void
AnisotropicViscoplasticModel<EvalT, Traits>::computeState(
typename Traits::EvalData workset,
DepFieldMap dep_fields,
FieldMap eval_fields)
{
std::string cauchy_string = (*field_name_map_)["Cauchy_Stress"];
std::string Fp_string = (*field_name_map_)["Fp"];
std::string eqps_string = (*field_name_map_)["eqps"];
std::string ess_string = (*field_name_map_)["ess"];
std::string kappa_string = (*field_name_map_)["iso_Hardening"];
std::string source_string = (*field_name_map_)["Mechanical_Source"];
std::string F_string = (*field_name_map_)["F"];
std::string J_string = (*field_name_map_)["J"];
// extract dependent MDFields
auto def_grad = *dep_fields[F_string];
auto J = *dep_fields[J_string];
auto poissons_ratio = *dep_fields["Poissons Ratio"];
auto elastic_modulus = *dep_fields["Elastic Modulus"];
auto yield_strength = *dep_fields["Yield Strength"];
auto hardening_modulus = *dep_fields["Hardening Modulus"];
auto recovery_modulus = *dep_fields["Recovery Modulus"];
auto flow_exp = *dep_fields["Flow Rule Exponent"];
auto flow_coeff = *dep_fields["Flow Rule Coefficient"];
auto delta_time = *dep_fields["Delta Time"];
// extract evaluated MDFields
auto stress = *eval_fields[cauchy_string];
auto Fp = *eval_fields[Fp_string];
auto eqps = *eval_fields[eqps_string];
auto ess = *eval_fields[ess_string];
auto kappa = *eval_fields[kappa_string];
PHX::MDField<ScalarT> source;
if (have_temperature_) { source = *eval_fields[source_string]; }
// get State Variables
Albany::MDArray Fpold = (*workset.stateArrayPtr)[Fp_string + "_old"];
Albany::MDArray eqpsold = (*workset.stateArrayPtr)[eqps_string + "_old"];
ScalarT bulk, mu, mubar, K, Y;
ScalarT Jm23, trace, smag2, smag, f, p, dgam;
ScalarT sq23(std::sqrt(2. / 3.));
minitensor::Tensor<ScalarT> F(num_dims_), be(num_dims_), s(num_dims_),
sigma(num_dims_);
minitensor::Tensor<ScalarT> N(num_dims_), A(num_dims_), expA(num_dims_),
Fpnew(num_dims_);
minitensor::Tensor<ScalarT> I(minitensor::eye<ScalarT>(num_dims_));
minitensor::Tensor<ScalarT> Fpn(num_dims_), Cpinv(num_dims_), Fe(num_dims_);
minitensor::Tensor<ScalarT> tau(num_dims_), M(num_dims_);
for (int cell(0); cell < workset.numCells; ++cell) {
for (int pt(0); pt < num_pts_; ++pt) {
bulk = elastic_modulus(cell, pt) /
(3. * (1. - 2. * poissons_ratio(cell, pt)));
mu = elastic_modulus(cell, pt) / (2. * (1. + poissons_ratio(cell, pt)));
K = hardening_modulus(cell, pt);
Y = yield_strength(cell, pt);
Jm23 = std::pow(J(cell, pt), -2. / 3.);
// fill local tensors
F.fill(def_grad, cell, pt, 0, 0);
// Mechanical deformation gradient
auto Fm = minitensor::Tensor<ScalarT>(F);
if (have_temperature_) {
// Compute the mechanical deformation gradient Fm based on the
// multiplicative decomposition of the deformation gradient
//
// F = Fm.Ft => Fm = F.inv(Ft)
//
// where Ft is the thermal part of F, given as
//
// Ft = Le * I = exp(alpha * dtemp) * I
//
// Le is the thermal stretch and alpha the coefficient of thermal
// expansion.
ScalarT dtemp = temperature_(cell, pt) - ref_temperature_;
ScalarT thermal_stretch = std::exp(expansion_coeff_ * dtemp);
Fm /= thermal_stretch;
}
// Fpn.fill( &Fpold(cell,pt,int(0),int(0)) );
for (int i(0); i < num_dims_; ++i) {
for (int j(0); j < num_dims_; ++j) {
Fpn(i, j) = ScalarT(Fpold(cell, pt, i, j));
}
}
// compute trial state
// compute the Kirchhoff stress in the current configuration
//
Fe = Fm * minitensor::inverse(Fpn);
Cpinv = minitensor::inverse(Fpn) *
minitensor::transpose(minitensor::inverse(Fpn));
be = Fm * Cpinv * minitensor::transpose(Fm);
ScalarT Je = std::sqrt(minitensor::det(be));
s = mu * minitensor::dev(be);
p = 0.5 * bulk * (Je * Je - 1.);
tau = p * I + s;
// pull back the Kirchhoff stress to the intermediate configuration
// this is the Mandel stress
//
M = minitensor::transpose(Fe) * tau *
minitensor::inverse(minitensor::transpose(Fe));
// check yield condition
smag = minitensor::norm(s);
f = smag - sq23 * (Y + K * eqpsold(cell, pt));
if (f > 1E-12) {
// return mapping algorithm
bool converged = false;
ScalarT g = f;
ScalarT H = 0.0;
ScalarT dH = 0.0;
ScalarT alpha = 0.0;
ScalarT res = 0.0;
int count = 0;
dgam = 0.0;
LocalNonlinearSolver<EvalT, Traits> solver;
std::vector<ScalarT> F(1);
std::vector<ScalarT> dFdX(1);
std::vector<ScalarT> X(1);
F[0] = f;
X[0] = 0.0;
dFdX[0] = (-2. * mubar) * (1. + H / (3. * mubar));
while (!converged && count <= 30) {
count++;
solver.solve(dFdX, X, F);
alpha = eqpsold(cell, pt) + sq23 * X[0];
H = K * alpha;
dH = K;
F[0] = smag - (2. * mubar * X[0] + sq23 * (Y + H));
dFdX[0] = -2. * mubar * (1. + dH / (3. * mubar));
res = std::abs(F[0]);
if (res < 1.e-11 || res / f < 1.E-11) converged = true;
TEUCHOS_TEST_FOR_EXCEPTION(
count == 30,
std::runtime_error,
std::endl
<< "Error in return mapping, count = " << count
<< "\nres = " << res << "\nrelres = " << res / f
<< "\ng = " << F[0] << "\ndg = " << dFdX[0]
<< "\nalpha = " << alpha << std::endl);
}
solver.computeFadInfo(dFdX, X, F);
dgam = X[0];
// plastic direction
N = (1 / smag) * s;
// update s
s -= 2 * mubar * dgam * N;
// update eqps
eqps(cell, pt) = alpha;
// mechanical source
if (have_temperature_ && delta_time(0) > 0) {
source(cell, pt) =
(sq23 * dgam / delta_time(0) * (Y + H + temperature_(cell, pt))) /
(density_ * heat_capacity_);
}
// exponential map to get Fpnew
A = dgam * N;
expA = minitensor::exp(A);
Fpnew = expA * Fpn;
for (int i(0); i < num_dims_; ++i) {
for (int j(0); j < num_dims_; ++j) {
Fp(cell, pt, i, j) = Fpnew(i, j);
}
}
} else {
eqps(cell, pt) = eqpsold(cell, pt);
if (have_temperature_) source(cell, pt) = 0.0;
for (int i(0); i < num_dims_; ++i) {
for (int j(0); j < num_dims_; ++j) { Fp(cell, pt, i, j) = Fpn(i, j); }
}
}
// compute pressure
p = 0.5 * bulk * (J(cell, pt) - 1. / (J(cell, pt)));
// compute stress
sigma = p * I + s / J(cell, pt);
for (int i(0); i < num_dims_; ++i) {
for (int j(0); j < num_dims_; ++j) {
stress(cell, pt, i, j) = sigma(i, j);
}
}
}
}
}
void ElastoViscoplasticModel<EvalT, Traits>::
computeState(typename Traits::EvalData workset,
std::map<std::string, Teuchos::RCP<PHX::MDField<ScalarT> > > dep_fields,
std::map<std::string, Teuchos::RCP<PHX::MDField<ScalarT> > > eval_fields)
{
std::string cauchy_string = (*field_name_map_)["Cauchy_Stress"];
std::string Fp_string = (*field_name_map_)["Fp"];
std::string eqps_string = (*field_name_map_)["eqps"];
std::string eps_ss_string = (*field_name_map_)["eps_ss"];
std::string kappa_string = (*field_name_map_)["isotropic_hardening"];
std::string source_string = (*field_name_map_)["Mechanical_Source"];
std::string F_string = (*field_name_map_)["F"];
std::string J_string = (*field_name_map_)["J"];
// extract dependent MDFields
PHX::MDField<ScalarT> def_grad_field = *dep_fields[F_string];
PHX::MDField<ScalarT> J = *dep_fields[J_string];
PHX::MDField<ScalarT> poissons_ratio = *dep_fields["Poissons Ratio"];
PHX::MDField<ScalarT> elastic_modulus = *dep_fields["Elastic Modulus"];
PHX::MDField<ScalarT> yield_strength = *dep_fields["Yield Strength"];
PHX::MDField<ScalarT> hardening_modulus = *dep_fields["Hardening Modulus"];
PHX::MDField<ScalarT> recovery_modulus = *dep_fields["Recovery Modulus"];
PHX::MDField<ScalarT> flow_exp = *dep_fields["Flow Rule Exponent"];
PHX::MDField<ScalarT> flow_coeff = *dep_fields["Flow Rule Coefficient"];
PHX::MDField<ScalarT> delta_time = *dep_fields["Delta Time"];
// extract evaluated MDFields
PHX::MDField<ScalarT> stress_field = *eval_fields[cauchy_string];
PHX::MDField<ScalarT> Fp_field = *eval_fields[Fp_string];
PHX::MDField<ScalarT> eqps_field = *eval_fields[eqps_string];
PHX::MDField<ScalarT> eps_ss_field = *eval_fields[eps_ss_string];
PHX::MDField<ScalarT> kappa_field = *eval_fields[kappa_string];
PHX::MDField<ScalarT> source_field;
if (have_temperature_) {
source_field = *eval_fields[source_string];
}
// get State Variables
Albany::MDArray Fp_field_old = (*workset.stateArrayPtr)[Fp_string + "_old"];
Albany::MDArray eqps_field_old = (*workset.stateArrayPtr)[eqps_string + "_old"];
Albany::MDArray eps_ss_field_old = (*workset.stateArrayPtr)[eps_ss_string + "_old"];
Albany::MDArray kappa_field_old = (*workset.stateArrayPtr)[kappa_string + "_old"];
// define constants
RealType sq23(std::sqrt(2. / 3.));
RealType sq32(std::sqrt(3. / 2.));
// pre-define some tensors that will be re-used below
Intrepid::Tensor<ScalarT> F(num_dims_), be(num_dims_);
Intrepid::Tensor<ScalarT> s(num_dims_), sigma(num_dims_);
Intrepid::Tensor<ScalarT> N(num_dims_), A(num_dims_);
Intrepid::Tensor<ScalarT> expA(num_dims_), Fpnew(num_dims_);
Intrepid::Tensor<ScalarT> I(Intrepid::eye<ScalarT>(num_dims_));
Intrepid::Tensor<ScalarT> Fpn(num_dims_), Cpinv(num_dims_), Fpinv(num_dims_);
for (std::size_t cell(0); cell < workset.numCells; ++cell) {
for (std::size_t pt(0); pt < num_pts_; ++pt) {
ScalarT bulk = elastic_modulus(cell, pt)
/ (3. * (1. - 2. * poissons_ratio(cell, pt)));
ScalarT mu = elastic_modulus(cell, pt) / (2. * (1. + poissons_ratio(cell, pt)));
ScalarT Y = yield_strength(cell, pt);
ScalarT Jm23 = std::pow(J(cell, pt), -2. / 3.);
// assign local state variables
//
//ScalarT kappa = kappa_field(cell,pt);
ScalarT kappa_old = kappa_field_old(cell,pt);
ScalarT eps_ss = eps_ss_field(cell,pt);
ScalarT eps_ss_old = eps_ss_field_old(cell,pt);
ScalarT eqps_old = eqps_field_old(cell,pt);
// fill local tensors
//
F.fill(&def_grad_field(cell, pt, 0, 0));
for (std::size_t i(0); i < num_dims_; ++i) {
for (std::size_t j(0); j < num_dims_; ++j) {
Fpn(i, j) = ScalarT(Fp_field_old(cell, pt, i, j));
}
}
// compute trial state
// compute the Kirchhoff stress in the current configuration
//
Cpinv = Intrepid::inverse(Fpn) * Intrepid::transpose(Intrepid::inverse(Fpn));
be = Jm23 * F * Cpinv * Intrepid::transpose(F);
s = mu * Intrepid::dev(be);
ScalarT smag = Intrepid::norm(s);
ScalarT mubar = Intrepid::trace(be) * mu / (num_dims_);
// check yield condition
//
ScalarT Phi = sq32 * smag - ( Y + kappa_old );
std::cout << "======== Phi: " << Phi << std::endl;
std::cout << "======== eps: " << std::numeric_limits<RealType>::epsilon() << std::endl;
if (Phi > std::numeric_limits<RealType>::epsilon()) {
// return mapping algorithm
//
bool converged = false;
int iter = 0;
RealType max_norm = std::numeric_limits<RealType>::min();
// hardening and recovery parameters
//
ScalarT H = hardening_modulus(cell, pt);
ScalarT Rd = recovery_modulus(cell, pt);
// flow rule temperature dependent parameters
//
ScalarT f = flow_coeff(cell,pt);
ScalarT n = flow_exp(cell,pt);
// This solver deals with Sacado type info
//
LocalNonlinearSolver<EvalT, Traits> solver;
// create some vectors to store solver data
//
std::vector<ScalarT> R(2);
std::vector<ScalarT> dRdX(4);
std::vector<ScalarT> X(2);
// initial guess
X[0] = 0.0;
X[1] = eps_ss_old;
// create a copy of be as a Fad
Intrepid::Tensor<Fad> beF(num_dims_);
for (std::size_t i = 0; i < num_dims_; ++i) {
for (std::size_t j = 0; j < num_dims_; ++j) {
beF(i, j) = be(i, j);
}
}
Fad two_mubarF = 2.0 * Intrepid::trace(beF) * mu / (num_dims_);
//Fad sq32F = std::sqrt(3.0/2.0);
// FIXME this seems to be necessary to get PhiF to compile below
// need to look into this more, it appears to be a conflict
// between the Intrepid::norm and FadType operations
//
Fad smagF = smag;
while (!converged) {
// set up data types
//
std::vector<Fad> XFad(2);
std::vector<Fad> RFad(2);
std::vector<ScalarT> Xval(2);
for (std::size_t i = 0; i < 2; ++i) {
Xval[i] = Sacado::ScalarValue<ScalarT>::eval(X[i]);
XFad[i] = Fad(2, i, Xval[i]);
}
// get solution vars
//
Fad dgamF = XFad[0];
Fad eps_ssF = XFad[1];
// compute yield function
//
Fad eqps_rateF = 0.0;
if (delta_time(0) > 0) eqps_rateF = sq23 * dgamF / delta_time(0);
Fad rate_termF = 1.0 + std::asinh( std::pow(eqps_rateF / f, n));
Fad kappaF = two_mubarF * eps_ssF;
Fad PhiF = sq32 * (smagF - two_mubarF * dgamF) - ( Y + kappaF ) * rate_termF;
// compute the hardening residual
//
Fad eps_resF = eps_ssF - eps_ss_old - (H - Rd*eps_ssF) * dgamF;
// for convenience put the residuals into a container
//
RFad[0] = PhiF;
RFad[1] = eps_resF;
// extract the values of the residuals
//
for (int i = 0; i < 2; ++i)
R[i] = RFad[i].val();
// extract the sensitivities of the residuals
//
for (int i = 0; i < 2; ++i)
for (int j = 0; j < 2; ++j)
dRdX[i + 2 * j] = RFad[i].dx(j);
// this call invokes the solver and updates the solution in X
//
solver.solve(dRdX, X, R);
// compute the norm of the residual
//
RealType R0 = Sacado::ScalarValue<ScalarT>::eval(R[0]);
RealType R1 = Sacado::ScalarValue<ScalarT>::eval(R[1]);
RealType norm_res = std::sqrt(R0*R0 + R1*R1);
max_norm = std::max(norm_res, max_norm);
// check against too many inerations
//
TEUCHOS_TEST_FOR_EXCEPTION(iter == 30, std::runtime_error,
std::endl <<
"Error in ElastoViscoplastic return mapping\n" <<
"iter count = " << iter << "\n" << std::endl);
// check for a sufficiently small residual
//
std::cout << "======== norm_res : " << norm_res << std::endl;
if ( (norm_res/max_norm < 1.e-12) || (norm_res < 1.e-12) )
converged = true;
// increment the iteratio counter
//
iter++;
}
solver.computeFadInfo(dRdX, X, R);
ScalarT dgam = X[0];
ScalarT eps_ss = X[1];
ScalarT kappa = 2.0 * mubar * eps_ss;
std::cout << "======== dgam : " << dgam << std::endl;
std::cout << "======== e_ss : " << eps_ss << std::endl;
std::cout << "======== kapp : " << kappa << std::endl;
// plastic direction
N = (1 / smag) * s;
// update s
s -= 2 * mubar * dgam * N;
// update state variables
eps_ss_field(cell, pt) = eps_ss;
eqps_field(cell,pt) = eqps_old + sq23 * dgam;
kappa_field(cell,pt) = kappa;
// mechanical source
// FIXME this is not correct, just a placeholder
//
if (have_temperature_ && delta_time(0) > 0) {
source_field(cell, pt) = (sq23 * dgam / delta_time(0))
* (Y + kappa) / (density_ * heat_capacity_);
}
// exponential map to get Fpnew
//
A = dgam * N;
expA = Intrepid::exp(A);
Fpnew = expA * Fpn;
for (std::size_t i(0); i < num_dims_; ++i) {
for (std::size_t j(0); j < num_dims_; ++j) {
Fp_field(cell, pt, i, j) = Fpnew(i, j);
}
}
} else {
// we are not yielding, variables do not evolve
//
eps_ss_field(cell, pt) = eps_ss_old;
eqps_field(cell,pt) = eqps_old;
kappa_field(cell,pt) = kappa_old;
if (have_temperature_) source_field(cell, pt) = 0.0;
for (std::size_t i(0); i < num_dims_; ++i) {
for (std::size_t j(0); j < num_dims_; ++j) {
Fp_field(cell, pt, i, j) = Fpn(i, j);
}
}
}
// compute pressure
ScalarT p = 0.5 * bulk * (J(cell, pt) - 1. / (J(cell, pt)));
// compute stress
sigma = p * I + s / J(cell, pt);
for (std::size_t i(0); i < num_dims_; ++i) {
for (std::size_t j(0); j < num_dims_; ++j) {
stress_field(cell, pt, i, j) = sigma(i, j);
}
}
}
}
if (have_temperature_) {
for (std::size_t cell(0); cell < workset.numCells; ++cell) {
for (std::size_t pt(0); pt < num_pts_; ++pt) {
F.fill(&def_grad_field(cell,pt,0,0));
ScalarT J = Intrepid::det(F);
sigma.fill(&stress_field(cell,pt,0,0));
sigma -= 3.0 * expansion_coeff_ * (1.0 + 1.0 / (J*J))
* (temperature_(cell,pt) - ref_temperature_) * I;
for (std::size_t i = 0; i < num_dims_; ++i) {
for (std::size_t j = 0; j < num_dims_; ++j) {
stress_field(cell, pt, i, j) = sigma(i, j);
}
}
}
}
}
}