Cell 1.\n      force         \
            A list specifying the applied forces.  In this problem, force is \
same as q.\n      sol                        A list of entities at the last \
computed solution:\n                                      n                   \
   current increment number (n=0 for first step)\n                            \
          lambdan           the stage control parameter \n                    \
                  un ={uXn,uYn}    the crown displacement components\n        \
                              vn={vXn,vYn}     incremental velocity over \
previous step\n                                      qn={qXn,qYn}     \
incremental force over previous step\n                                      \
Kdetn                the determinant of Kn\n                                  \
    elln                    same as ell for a fixed stepsize incrementation\n \
                                     an                      cosine of angle \
between last 2 incremental directions\n                                      \
kappan              the limit point sensor\n                                  \
    kappa0              kappa at n=0\n                                      \
rem                    a char string storing remarks about any remarkable \
happenings\n      solpar                   A list of solution control \
parameters: \n                                       nmax             maximum \
number of incremental steps\n                                      lambdamax  \
  bound on abs(lambda)\n                                      umax            \
 {uXmax,uYmax}   bounds on node displacements\n                               \
       ell                   dimensionless length of increment\n              \
                        ellmin             not used here\n                    \
                  ellmax            not used here\n                           \
           acctol              not used here\n                                \
      lref                 a reference length for scaling u if ics is DC or \
AC\n\n                                   If n>nmax, abs(lambda)>lambdamax, \
abs(uX)>uXmax or\n                                   abs(uY)>uYmax, the \
solution is terminated.   If a variable stepsize\n                            \
       implementation is adopted,  ellmin, elklmax and epsacc are used\n      \
                             In corrective and perturbation-injection method, \
this list is\n                                   expanded.\n                  \
                 \n The function returns {solnext,status}  where\n      \
solnext                  Updated values of sol at the next solution if status \
is blank\n      status                    A character string status \
indicator.  If nonblank, the solution process has been\n                      \
             terminated for the reason stated in this string.\n               \
                    \n  In essence ",
  StyleBox["IncSolution ",
    FontWeight->"Bold"],
  "is a driver that  checks whether certain bounds are exceeded (for example\n\
  number of steps) and if satisfied simply calls the appropriate integrator.  \
If termination is\n  indicated by bound violation(s) or other reasons, it \
returns Null; else it reports the next solution."
}], "Text",
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Cell["\<\
IncSolution[method_,sprop_,force_,sol_,solpar_]:=
   Module[ {integ,ics,n=sol[[1]],lambda=sol[[2]],u=sol[[3]],
    nmax=solpar[[1]],lambdamax=solpar[[2]],umax=solpar[[3]],
    solnext,status},
   If [n>nmax, Return[{Null,
       \"IncSolution: Max steps reached\"}]];
   If [Abs[lambda]>Abs[lambdamax], Return[{Null,
       \"IncSolution: lambda exceeds bound\"}]];
   If [Abs[u[[1]]]>Abs[umax[[1]]], Return[{Null,
       \"IncSolution: uX exceeds bound\"}]];
   If [Abs[u[[2]]]>Abs[umax[[2]]], Return[{Null,
       \"IncSolution: uY exceeds bound\"}]]; 
   {integ,ics}=method;
   If [integ==\"FE\", 
      {solnext,status}=FEstep [ics,sprop,force,sol,solpar]];
   If [integ==\"MR\", 
      {solnext,status}=MRstep [ics,sprop,force,sol,solpar]];
   If [integ==\"RK4\", 
      {solnext,status}=RK4step[ics,sprop,force,sol,solpar]];
   Return[{solnext,status}]];\
\>", "Input",
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  InitializationCell->True,
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
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Cell[TextData[{
  "Cell 5.   ",
  StyleBox["FEstep",
    FontWeight->"Bold"],
  " is the Forward Euler (FE) driver module.  It simply calls FELCstep, \
FEDCstep or\nFEACstep according to the increment control strategy specified \
in ics."
}], "Text",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
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  Background->RGBColor[0, 1, 1]],

Cell["\<\
FEstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
   {solnext,status},
   If [ics==\"LC\", 
      {solnext,status}=FELCstep[ics,sprop,force,sol,solpar]];
   If [ics==\"DC\", 
      {solnext,status}=FEDCstep[ics,sprop,force,sol,solpar]];
   If [ics==\"AC\", 
      {solnext,status}=FEACstep[ics,sprop,force,sol,solpar]];
   Return[{solnext,status}]];\
\>", "Input",
  CellFrame->True,
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  InitializationCell->True,
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
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  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[TextData[{
  "Cell 6.   ",
  StyleBox["FELCstep, FEDCstep ",
    FontWeight->"Bold"],
  "and",
  StyleBox[" FEACstep ",
    FontWeight->"Bold"],
  " implement the Forward Euler (FE) integrator for Load\nControl, \
Displacement Control and ArcLength Control, respectively."
}], "Text",
  CellFrame->True,
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Cell["\<\
FELCstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v,q,status}=IncVelocity[sprop,force,{lambdan,un}]; 
   If [status!=\" \", rem=status; Return[{Null,status}]];
   n++; qv=q.v; f=Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];
      
FEDCstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v,q,status}=IncVelocity[sprop,force,{lambdan,un}];
   If [status!=\" \", rem=status; Return[{Null,status}]];
   n++; qv=q.v;  f=Sqrt[(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];
   
FEACstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v,q,status}=IncVelocity[sprop,force,{lambdan,un}];
   If [status!=\" \", rem=status; Return[{Null,status}]];
   n++; qv=q.v;  f=Sqrt[1+(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];\
\>", \
"Input",
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Cell[TextData[{
  "Cell 7. ",
  StyleBox[" MRstep",
    FontWeight->"Bold"],
  " is the Midpoint Rule (MR) driver module.  It simply calls MRLCstep, \
MRDCstep or\nMRACstep according to the increment control strategy specified \
in ics."
}], "Text",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
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  Background->RGBColor[0, 1, 1]],

Cell["\<\
MRstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
   {solnext,status},
   If [ics==\"LC\", 
      {solnext,status}=MRLCstep[ics,sprop,force,sol,solpar]];
   If [ics==\"DC\", 
      {solnext,status}=MRDCstep[ics,sprop,force,sol,solpar]];
   If [ics==\"AC\", 
      {solnext,status}=MRACstep[ics,sprop,force,sol,solpar]];
   Return[{solnext,status}]];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  InitializationCell->True,
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
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  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[TextData[{
  StyleBox["Cell 8.",
    FontFamily->"Times",
    FontWeight->"Plain"],
  StyleBox["  MRLCstep, MRDCstep ",
    FontFamily->"Times"],
  StyleBox["and ",
    FontFamily->"Times",
    FontWeight->"Plain"],
  StyleBox["MRACstep ",
    FontFamily->"Times"],
  StyleBox[" implement the Midpoint Rule (MR) integrator for Load\nControl, \
Displacement Control and ArcLength Control, respectively.",
    FontFamily->"Times",
    FontWeight->"Plain"]
}], "Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell["\<\
MRLCstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v,q,status}=IncVelocity[sprop,force,{lambdan,un}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v;      f=Sign[qv]; ell=elln; 
   lambdaM=lambdan+ell/(2*f); uM=un+v*ell/(2*f);
  {v,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", rem=status; Return[{Null,status}]];
   n++; qv=q.v; f=Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];
      
MRDCstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rem}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v,q,status}=IncVelocity[sprop,force,{lambdan,un}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v; f=Sqrt[(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(2*f); uM=un+v*ell/(2*f);
  {v,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", rem=status; Return[{Null,status}]];
   n++; qv=q.v; f=Sqrt[(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];
   
MRACstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v,q,status}=IncVelocity[sprop,force,{lambdan,un}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v; f=Sqrt[1+(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(2*f); uM=un+v*ell/(2*f);
  {v,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", rem=status; Return[{Null,status}]];
   n++; qv=q.v; f=Sqrt[1+(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];  \
\>", \
"Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  InitializationCell->True,
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[TextData[{
  "Cell 9. ",
  StyleBox[" RK4step",
    FontWeight->"Bold"],
  " is the 4th order Runge Kutta (RK4) driver module.  It simply calls \
RK4LCstep, \nRK4DCstep or RK4ACstep according to the increment control \
strategy specified in ics."
}], "Text",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 0, 0]],

Cell["\<\
RK4step[ics_,sprop_,force_,sol_,solpar_]:=Module[
   {solnext,status},
   If [ics==\"LC\", 
      {solnext,status}=RK4LCstep[ics,sprop,force,sol,solpar]];
   If [ics==\"DC\", 
      {solnext,status}=RK4DCstep[ics,sprop,force,sol,solpar]];
   If [ics==\"AC\", 
      {solnext,status}=RK4ACstep[ics,sprop,force,sol,solpar]];
   If [status!=\" \", Return[{Null,status}]];
   Return[{solnext,\" \"}]];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  InitializationCell->True,
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell[TextData[{
  StyleBox["Cell 10.  ",
    FontFamily->"Times",
    FontWeight->"Plain"],
  StyleBox["RK4LCstep, RK4DCstep ",
    FontFamily->"Times"],
  StyleBox["and ",
    FontFamily->"Times",
    FontWeight->"Plain"],
  StyleBox["RK4ACstep ",
    FontFamily->"Times"],
  StyleBox[" implement the classical Runge-Kutta (RK4) integrator\n for Load \
Control, Displacement Control and ArcLength Control, respectively.  ",
    FontFamily->"Times",
    FontWeight->"Plain"]
}], "Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 0, 0]],

Cell["\<\
RK4LCstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,v1,v2,v3,v4,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v1,q,status}=IncVelocity[sprop,force,{lambdan,un}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v1; f=Sqrt[(v1.v1)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(2*f); uM=un+v1*ell/(2*f);
   {v2,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v2; f=Sqrt[(v2.v2)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(2*f); uM=un+v2*ell/(2*f);
   {v3,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v3; f=Sqrt[(v3.v3)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(f); uM=un+v3*ell/(f);
   {v4,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", rem=status; Return[{Null,status}]];
   v=(v1+2*v2+2*v3+v4)/6;
   n++; qv=q.v; f=Sqrt[(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];
      
RK4DCstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,v1,v2,v3,v4,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rem}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v1,q,status}=IncVelocity[sprop,force,{lambdan,un}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v1; f=Sqrt[(v1.v1)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(2*f); uM=un+v1*ell/(2*f);
   {v2,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v2; f=Sqrt[(v2.v2)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(2*f); uM=un+v2*ell/(2*f);
   {v3,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v3; f=Sqrt[(v3.v3)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(f); uM=un+v3*ell/(f);
   {v4,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", rem=status; Return[{Null,status}]];
   v=(v1+2*v2+2*v3+v4)/6;
   n++; qv=q.v; f=Sqrt[(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];
      
   
RK4ACstep[ics_,sprop_,force_,sol_,solpar_]:=Module[
  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn,
   rem=\" \",v,v1,v2,v3,v4,q,qv,lambda,u,Kdet,a,kappa,nmax,lambdamax,umax,
   ell,ellmin,ellmax,acctol,lref,f,status},

  {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rn}=sol;
  {nmax,lambdamax,umax,ell,ellmin,ellmax,acctol,lref}=solpar;
  {v1,q,status}=IncVelocity[sprop,force,{lambdan,un}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v1; f=Sqrt[(v1.v1)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(2*f); uM=un+v1*ell/(2*f);
   {v2,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v2; f=Sqrt[(v2.v2)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(2*f); uM=un+v2*ell/(2*f);
   {v3,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", Return[{Null,status}]];
   qv=q.v3; f=Sqrt[(v3.v3)/lref^2]*Sign[qv]; ell=elln;
   lambdaM=lambdan+ell/(f); uM=un+v3*ell/(f);
   {v4,q,status}=IncVelocity[sprop,force,{lambdaM,uM}];
   If [status!=\" \", rem=status; Return[{Null,status}]];
   v=(v1+2*v2+2*v3+v4)/6;
   n++; qv=q.v; f=Sqrt[(v.v)/lref^2]*Sign[qv]; ell=elln;
   lambda=lambdan+ell/f; u=un+v*ell/f;
   Kdet=DetTanStiffness[sprop,u];
   a=(v/Sqrt[v.v]).(vn/Sqrt[vn.vn]); kappa=((q.v)/(v.v))/kappa0; 
   Return[{{n,lambda,u,v,q,Kdet,ell,a,kappa,kappa0,rem},\" \"}]];
      \
\>", "Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  InitializationCell->True,
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell["\<\
Cell 11.  Module to print computed solution table.  Looks weird \
because it is adapted from Fortran.\
\>", "Text",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell["\<\
PrintSolutionTable[soltab_]:= Module[{numsteps,
   n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rem,t},
   numsteps=Length[soltab]; t=Table[\"    \",{numsteps+1},{8}];
   For [i=1, i<=numsteps, i++,
     {n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,rem}=
        Chop[soltab[[i]],.00000001];
      t[[i+1,1]]=ToString[n];
      t[[i+1,2]]=ToString[lambdan]; 
      t[[i+1,3]]=ToString[un];
      t[[i+1,4]]=ToString[vn];
      t[[i+1,5]]=ToString[Kdetn];
      t[[i+1,6]]=ToString[an];
      t[[i+1,7]]=ToString[kappan];
      t[[i+1,8]]=rem];
   t[[1]] = {\"n\",\"lambda\",\"u={uX,uY}\",\"v={vX,vY}\",
             \"Kdet\", \"a\", \"kappa\", \"remarks\"};
   Print[TableForm[t,TableAlignments->{Left},
         TableDirections->{Column,Row},TableSpacing->{0,1}]];
];

q={0,-1};  soltab=
{ {0,0.0,{ 0.0,0.},{0.42,0.0},q,45.,0.02,1.0,1.0,2.5,\"ref state\"}, 
  {1,0.2,{-0.3,0.},{0.35,0.0},q,43.,0.02,1.0,0.9,2.5,\"step 1\"} };
PrintSolutionTable[soltab];\
\>", "Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  InitializationCell->True,
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[1, 0.647715, 0.521981]],

Cell["\<\
Cell 12.  The main program to run the two-bar arch structure.   \
S=2, H=1, Em=1, A0=1,
q={0,-1},  ell=0.010.  Method: Forward Euler (FE) with Load Control \
(LC)\
\>", "Text",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell["\<\
ClearAll[S,H,Em,A0]; 

(*  Define solution method via method={integ,ics}  *)

method={\"FE\",\"LC\"};

(*  Define structural properties: {S,H,Em,A0} (see Cell 1) *)

S=2; H=1; Em=1; A0=1; sprop={S,H,Em,A0};

(*  Define applied force reference: fx=0, fy=-lambda *)

force={0,-1};

(*  Define solution parameters  *)  

nmax=50; lambdamax=2; umax={2*S,2*H}; ell=0.010; acctol=0; lref=H; 
solpar={nmax,lambdamax,umax,ell,ell,ell,acctol,H};

(*  Define reference state solution *)

lambda0=0.; u0={0.,0.}; 
{v0,q0,status}=IncVelocity[sprop,force,{lambda0,u0}];
If[status!=\" \", Print[\"status=\",status]];
kappa0=(q0.v0)/(v0.v0); Kdet=DetTanStiffness[sprop,u0];
sol={0,lambda0,u0,v0,q0,Kdet,ell,1,1,kappa0,\"C0 cfg\"};

(*  Prepare Table to save solutions and store initial one *)

solnull={0,0,0,0,0,0,0,0,0,0,\" \"};
soltab=Table[solnull,{nmax+1}];
soltab[[1]]=sol; 

(*  Now do nmax steps and record solutions in soltab  *)

n=0;
While [status==\" \" && n<nmax, 
 {sol,status}=IncSolution[method,sprop,force,sol,solpar];
  If [status==\" \", n++; soltab[[n+1]]=sol]
      ];
  
(*  Print solution table *)

PrintSolutionTable[Take[soltab,n]];
If [status!=\" \",Print[status]];

(*  Response plot of deflection versus lambda & kappa *)

uXvslambda=Table[{soltab[[i,3,1]],soltab[[i,2]]},{i,1,n+1}];
uYvslambda=Table[{soltab[[i,3,2]],soltab[[i,2]]},{i,1,n+1}];
uYvskappa=Table[{soltab[[i,3,2]],soltab[[i,9]]},{i,1,n+1}];
ListPlot[uYvslambda,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uY\",\"lambda\"},
         PlotLabel->\"Response: lambda vs uY, \\nMethod:\"<>method];
ListPlot[uYvskappa,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uY\",\"kappa\"},
         PlotLabel->\"LP sensor kappa vs uY, \\nMethod:\"<>method];\
\>", \
"Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]],

Cell["\<\
Cell 13.  The main program to run the two-bar arch structure.   \
S=2, H=1, Em=1, A0=1,
q={0,-1},  ell=0.010.  Method: Midpoint Rule (MR) with Load Control \
(LC)\
\>", "Text",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell["\<\
ClearAll[S,H,Em,A0]; 

(*  Define solution method via method={integ,ics}  *)

method={\"RK4\",\"AC\"};

(*  Define structural properties: {S,H,Em,A0} (see Cell 1) *)

S=2; H=1; Em=1; A0=1; sprop={S,H,Em,A0};

(*  Define applied force reference: fx=0, fy=-lambda *)

force={0,-1};

(*  Define solution parameters  *)  

nmax=50; lambdamax=2; umax={2*S,2*H}; ell=0.010; acctol=0; lref=H; 
solpar={nmax,lambdamax,umax,ell,ell,ell,acctol,H};

(*  Define reference state solution *)

lambda0=0.; u0={0.,0.}; 
{v0,q0,status}=IncVelocity[sprop,force,{lambda0,u0}];
If[status!=\" \", Print[\"status=\",status]];
kappa0=(q0.v0)/(v0.v0); Kdet=DetTanStiffness[sprop,u0];
sol={0,lambda0,u0,v0,q0,Kdet,ell,1,1,kappa0,\"C0 cfg\"};

(*  Prepare Table to save solutions and store initial one *)

solnull={0,0,0,0,0,0,0,0,0,0,\" \"};
soltab=Table[solnull,{nmax+1}];
soltab[[1]]=sol; 

(*  Now do nmax steps and record solutions in soltab  *)

n=0;
While [status==\" \" && n<nmax, 
 {sol,status}=IncSolution[method,sprop,force,sol,solpar];
  If [status==\" \", n++; soltab[[n+1]]=sol]
      ];
  
(*  Print solution table *)

PrintSolutionTable[Take[soltab,n]];
If [status!=\" \",Print[status]];

(*  Response plot of deflection versus lambda & kappa *)

uXvslambda=Table[{soltab[[i,3,1]],soltab[[i,2]]},{i,1,n+1}];
uYvslambda=Table[{soltab[[i,3,2]],soltab[[i,2]]},{i,1,n+1}];
uYvskappa=Table[{soltab[[i,3,2]],soltab[[i,9]]},{i,1,n+1}];
ListPlot[uYvslambda,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uY\",\"lambda\"},
         PlotLabel->\"Response: lambda vs uY\\nMethod:\"<>method];
ListPlot[uYvskappa,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uY\",\"kappa\"},
         PlotLabel->\"LP sensor kappa vs uY\\nMethod:\"<>method];\
\>", \
"Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]],

Cell["\<\
Cell 14.  As in cell 12, but with arclength control.  Note the much \
larger ell.\
\>", "Text",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell["\<\
ClearAll[S,H,Em,A0]; 

(*  Define solution method via method={integ,ics}  *)

method={\"FE\",\"AC\"};

(*  Define structural properties: {S,H,Em,A0} (see Cell 1) *)

S=2; H=5; Em=1; A0=1; sprop={S,H,Em,A0};

(*  Define applied force reference: fx=0, fy=-lambda *)

force={-0.01,-1};

(*  Define solution parameters  *)  

nmax=80; lambdamax=2; umax={2*S,2*H}; ell=0.01; acctol=0; lref=H; 
solpar={nmax,lambdamax,umax,ell,ell,ell,acctol,H};

(*  Define reference state solution *)

lambda0=0.; u0={0.,0.}; 
{v0,q0,status}=IncVelocity[sprop,force,{lambda0,u0}];
If[status!=\" \", Print[\"status=\",status]];
kappa0=(q0.v0)/(v0.v0); Kdet=DetTanStiffness[sprop,u0];
sol={0,lambda0,u0,v0,q0,Kdet,ell,1,1,kappa0,\"C0 cfg\"};

(*  Prepare Table to save solutions and store initial one *)

solnull={0,0,0,0,0,0,0,0,0,0,\" \"};
soltab=Table[solnull,{nmax+1}];
soltab[[1]]=sol; 

(*  Now do nmax steps and record solutions in soltab  *)

n=0;
While [status==\" \" && n<nmax, 
 {sol,status}=IncSolution[method,sprop,force,sol,solpar];
  If [status==\" \", n++; soltab[[n+1]]=sol]
      ];
  
(*  Print solution table *)

PrintSolutionTable[Take[soltab,n]];
If [status!=\" \",Print[status]];

(*  Response plot of deflection versus lambda & kappa *)

uXvslambda=Table[{soltab[[i,3,1]],soltab[[i,2]]},{i,1,n+1}];
uYvslambda=Table[{soltab[[i,3,2]],soltab[[i,2]]},{i,1,n+1}];
uYvskappa=Table[{soltab[[i,3,2]],soltab[[i,9]]},{i,1,n+1}];
ListPlot[uYvslambda,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uY\",\"lambda\"},
         PlotLabel->\"Response: lambda vs uY, \\nMethod:\"<>method];
ListPlot[uXvslambda,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uX\",\"lambda\"},
         PlotLabel->\"Response: lambda vs uX, \\nMethod:\"<>method];
ListPlot[uYvskappa,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uY\",\"kappa\"},
         PlotLabel->\"LP sensor kappa vs uY, \\nMethod:\"<>method];\
\>", \
"Input",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 0]],

Cell["\<\
Cell 15. As in Cell 13 but with arclegth control.  Note the much \
larger ell.\
\>", "Text",
  CellFrame->True,
  CellMargins->{{14, 37}, {Inherited, Inherited}},
  CellLabelMargins->{{8, Inherited}, {Inherited, Inherited}},
  ImageRegion->{{-0, 1}, {0, 1}},
  Background->RGBColor[0, 1, 1]],

Cell["\<\
ClearAll[S,H,Em,A0]; 

(*  Define solution method via method={integ,ics}  *)

method={\"FE\",\"AC\"};

(*  Define structural properties: {S,H,Em,A0} (see Cell 1) *)

S=2; H=2; Em=1; A0=1; sprop={S,H,Em,A0};

(*  Define applied force reference: fx=0, fy=-lambda *)

force={0.001,-1};

(*  Define solution parameters  *)  

nmax=400; lambdamax=2; umax={4*S,2*H}; ell=0.01; acctol=0; lref=H; 
solpar={nmax,lambdamax,umax,ell,ell,ell,acctol,H};

(*  Define reference state solution *)

lambda0=0.; u0={0.,0.}; 
{v0,q0,status}=IncVelocity[sprop,force,{lambda0,u0}];
If[status!=\" \", Print[\"status=\",status]];
kappa0=(q0.v0)/(v0.v0); Kdet=DetTanStiffness[sprop,u0];
sol={0,lambda0,u0,v0,q0,Kdet,ell,1,1,kappa0,\"C0 cfg\"};

(*  Prepare Table to save solutions and store initial one *)

solnull={0,0,0,0,0,0,0,0,0,0,\" \"};
soltab=Table[solnull,{nmax+1}];
soltab[[1]]=sol; 

(*  Now do nmax steps and record solutions in soltab  *)

n=0;
While [status==\" \" && n<nmax, 
 {sol,status}=IncSolution[method,sprop,force,sol,solpar];
  If [status==\" \", n++; soltab[[n+1]]=sol]
      ];
  
(*  Print solution table *)

PrintSolutionTable[Take[soltab,n]];
If [status!=\" \",Print[status]];

(*  Response plot of deflection versus lambda & kappa *)

uXvslambda=Table[{soltab[[i,3,1]],soltab[[i,2]]},{i,1,n+1}];
uYvslambda=Table[{soltab[[i,3,2]],soltab[[i,2]]},{i,1,n+1}];
uYvskappa=Table[{soltab[[i,3,2]],soltab[[i,9]]},{i,1,n+1}];
ListPlot[uYvslambda,PlotJoined->True,PlotRange->All,
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         PlotLabel->\"Response: lambda vs uY\\nMethod:\"<>method];
ListPlot[uXvslambda,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uX\",\"lambda\"},
         PlotLabel->\"Response: lambda vs uX\\nMethod:\"<>method];
ListPlot[uYvskappa,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uY\",\"kappa\"},
         PlotLabel->\"LP sensor kappa vs uY\\nMethod\"<>method];\
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