void ESDIRK::solveTimeStep( const scalar t0 )
{
if ( adaptiveTimeStepper->isPreviousStepAccepted() )
solver->nextTimeStep();
fsi::matrix solStages( nbStages, N ), F( nbStages, N );
F.setZero();
fsi::vector sol( N ), f( N ), qold( N ), result( N ), rhs( N );
solver->getSolution( sol, f );
solStages.row( 0 ) = sol;
qold = sol;
scalar t = t0;
solver->evaluateFunction( 0, sol, t, f );
F.row( 0 ) = f;
// Keep the solution of the first stage and function evaluate
// in memory in case the time step is rejected
fsi::vector solOld = sol;
fsi::vector fOld = f;
// Loop over the stages
solver->initTimeStep();
int iImplicitStage = 0;
for ( int j = 0; j < nbStages; j++ )
{
if ( !isStageImplicit( A( j, j ) ) )
continue;
t = t0 + C( j ) * dt;
Info << "\nTime = " << t << ", ESDIRK stage = " << j + 1 << "/" << nbStages << nl << endl;
rhs.setZero();
// Calculate sum of the stage residuals
for ( int iStage = 0; iStage < j; iStage++ )
rhs += A( j, iStage ) * F.row( iStage ).transpose();
rhs.array() *= dt;
solver->implicitSolve( false, iImplicitStage, 0, t, A( j, j ) * dt, qold, rhs, f, result );
assert( (1.0 / (A( j, j ) * dt) * (result - qold - rhs) - f).array().abs().maxCoeff() < 1.0e-8 );
solStages.row( j ) = result;
F.row( j ) = f;
iImplicitStage++;
}
if ( adaptiveTimeStepper->isEnabled() )
{
scalar newTimeStep = 0;
fsi::vector errorEstimate( N );
errorEstimate.setZero();
for ( int iStage = 0; iStage < nbStages; iStage++ )
errorEstimate += ( B( iStage ) - Bhat( iStage ) ) * F.row( iStage );
errorEstimate *= dt;
bool accepted = adaptiveTimeStepper->determineNewTimeStep( errorEstimate, result, dt, newTimeStep );
dt = newTimeStep;
if ( not accepted )
solver->setSolution( solOld, fOld );
}
if ( adaptiveTimeStepper->isAccepted() )
solver->finalizeTimeStep();
}
void umat_plasticity_kin_iso_CCP(const vec &Etot, const vec &DEtot, vec &sigma, mat &Lt, mat &L, vec &sigma_in, const mat &DR, const int &nprops, const vec &props, const int &nstatev, vec &statev, const double &T, const double &DT, const double &Time, const double &DTime, double &Wm, double &Wm_r, double &Wm_ir, double &Wm_d, const int &ndi, const int &nshr, const bool &start, const int &solver_type, double &tnew_dt)
{
UNUSED(nprops);
UNUSED(nstatev);
UNUSED(Time);
UNUSED(DTime);
UNUSED(nshr);
UNUSED(tnew_dt);
//From the props to the material properties
double E = props(0);
double nu= props(1);
double alpha_iso = props(2);
double sigmaY = props(3);
double k=props(4);
double m=props(5);
double kX = props(6);
//definition of the CTE tensor
vec alpha = alpha_iso*Ith();
///@brief Temperature initialization
double T_init = statev(0);
//From the statev to the internal variables
double p = statev(1);
vec EP(6);
EP(0) = statev(2);
EP(1) = statev(3);
EP(2) = statev(4);
EP(3) = statev(5);
EP(4) = statev(6);
EP(5) = statev(7);
///@brief a is the internal variable associated with kinematical hardening
vec a = zeros(6);
a(0) = statev(8);
a(1) = statev(9);
a(2) = statev(10);
a(3) = statev(11);
a(4) = statev(12);
a(5) = statev(13);
//Rotation of internal variables (tensors)
EP = rotate_strain(EP, DR);
a = rotate_strain(a, DR);
///@brief Initialization
if(start)
{
//Elstic stiffness tensor
L = L_iso(E, nu, "Enu");
T_init = T;
vec vide = zeros(6);
sigma = vide;
EP = vide;
a = vide;
p = 0.;
Wm = 0.;
Wm_r = 0.;
Wm_ir = 0.;
Wm_d = 0.;
}
//Additional parameters and variables
vec X = kX*(a%Ir05());
double Hp=0.;
double dHpdp=0.;
if (p > iota) {
dHpdp = m*k*pow(p, m-1);
Hp = k*pow(p, m);
}
else {
dHpdp = 0.;
Hp = 0.;
}
//Variables values at the start of the increment
vec sigma_start = sigma;
vec EP_start = EP;
vec a_start = a;
vec X_start = X;
double A_p_start = -Hp;
vec A_a_start = -X_start;
//Variables required for the loop
vec s_j = zeros(1);
s_j(0) = p;
vec Ds_j = zeros(1);
vec ds_j = zeros(1);
///Elastic prediction - Accounting for the thermal prediction
vec Eel = Etot + DEtot - alpha*(T+DT-T_init) - EP;
sigma = el_pred(L, Eel, ndi);
//Define the plastic function and the stress
vec Phi = zeros(1);
mat B = zeros(1,1);
vec Y_crit = zeros(1);
double dPhidp=0.;
vec dPhida = zeros(6);
vec dPhidsigma = zeros(6);
double dPhidtheta = 0.;
//Compute the explicit flow direction
vec Lambdap = eta_stress(sigma-X);
vec Lambdaa = eta_stress(sigma-X);
std::vector<vec> kappa_j(1);
kappa_j[0] = L*Lambdap;
mat K = zeros(1,1);
//Loop parameters
int compteur = 0;
double error = 1.;
//Loop
for (compteur = 0; ((compteur < maxiter_umat) && (error > precision_umat)); compteur++) {
p = s_j(0);
if (p > iota) {
dHpdp = m*k*pow(p, m-1);
Hp = k*pow(p, m);
}
else {
dHpdp = 0.;
Hp = 0.;
}
dPhidsigma = eta_stress(sigma-X);
dPhidp = -1.*dHpdp;
dPhida = -1.*kX*(eta_stress(sigma - X)%Ir05());
//compute Phi and the derivatives
Phi(0) = Mises_stress(sigma-X) - Hp - sigmaY;
Lambdap = eta_stress(sigma-X);
Lambdaa = eta_stress(sigma-X);
kappa_j[0] = L*Lambdap;
K(0,0) = dPhidp + sum(dPhida%Lambdaa);
B(0, 0) = -1.*sum(dPhidsigma%kappa_j[0]) + K(0,0);
Y_crit(0) = sigmaY;
Fischer_Burmeister_m(Phi, Y_crit, B, Ds_j, ds_j, error);
s_j(0) += ds_j(0);
EP = EP + ds_j(0)*Lambdap;
a = a + ds_j(0)*Lambdaa;
X = kX*(a%Ir05());
//the stress is now computed using the relationship sigma = L(E-Ep)
Eel = Etot + DEtot - alpha*(T + DT - T_init) - EP;
sigma = el_pred(L, Eel, ndi);
}
//Computation of the increments of variables
vec Dsigma = sigma - sigma_start;
vec DEP = EP - EP_start;
double Dp = Ds_j[0];
vec Da = a - a_start;
if (solver_type == 0) {
//Computation of the tangent modulus
mat Bhat = zeros(1, 1);
Bhat(0, 0) = sum(dPhidsigma%kappa_j[0]) - K(0,0);
vec op = zeros(1);
mat delta = eye(1,1);
for (int i=0; i<1; i++) {
if(Ds_j[i] > iota)
op(i) = 1.;
}
mat Bbar = zeros(1,1);
for (int i = 0; i < 1; i++) {
for (int j = 0; j < 1; j++) {
Bbar(i, j) = op(i)*op(j)*Bhat(i, j) + delta(i,j)*(1-op(i)*op(j));
}
}
mat invBbar = zeros(1, 1);
mat invBhat = zeros(1, 1);
invBbar = inv(Bbar);
for (int i = 0; i < 1; i++) {
for (int j = 0; j < 1; j++) {
invBhat(i, j) = op(i)*op(j)*invBbar(i, j);
}
}
std::vector<vec> P_epsilon(1);
P_epsilon[0] = invBhat(0, 0)*(L*dPhidsigma);
std::vector<double> P_theta(1);
P_theta[0] = dPhidtheta - sum(dPhidsigma%(L*alpha));
Lt = L - (kappa_j[0]*P_epsilon[0].t());
}
else if(solver_type == 1) {
sigma_in = -L*EP;
}
double A_p = -Hp;
vec A_a = -X;
double Dgamma_loc = 0.5*sum((sigma_start+sigma)%DEP) + 0.5*(A_p_start + A_p)*Dp + 0.5*sum((A_a_start + A_a)%Da);
//Computation of the mechanical and thermal work quantities
Wm += 0.5*sum((sigma_start+sigma)%DEtot);
Wm_r += 0.5*sum((sigma_start+sigma)%(DEtot-DEP)) - 0.5*sum((A_a_start + A_a)%Da);
Wm_ir += -0.5*(A_p_start + A_p)*Dp;
Wm_d += Dgamma_loc;
///@brief statev evolving variables
//statev
statev(0) = T_init;
statev(1) = p;
statev(2) = EP(0);
statev(3) = EP(1);
statev(4) = EP(2);
statev(5) = EP(3);
statev(6) = EP(4);
statev(7) = EP(5);
statev(8) = a(0);
statev(9) = a(1);
statev(10) = a(2);
statev(11) = a(3);
statev(12) = a(4);
statev(13) = a(5);
}
void ESDIRK::initializeButcherTableau( std::string method )
{
assert( method == "SDIRK2" || method == "SDIRK3" || method == "SDIRK4" || method == "ESDIRK3" || method == "ESDIRK4" || method == "ESDIRK5" || method == "ESDIRK53PR" || method == "ESDIRK63PR" || method == "ESDIRK74PR" );
// source: Ellsiepen. Habilitation thesis Philipp Birken
if ( method == "SDIRK2" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 1 );
scalar alpha = 1.0 - 0.5 * std::sqrt( 2 );
scalar alphahat = 2.0 - 5.0 / 4.0 * std::sqrt( 2 );
A.resize( 2, 2 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
A( 0, 0 ) = alpha;
A( 1, 0 ) = 1.0 - alpha;
A( 1, 1 ) = alpha;
Bhat( 0 ) = 1 - alphahat;
Bhat( 1 ) = alphahat;
}
// source: Cash. Habilitation thesis Philipp Birken
if ( method == "SDIRK3" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 2 );
A.resize( 3, 3 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
scalar alpha = 1.2084966491760101;
scalar beta = -0.6443631706844691;
scalar gamma = 0.4358665215084580;
scalar delta = 0.7179332607542295;
scalar alphahat = 0.7726301276675511;
scalar betahat = 0.2273698723324489;
A( 0, 0 ) = gamma;
A( 1, 0 ) = delta - gamma;
A( 1, 1 ) = gamma;
A( 2, 0 ) = alpha;
A( 2, 1 ) = beta;
A( 2, 2 ) = gamma;
Bhat( 0 ) = alphahat;
Bhat( 1 ) = betahat;
Bhat( 2 ) = 0;
}
// source: Cash-5-3-4 http://runge.math.smu.edu/arkode_dev/doc/guide/build/html/Butcher.html
if ( method == "SDIRK4" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 3 );
A.resize( 5, 5 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
A( 0, 0 ) = 0.435866521508;
A( 1, 0 ) = -1.13586652150;
A( 1, 1 ) = 0.435866521508;
A( 2, 0 ) = 1.08543330679;
A( 2, 1 ) = -0.721299828287;
A( 2, 2 ) = 0.435866521508;
A( 3, 0 ) = 0.416349501547;
A( 3, 1 ) = 0.190984004184;
A( 3, 2 ) = -0.118643265417;
A( 3, 3 ) = 0.435866521508;
A( 4, 0 ) = 0.896869652944;
A( 4, 1 ) = 0.0182725272734;
A( 4, 2 ) = -0.0845900310706;
A( 4, 3 ) = -0.266418670647;
A( 4, 4 ) = 0.435866521508;
Bhat( 0 ) = 0.776691932910;
Bhat( 1 ) = 0.0297472791484;
Bhat( 2 ) = -0.0267440239074;
Bhat( 3 ) = 0.220304811849;
Bhat( 4 ) = 0;
}
if ( method == "ESDIRK3" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 2 );
A.resize( 4, 4 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
A( 0, 0 ) = 0.0;
A( 1, 0 ) = 1767732205903.0 / 4055673282236.0;
A( 1, 1 ) = 1767732205903.0 / 4055673282236.0;
A( 2, 0 ) = 2746238789719.0 / 10658868560708.0;
A( 2, 1 ) = -640167445237.0 / 6845629431997.0;
A( 2, 2 ) = 1767732205903.0 / 4055673282236.0;
A( 3, 0 ) = 1471266399579.0 / 7840856788654.0;
A( 3, 1 ) = -4482444167858.0 / 7529755066697.0;
A( 3, 2 ) = 11266239266428.0 / 11593286722821.0;
A( 3, 3 ) = 1767732205903.0 / 4055673282236.0;
Bhat( 0 ) = 2756255671327.e0 / 12835298489170.e0;
Bhat( 1 ) = -10771552573575.e0 / 22201958757719.e0;
Bhat( 2 ) = 9247589265047.e0 / 10645013368117.e0;
Bhat( 3 ) = 2193209047091.e0 / 5459859503100.e0;
}
if ( method == "ESDIRK4" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 3 );
A.resize( 6, 6 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
A( 1, 0 ) = 1.e0 / 4.e0;
A( 1, 1 ) = 1.e0 / 4.e0;
A( 2, 0 ) = 8611.e0 / 62500.e0;
A( 2, 1 ) = -1743.e0 / 31250.e0;
A( 2, 2 ) = 1.e0 / 4.e0;
A( 3, 0 ) = 5012029.e0 / 34652500.e0;
A( 3, 1 ) = -654441.e0 / 2922500.e0;
A( 3, 2 ) = 174375.e0 / 388108.e0;
A( 3, 3 ) = 1.e0 / 4.e0;
A( 4, 0 ) = 15267082809.e0 / 155376265600.e0;
A( 4, 1 ) = -71443401.e0 / 120774400.e0;
A( 4, 2 ) = 730878875.e0 / 902184768.e0;
A( 4, 3 ) = 2285395.e0 / 8070912.e0;
A( 4, 4 ) = 1.e0 / 4.e0;
A( 5, 0 ) = 82889.e0 / 524892.e0;
A( 5, 1 ) = 0.e0;
A( 5, 2 ) = 15625.e0 / 83664.e0;
A( 5, 3 ) = 69875.e0 / 102672.e0;
A( 5, 4 ) = -2260.e0 / 8211.e0;
A( 5, 5 ) = 1.e0 / 4.e0;
Bhat( 0 ) = 4586570599.e0 / 29645900160.e0;
Bhat( 1 ) = 0.e0;
Bhat( 2 ) = 178811875.e0 / 945068544.e0;
Bhat( 3 ) = 814220225.e0 / 1159782912.e0;
Bhat( 4 ) = -3700637.e0 / 11593932.e0;
Bhat( 5 ) = 61727.e0 / 225920.e0;
}
if ( method == "ESDIRK5" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 4 );
A.resize( 8, 8 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
A( 0, 0 ) = 0.e0;
A( 1, 0 ) = 41.e0 / 200.e0;
A( 1, 1 ) = 41.e0 / 200.e0;
A( 2, 0 ) = 41.e0 / 400.e0;
A( 2, 1 ) = -567603406766.e0 / 11931857230679.e0;
A( 2, 2 ) = 41.e0 / 200.e0;
A( 3, 0 ) = 683785636431.e0 / 9252920307686.e0;
A( 3, 1 ) = 0.e0;
A( 3, 2 ) = -110385047103.e0 / 1367015193373.e0;
A( 3, 3 ) = 41.e0 / 200.e0;
A( 4, 0 ) = 3016520224154.e0 / 10081342136671.e0;
A( 4, 1 ) = 0.e0;
A( 4, 2 ) = 30586259806659.e0 / 12414158314087.e0;
A( 4, 3 ) = -22760509404356.e0 / 11113319521817.e0;
A( 4, 4 ) = 41.e0 / 200.e0;
A( 5, 0 ) = 218866479029.e0 / 1489978393911.e0;
A( 5, 1 ) = 0.e0;
A( 5, 2 ) = 638256894668.e0 / 5436446318841.e0;
A( 5, 3 ) = -1179710474555.e0 / 5321154724896.e0;
A( 5, 4 ) = -60928119172.e0 / 8023461067671.e0;
A( 5, 5 ) = 41.e0 / 200.e0;
A( 6, 0 ) = 1020004230633.e0 / 5715676835656.e0;
A( 6, 1 ) = 0.e0;
A( 6, 2 ) = 25762820946817.e0 / 25263940353407.e0;
A( 6, 3 ) = -2161375909145.e0 / 9755907335909.e0;
A( 6, 4 ) = -211217309593.e0 / 5846859502534.e0;
A( 6, 5 ) = -4269925059573.e0 / 7827059040749.e0;
A( 6, 6 ) = 41.e0 / 200.e0;
A( 7, 0 ) = -872700587467.e0 / 9133579230613.e0;
A( 7, 1 ) = 0.e0;
A( 7, 2 ) = 0.e0;
A( 7, 3 ) = 22348218063261.e0 / 9555858737531.e0;
A( 7, 4 ) = -1143369518992.e0 / 8141816002931.e0;
A( 7, 5 ) = -39379526789629.e0 / 19018526304540.e0;
A( 7, 6 ) = 32727382324388.e0 / 42900044865799.e0;
A( 7, 7 ) = 41.e0 / 200.e0;
Bhat( 0 ) = -975461918565.e0 / 9796059967033.e0;
Bhat( 1 ) = 0.e0;
Bhat( 2 ) = 0.e0;
Bhat( 3 ) = 78070527104295.e0 / 32432590147079.e0;
Bhat( 4 ) = -548382580838.e0 / 3424219808633.e0;
Bhat( 5 ) = -33438840321285.e0 / 15594753105479.e0;
Bhat( 6 ) = 3629800801594.e0 / 4656183773603.e0;
Bhat( 7 ) = 4035322873751.e0 / 18575991585200.e0;
}
// source: http://www.sciencedirect.com/science/article/pii/S0168927415000501
if ( method == "ESDIRK53PR" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 2 );
A.resize( 5, 5 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
A( 0, 0 ) = 0.0;
A( 1, 0 ) = 2.777777777777778e-01;
A( 1, 1 ) = 2.777777777777778e-01;
A( 2, 0 ) = 3.456552483519272e-01;
A( 2, 1 ) = 1.681740315717733e-01;
A( 2, 2 ) = 2.777777777777778e-01;
A( 3, 0 ) = 3.965643047257401e-01;
A( 3, 1 ) = 1.001154404932533e-01;
A( 3, 2 ) = 1.255424770032288e-01;
A( 3, 3 ) = 2.777777777777778e-01;
A( 4, 0 ) = 2.481479828780141e-01;
A( 4, 1 ) = 2.139473588935955e-01;
A( 4, 2 ) = 1.206274239267400e+00;
A( 4, 3 ) = -9.461473588167871e-01;
A( 4, 4 ) = 2.777777777777778e-01;
Bhat( 0 ) = 4.445537532713554e-01;
Bhat( 1 ) = -1.065203443758999e-01;
Bhat( 2 ) = 2.533129069755295e-01;
Bhat( 3 ) = 5.000000000000000e-01;
Bhat( 4 ) = -9.134631587098500e-02;
}
// source: http://www.sciencedirect.com/science/article/pii/S0168927415000501
if ( method == "ESDIRK63PR" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 2 );
A.resize( 6, 6 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
A( 0, 0 ) = 0.0;
A( 1, 0 ) = 4.166666666666667e-01;
A( 1, 1 ) = 4.166666666666667e-01;
A( 2, 0 ) = 3.640473915723038e-01;
A( 2, 1 ) = -4.189886135331312e-02;
A( 2, 2 ) = 4.166666666666667e-01;
A( 3, 0 ) = -2.894969214392781e+00;
A( 3, 1 ) = -2.256341718064659e+01;
A( 3, 2 ) = 2.534171972837271e+01;
A( 3, 3 ) = 4.166666666666667e-01;
A( 4, 0 ) = 2.309551022782098e-01;
A( 4, 1 ) = -1.849667242832423e+00;
A( 4, 2 ) = 2.197073089164931e+00;
A( 4, 3 ) = 4.972384722615363e-03;
A( 4, 4 ) = 4.166666666666667e-01;
A( 5, 0 ) = 3.054968378466108e-01;
A( 5, 1 ) = 4.057983152922798e+00;
A( 5, 2 ) = -2.202162095667910e+00;
A( 5, 3 ) = 1.333484429273537e-01;
A( 5, 4 ) = -1.711333004695519e+00;
A( 5, 5 ) = 4.166666666666667e-01;
Bhat( 0 ) = 2.309551022782098e-01;
Bhat( 1 ) = -1.849667242832423e+00;
Bhat( 2 ) = 2.197073089164931e+00;
Bhat( 3 ) = 4.972384722615363e-03;
Bhat( 4 ) = 4.166666666666667e-01;
Bhat( 5 ) = 0.0;
}
// source: http://www.sciencedirect.com/science/article/pii/S0168927415000501
if ( method == "ESDIRK74PR" )
{
if ( adaptiveTimeStepper )
adaptiveTimeStepper->setOrderEmbeddedMethod( 3 );
A.resize( 7, 7 );
A.setZero();
Bhat.resize( A.cols() );
Bhat.setZero();
A( 0, 0 ) = 0.0;
A( 1, 0 ) = 1.666666666666667e-01;
A( 1, 1 ) = 1.666666666666667e-01;
A( 2, 0 ) = 4.166666666666666e-02;
A( 2, 1 ) = -4.166666666666666e-02;
A( 2, 2 ) = 1.666666666666667e-01;
A( 3, 0 ) = -1.500000000000000e+00;
A( 3, 1 ) = -1.333333333333333e+00;
A( 3, 2 ) = 3.333333333333333e+00;
A( 3, 3 ) = 1.666666666666667e-01;
A( 4, 0 ) = -1.580729166666667e+00;
A( 4, 1 ) = -1.349609375000000e+00;
A( 4, 2 ) = 3.472656250000000e+00;
A( 4, 3 ) = 4.101562500000000e-02;
A( 4, 4 ) = 1.666666666666667e-01;
A( 5, 0 ) = -2.005366150605651e+00;
A( 5, 1 ) = -1.768688648609954e+00;
A( 5, 2 ) = 4.341269295345690e+00;
A( 5, 3 ) = 2.326169434610579e-02;
A( 5, 4 ) = 1.000000000000000e-01;
A( 5, 5 ) = 1.666666666666667e-01;
A( 6, 0 ) = 1.684854267805816e-01;
A( 6, 1 ) = 7.501080898831836e-01;
A( 6, 2 ) = -2.255843889686931e-01;
A( 6, 3 ) = -9.134421504267402e-01;
A( 6, 4 ) = 1.618140253772232e+00;
A( 6, 5 ) = -5.643738977072310e-01;
A( 6, 6 ) = 1.666666666666667e-01;
Bhat( 0 ) = -3.930182461751728e-01;
Bhat( 1 ) = 1.000000000000000e-01;
Bhat( 2 ) = 9.916346405575472e-01;
Bhat( 3 ) = 0.0;
Bhat( 4 ) = -2.511232158528943e-01;
Bhat( 5 ) = 4.393912810497486e-01;
Bhat( 6 ) = 1.131155404207712e-01;
}
nbStages = A.cols();
C.resize( nbStages );
for ( int i = 0; i < nbStages; i++ )
C( i ) = A.row( i ).sum();
B.resize( nbStages );
for ( int i = 0; i < nbStages; i++ )
B( i ) = A( nbStages - 1, i );
}
