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FeFpGursonImplicitViscoPlasticity.mfront
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FeFpGursonImplicitViscoPlasticity.mfront
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@DSL ImplicitFiniteStrain;
@Behaviour GursonImplicitFeFpViscoPlasticity;
@Author Thomas Helfer;
@Date 13 / 09 / 2021;
@Description {
}
@Includes {
#include "TFEL/Math/ScalarNewtonRaphson.hxx"
#include "TFEL/Math/tensor.hxx"
}
//@UseQt true;
//@Algorithm NewtonRaphson;
@Algorithm NewtonRaphson_NumericalJacobian;
@PerturbationValueForNumericalJacobianComputation 1e-8;
@Epsilon 1.e-14;
@Theta 1;
@ModellingHypothesis Tridimensional;
@ElasticMaterialProperties {200e3, 0.3};
//@IntegrationVariable StrainStensor eel;
@StateVariable DeformationGradientTensor Fe;
Fe.setEntryName("ElasticPartOfTheDeformationGradient");
@StateVariable real p;
@StateVariable real f;
//f.setGlossaryName("f");
@AuxiliaryStateVariable StrainStensor eel;
@AuxiliaryStateVariable DeformationGradientTensor Fp;
Fp.setEntryName("PlasticPartOfTheDeformationGradient");
@AuxiliaryStateVariable real fstar;
@AuxiliaryStateVariable real sigstar;
@AuxiliaryStateVariable real broken;
@LocalVariable DeformationGradientTensor Fe_tr;
@LocalVariable DeformationGradientTensor Fm1;
@LocalVariable DeformationGradientTensor iFp;
@LocalVariable DeformationGradientTensor idFp;
@LocalVariable StiffnessTensor De;
@LocalVariable Stensor4 Je;
@LocalVariable tfel::math::st2tot2<N,real> didFp_ddeel;
@LocalVariable Tensor normal;
@LocalVariable real df_e;
@LocalVariable bool bp;
@LocalVariable bool bf;
@LocalVariable size_t iter_f;
@MaterialProperty real q_1 ;
q_1.setEntryName("q_1");
@MaterialProperty real q_2 ;
q_2.setEntryName("q_2");
@MaterialProperty real q_3 ;
q_3.setEntryName("q_3");
@MaterialProperty real A0 ;
A0.setEntryName("A0");
@MaterialProperty real Q1 ;
Q1.setEntryName("Q1");
@MaterialProperty real b1 ;
b1.setEntryName("b1");
@MaterialProperty real Q2 ;
Q2.setEntryName("Q2");
@MaterialProperty real b2 ;
b2.setEntryName("b2");
@MaterialProperty real H ;
H.setEntryName("H");
@MaterialProperty real f_c ;
f_c.setEntryName("f_c");
@MaterialProperty real f_r ;
f_r.setEntryName("f_r");
@MaterialProperty real gamma0 ;
gamma0.setEntryName("gamma0");
@MaterialProperty real K ;
K.setEntryName("K");
@MaterialProperty real n ;
n.setEntryName("n");
@InitializeLocalVariables {
De = lambda * Stensor4::IxI() + 2 * mu * Stensor4::Id();
iFp = invert(Fp);
const auto Fe_ = F1 * iFp; //F0 * iFp;
eel = computeGreenLagrangeTensor(Fe_);
//Fe_tr = F1 * iFp;
// elastic prediction
df_e = real{};
iter_f = size_t{};
bf = false;
if (broken < 0.5) {
constexpr const auto eeps = real(1.e-14);
const auto seps = young * eeps;
// to be factorized
const auto S = De * eel;
const auto id = Stensor::Id();
const auto uid = Tensor::Id();
const auto M = eval((id + 2 * eel) * S);
const auto Mm = trace(M) / 3;
const auto Mdev = M - Mm * uid;
const auto Meq = sqrt(3 * (Mdev | Mdev) / 2);
const auto iMeq = 1 / max(Meq, seps);
const auto Mm2 = Mm * Mm;
const auto Meq2 = Meq * Meq;
const auto delta = ((1. / q_1) - f_c) / (f_r - f_c);
fstar = (f < f_c) ? f : (f_c + delta * (f - f_c));
if (std::abs(Mm) < seps) {
const auto iomf = 1 / std::max(1 - 2 * q_1 * fstar + q_3 * fstar * fstar, real(1.e-12));
sigstar = Meq * std::sqrt(iomf);
}
else if (Meq < seps) {
const auto argach = (1 + q_3 * fstar * fstar) / std::max(2 * q_1 * fstar, real(1.e-12));
const auto den = 2 * std::acosh(argach);
sigstar = (3 * q_2 * Mm) / den;
} else {
//const auto SdS = [Meq2, Mm, fstar](const auto x) {
const auto SdS = [&](const auto x) {
const auto fstar2 = fstar * fstar;
const auto e = std::exp(3 * q_2 * Mm * (x / 2));
const auto ch = (e + 1 / e) / 2;
const auto sh = (e - 1 / e) / 2;
const auto S = Meq2 * x * x + 2 * q_1 * fstar * ch - 1 - q_3 * fstar2;
const auto dS = 2 * Meq2 * x + 3 * q_1 * q_2 * fstar * Mm * sh;
return std::make_tuple(S, dS);
};
const auto c = [seps, S](const real, const auto dx, const auto x,
const size_t) {
// First criterion based on the value of the yield surface
//if (std::abs(S) < seps) {
// return true;
//}
// d(1/y) = -dy/(y*y) => y*y*d(1/y) = -dy
// so if I want |dy|<eps, |y*y*d(1/y)| must be lower than eps
// Here, x is the inverse of the equivalent stress, so
return 10 * std::abs(dx) < seps * std::abs(x * x);
};
const auto x0 = std::sqrt((4 * (1 + q_3 * fstar * fstar - 2 * q_1 * fstar)) / (4 * Meq2 + 9 * q_1 * q_2 * q_2 * fstar * Mm2));
//std::cout << "x0 = " << x0 << '\n';
const auto r = tfel::math::scalarNewtonRaphson(SdS, c, x0, size_t{100});
//std::cout << "r = " << (1/std::get<1>(r)) << '\n';
if (!std::get<0>(r)) {
throw(DivergenceException());
};
sigstar = 1 / std::get<1>(r);
}
//
double R = A0 + Q1 * (1. - exp(-b1 * p)) + Q2 * (1. - exp(-b2 * p)) + H * p ;
bp = sigstar > R;
//std::cout << "bp = " << bp << '\n';
} else {
bp = true;
}
}
@Integrator {
/**/
if (!bp) {
// elastic loading, nothing to be done
fFe = dFe - (F1 - F0);
return true;
}
/**/
if (broken > 0.5) {
fFe = Fe + dFe;
} else {
Fm1 = invert(F1);
//const auto seps = strain{1e-12} * young;
const auto f_ = min(max(f + theta * df, real(0)), real(1));
const double pt = max(p + theta * dp, strain(0));
const auto Fe_ets = Fe + theta * dFe;
const auto eel_ets = computeGreenLagrangeTensor(Fe_ets);
const auto S = De * eel_ets;
constexpr const auto eeps = real(1.e-14);
const auto seps = young * eeps;
// to be factorized
//
const auto id = Stensor::Id();
const auto uid = Tensor::Id();
const auto M = eval((uid + 2 * eel_ets) * S);
const auto Mm = trace(M) / 3;
const auto Mdev = M - Mm * uid;
const auto Meq = sqrt(3 * (Mdev | Mdev) / 2);
const auto iMeq = 1 / max(Meq, seps);
const auto Mm2 = Mm * Mm;
const auto Meq2 = Meq * Meq;
const auto delta = ((1. / q_1) - f_c) / (f_r - f_c);
fstar = (f < f_c) ? f : (f_c + delta * (f - f_c));
const auto dfstar = (f < f_c) ? 1. : delta;
//compute sigstar
fstar = (f < f_c) ? f : (f_c + delta * (f - f_c));
if (std::abs(Mm) < seps) {
const auto iomf = 1 / std::max(1 - 2 * q_1 * fstar + q_3 * fstar * fstar, real(1.e-12));
sigstar = Meq * std::sqrt(iomf);
}
else if (Meq < seps) {
const auto argach = (1 + q_3 * fstar * fstar) / std::max(2 * q_1 * fstar, real(1.e-12));
const auto den = 2 * std::acosh(argach);
sigstar = (3 * q_2 * Mm) / den;
} else {
//const auto SdS = [Meq2, Mm, fstar](const auto x) {
const auto SdS = [&](const auto x) {
const auto fstar2 = fstar * fstar;
const auto e = std::exp(3 * q_2 * Mm * (x / 2));
const auto ch = (e + 1 / e) / 2;
const auto sh = (e - 1 / e) / 2;
const auto S = Meq2 * x * x + 2 * q_1 * fstar * ch - 1 - q_3 * fstar2;
const auto dS = 2 * Meq2 * x + 3 * q_1 * q_2 * fstar * Mm * sh;
return std::make_tuple(S, dS);
};
const auto c = [seps, S](const real, const auto dx, const auto x,
const size_t) {
// First criterion based on the value of the yield surface
//if (std::abs(S) < seps) {
// return true;
//}
// d(1/y) = -dy/(y*y) => y*y*d(1/y) = -dy
// so if I want |dy|<eps, |y*y*d(1/y)| must be lower than eps
// Here, x is the inverse of the equivalent stress, so
return 10 * std::abs(dx) < seps * std::abs(x * x);
};
const auto x0 = std::sqrt((4 * (1 + q_3 * fstar * fstar - 2 * q_1 * fstar)) / (4 * Meq2 + 9 * q_1 * q_2 * q_2 * fstar * Mm2));
const auto r = tfel::math::scalarNewtonRaphson(SdS, c, x0, size_t{100});
if (!std::get<0>(r)) {
throw(DivergenceException());
};
sigstar = 1 / std::get<1>(r);
}
//compute normal
const auto iss = 1 / std::max(sigstar, seps);
const auto e = std::exp(3 * q_2 * Mm * iss / 2);
const auto ch = (e + 1 / e) / 2;
const auto sh = (e - 1 / e) / 2;
const auto q1q2 = q_1 * q_2;
const auto dphi_dss =
(-2 * power<2>(Meq) * iss - 3 * q1q2 * fstar * Mm * sh) * power<2>(iss);
const auto sgn = (dphi_dss < 0) ? -1 : 1;
const auto idphi_dss =
sgn * ((tfel::math::ieee754::fpclassify(dphi_dss) == FP_ZERO)
? 1 / seps
: 1 / std::abs(dphi_dss));
const auto dphi_dsig = 3 * power<2>(iss) * Mdev + //
q1q2 * fstar * iss * sh * uid;
normal = -dphi_dsig * idphi_dss;
//normal = eval(-dphi_dsig * idphi_dss);
// second derivative of the normal
const auto MM = 1.5 * (t2tot2<N, real>::Id() - ((1./3.) * (uid ^ uid))) ; // tfel::math::t2tot2<N,double>::M();
const auto d2phi_dsig2 = (2. * MM + (q1q2 * q_2 * (fstar / 2) * ch * (uid ^ uid))) * power<2>(iss) ;
const auto d2phi_dsigdss = (-6. * Mdev * iss - q1q2 * fstar * sh * uid - //
3 * q1q2 * q_2 * (fstar / 2) * Mm * iss * ch * uid) * power<2>(iss) ;
const auto d2phi_dss2 = ((6 * Meq2 * iss) + (3 * q1q2 * fstar * Mm) * //
(2 * sh + 3 * q_2 * Mm * (iss / 2) * ch)) * power<3>(iss);
const auto dn_dsig = - idphi_dss * (d2phi_dsig2 + (d2phi_dsigdss ^ normal)) + //
power<2>(idphi_dss) * (dphi_dsig ^ (d2phi_dsigdss + (d2phi_dss2 * normal)));
// derivative of \(\sigma^{\star}\) with respect to \(f\)
const auto dphi_df = (2 * q_1 * ch - 2 * q_3 * fstar) * dfstar;
const auto dss_df = -dphi_df * idphi_dss;
// derivatives with respect to \(f\)
const auto d2phi_dssdf = (-3. * q1q2 * Mm * power<2>(iss) * sh) * dfstar;
const auto d2phi_dsigdf = (q1q2 * iss * sh * id) * dfstar;
const auto dn_df = -idphi_dss * (d2phi_dsigdf + (d2phi_dssdf * normal)) + //
power<2>(idphi_dss) * ((d2phi_dsigdss + (d2phi_dss2 * normal)) * dphi_df);
//
const auto dM_ddeel =
st2tot2<N, real>{2 * st2tot2<N, real>::tpld(S) +
st2tot2<N, real>::tprd(id + 2 * eel_ets, De)};
//const auto vp = de0 * pow(Meq/M0, E);
//idFp = uid - dt * vp * n;
// current estimate of the elastic part of the deformation gradient at the end
// of the time step
//const auto Fe = Fe_tr * (idFp / cbrt(det(idFp)));
double R = A0 + Q1 * (1. - exp(-b1 * pt)) + Q2 * (1. - exp(-b2 * pt)) + H * pt ;
fFe = dFe - (F1 - F0) * Fm1 * Fe_ets + Fe_ets * (1 - f_) * dp * normal;
fp = (dp - dt * gamma0 * pow( max( (sigstar - R) / K, 0.), n )) / young;
// jacobian
//const auto degl_didFp =
// t2tost2<N, real>::dCdF(Fe) * t2tot2<N, real>::tprd(Fe_tr) / 2;
//const auto dvp_dMeq = E * vp * iMeq;
//const auto uM = (3 * t2tot2<N, real>::K()) / 2;
//const auto dn_dM = iMeq * (uM - (n ^ n));
//const auto didFp_dM = -dt * dvp_dMeq * (n ^ n) - dt * vp * dn_dM;
//didFp_ddeel = didFp_dM * dM_ddeel;
//Je = Stensor4::Id() - degl_didFp * didFp_ddeel;
//dfeel_ddeel = Je;
/*
dfFe_ddFe = t2tot2<N, real>::Id() - t2tot2<N, real>::tprd ( (F1 - F0) * Fm1 ) + t2tot2<N, real>::tpld( (1 - f_) * dp * normal ) ;
dfFe_ddp = Fe_ets * (1 - f_) * normal;
dfFe_ddf = -theta * dp * normal + theta * (1 - f_) * dp * dn_df ;
dfp_ddFe = 1.e8*Tensor::Id() ; //to be implemented
dfp_ddp = 1 ;
dfp_ddf = 1 ; //to be implemented
*/
if (bf) {
ff -= power<2>(1 - f_) * dp * trace(normal);
/*
dff_ddf += 2 * theta * (1 - f_) * dp * trace(normal) - power<2>(1 - f_) * dp * trace(dn_df);
dff_ddp = -power<2>(1 - f_) * trace(n);
dff_ddFe -= ; // to be implemented // theta * power<2>(1 - f_) * dp * (Stensor::Id() | (dn_ds * D));
*/
} else {
ff -= df_e;
}
// std::cout << "normal = " << normal << '\n';
/*
std::cout << "sigstar = " << sigstar << '\n';
std::cout << "fp = " << fp << '\n';
std::cout << "ff = " << ff << '\n';
std::cout << "fFe = " << fFe << '\n';
std::cout << "sig = " << sig << '\n';
std::cout << "S = " << S << '\n';
std::cout << "Fe = " << Fe << '\n';
*/
}
}
@ComputeFinalStress {
//const auto Fe = Fe_tr * idFp;
Fp = invert(Fe)*F1;
eel = computeGreenLagrangeTensor(Fe);
const auto S = De * eel;
sig = convertSecondPiolaKirchhoffStressToCauchyStress(S,Fe);
/*
std::cout << "eel = " << eel << '\n';
std::cout << "cauchy = " << sig << '\n';
*/
}
/*
@UpdateAuxiliaryStateVariables {
Fp = eval(invert(idFp) * Fp);
Fp /= cbrt(det(Fp));
}
*/
@TangentOperator<DTAU_DF> {
//const auto Fe = Fe_tr * idFp;
const auto S = De * eel;
const auto dCe_dFe = t2tost2<N, real>::dCdF(Fe);
const auto dS_dFe = De * dCe_dFe / 2;
t2tost2<N, stress> dtau_dFe;
computePushForwardDerivative(dtau_dFe, dS_dFe, S, Fe);
// now we need dFe_dF
const auto dFe_tr_dF = t2tot2<N, real>::tpld(iFp);
const auto dfeel_dF = -dCe_dFe * t2tot2<N, real>::tpld(idFp, dFe_tr_dF) / 2;
const auto ddeel_dF = -invert(Je) * dfeel_dF;
const auto didFp_dF = didFp_ddeel * ddeel_dF;
const auto dFe_dF = t2tot2<N, real>::tpld(idFp, dFe_tr_dF) +
t2tot2<N, real>::tprd(Fe_tr, didFp_dF);
Dt = dtau_dFe * dFe_dF;
}
@AdditionalConvergenceChecks{
if (converged && bp && (broken < 0.5)) {
if (!bf) {
/*
// new porosity increment estimate;
const auto dfg_1 = [&, this]() -> real {
const auto p_ = p + theta * dp;
if (p_ < p_n) {
return 0;
}
const auto vp = sig.computeEigenValues();
const auto s1 = *(tfel::fsalgo::max_element<3u>::exe(vp.begin()));
if (s1 < s1_n) {
return 0;
}
return An0 * pow(s1 / s1_n - 1, ng) * dp;
}();
dfg_e = min(dfg_1, fg_max - fg);
*/
const auto f_ = f + df_e;
auto df_n = power<2>(1 - f_) * dp * trace(normal) ; // + dfg_e;
if (f_ + df_n > 0.99 * f_r) {
df_n = (f_ + 0.99 * f_r) / 2 - f;
}
auto ddf = df_n - df_e;
if (abs(ddf) < 1e-10) {
bf = true;
if (f_ > 0.985 * f_r) {
broken = true;
}
} else {
// std::cout << "ddf: " << ddf << '\n';
if (iter_f > 100) {
throw(DivergenceException());
}
iter = 0;
++iter_f;
df_e = df_n;
converged = false;
}
} else {
std::cout << "iter: " << iter << '\n';
}
}
//std::cout << "converged = " << converged << '\n';
}