complex_t BiquadBase::response (double normalizedFrequency) const
{
const double a0 = getA0 ();
const double a1 = getA1 ();
const double a2 = getA2 ();
const double b0 = getB0 ();
const double b1 = getB1 ();
const double b2 = getB2 ();
const double w = 2 * doublePi * normalizedFrequency;
const complex_t czn1 = std::polar (1., -w);
const complex_t czn2 = std::polar (1., -2 * w);
complex_t ch (1);
complex_t cbot (1);
complex_t ct (b0/a0);
complex_t cb (1);
ct = addmul (ct, b1/a0, czn1);
ct = addmul (ct, b2/a0, czn2);
cb = addmul (cb, a1/a0, czn1);
cb = addmul (cb, a2/a0, czn2);
ch *= ct;
cbot *= cb;
return ch / cbot;
}
//================================================================================
const tensor& DM04_PF::Dm_Dkin(const stresstensor &Stre,
const straintensor &Stra,
const MaterialParameter &MaterialParameter_in) const
{
const double oneOver3 = 1.0/3.0;
const double rt23 = sqrt(2.0/3.0);
tensor I2("I", 2, def_dim_2);
tensor I4 = I2("ij")*I2("kl");
tensor I4s = ( I4.transpose0110() + I4.transpose0111() ) *0.5;
double e0 = gete0(MaterialParameter_in);
double e_r = gete_r(MaterialParameter_in);
double lambda_c = getlambda_c(MaterialParameter_in);
double xi = getxi(MaterialParameter_in);
double Pat = getPat(MaterialParameter_in);
//double m = getm(MaterialParameter_in);
double M_cal = getM_cal(MaterialParameter_in);
double cc = getcc(MaterialParameter_in);
double A0 = getA0(MaterialParameter_in);
double nd = getnd(MaterialParameter_in);
stresstensor alpha = getalpha(MaterialParameter_in);
stresstensor z = getz(MaterialParameter_in);
stresstensor n;
stresstensor alpha_d;
stresstensor alpha_d_alpha;
double g = 0.0;
double ec = e_r;
double stateParameter = 0.0;
double expnd = 1.0;
//double ad = 0.0;
double A_d = 0.0;
double B = 1.0;
double C = 0.0;
double D0 = 0.0;
stresstensor s_bar;
double norm_s = 0.0;
double epsilon_v = 0.0;
double e = e0;
double J3D;
double cos3theta = 0.0;
double z_n = 0.0;
double alpha_n = 0.0;
double s_n = 0.0;
double p = Stre.p_hydrostatic();
stresstensor s = Stre.deviator();
s_bar = s - (alpha *p);
norm_s = sqrt( (s_bar("ij")*s_bar("ij")).trace() );
if (p > 0.0 && norm_s > 0.0)
n = s_bar * (1.0/norm_s);
J3D = n.Jinvariant3();
cos3theta = -3.0*sqrt(6.0) *J3D;
if (cos3theta > 1.0)
cos3theta = 1.0;
if (cos3theta < -1.0)
cos3theta = -1.0;
g = getg(cc, cos3theta);
if ( (p/Pat) >= 0.0 )
ec = getec(e_r, lambda_c, xi, Pat, p);
epsilon_v = Stra.Iinvariant1();
e = e0 + (1.0 + e0) *epsilon_v;
stateParameter = e - ec;
expnd = exp(nd*stateParameter);
alpha_n = (alpha("ij")*n("ij")).trace();
s_n = (s("ij")*n("ij")).trace();
// way 1
//ad = g*M_cal*expnd - m;
//D0 = rt23 * ad - alpha_n;
// way 2
D0 = rt23*g*M_cal*expnd - s_n /p;
z_n = (z("ij")*n("ij")).trace();
if (z_n < 0.0)
z_n = 0.0;
A_d = A0 * (1.0 + z_n);
B = 1.0 + 1.5 *((1.0-cc)/cc) *g *cos3theta;
C = 3.0 *sqrt(1.5) *((1.0-cc)/cc) *g;
tensor n_n = n("ik")*n("kj");
n_n.null_indices();
tensor nt_nt = n("ij")*n("kl");
nt_nt.null_indices();
tensor alpha_I = alpha("ij")*I2("kl");
alpha_I.null_indices();
tensor n_I = n("ij")*I2("kl");
n_I.null_indices();
// dn_dalpha:
tensor dn_da = nt_nt - I4s;
dn_da = dn_da *(p/norm_s);
// dcos3theta_dalpha:
tensor dcos3theta_da = dn_da("ijmn")*n_n("ji");
dcos3theta_da.null_indices();
dcos3theta_da = dcos3theta_da *(-3.0*sqrt(6.0));
// dg_da:
tensor dg_da = dcos3theta_da *(g*g*(1.0-cc)/(2.0*cc));
// dB_da:
tensor dB_da = (dg_da*cos3theta + dcos3theta_da*g) *(1.5*(1.0-cc)/cc);
// dC_ds:
tensor dC_da = dg_da *(3.0*sqrt(1.5)*(1.0-cc)/cc);
// dR_da:
tensor tensor1 = n("ij")*dB_da("mn");
tensor1.null_indices();
tensor tensor2 = n_n - I2 *oneOver3;
tensor tensor3 = tensor2("ij")*dC_da("mn");
tensor3.null_indices();
tensor tensor4 = n("kj")*dn_da("ikmn");
tensor4.null_indices();
tensor4.transpose1100();
tensor dR_da = dn_da *B + tensor1 + tensor4 *(2.0*C) + tensor3;
// dad_da:
tensor dad_da = dg_da *(M_cal*expnd);
// dD_da:
// way 1
//tensor tensor5 = alpha("pq")*dn_da("pqmn");
// tensor5.null_indices();
//tensor dD_da = (dad_da *rt23 - n - tensor5) *(-A_d);
// way 2
tensor tensor5 = s("pq")*dn_da("pqmn");
tensor5.null_indices();
tensor dD_da = (dad_da *rt23 - tensor5 *(1.0/p)) *(-A_d);
if (z_n > 0.0) {
tensor tensor6 = z("pq")*dn_da("pqmn");
tensor6.null_indices();
dD_da += tensor6 *(-A0*D0);
}
// dm_da:
tensor tensor7 = I2("ij")*dD_da("mn");
tensor7.null_indices();
PlasticFlow::PF_tensorR4 = dR_da + tensor7 *oneOver3;
return PlasticFlow::PF_tensorR4;
}
//================================================================================
const tensor& DM04_PF::Dm_Dkin2(const stresstensor &Stre,
const straintensor &Stra,
const MaterialParameter &MaterialParameter_in) const
{
const double oneOver3 = 1.0/3.0;
const double rt23 = sqrt(2.0/3.0);
tensor I2("I", 2, def_dim_2);
double e0 = gete0(MaterialParameter_in);
double e_r = gete_r(MaterialParameter_in);
double lambda_c = getlambda_c(MaterialParameter_in);
double xi = getxi(MaterialParameter_in);
double Pat = getPat(MaterialParameter_in);
//double m = getm(MaterialParameter_in);
double M_cal = getM_cal(MaterialParameter_in);
double cc = getcc(MaterialParameter_in);
double A0 = getA0(MaterialParameter_in);
double nd = getnd(MaterialParameter_in);
stresstensor alpha = getalpha(MaterialParameter_in);
stresstensor z = getz(MaterialParameter_in);
stresstensor n;
stresstensor alpha_d;
stresstensor alpha_d_alpha;
double g = 0.0;
double ec = e_r;
double stateParameter = 0.0;
double expnd = 1.0;
//double ad = 0.0;
double D0 = 0.0;
stresstensor s_bar;
double norm_s = 0.0;
double epsilon_v = 0.0;
double e = e0;
double J3D;
double cos3theta = 0.0;
double z_n = 0.0;
double alpha_n = 0.0;
double s_n = 0.0;
double p = Stre.p_hydrostatic();
stresstensor s = Stre.deviator();
s_bar = s - (alpha *p);
norm_s = sqrt( (s_bar("ij")*s_bar("ij")).trace() );
if (p > 0.0 && norm_s > 0.0)
n = s_bar * (1.0/norm_s);
J3D = n.Jinvariant3();
cos3theta = -3.0*sqrt(6.0) *J3D;
if (cos3theta > 1.0)
cos3theta = 1.0;
if (cos3theta < -1.0)
cos3theta = -1.0;
g = getg(cc, cos3theta);
if ( (p/Pat) >= 0.0 )
ec = getec(e_r, lambda_c, xi, Pat, p);
epsilon_v = Stra.Iinvariant1();
e = e0 + (1.0 + e0) *epsilon_v;
stateParameter = e - ec;
expnd = exp(nd*stateParameter);
alpha_n = (alpha("ij")*n("ij")).trace();
s_n = (s("ij")*n("ij")).trace();
// way 1
//ad = g*M_cal*expnd - m;
//D0 = rt23 * ad - alpha_n;
// way 2
D0 = rt23*g*M_cal*expnd - s_n /p;
tensor dD_dz(2, def_dim_2, 0.0);
// dD_dz:
if (z_n > 0.0)
dD_dz = n *A0;
// dm_da:
tensor tensor1 = I2("ij")*dD_dz("mn");
tensor1.null_indices();
PlasticFlow::PF_tensorR4 = tensor1 *(-D0*oneOver3);
return PlasticFlow::PF_tensorR4;
}
//================================================================================
const straintensor& DM04_PF::PlasticFlowTensor(const stresstensor& Stre,
const straintensor& Stra,
const MaterialParameter &MaterialParameter_in) const
{
const double oneOver3 = 1.0/3.0;
const double rt23 = sqrt(2.0/3.0);
tensor I2("I", 2, def_dim_2);
double e0 = gete0(MaterialParameter_in);
double e_r = gete_r(MaterialParameter_in);
double lambda_c = getlambda_c(MaterialParameter_in);
double xi = getxi(MaterialParameter_in);
double Pat = getPat(MaterialParameter_in);
//double m = getm(MaterialParameter_in);
double M_cal = getM_cal(MaterialParameter_in);
double cc = getcc(MaterialParameter_in);
double A0 = getA0(MaterialParameter_in);
double nd = getnd(MaterialParameter_in);
stresstensor alpha = getalpha(MaterialParameter_in);
stresstensor z = getz(MaterialParameter_in);
stresstensor n;
stresstensor alpha_d;
stresstensor alpha_d_alpha;
double g = 0.0;
double ec = e_r;
double stateParameter = 0.0;
double expnd = 1.0;
//double ad = 0.0;
double A_d = 0.0;
double B = 1.0;
double C = 0.0;
double D = 0.0;
double D0 = 0.0;
stresstensor s_bar;
double norm_s = 0.0;
double epsilon_v = 0.0;
double e = e0;
double J3D;
double cos3theta = 0.0;
double z_n = 0.0;
double alpha_n = 0.0;
double s_n = 0.0;
double p = Stre.p_hydrostatic();
stresstensor s = Stre.deviator();
s_bar = s - (alpha *p);
norm_s = sqrt( (s_bar("ij")*s_bar("ij")).trace() );
if (p > 0.0 && norm_s > 0.0)
n = s_bar * (1.0/norm_s);
J3D = n.Jinvariant3();
cos3theta = -3.0*sqrt(6.0) *J3D;
if (p <= 0.0)
cos3theta = 1.0;
if (cos3theta > 1.0)
cos3theta = 1.0;
if (cos3theta < -1.0)
cos3theta = -1.0;
g = getg(cc, cos3theta);
if ( (p/Pat) >= 0.0 )
ec = getec(e_r, lambda_c, xi, Pat, p);
epsilon_v = Stra.Iinvariant1();
e = e0 + (1.0 + e0) *epsilon_v;
stateParameter = e - ec;
expnd = exp(nd*stateParameter);
alpha_n = (alpha("ij")*n("ij")).trace();
s_n = (s("ij")*n("ij")).trace();
// way 1
//ad = g*M_cal*expnd - m;
//D0 = rt23 *ad - alpha_n;
// way 2, better use this when "p" is small
D0 = rt23*g*M_cal*expnd - s_n /p;
z_n = (z("ij")*n("ij")).trace();
if (z_n < 0.0)
z_n = 0.0;
A_d = A0 * (1.0 + z_n);
D = D0 *(-A_d);
B = 1.0 + 1.5 *((1.0-cc)/cc) *g *cos3theta;
C = 3.0 *sqrt(1.5) *((1.0-cc)/cc) *g;
stresstensor n_n = n("ik")*n("kj");
n_n.null_indices();
// note different 'positive-negative' since we assume extension (dilation) positive
// which is different from the Ref.
DM04_PF::DM04m = n *B + n_n *C + I2 *((D-C)*oneOver3);
return DM04_PF::DM04m;
}
