(************** Content-type: application/mathematica **************
                     CreatedBy='Mathematica 4.2'

                    Mathematica-Compatible Notebook

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Cell[CellGroupData[{
Cell["\<\
 Incremental/Iterative Solution Methods For Two-Bar Arch Modeled by \
the Total Lagrangian Formulation. \
\>", "Section",
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"This Notebook is a testbed of incremental /iterative (predictor/\ncorrector) \
solution methods, converted from ancient Fortran.  The methods \nare applied \
to the 2-bar arch problem of Chapter 8.  \nIt can used for the Exercises of \
Chapter 20.  \n\nThe present implementation includes three predictors\n    \
FE:      Forward Euler\n    MR:     Midpoint Rule (a 2-step Runge Kutta)\n    \
RK4:    Classical 4-step Runge Kutta\n with three Increment Control \
Strategies\n    LC         Lambda Control, also called Load Control\n    DC   \
    Displacement Control\n    AC       ArcLength Control\n The \
implementations below use a constant stepsize ell and a positive-work \
criterion to\n traverse limit points.\n \n To run the main programs in Cells \
12ff, Cells 1 through 12 must be initialized.  "], "Text",
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  "\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",
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  " 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  ",
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  "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",
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  "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\
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  "\nClearAll[uX,uY];\nK= FormTanStiffnessOfTwoBarArch[{S,H,Em,A0},{uX,uY}];\n\
p= FormInternalForceOfTwoBarArch[{S,H,Em,A0},{uX,uY}];\nKK={D[p,uX],D[p,uY]};\
\nPrint[\"Check K=grad(p): \",Simplify[K-KK]];\n\
Print[Simplify[Together[DetTanStiffness[{S,H,Em,A0},{0,uY}]]]];\n",
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  "  is an ad-hoc module that solves a system of two linear equations: \n\
Ax=b.   It returns x and a regularity/singularity indicator.",
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  "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       sing=True; Return[{Null,False}]];\n   \
x={A22*b1-A12*b2,-A21*b1+A11*b2}/Adet; \n   Return[{x,True}]];\n",
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Print[SolveTwoLinearEqs[{{2,5},{1,3}},{7,4}]];",
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  "computes of 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.\n \n ",
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  StyleBox["is similar to ",
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  " but it computes and returns the residual r and the\n residual velocity   \
dr=-Kinv.r .  It supports a correctrive process.",
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  StyleBox[
  "IncVelocity[sprop_,force_,{lambda_,u_}]:= \n  \
Module[{K,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    ",
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{v,OK}=SolveTwoLinearEqs[K,q]; \n      ",
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  "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\nResVelocity[sprop_,force_,{lambda_,u_}]:= \n  \
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  "\n    While [nudge<=10,\n      K=FormTanStiffnessOfTwoBarArch[sprop,un]; \
K=N[K];\n      p=FormInternalForceOfTwoBarArch[sprop,un]; p=N[p];\n      \
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  "If [OK, Break[]];\n      nudge++; un+=Table[Random[]*ueps,{2}] ];\n    If \
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  "\nClearAll[uX,uY];\n{v,q,status}= IncVelocity[{2,3,1,1},{4,0},{1,{1,1}}];\n\
Print[\"v=\",v];  Print[\"q=\",q]; \nIf[status!=\" \", \
Print[\"status=\",status]];\n{dr,r,status}= \
ResVelocity[{2,3,1,1},{0,-0.33},{1,{0,-1}}];\nPrint[\"dr=\",dr];  \
Print[\"r=\",r]; \nIf[status!=\" \", Print[\"status=\",status]];",
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  "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 ",
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  number of steps) and if satisfied simply calls the appropriate integrator.  \
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  "IncSolution[method_,sprop_,force_,sol_,solpar_]:=\n   Module[ \
{integ,ics,n=sol[[1]],lambda=sol[[2]],u=sol[[3]],\n    \
nmax=solpar[[1]],lambdamax=solpar[[2]],umax=solpar[[3]],\n    \
solnext,status},\n   If [n>nmax, Return[{Null,\n       \"IncSolution: Max \
steps reached\"}]];\n   If [Abs[lambda]>Abs[lambdamax], Return[{Null,\n       \
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bound\"}]]; \n   {integ,ics}=method;\n   If [integ==\"FE\", \n      \
{solnext,status}=FEstep [ics,sprop,force,sol,solpar]];\n   If [integ==\"MR\", \
\n      {solnext,status}=MRstep [ics,sprop,force,sol,solpar]];\n   If \
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IncIterSolution[method_,sprop_,force_,sol_,solpar_]:=\n   Module[ \
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nmax=solpar[[1]],lambdamax=solpar[[2]],umax=solpar[[3]],\n    \
solpred,solnext,status,sol0}, \n   If [n>nmax, Return[{Null,\n       \
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bound\"}]];\n   If [Abs[u[[2]]]>Abs[umax[[2]]], Return[{Null,\n       \
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  " is the Forward Euler (FE) driver module.  It simply calls FELCstep, \
FEDCstep or\nFEACstep according to the increment control strategy specified \
in ics.",
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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",
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  " implement the Forward Euler (FE) integrator for Load\nControl, \
Displacement Control and ArcLength Control, respectively.",
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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},\" \"}]];\
\>", 
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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",
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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},\" \"}]];  \
\>", 
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  StyleBox["Cell 9. ",
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  StyleBox[" RK4step",
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  StyleBox[
  " 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.",
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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",
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  StyleBox["Cell 10.  ",
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  StyleBox["RK4LCstep, RK4DCstep ",
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  " implement the classical Runge-Kutta (RK4) integrator\n for Load Control, \
Displacement Control and ArcLength Control, respectively.  ",
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Cell["\<\
   
   
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",
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"Cell 11.  Module to print computed solution table.  Adapted from Fortran."], 
  "Text",
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Cell[CellGroupData[{

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->{0,2}]];
];

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",
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    TagBox[GridBox[{
          {"\<\"n\"\>", "\<\"lambda\"\>", "\<\"u={uX,uY}\"\>", 
            "\<\"v={vX,vY}\"\>", "\<\"Kdet\"\>", "\<\"a\"\>", 
            "\<\"kappa\"\>", "\<\"remarks\"\>"},
          {"\<\"0\"\>", "\<\"0\"\>", "\<\"{0, 0}\"\>", "\<\"{0.42, 0}\"\>", 
            "\<\"45.\"\>", "\<\"1.\"\>", "\<\"1.\"\>", "\<\"ref state\"\>"},
          {"\<\"1\"\>", "\<\"0.2\"\>", "\<\"{-0.3, 0}\"\>", 
            "\<\"{0.35, 0}\"\>", "\<\"43.\"\>", "\<\"1.\"\>", "\<\"0.9\"\>", 
            "\<\"step 1\"\>"}
          },
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        ColumnSpacings->2,
        RowAlignments->Baseline,
        ColumnAlignments->{Left}],
      (TableForm[ #, TableAlignments -> {Left}, TableDirections -> {Column, 
        Row}, TableSpacing -> {0, 2}]&)]], "Print"]
}, Open  ]],

Cell[TextData[
"Cell 12. Driver of Newton iteration for corrective process.  The following \
solution parameters are\nbuilt in:  \n\n          mNiter            max \
number of Newton iterations, set to 8\n          epsconv           \
convergence tolerance for  residual norm ratio, set to 0.00001\n          \
facdiv             divergence factor for residual norm ratio, set to 10000."],
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  StyleBox[
  "NewtonIteration[ics_,sprop_,force_,sol_,solpar_,sol0_]:=Module[\n  \
{n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,remn,\n   rem=\" \
\",state,mNiter,epsconv,facdiv,r0,r0norm,rk,rknorm,\n   \
statenext,converged,status,lambda0,u0},\n  \
{n,lambdan,un,vn,qn,Kdetn,elln,an,kappan,kappa0,remn}=sol;\n   mNiter=2; \
epsconv=0.00001; facdiv=10000.;\n   \
p=FormInternalForceOfTwoBarArch[sprop,un]; p=N[p];\n",
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  "   (*Print[\"NewtonIteration entered\"];\n   Print[\"sprop=\",sprop,\"  \
un=\",un, \"  p=\",p]; *)",
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  StyleBox["\n   rn=p-lambdan*force; rn=N[rn];\n   rnnorm=Sqrt[rn.rn];",
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  "\n   state={lambdan,un,rn}; lambda0=sol0[[2]]; u0=sol0[[3]];\n   If \
[rnnorm==0, Return[sol,status]]; converged=False;\n   For [k=0, k<=mNiter, \
k++, \n     If [ics==\"LC\", \n      \
{statenext,status}=LCNewtonCycle[sprop,force,state]];\n     If [ics==\"DC\", \
\n      {statenext,status}=DCNewtonCycle[sprop,force,state,\n       \
{lambda0,u0,elln}]];\n     If [ics==\"AC\", \n      \
{statenext,status}=ACNewtonCycle[sprop,force,state,\n       \
{lambda0,u0,v0,elln} ]];\n     If [status!=\" \",Break[]];\n     \
{lambda,u,r}=statenext; rnorm=Sqrt[r.r];\n     ",
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  "Print[\"lambda=\",lambda,\"  u=\",u,\" r=\",r,\n           \" \
rnorm=\",rnorm//InputForm];\n     (*Print[\"norm ratio=\",rnorm/rnnorm,\" vs \
\",epsconv];*)",
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  "\n     If [rnorm/rnnorm<epsconv, converged=True; Break[]];\n     If \
[rnorm/rnnorm>facdiv, status=\"Newton iteration diverges\";\n         \
Return[Null,status]];\n     state=statenext;\n     ];\n   If [!converged, \
status=\"Newton didnt converge in max iters\"];\n   If [converged, \
Print[\"Newton converged after \",k,\" cycles\"]];\n   {lambda,u,r}=state;\n  \
 solnext={n,lambda,u,vn,qn,Kdetn,elln,an,kappan,kappa0,remn};\n   \
Return[{solnext,status}]];\n  ",
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Cell[TextData[{
  StyleBox[
  "Cell 13.  These modules perform one Newton iteration cycle under different \
increment \ncontrol constraints.  They are called by ",
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  "\nOnly the Load Control (LC) and Displacement Control (DC)\nare \
implemented in modules ",
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  StyleBox[" DCNewtonCycle.",
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  StyleBox["\nThe arguments of ",
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  " are\n       sprop                      structural properties\n       \
force                       reference force\n      {lambdak,uk,rk}     \
lambda, u and r from the previous (kth) iteration cycle  \n It returns \
{{lambda,u,r},\" \"}   if no error detected:  {lambda,u,r} are the updated \
values;\n otherwise  {Null,Status}   where status has an error message\n \n \
The arguments of",
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  StyleBox[
  " are\n       sprop                      structural properties\n       \
force                       reference force\n      {lambdak,uk,rk}     \
lambda, u, and r  from the previous (kth) iteration cycle \n      \
{lambda0,u0,ell}     information needed to specify the increment control \
constraint c=0.\n                                      {lambda0,u0} is the \
previous converged solution, ell the increment length\n It returns the same \
information as LCNewtonCycle",
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  StyleBox[
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{v,q,status,dr,r,lambda,u},\n",
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  StyleBox[
  "   (*Print[\"Enter LCNewtoncycle\"];\n   Print[\"sprop=\",sprop,\" force =\
\",force];\n   Print[\"entry lambdak=\",lambdak,\"  uk=\",uk,\"  \
rk=\",rk];*)",
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DCNewtonCycle[sprop_,force_,{lambdak_,uk_,rk_},\n   \
{lambda0_,u0_,ell_}]:=Module[ {v,q,status,dr,r,lambda,u,\n   eta,ck},\n",
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  "\n   If [status!=\" \",Return[Null,status]];\n   ck=(uk-u0).(uk-u0)-ell^2; \
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(uk-u0).dr,v=\",(uk-u0).dr,\n    \" (uk-u0).v=\",(uk-u0).v];*)",
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r=FormInternalForceOfTwoBarArch[sprop,u]-lambda*force; r=N[r];\n",
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  "\n   Return[{{lambda,u,r},\" \"}]];\n   \n\
ACNewtonCycle[sprop_,force_,{lambdak_,uk_,rk_},\n   \
{lambda0_,u0_,v0_,ell_}]:=Module[ {v,q,status,dr,r,lambda,u,\n   eta,ck,f0},  \
f0=Sqrt[1+v0.v0];\n",
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  "   Print[\"Enter ACNewtoncycle\"];\n   Print[\"sprop=\",sprop,\" force \
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   Print[\"entry lambda0=\",lambda0,\"  u0=\",u0,\"  f0=\",f0];",
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  "\n   If [status!=\" \",Return[Null,status]];\n   \
ck=(v0.(uk-u0)+(lambdak-lambda0))/f0-ell; ",
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r=FormInternalForceOfTwoBarArch[sprop,u]-lambda*force; r=N[r];\n",
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  StyleBox[
  "Cell 13.  The main program to run the two-bar arch structure.   S=2, H=1, \
Em=1, A0=1,\nq={0,-1},  ell=0.010.  Method: Forward Euler (FE), with",
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  StyleBox["(LC) and Conventional Newton",
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Cell[TextData[
"ClearAll[S,H,Em,A0]; \n\n(*  Define solution method via method={integ,ics}  \
*)\n\nmethod={\"FE\",\"LC\"};\n\n(*  Define structural properties: \
{S,H,Em,A0} (see Cell 1) *)\n\nS=2; H=1; Em=1; A0=1; sprop={S,H,Em,A0};\n\n(* \
 Define applied force reference: fx=0, fy=-lambda *)\n\nforce={0,-1};\n\n(*  \
Define solution parameters  *)  \n\nnmax=20; lambdamax=2; umax={2*S,2*H}; \
ell=0.010; acctol=0; lref=H; \n\
solpar={nmax,lambdamax,umax,ell,ell,ell,acctol,H};\n\n(*  Define reference \
state solution *)\n\nlambda0=0.; u0={0.,0.}; \n\
{v0,q0,status}=IncVelocity[sprop,force,{lambda0,u0}];\nIf[status!=\" \", \
Print[\"status=\",status]];\nkappa0=(q0.v0)/(v0.v0); \
Kdet=DetTanStiffness[sprop,u0];\nsol={0,lambda0,u0,v0,q0,Kdet,ell,1,1,kappa0,\
\"C0 cfg\"};\n\n(*  Prepare Table to save solutions and store initial one *)\n\
\nsolnull={0,0,0,0,0,0,0,0,0,0,\" \"};\nsoltab=Table[solnull,{nmax+1}];\n\
soltab[[1]]=sol; \n\n(*  Now do nmax steps and record solutions in soltab  *)\
\n\nn=0;\nWhile [status==\" \" && n<nmax, \n \
{sol,status}=IncIterSolution[method,sprop,force,sol,solpar];\n  If \
[status==\" \", n++; soltab[[n+1]]=sol]\n      ];\n  \n(*  Print solution \
table *)\n\nPrintSolutionTable[Take[soltab,n]];\nIf [status!=\" \
\",Print[status]];\n\n(*  Response plot of deflection versus lambda & kappa \
*)\n\nuXvslambda=Table[{soltab[[i,3,1]],soltab[[i,2]]},{i,1,n+1}];\n\
uYvslambda=Table[{soltab[[i,3,2]],soltab[[i,2]]},{i,1,n+1}];\n\
uYvskappa=Table[{soltab[[i,3,2]],soltab[[i,9]]},{i,1,n+1}];\n\
ListPlot[uYvslambda,PlotJoined->True,PlotRange->All,\n         \
AxesOrigin->{0,0},AxesLabel->{\"uY\",\"lambda\"},\n         \
PlotLabel->\"Response: lambda vs uY\"];\n\
ListPlot[uYvskappa,PlotJoined->True,PlotRange->All,\n         \
AxesOrigin->{0,0},AxesLabel->{\"uY\",\"kappa\"},\n         PlotLabel->\"LP \
sensor kappa vs uY\"];"], "Input",
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"                                                 -7\nlambda=0.01  u={0., \
-0.0144539} r={0., 1.01594 10  } rnorm=1.015936701900116*10^-7\n              \
                                 -14\nlambda=0.01  u={0., -0.014454} r={0., \
2.3561 10   } rnorm=2.356101425071699*10^-14\nNewton converged after 1 cycles\
\n                                                -7\nlambda=0.02  u={0., \
-0.029584} r={0., 1.27413 10  } rnorm=1.274128222129855*10^-7\n               \
                                  -14\nlambda=0.02  u={0., -0.0295842} r={0., \
4.01311 10   } rnorm=4.013109289324746*10^-14\nNewton converged after 1 \
cycles\n                                                 -7\nlambda=0.03  \
u={0., -0.0454821} r={0., 1.62766 10  } rnorm=1.627655716873733*10^-7\n       \
                                          -14\nlambda=0.03  u={0., \
-0.0454823} r={0., 7.14186 10   } rnorm=7.141856550596515*10^-14\nNewton \
converged after 1 cycles\n                                                 -7\
\nlambda=0.04  u={0., -0.0622625} r={0., 2.12721 10  } \
rnorm=2.127209427146837*10^-7\n                                               \
  -13\nlambda=0.04  u={0., -0.0622629} r={0., 1.34191 10   } \
rnorm=1.341912692076619*10^-13\nNewton converged after 1 cycles\n             \
                                    -7\nlambda=0.05  u={0., -0.0800705} \
r={0., 2.85757 10  } rnorm=2.857566261255107*10^-7\n                          \
                       -13\nlambda=0.05  u={0., -0.0800711} r={0., 2.69181 10 \
  } rnorm=2.691805112142731*10^-13\nNewton converged after 1 cycles\n         \
                                        -7\nlambda=0.06  u={0., -0.0990956} \
r={0., 3.97067 10  } rnorm=3.970674723532963*10^-7\n                          \
                       -13\nlambda=0.06  u={0., -0.0990964} r={0., 5.85393 10 \
  } rnorm=5.853928453092295*10^-13\nNewton converged after 1 cycles\n         \
                                       -7\nlambda=0.07  u={0., -0.119592} \
r={0., 5.75742 10  } rnorm=5.757420976637562*10^-7\n                          \
                      -12\nlambda=0.07  u={0., -0.119594} r={0., 1.40976 10   \
} rnorm=1.409761196669024*10^-12\nNewton converged after 1 cycles\n           \
                                     -7\nlambda=0.08  u={0., -0.141918} \
r={0., 8.82291 10  } rnorm=8.82291380427391*10^-7\n                           \
                    -12\nlambda=0.08  u={0., -0.14192} r={0., 3.87816 10   } \
rnorm=3.878161680681558*10^-12\nNewton converged after 1 cycles\n             \
                                 -6\nlambda=0.09  u={0., -0.166597} r={0., \
1.457 10  } rnorm=1.45699628448237*10^-6\n                                    \
            -11\nlambda=0.09  u={0., -0.166601} r={0., 1.27831 10   } \
rnorm=1.278305239438282*10^-11\nNewton converged after 1 cycles\n             \
                                  -6\nlambda=0.1  u={0., -0.194466} r={0., \
2.67688 10  } rnorm=2.676875172219484*10^-6\n                                 \
             -11\nlambda=0.1  u={0., -0.194474} r={0., 5.4654 10   } \
rnorm=5.465397578952037*10^-11\nNewton converged after 1 cycles\n             \
                                   -6\nlambda=0.11  u={0., -0.227001} r={0., \
5.79878 10  } rnorm=5.798782686930461*10^-6\n                                 \
               -10\nlambda=0.11  u={0., -0.227021} r={0., 3.51095 10   } \
rnorm=3.510948892726518*10^-10\nNewton converged after 1 cycles\nlambda=0.12  \
u={0., -0.267322} r={0., 0.0000167657} rnorm=0.00001676568659284006\n         \
                                       -9\nlambda=0.12  u={0., -0.267399} \
r={0., 4.68925 10  } rnorm=4.689254123890763*10^-9\nNewton converged after 1 \
cycles\nlambda=0.13  u={0., -0.324949} r={0., 0.0000923426} \
rnorm=0.0000923425802657474\n                                               \
-7\nlambda=0.13  u={0., -0.32566} r={0., 3.62349 10  } \
rnorm=3.623493324556026*10^-7\n                                               \
 -12\nlambda=0.13  u={0., -0.325663} r={0., 5.66389 10   } \
rnorm=5.663886026852083*10^-12\nNewton converged after 2 cycles\nlambda=0.14  \
u={0., -0.575685} r={0., 0.0169917} rnorm=0.0169917169583524\nlambda=0.14  \
u={0., -0.471178} r={0., 0.00531894} rnorm=0.005318943076695671\nlambda=0.14  \
u={0., -0.377759} r={0., 0.00518327} rnorm=0.005183272960622098\nlambda=0.14  \
u={0., -0.468506} r={0., 0.00517086} rnorm=0.00517086240471476\nlambda=0.14  \
u={0., -0.37263} r={0., 0.00549365} rnorm=0.005493648493994085\nlambda=0.14  \
u={0., -0.458581} r={0., 0.00469145} rnorm=0.004691445952969803\nlambda=0.14  \
u={0., -0.34855} r={0., 0.00742351} rnorm=0.007423507500187859\nlambda=0.14  \
u={0., -0.425416} r={0., 0.00392191} rnorm=0.003921913521463483\nlambda=0.14  \
u={0., 0.735152} r={0., 1.37354} rnorm=1.373535016237653\nn   lambda  \
u={uX,uY}        v={vX,vY}      Kdet      a   kappa     remarks\n0   0       \
{0, 0}           {0, -1.41421}  0.5       1   1         C0 cfg\n1   0.01    \
{0, -0.0144539}  {0, -1.41421}  0.472212  1.  1.         \n2   0.02    {0, \
-0.029584}   {0, -1.47783}  0.443635  1.  0.956952   \n3   0.03    {0, \
-0.0454821}  {0, -1.54972}  0.414783  1.  0.912561   \n4   0.04    {0, \
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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=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=10; lambdamax=2; umax={2*S,2*H}; ell=0.100; 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}=IncIterSolution[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\"];
ListPlot[uYvskappa,PlotJoined->True,PlotRange->All,
         AxesOrigin->{0,0},AxesLabel->{\"uY\",\"kappa\"},
         PlotLabel->\"LP sensor kappa vs uY\"];\
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(*******************************************************************
End of Mathematica Notebook file.
*******************************************************************)