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laghost_function.hpp
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#include "mfem.hpp"
#include <ctime>
#include <cstdlib>
namespace mfem
{
double e0(const Vector &);
double p0(const Vector &);
double depth0(const Vector &);
double rho0(const Vector &);
double gamma_func(const Vector &);
double zero_func(const Vector &);
void v0(const Vector &, Vector &);
void xyz0(const Vector &, Vector &);
double x_l2(const Vector &);
double y_l2(const Vector &);
double z_l2(const Vector &);
static void principal_stresses2(const double* s, double p[2],
double& cos2t, double& sin2t)
{
/* 's' is a flattened stress vector, with the components {XX, ZZ, XZ}.
* Returns the eigenvalues 'p', and the direction cosine of the
* eigenvectors in the X-Z plane.
* The eigenvalues are ordered such that p[0] <= p[1].
*/
// center and radius of Mohr circle
double s0 = 0.5 * (s[0] + s[1]);
double rad = std::sqrt(0.25*(s[0] - s[1])*(s[0] - s[1]) + s[2]*s[2]);
// principal stresses in the X-Z plane
p[0] = s0 - rad;
p[1] = s0 + rad;
{
// direction cosine and sine of 2*theta
const double eps = 1e-15;
double a = 0.5 * (s[0] - s[1]);
double b = - rad; // always negative
if (b < -eps) {
cos2t = a / b;
sin2t = s[2] / b;
}
else {
cos2t = 1;
sin2t = 0;
}
}
}
class PlasticCoefficient : public VectorCoefficient
{
private:
ParGridFunction &xyz;
int dim;
Vector location;
double rad, ini_pls;
public:
PlasticCoefficient (int &_dim, ParGridFunction &_xyz, Vector &_location, double &_rad, double &_ini_pls)
: VectorCoefficient(1), xyz(_xyz)
{
dim=_dim; location = _location; rad = _rad; ini_pls = _ini_pls;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(1);
double r = 0.0;
double xc = xyz.GetValue(T, ip,1);
double yc = xyz.GetValue(T, ip,2);
double zc = 0.0;
double randomNumber;
double xc_0 = 0.0;
double yc_0 = 0.0;
double zc_0 = 0.0;
double dist = 0.0;
if(dim == 3){zc = xyz.GetValue(T, ip,3);}
if(dim == 2){ r = sqrt(pow((xc-location[0]), 2) + pow((yc-location[1]), 2));}
else if(dim == 3){r = sqrt(pow((xc-location[0]), 2) + pow((yc-location[1]), 2) + pow((zc-location[2]), 2));}
if(r <= rad)
{
K(0) = ini_pls;
}
else
{
K(0) = 0.0;
}
// // Define the line's angle (in radians)
// double angle_rad = M_PI / 3; // 60 degrees in radians
// // Define the slope and intercept of the line
// double m = tan(angle_rad);
// double b = 0; // Assuming the line passes through the origin
// // Define the width of the band
// double width = 3000;
// xc_0 = xc-location[0]; yc_0 = yc-location[1];
// dist = fabs((m * xc_0 - yc_0 + b) / sqrt(m * m + 1)); // distance between points and line
// if ((dist <= width / 2) & (yc <= 5000))
// {
// K(0) = ini_pls;
// }
// else
// {
// K(0) = 0.0;
// }
// if((xc >= 49375.0) & (xc <= 50625.0) & (yc <= 1250.0))
// {
// K(0) = ini_pls;
// }
// else
// {
// K(0) = 0.0;
// }
// srand(time(0));
// randomNumber = (rand() % 101)/100.0;
// if((xc >= 97.5e3) & (xc <= 102.5e3) & (yc <= 5e3))
// // if((xc >= 80e3) & (xc <= 120e3))
// // if(xc <= 100e3)
// {
// // K(0) = ini_pls*randomNumber;
// K(0) = ini_pls;
// // std::cout << randomNumber << std::endl;
// }
// else
// {
// K(0) = 0.0;
// }
// K(0) = ini_pls*randomNumber;
}
virtual ~PlasticCoefficient() { }
};
class LithostaticCoefficient : public VectorCoefficient
{
private:
ParGridFunction &xyz, ρ
int dim;
double gravity, thickness;
public:
LithostaticCoefficient (int &_dim, ParGridFunction &_xyz, ParGridFunction &_rho, double &_gravity, double &_thickness)
: VectorCoefficient(3*(_dim-1)), xyz(_xyz), rho(_rho)
{
dim=_dim; gravity = _gravity; thickness = _thickness;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(3*(dim-1));
double xc = xyz.GetValue(T, ip, 1);
double zc = xyz.GetValue(T, ip, dim);
double denc = rho.GetValue(T, ip);
double atm = -101325; // 1 atm in Pa
K = 0.0;
for (int i = 0; i < dim; i++)
K(i) = -1.0*fabs(thickness - zc)*denc*gravity + atm;
}
virtual ~LithostaticCoefficient() { }
};
class ATMCoefficient : public VectorCoefficient
{
private:
ParGridFunction &xyz, ρ
int dim;
double gravity, thickness;
public:
ATMCoefficient (int &_dim, ParGridFunction &_xyz, ParGridFunction &_rho, double &_gravity, double &_thickness)
: VectorCoefficient(3*(_dim-1)), xyz(_xyz), rho(_rho)
{
dim=_dim; gravity = _gravity; thickness = _thickness;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(3*(dim-1));
// K.SetSize(dim);
double atm = -101325; // 1 atm in Pa
// atm =0.0;
double zc = xyz.GetValue(T, ip, dim);
//
K = 0.0;
for (int i = 0; i < dim; i++)
K(i) = atm;
}
virtual ~ATMCoefficient() { }
};
class CompoCoefficient : public VectorCoefficient
{
private:
ParGridFunction &mat;
int mat_num;
int comp;
public:
CompoCoefficient (int &_mat_num, ParGridFunction &_mat)
: VectorCoefficient(_mat_num), mat(_mat)
{
mat_num=_mat_num; comp = 0;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(mat_num);
K = 0.0;
comp = mat.GetValue(T, ip);
for (int i = 0; i < mat_num; i++)
{
if(comp == i)
{
K(i) = 1.0;
}
}
}
virtual ~CompoCoefficient() { }
};
class CompMassCoefficient : public VectorCoefficient
{
private:
ParGridFunction &comp_ref_gf, vol_ini_gf, quality;
int mat_num;
public:
CompMassCoefficient (int &_mat_num, ParGridFunction &_comp_ref_gf, ParGridFunction &_vol_ini_gf, ParGridFunction &_quality)
: VectorCoefficient(_mat_num), comp_ref_gf(_comp_ref_gf), vol_ini_gf(_vol_ini_gf), quality(_quality)
{
mat_num=_mat_num;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(mat_num);
K = 0.0;
for (int i = 0; i < mat_num; i++)
{
K(i) = comp_ref_gf.GetValue(T, ip, i+1) * vol_ini_gf.GetValue(T, ip) / quality.GetValue(T, ip, 1);
}
}
virtual ~CompMassCoefficient() { }
};
class Temp_steessCoefficient : public VectorCoefficient
{
private:
ParGridFunction &s_gf;
int component;
public:
Temp_steessCoefficient (int &_component, ParGridFunction &_s_gf)
: VectorCoefficient(_component), s_gf(_s_gf)
{
component=_component;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K(0) = s_gf.GetValue(T, ip, component);
}
virtual ~Temp_steessCoefficient() { }
};
class Temp_compCoefficient : public VectorCoefficient
{
private:
ParGridFunction &comp_gf;
int component;
public:
Temp_compCoefficient (int &_component, ParGridFunction &_comp_gf)
: VectorCoefficient(_component), comp_gf(_comp_gf)
{
component=_component;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K(0) = comp_gf.GetValue(T, ip, component+1);
}
virtual ~Temp_compCoefficient() { }
};
class StressMappingCoefficient : public VectorCoefficient
{
private:
ParGridFunction &temp_gf;
int dim;
public:
StressMappingCoefficient (int &_dim, ParGridFunction &_temp_gf)
: VectorCoefficient(_dim), temp_gf(_temp_gf)
{
dim = _dim;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(3*(dim-1));
if(dim == 2)
{
K(0) = temp_gf.GetValue(T, ip, 1);
K(1) = temp_gf.GetValue(T, ip, 2);
K(2) = temp_gf.GetValue(T, ip, 3);
}
else
{
K(0) = temp_gf.GetValue(T, ip, 1);
K(1) = temp_gf.GetValue(T, ip, 2);
K(2) = temp_gf.GetValue(T, ip, 3);
K(3) = temp_gf.GetValue(T, ip, 4);
K(4) = temp_gf.GetValue(T, ip, 5);
K(5) = temp_gf.GetValue(T, ip, 6);
}
}
virtual ~StressMappingCoefficient() { }
};
class PlasticityMappingCoefficient : public VectorCoefficient
{
private:
ParGridFunction &temp_gf;
int dim;
public:
PlasticityMappingCoefficient (int &_dim, ParGridFunction &_temp_gf)
: VectorCoefficient(_dim), temp_gf(_temp_gf)
{
dim = _dim;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(1);
if(dim == 2)
{
K(0) = temp_gf.GetValue(T, ip, 4);
}
else
{
K(0) = temp_gf.GetValue(T, ip, 7);
}
}
virtual ~PlasticityMappingCoefficient() { }
};
class ReturnMapping3DCoefficient : public VectorCoefficient
{
private:
ParGridFunction &comp_gf, &s_gf, &s_old_gf, &p_gf, &mat_gf;
int dim;
double h_min, dt_old;
bool viscoplastic;
Vector rho, lambda, mu, tension_cutoff, cohesion0, cohesion1, pls0, pls1, friction_angle0, friction_angle1, dilation_angle0, dilation_angle1, plastic_viscosity;
public:
ReturnMapping3DCoefficient (int &_dim, double &_h_min, double &_dt_old, bool &_viscoplastic, ParGridFunction &_comp_gf, ParGridFunction &_s_gf, ParGridFunction &_s_old_gf, ParGridFunction &_p_gf, ParGridFunction &_mat_gf, Vector &_rho, Vector &_lambda, Vector &_mu, Vector &_tension_cutoff, Vector &_cohesion0, Vector &_cohesion1, Vector &_pls0, Vector &_pls1, Vector &_friction_angle0, Vector &_friction_angle1, Vector &_dilation_angle0, Vector &_dilation_angle1, Vector &_plastic_viscosity)
: VectorCoefficient(_dim), comp_gf(_comp_gf), s_gf(_s_gf), s_old_gf(_s_old_gf), p_gf(_p_gf), mat_gf(_mat_gf)
{
dim = _dim; h_min = _h_min; dt_old = _dt_old; viscoplastic = _viscoplastic;
rho = _rho; lambda = _lambda; mu = _mu; tension_cutoff = _tension_cutoff; cohesion0 = _cohesion0; cohesion1 = _cohesion1; pls0 = _pls0; pls1 = _pls1; friction_angle0 = _friction_angle0; friction_angle1 = _friction_angle1; dilation_angle0 = _dilation_angle0; dilation_angle1 = _dilation_angle1; plastic_viscosity = _plastic_viscosity;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(3*(dim-1)+1);
K = 0.0;
DenseMatrix esig(3);
DenseMatrix esig_old(3);
DenseMatrix esig_inc(3);
DenseMatrix plastic_sig(3);
DenseMatrix plastic_str(3);
esig=0.0; plastic_sig=0.0; plastic_str=0.0;
double eig_sig_var[3], eig_sig_vec[9];
double sig1{0.0};
double sig3{0.0};
double syy{0.0}; // Syy is non-zero value in plane strain condition
double msig{0.0}; // mean stress
double evol{0.0}; // volumetric strain
double DEG2RAD{M_PI/180.0};
double depls{0.0}; // 2nd invariant of plastic strain
double fs{0.0};
double ft{0.0};
double fh{0.0};
double N_phi{0.0};
double st_N_phi{0.0};
double N_psi{0.0};
double beta{0.0};
double viscosity{0.0};
double relax{0.0};
// double relax_limit{1.0};
double dt_scaled{0.0};
double numerator = {0.0};
double denominator = {0.0};
double pls_old = {0.0}; // cumulative 2nd invariant of plastic strain
double p_slope = {0.0};
double fri_str = {0.0}; // strain_dependent friction angle
double dil_str = {0.0}; // strain_dependent dilation angle
double coh_str = {0.0}; // strain_dependent cohesion
double ten_cut = {0.0};
int mat{0};
int nsize{mat_gf.Size()};
int mat_num{lambda.Size()};
double pls0_c = {0.0};
double pls1_c = {0.0};
double rho_c = {0.0};
double lambda_c = {0.0};
double mu_c = {0.0};
double tension_cutoff_c = {0.0};
double cohesion0_c = {0.0};
double cohesion1_c = {0.0};
double friction_angle0_c ={0.0};
double friction_angle1_c ={0.0};
double dilation_angle0_c ={0.0};
double dilation_angle1_c ={0.0};
double plastic_viscosity_c ={0.0};
double pwave_speed ={0.0};
double time_scale ={1.0};
// element-wise loop
esig=0.0; plastic_sig=0.0; plastic_str=0.0;
esig_old=0.0; esig_inc=0.0;
pls_old = p_gf.GetValue(T, ip);
if(pls_old < 0.0){pls_old =0.0;}
pls0_c =0.0; pls1_c =0.0; rho_c = 0.0; lambda_c = 0.0; mu_c = 0.0; time_scale = 1.0;
tension_cutoff_c = 0.0; cohesion0_c = 0.0; cohesion1_c = 0.0; friction_angle0_c = 0.0; friction_angle1_c = 0.0;
dilation_angle0_c = 0.0; dilation_angle1_c = 0.0; plastic_viscosity_c = 0.0;
for( int i = 0; i < mat_num; i++ )
{
pls0_c = pls0_c + comp_gf.GetValue(T, ip, i+1)*pls0[i];
pls1_c = pls1_c + comp_gf.GetValue(T, ip, i+1)*pls1[i];
rho_c = rho_c + comp_gf.GetValue(T, ip, i+1)*rho[i];
lambda_c = lambda_c + comp_gf.GetValue(T, ip, i+1)*lambda[i];
mu_c = mu_c + comp_gf.GetValue(T, ip, i+1)*mu[i];
tension_cutoff_c = tension_cutoff_c + comp_gf.GetValue(T, ip, i+1)*tension_cutoff[i];
cohesion0_c = cohesion0_c + comp_gf.GetValue(T, ip, i+1)*cohesion0[i];
cohesion1_c = cohesion1_c + comp_gf.GetValue(T, ip, i+1)*cohesion1[i];
friction_angle0_c = friction_angle0_c + comp_gf.GetValue(T, ip, i+1)*friction_angle0[i];
friction_angle1_c = friction_angle1_c + comp_gf.GetValue(T, ip, i+1)*friction_angle1[i];
dilation_angle0_c = dilation_angle0_c + comp_gf.GetValue(T, ip, i+1)*dilation_angle0[i];
dilation_angle1_c = dilation_angle1_c + comp_gf.GetValue(T, ip, i+1)*dilation_angle1[i];
}
// linear weakening
p_slope = (pls_old - pls0_c)/(pls1_c - pls0_c);
pwave_speed = sqrt((lambda_c + 2*mu_c)/rho_c);
if(h_min > 0){time_scale = h_min / pwave_speed;}
plastic_viscosity_c = time_scale * mu_c;
if(dim ==2)
{
// 2D in plane strain condition
// sxx, syy, szz, sxz are non zeros.
// sxy, syz are zero.
msig = (s_gf.GetValue(T, ip, 1) + s_gf.GetValue(T, ip, 2))*0.5;
evol = msig/(lambda_c+mu_c);
syy = evol * lambda_c;
esig(0,0) = s_gf.GetValue(T, ip, 1); esig(0,1) = 0.0; esig(0,2) = s_gf.GetValue(T, ip, 3);
esig(1,0) = 0.0; esig(1,1) = syy; esig(1,2) = 0.0;
esig(2,0) = s_gf.GetValue(T, ip, 3); esig(2,1) = 0.0; esig(2,2) = s_gf.GetValue(T, ip, 2);
// Caushy stress at previous time step
msig = (s_old_gf.GetValue(T, ip, 1) + s_old_gf.GetValue(T, ip, 2))*0.5;
evol = msig/(lambda_c+mu_c);
syy = evol * lambda_c;
esig_old(0,0) = s_old_gf.GetValue(T, ip, 1); esig_old(0,1) = 0.0; esig_old(0,2) = s_old_gf.GetValue(T, ip, 3);
esig_old(1,0) = 0.0; esig_old(1,1) = syy; esig_old(1,2) = 0.0;
esig_old(2,0) = s_old_gf.GetValue(T, ip, 3); esig_old(2,1) = 0.0; esig_old(2,2) = s_old_gf.GetValue(T, ip, 2);
}
else
{
esig(0,0) = s_gf.GetValue(T, ip, 1); esig(0,1) = s_gf.GetValue(T, ip, 4); esig(0,2) = s_gf.GetValue(T, ip, 5);
esig(1,0) = s_gf.GetValue(T, ip, 4); esig(1,1) = s_gf.GetValue(T, ip, 2); esig(1,2) = s_gf.GetValue(T, ip, 6);
esig(2,0) = s_gf.GetValue(T, ip, 5); esig(2,1) = s_gf.GetValue(T, ip, 6); esig(2,2) = s_gf.GetValue(T, ip, 3);
// Caushy stress at previous time step
esig_old(0,0) = s_gf.GetValue(T, ip, 1); esig_old(0,1) = s_gf.GetValue(T, ip, 4); esig_old(0,2) = s_gf.GetValue(T, ip, 5);
esig_old(1,0) = s_gf.GetValue(T, ip, 4); esig_old(1,1) = s_gf.GetValue(T, ip, 2); esig_old(1,2) = s_gf.GetValue(T, ip, 6);
esig_old(2,0) = s_gf.GetValue(T, ip, 5); esig_old(2,1) = s_gf.GetValue(T, ip, 6); esig_old(2,2) = s_gf.GetValue(T, ip, 3);
}
// Elastic stress increment
esig_inc(0,0) = esig(0,0) - esig_old(0,0); esig_inc(0,1) = esig(0,1) - esig_old(0,1); esig_inc(0,2) = esig(0,2) - esig_old(0,2);
esig_inc(1,0) = esig(1,0) - esig_old(1,0); esig_inc(1,1) = esig(1,1) - esig_old(1,1); esig_inc(1,2) = esig(1,2) - esig_old(1,2);
esig_inc(2,0) = esig(2,0) - esig_old(2,0); esig_inc(2,1) = esig(2,1) - esig_old(2,1); esig_inc(2,2) = esig(2,2) - esig_old(2,2);
esig.CalcEigenvalues(eig_sig_var, eig_sig_vec);
Vector sig_var(eig_sig_var, 3);
Vector sig_dir(eig_sig_vec, 3);
auto max_it = std::max_element(sig_var.begin(), sig_var.end()); // find iterator to max element
auto min_it = std::min_element(sig_var.begin(), sig_var.end()); // find iterator to min element
int max_index = std::distance(sig_var.begin(), max_it); // calculate index of max element
int min_index = std::distance(sig_var.begin(), min_it); // calculate index of min element
int itm_index = 0; // calculate index of intermediate element
if (max_index + min_index == 1) {itm_index = 2;}
else if(max_index + min_index == 2) {itm_index = 1;}
else {itm_index = 0;}
sig1 = sig_var[min_index]; // most compressive pincipal stress
sig3 = sig_var[max_index]; // least compressive pincipal stress
// linear strain weaking on cohesion, friction and dilation angles.
coh_str = cohesion0_c; fri_str = friction_angle0_c; dil_str = dilation_angle0_c;
if (pls_old < pls0_c) {
// no weakening yet
coh_str = cohesion0_c;
fri_str = friction_angle0_c;
dil_str = dilation_angle0_c;
}
else if (pls_old < pls1_c) {
// linear weakening
coh_str = cohesion0_c + p_slope * (cohesion1_c - cohesion0_c);
fri_str = friction_angle0_c + p_slope * (friction_angle1_c - friction_angle0_c);
dil_str = dilation_angle0_c + p_slope * (dilation_angle1_c - dilation_angle0_c);
}
else {
// saturated weakening
coh_str = cohesion1_c; fri_str = friction_angle1_c; dil_str = dilation_angle1_c;
}
N_phi = (1+sin(DEG2RAD*fri_str))/(1-sin(DEG2RAD*fri_str));
st_N_phi = cos(DEG2RAD*fri_str)/(1-sin(DEG2RAD*fri_str));
N_psi = -1*(1+sin(DEG2RAD*dil_str))/(1-sin(DEG2RAD*dil_str)); // partial_g/partial_sig3
if(tension_cutoff_c == 0)
{
ten_cut = coh_str/tan(DEG2RAD*fri_str);
}
else{ten_cut = tension_cutoff_c;}
// shear failure function
fs = sig1 - N_phi*sig3 + 2*coh_str*st_N_phi;
// tension failure function
ft = sig3 - ten_cut;
// bisects the obtuse angle made by two yield function
fh = sig3 - ten_cut + (sqrt(N_phi*N_phi + 1.0)+ N_phi)*(sig1 - N_phi*ten_cut + 2*coh_str*st_N_phi);
depls = 0.0;
if( ( (fs < 0) & (fh < 0) ) | ( (ft > 0) & (fh > 0) ) )
{
if(fs < 0 & fh < 0) // stress correction at shear failure
{
// Equations 28 and 30 from Choi et al. (2013; DynEarthSol2D: An efficient unstructured finite element method to study long-term tectonic deformation).
beta = fs;
beta = beta / (((lambda_c+2*mu_c)*1 - N_phi*lambda_c*1) + (lambda_c*N_psi - N_phi*(lambda_c+2*mu_c)*N_psi));
plastic_str(0,0) = (lambda_c + 2*mu_c + lambda_c*N_psi) * beta;
plastic_str(1,1) = (lambda_c + lambda_c*N_psi) * beta;
plastic_str(2,2) = (lambda_c + (lambda_c+2*mu_c)*N_psi) * beta;
// reduced form of 2nd invariant
if(dim ==2)
{
depls = std::fabs(beta) * std::sqrt((3 - 2*N_psi + 3*N_psi*N_psi) / 8);
}
else
{
depls = std::fabs(beta) * std::sqrt((7 - 4*N_psi + 7*N_psi*N_psi) / 18);
}
}
else if (ft > 0 & fh > 0) // stress correction at tension failure
{
beta = ft;
beta = beta/(lambda_c+2*mu_c);
plastic_str(0,0) = lambda_c * beta * 1;
plastic_str(1,1) = lambda_c * beta * 1;
plastic_str(2,2) = (lambda_c+2*mu_c) * beta * 1;
// reduced form of 2nd invariant
if(dim ==2)
{
depls = std::fabs(beta) * std::sqrt(3. / 8);
}
else
{
depls = std::fabs(beta) * std::sqrt(7. / 18);
}
}
// Rotating Principal axis to XYZ axis
plastic_sig(0,0) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[0+min_index*3]*sig_dir[0+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[0+itm_index*3]*sig_dir[0+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[0+max_index*3]*sig_dir[0+max_index*3]);
plastic_sig(0,1) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[0+min_index*3]*sig_dir[1+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[0+itm_index*3]*sig_dir[1+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[0+max_index*3]*sig_dir[1+max_index*3]);
plastic_sig(0,2) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[0+min_index*3]*sig_dir[2+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[0+itm_index*3]*sig_dir[2+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[0+max_index*3]*sig_dir[2+max_index*3]);
plastic_sig(1,0) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[1+min_index*3]*sig_dir[0+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[1+itm_index*3]*sig_dir[0+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[1+max_index*3]*sig_dir[0+max_index*3]);
plastic_sig(1,1) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[1+min_index*3]*sig_dir[1+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[1+itm_index*3]*sig_dir[1+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[1+max_index*3]*sig_dir[1+max_index*3]);
plastic_sig(1,2) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[1+min_index*3]*sig_dir[2+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[1+itm_index*3]*sig_dir[2+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[1+max_index*3]*sig_dir[2+max_index*3]);
plastic_sig(2,0) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[2+min_index*3]*sig_dir[0+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[2+itm_index*3]*sig_dir[0+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[2+max_index*3]*sig_dir[0+max_index*3]);
plastic_sig(2,1) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[2+min_index*3]*sig_dir[1+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[2+itm_index*3]*sig_dir[1+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[2+max_index*3]*sig_dir[1+max_index*3]);
plastic_sig(2,2) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[2+min_index*3]*sig_dir[2+min_index*3] + (sig_var[itm_index]-plastic_str(1,1))*sig_dir[2+itm_index*3]*sig_dir[2+itm_index*3] + (sig_var[max_index]-plastic_str(2,2))*sig_dir[2+max_index*3]*sig_dir[2+max_index*3]);
// Updating new stress to grid function
viscosity = plastic_viscosity_c;
relax = exp(-dt_old/viscosity);
numerator = fabs(1 - relax);
denominator= fabs(dt_old/viscosity);
dt_scaled = dt_old/viscosity;
// Based on L'Hôpital's rule, 0/0 become 1
// if((numerator > 1e-15) & (denominator > 1e-15)){relax_limit = (1 - relax)/(dt_old/viscosity);}
if(dim ==2)
{
if(viscoplastic)
{
// Closed-form algorithm for viscoplasticity from Computational Inelasticity on p. 218 (Simo and Hughes, 1998)
// s_gf[i+nsize*0]=esig_old(0,0)*relax + (1.0 - relax)*plastic_sig(0,0) + relax_limit*esig_inc(0,0);
// s_gf[i+nsize*2]=esig_old(0,2)*relax + (1.0 - relax)*plastic_sig(0,2) + relax_limit*esig_inc(0,2);
// s_gf[i+nsize*2]=esig_old(2,0)*relax + (1.0 - relax)*plastic_sig(2,0) + relax_limit*esig_inc(2,0);
// s_gf[i+nsize*1]=esig_old(2,2)*relax + (1.0 - relax)*plastic_sig(2,2) + relax_limit*esig_inc(2,2);
// Implicit backward Euler algorithm
K(0)=((esig_old(0,0) + esig_inc(0,0)) + dt_scaled*plastic_sig(0,0))/(1.0+dt_scaled);
K(1)=((esig_old(2,2) + esig_inc(2,2)) + dt_scaled*plastic_sig(2,2))/(1.0+dt_scaled);
K(2)=((esig_old(2,0) + esig_inc(2,0)) + dt_scaled*plastic_sig(2,0))/(1.0+dt_scaled);
depls = dt_scaled*depls/(1.0+dt_scaled); K(3) = depls;
}
else
{
K(0)=plastic_sig(0,0);
K(1)=plastic_sig(2,2);
K(2)=plastic_sig(0,2);
K(3)=depls;
}
}
else
{
if(viscoplastic)
{
// // Closed-form algorithm for viscoplasticity from Computational Inelasticity on p. 218 (Simo and Hughes, 1998)
// s_gf[i+nsize*0]=esig_old(0,0)*relax + (1.0 - relax)*plastic_sig(0,0) + relax_limit*esig_inc(0,0);
// s_gf[i+nsize*3]=esig_old(0,1)*relax + (1.0 - relax)*plastic_sig(0,1) + relax_limit*esig_inc(0,1);
// s_gf[i+nsize*4]=esig_old(0,2)*relax + (1.0 - relax)*plastic_sig(0,2) + relax_limit*esig_inc(0,2);
// s_gf[i+nsize*3]=esig_old(1,0)*relax + (1.0 - relax)*plastic_sig(1,0) + relax_limit*esig_inc(1,0);
// s_gf[i+nsize*1]=esig_old(1,1)*relax + (1.0 - relax)*plastic_sig(1,1) + relax_limit*esig_inc(1,1);
// s_gf[i+nsize*5]=esig_old(1,2)*relax + (1.0 - relax)*plastic_sig(1,2) + relax_limit*esig_inc(1,2);
// s_gf[i+nsize*4]=esig_old(2,0)*relax + (1.0 - relax)*plastic_sig(2,0) + relax_limit*esig_inc(2,0);
// s_gf[i+nsize*5]=esig_old(2,1)*relax + (1.0 - relax)*plastic_sig(2,1) + relax_limit*esig_inc(2,1);
// s_gf[i+nsize*2]=esig_old(2,2)*relax + (1.0 - relax)*plastic_sig(2,2) + relax_limit*esig_inc(2,2);
// Implicit backward Euler algorithm
K(0)=((esig_old(0,0) + esig_inc(0,0)) + dt_scaled*plastic_sig(0,0))/(1.0+dt_scaled);
K(1)=((esig_old(1,1) + esig_inc(1,1)) + dt_scaled*plastic_sig(1,1))/(1.0+dt_scaled);
K(2)=((esig_old(2,2) + esig_inc(2,2)) + dt_scaled*plastic_sig(2,2))/(1.0+dt_scaled);
K(3)=((esig_old(0,1) + esig_inc(0,1)) + dt_scaled*plastic_sig(0,1))/(1.0+dt_scaled);
K(4)=((esig_old(0,2) + esig_inc(0,2)) + dt_scaled*plastic_sig(0,2))/(1.0+dt_scaled);
K(5)=((esig_old(1,2) + esig_inc(1,2)) + dt_scaled*plastic_sig(1,2))/(1.0+dt_scaled);
depls = dt_scaled*depls/(1.0+dt_scaled); K(6) = depls;
}
else
{
K(0)=plastic_sig(0,0);
K(1)=plastic_sig(1,1);
K(2)=plastic_sig(2,2);
K(3)=plastic_sig(0,1);
K(4)=plastic_sig(0,2);
K(5)=plastic_sig(1,2);
K(6)=depls;
}
}
}
else
{
if(dim ==2)
{
K(0)=esig(0,0);
K(1)=esig(2,2);
K(2)=esig(0,2);
K(3)=0.0;
}
else
{
K(0)=esig(0,0);
K(1)=esig(1,1);
K(2)=esig(2,2);
K(3)=esig(0,1);
K(4)=esig(0,2);
K(5)=esig(1,2);
K(6)=0.0;
}
}
}
virtual ~ReturnMapping3DCoefficient() { }
};
class ReturnMapping2DCoefficient : public VectorCoefficient
{
private:
ParGridFunction &comp_gf, &s_gf, &s_old_gf, &p_gf, &mat_gf;
int dim;
double h_min, dt_old;
bool viscoplastic;
Vector rho, lambda, mu, tension_cutoff, cohesion0, cohesion1, pls0, pls1, friction_angle0, friction_angle1, dilation_angle0, dilation_angle1, plastic_viscosity;
public:
ReturnMapping2DCoefficient (int &_dim, double &_h_min, double &_dt_old, bool &_viscoplastic, ParGridFunction &_comp_gf, ParGridFunction &_s_gf, ParGridFunction &_s_old_gf, ParGridFunction &_p_gf, ParGridFunction &_mat_gf, Vector &_rho, Vector &_lambda, Vector &_mu, Vector &_tension_cutoff, Vector &_cohesion0, Vector &_cohesion1, Vector &_pls0, Vector &_pls1, Vector &_friction_angle0, Vector &_friction_angle1, Vector &_dilation_angle0, Vector &_dilation_angle1, Vector &_plastic_viscosity)
: VectorCoefficient(_dim), comp_gf(_comp_gf), s_gf(_s_gf), s_old_gf(_s_old_gf), p_gf(_p_gf), mat_gf(_mat_gf)
{
dim = _dim; h_min = _h_min; dt_old = _dt_old; viscoplastic = _viscoplastic;
rho = _rho; lambda = _lambda; mu = _mu; tension_cutoff = _tension_cutoff; cohesion0 = _cohesion0; cohesion1 = _cohesion1; pls0 = _pls0; pls1 = _pls1; friction_angle0 = _friction_angle0; friction_angle1 = _friction_angle1; dilation_angle0 = _dilation_angle0; dilation_angle1 = _dilation_angle1; plastic_viscosity = _plastic_viscosity;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(3*(dim-1)+1);
K = 0.0;
DenseMatrix esig(2);
DenseMatrix esig_old(2);
DenseMatrix esig_inc(2);
DenseMatrix plastic_sig(2);
DenseMatrix plastic_str(2);
esig=0.0; plastic_sig=0.0; plastic_str=0.0;
double eig_sig_var[2], eig_sig_vec[4];
double sig1{0.0};
double sig3{0.0};
double syy{0.0}; // Syy is non-zero value in plane strain condition
double syy_old{0.0}; // Syy is non-zero value in plane strain condition
double msig{0.0}; // mean stress
double evol{0.0}; // volumetric strain
double DEG2RAD{M_PI/180.0};
double depls{0.0}; // 2nd invariant of plastic strain
double fs{0.0};
double ft{0.0};
double fh{0.0};
double N_phi{0.0};
double st_N_phi{0.0};
double N_psi{0.0};
double beta{0.0};
double viscosity{0.0};
double relax{0.0};
// double relax_limit{1.0};
double dt_scaled{0.0};
double numerator = {0.0};
double denominator = {0.0};
double pls_old = {0.0}; // cumulative 2nd invariant of plastic strain
double p_slope = {0.0};
double fri_str = {0.0}; // strain_dependent friction angle
double dil_str = {0.0}; // strain_dependent dilation angle
double coh_str = {0.0}; // strain_dependent cohesion
double ten_cut = {0.0};
int mat{0};
int nsize{mat_gf.Size()};
int mat_num{lambda.Size()};
double pls0_c = {0.0};
double pls1_c = {0.0};
double rho_c = {0.0};
double lambda_c = {0.0};
double mu_c = {0.0};
double tension_cutoff_c = {0.0};
double cohesion0_c = {0.0};
double cohesion1_c = {0.0};
double friction_angle0_c ={0.0};
double friction_angle1_c ={0.0};
double dilation_angle0_c ={0.0};
double dilation_angle1_c ={0.0};
double plastic_viscosity_c ={0.0};
double pwave_speed ={0.0};
double time_scale ={1.0};
// element-wise loop
esig=0.0; plastic_sig=0.0; plastic_str=0.0;
esig_old=0.0; esig_inc=0.0;
pls_old = p_gf.GetValue(T, ip);
if(pls_old < 0.0){pls_old =0.0;}
pls0_c =0.0; pls1_c =0.0; rho_c = 0.0; lambda_c = 0.0; mu_c = 0.0; time_scale = 1.0;
tension_cutoff_c = 0.0; cohesion0_c = 0.0; cohesion1_c = 0.0; friction_angle0_c = 0.0; friction_angle1_c = 0.0;
dilation_angle0_c = 0.0; dilation_angle1_c = 0.0; plastic_viscosity_c = 0.0;
for( int i = 0; i < mat_num; i++ )
{
pls0_c = pls0_c + comp_gf.GetValue(T, ip, i+1)*pls0[i];
pls1_c = pls1_c + comp_gf.GetValue(T, ip, i+1)*pls1[i];
rho_c = rho_c + comp_gf.GetValue(T, ip, i+1)*rho[i];
lambda_c = lambda_c + comp_gf.GetValue(T, ip, i+1)*lambda[i];
mu_c = mu_c + comp_gf.GetValue(T, ip, i+1)*mu[i];
tension_cutoff_c = tension_cutoff_c + comp_gf.GetValue(T, ip, i+1)*tension_cutoff[i];
cohesion0_c = cohesion0_c + comp_gf.GetValue(T, ip, i+1)*cohesion0[i];
cohesion1_c = cohesion1_c + comp_gf.GetValue(T, ip, i+1)*cohesion1[i];
friction_angle0_c = friction_angle0_c + comp_gf.GetValue(T, ip, i+1)*friction_angle0[i];
friction_angle1_c = friction_angle1_c + comp_gf.GetValue(T, ip, i+1)*friction_angle1[i];
dilation_angle0_c = dilation_angle0_c + comp_gf.GetValue(T, ip, i+1)*dilation_angle0[i];
dilation_angle1_c = dilation_angle1_c + comp_gf.GetValue(T, ip, i+1)*dilation_angle1[i];
}
// linear weakening
p_slope = (pls_old - pls0_c)/(pls1_c - pls0_c);
pwave_speed = sqrt((lambda_c + 2*mu_c)/rho_c);
if(h_min > 0){time_scale = h_min / pwave_speed;}
plastic_viscosity_c = time_scale * mu_c;
msig = (s_gf.GetValue(T, ip, 1) + s_gf.GetValue(T, ip, 2))*0.5;
evol = msig/(lambda_c+mu_c);
syy = evol * lambda_c;
esig(0,0) = s_gf.GetValue(T, ip, 1); esig(0,1) = s_gf.GetValue(T, ip, 3);
esig(1,0) = s_gf.GetValue(T, ip, 3); esig(1,1) = s_gf.GetValue(T, ip, 2);
// std::cout << mat_gf.GetValue(T, ip) << " " << s_gf.GetValue(T, ip, 1) << " " << s_gf.GetValue(T, ip, 2) << " " << s_gf.GetValue(T, ip, 3) << std::endl;
// Caushy stress at previous time step
msig = (s_old_gf.GetValue(T, ip, 1) + s_old_gf.GetValue(T, ip, 2))*0.5;
evol = msig/(lambda_c+mu_c);
syy_old = evol * lambda_c;
esig_old(0,0) = s_old_gf.GetValue(T, ip, 1); esig_old(0,1) = s_old_gf.GetValue(T, ip, 3);
esig_old(1,0) = s_old_gf.GetValue(T, ip, 3); esig_old(1,1) = s_old_gf.GetValue(T, ip, 2);
// Elastic stress increment
esig_inc(0,0) = esig(0,0) - esig_old(0,0); esig_inc(0,1) = esig(0,1) - esig_old(0,1);
esig_inc(1,0) = esig(1,0) - esig_old(1,0); esig_inc(1,1) = esig(1,1) - esig_old(1,1);
esig.CalcEigenvalues(eig_sig_var, eig_sig_vec);
Vector sig_var(eig_sig_var, 2);
Vector sig_dir(eig_sig_vec, 2);
auto max_it = std::max_element(sig_var.begin(), sig_var.end()); // find iterator to max element
auto min_it = std::min_element(sig_var.begin(), sig_var.end()); // find iterator to min element
int max_index = std::distance(sig_var.begin(), max_it); // calculate index of max element
int min_index = std::distance(sig_var.begin(), min_it); // calculate index of min element
sig1 = sig_var[min_index]; // most compressive pincipal stress
sig3 = sig_var[max_index]; // least compressive pincipal stress
// linear strain weaking on cohesion, friction and dilation angles.
coh_str = cohesion0_c; fri_str = friction_angle0_c; dil_str = dilation_angle0_c;
if (pls_old < pls0_c) {
// no weakening yet
coh_str = cohesion0_c;
fri_str = friction_angle0_c;
dil_str = dilation_angle0_c;
}
else if (pls_old < pls1_c) {
// linear weakening
coh_str = cohesion0_c + p_slope * (cohesion1_c - cohesion0_c);
fri_str = friction_angle0_c + p_slope * (friction_angle1_c - friction_angle0_c);
dil_str = dilation_angle0_c + p_slope * (dilation_angle1_c - dilation_angle0_c);
}
else {
// saturated weakening
coh_str = cohesion1_c; fri_str = friction_angle1_c; dil_str = dilation_angle1_c;
}
// std::cout << coh_str <<"," << p_slope << "," << fri_str << std::endl;
N_phi = (1+sin(DEG2RAD*fri_str))/(1-sin(DEG2RAD*fri_str));
st_N_phi = cos(DEG2RAD*fri_str)/(1-sin(DEG2RAD*fri_str));
N_psi = -1*(1+sin(DEG2RAD*dil_str))/(1-sin(DEG2RAD*dil_str)); // partial_g/partial_sig3
if(tension_cutoff_c == 0)
{
ten_cut = coh_str/tan(DEG2RAD*fri_str);
}
else{ten_cut = tension_cutoff_c;}
// shear failure function
fs = sig1 - N_phi*sig3 + 2*coh_str*st_N_phi;
// tension failure function
ft = sig3 - ten_cut;
// bisects the obtuse angle made by two yield function
fh = sig3 - ten_cut + (sqrt(N_phi*N_phi + 1.0)+ N_phi)*(sig1 - N_phi*ten_cut + 2*coh_str*st_N_phi);
depls = 0.0;
if((fs < 0 & fh < 0) | (ft > 0 & fh > 0))
{
// std::cout <<"failure occurs" << std::endl;
if(fs < 0 & fh < 0) // stress correction at shear failure
{
// Equations 28 and 30 from Choi et al. (2013; DynEarthSol2D: An efficient unstructured finite element method to study long-term tectonic deformation).
beta = fs;
beta = beta / (((lambda_c+2*mu_c)*1 - N_phi*lambda_c*1) + (lambda_c*N_psi - N_phi*(lambda_c+2*mu_c)*N_psi));
plastic_str(0,0) = (lambda_c + 2*mu_c + lambda_c*N_psi) * beta;
syy -= (lambda_c + lambda_c*N_psi) * beta;
plastic_str(1,1) = (lambda_c + (lambda_c+2*mu_c)*N_psi) * beta;
// reduced form of 2nd invariant
depls = std::fabs(beta) * std::sqrt((3 - 2*N_psi + 3*N_psi*N_psi) / 8);
}
else if (ft > 0 & fh > 0) // stress correction at tension failure
{
beta = ft;
beta = beta/(lambda_c+2*mu_c);
plastic_str(0,0) = lambda_c * beta * 1;
syy -= lambda_c * beta * 1;
plastic_str(1,1) = (lambda_c+2*mu_c) * beta * 1;
// reduced form of 2nd invariant
depls = std::fabs(beta) * std::sqrt(7. / 18);
}
// Rotating Principal axis to XYZ axis
plastic_sig(0,0) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[0+min_index*2]*sig_dir[0+min_index*2] + (sig_var[max_index]-plastic_str(1,1))*sig_dir[0+max_index*2]*sig_dir[0+max_index*2]);
plastic_sig(0,1) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[0+min_index*2]*sig_dir[1+min_index*2] + (sig_var[max_index]-plastic_str(1,1))*sig_dir[0+max_index*2]*sig_dir[1+max_index*2]);
plastic_sig(1,0) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[1+min_index*2]*sig_dir[0+min_index*2] + (sig_var[max_index]-plastic_str(1,1))*sig_dir[1+max_index*2]*sig_dir[0+max_index*2]);
plastic_sig(1,1) = ((sig_var[min_index]-plastic_str(0,0))*sig_dir[1+min_index*2]*sig_dir[1+min_index*2] + (sig_var[max_index]-plastic_str(1,1))*sig_dir[1+max_index*2]*sig_dir[1+max_index*2]);
// Updating new stress to grid function
viscosity = plastic_viscosity_c;
relax = exp(-dt_old/viscosity);
numerator = fabs(1 - relax);
denominator= fabs(dt_old/viscosity);
dt_scaled = dt_old/viscosity;
// Based on L'Hôpital's rule, 0/0 become 1
// if((numerator > 1e-15) & (denominator > 1e-15)){relax_limit = (1 - relax)/(dt_old/viscosity);}
if(viscoplastic)
{
// Implicit backward Euler algorithm
K(0)=((esig_old(0,0) + esig_inc(0,0)) + dt_scaled*plastic_sig(0,0))/(1.0+dt_scaled);
K(1)=((esig_old(1,1) + esig_inc(1,1)) + dt_scaled*plastic_sig(1,1))/(1.0+dt_scaled);
K(2)=((esig_old(0,1) + esig_inc(0,1)) + dt_scaled*plastic_sig(0,1))/(1.0+dt_scaled);
depls = dt_scaled*depls/(1.0+dt_scaled);
K(3)=depls;
}
else
{
K(0)=plastic_sig(0,0);
K(1)=plastic_sig(1,1);
K(2)=plastic_sig(0,1);
K(3)=depls;
}
}
else
{
K(0)=esig(0,0);
K(1)=esig(1,1);
K(2)=esig(0,1);
K(3)=0.0;
}
}
virtual ~ReturnMapping2DCoefficient() { }
};
class ReturnMapping2D_simple_Coefficient : public VectorCoefficient
{
private:
ParGridFunction &comp_gf, &s_gf, &s_old_gf, &p_gf, &mat_gf;
int dim;
double h_min, dt_old;
bool viscoplastic;
Vector rho, lambda, mu, tension_cutoff, cohesion0, cohesion1, pls0, pls1, friction_angle0, friction_angle1, dilation_angle0, dilation_angle1, plastic_viscosity;
public:
ReturnMapping2D_simple_Coefficient (int &_dim, double &_h_min, double &_dt_old, bool &_viscoplastic, ParGridFunction &_comp_gf, ParGridFunction &_s_gf, ParGridFunction &_s_old_gf, ParGridFunction &_p_gf, ParGridFunction &_mat_gf, Vector &_rho, Vector &_lambda, Vector &_mu, Vector &_tension_cutoff, Vector &_cohesion0, Vector &_cohesion1, Vector &_pls0, Vector &_pls1, Vector &_friction_angle0, Vector &_friction_angle1, Vector &_dilation_angle0, Vector &_dilation_angle1, Vector &_plastic_viscosity)
: VectorCoefficient(_dim), comp_gf(_comp_gf), s_gf(_s_gf), s_old_gf(_s_old_gf), p_gf(_p_gf), mat_gf(_mat_gf)
{
dim = _dim; h_min = _h_min; dt_old = _dt_old; viscoplastic = _viscoplastic;
rho = _rho; lambda = _lambda; mu = _mu; tension_cutoff = _tension_cutoff; cohesion0 = _cohesion0; cohesion1 = _cohesion1; pls0 = _pls0; pls1 = _pls1; friction_angle0 = _friction_angle0; friction_angle1 = _friction_angle1; dilation_angle0 = _dilation_angle0; dilation_angle1 = _dilation_angle1; plastic_viscosity = _plastic_viscosity;
}
virtual void Eval(Vector &K, ElementTransformation &T, const IntegrationPoint &ip)
{
K.SetSize(3*(dim-1)+1);
K = 0.0;
DenseMatrix esig(2);
DenseMatrix esig_old(2);
DenseMatrix esig_inc(2);
DenseMatrix plastic_sig(2);
DenseMatrix plastic_str(2);
esig=0.0; plastic_sig=0.0; plastic_str=0.0;
double eig_sig_var[2], eig_sig_vec[4];
double sig1{0.0};
double sig3{0.0};
double syy{0.0}; // Syy is non-zero value in plane strain condition
double syy_old{0.0}; // Syy is non-zero value in plane strain condition
double msig{0.0}; // mean stress
double evol{0.0}; // volumetric strain
double DEG2RAD{M_PI/180.0};