double Tri31::shapeFunction(double xi, double eta)
{
const Vector &nd1Crds = theNodes[0]->getCrds();
const Vector &nd2Crds = theNodes[1]->getCrds();
const Vector &nd3Crds = theNodes[2]->getCrds();
shp[2][0] = xi; // N_1
shp[2][1] = eta; // N_2
shp[2][2] = 1-xi-eta; // N_3
double J[2][2];
// See p 180 "A First Course in Finite Elements" by Fish and Belytschko.
J[0][0] = (nd1Crds(0) - nd3Crds(0));
J[0][1] = (nd2Crds(0) - nd3Crds(0));
J[1][0] = (nd1Crds(1) - nd3Crds(1));
J[1][1] = (nd2Crds(1) - nd3Crds(1));
double detJ = J[0][0]*J[1][1] - J[0][1]*J[1][0];
double oneOverdetJ = 1.0/detJ;
double L[2][2];
// L = inv(J)
L[0][0] = J[1][1]*oneOverdetJ;
L[1][0] = -J[0][1]*oneOverdetJ;
L[0][1] = -J[1][0]*oneOverdetJ;
L[1][1] = J[0][0]*oneOverdetJ;
// See Cook, Malkus, Plesha p. 169 for the derivation of these terms
shp[0][0] = L[0][0]; // N_1,1
shp[0][1] = L[0][1]; // N_2,1
shp[0][2] = -(L[0][0] + L[0][1]); // N_3,1
shp[1][0] = L[1][0]; // N_1,2
shp[1][1] = L[1][1]; // N_2,2
shp[1][2] = -(L[1][0] + L[1][1]); // N_3,2
return detJ;
}
double FourNodeQuadWithSensitivity::shapeFunction(double xi, double eta)
{
const Vector &nd1Crds = theNodes[0]->getCrds();
const Vector &nd2Crds = theNodes[1]->getCrds();
const Vector &nd3Crds = theNodes[2]->getCrds();
const Vector &nd4Crds = theNodes[3]->getCrds();
double oneMinuseta = 1.0-eta;
double onePluseta = 1.0+eta;
double oneMinusxi = 1.0-xi;
double onePlusxi = 1.0+xi;
shp[2][0] = 0.25*oneMinusxi*oneMinuseta; // N_1
shp[2][1] = 0.25*onePlusxi*oneMinuseta; // N_2
shp[2][2] = 0.25*onePlusxi*onePluseta; // N_3
shp[2][3] = 0.25*oneMinusxi*onePluseta; // N_4
double J[2][2];
J[0][0] = 0.25 * (-nd1Crds(0)*oneMinuseta + nd2Crds(0)*oneMinuseta +
nd3Crds(0)*(onePluseta) - nd4Crds(0)*(onePluseta));
J[0][1] = 0.25 * (-nd1Crds(0)*oneMinusxi - nd2Crds(0)*onePlusxi +
nd3Crds(0)*onePlusxi + nd4Crds(0)*oneMinusxi);
J[1][0] = 0.25 * (-nd1Crds(1)*oneMinuseta + nd2Crds(1)*oneMinuseta +
nd3Crds(1)*onePluseta - nd4Crds(1)*onePluseta);
J[1][1] = 0.25 * (-nd1Crds(1)*oneMinusxi - nd2Crds(1)*onePlusxi +
nd3Crds(1)*onePlusxi + nd4Crds(1)*oneMinusxi);
double detJ = J[0][0]*J[1][1] - J[0][1]*J[1][0];
double oneOverdetJ = 1.0/detJ;
double L[2][2];
// L = inv(J)
L[0][0] = J[1][1]*oneOverdetJ;
L[1][0] = -J[0][1]*oneOverdetJ;
L[0][1] = -J[1][0]*oneOverdetJ;
L[1][1] = J[0][0]*oneOverdetJ;
double L00 = 0.25*L[0][0];
double L10 = 0.25*L[1][0];
double L01 = 0.25*L[0][1];
double L11 = 0.25*L[1][1];
double L00oneMinuseta = L00*oneMinuseta;
double L00onePluseta = L00*onePluseta;
double L01oneMinusxi = L01*oneMinusxi;
double L01onePlusxi = L01*onePlusxi;
double L10oneMinuseta = L10*oneMinuseta;
double L10onePluseta = L10*onePluseta;
double L11oneMinusxi = L11*oneMinusxi;
double L11onePlusxi = L11*onePlusxi;
// See Cook, Malkus, Plesha p. 169 for the derivation of these terms
shp[0][0] = -L00oneMinuseta - L01oneMinusxi; // N_1,1
shp[0][1] = L00oneMinuseta - L01onePlusxi; // N_2,1
shp[0][2] = L00onePluseta + L01onePlusxi; // N_3,1
shp[0][3] = -L00onePluseta + L01oneMinusxi; // N_4,1
shp[1][0] = -L10oneMinuseta - L11oneMinusxi; // N_1,2
shp[1][1] = L10oneMinuseta - L11onePlusxi; // N_2,2
shp[1][2] = L10onePluseta + L11onePlusxi; // N_3,2
shp[1][3] = -L10onePluseta + L11oneMinusxi; // N_4,2
return detJ;
}
void FourNodeQuadUP::shapeFunction(void)
{
double xi, eta, oneMinuseta, onePluseta, oneMinusxi, onePlusxi,
detJ, oneOverdetJ, J[2][2], L[2][2], L00, L01, L10, L11,
L00oneMinuseta, L00onePluseta, L01oneMinusxi, L01onePlusxi,
L10oneMinuseta, L10onePluseta, L11oneMinusxi, L11onePlusxi,
vol = 0.0;
int k, l;
for (k=0; k<3; k++) {
for (l=0; l<4; l++) {
shpBar[k][l] = 0.0;
}
}
// loop over integration points
for (int i=0; i<4; i++) {
xi = pts[i][0];
eta = pts[i][1];
const Vector &nd1Crds = nd1Ptr->getCrds();
const Vector &nd2Crds = nd2Ptr->getCrds();
const Vector &nd3Crds = nd3Ptr->getCrds();
const Vector &nd4Crds = nd4Ptr->getCrds();
oneMinuseta = 1.0-eta;
onePluseta = 1.0+eta;
oneMinusxi = 1.0-xi;
onePlusxi = 1.0+xi;
shp[2][0][i] = 0.25*oneMinusxi*oneMinuseta; // N_1
shp[2][1][i] = 0.25*onePlusxi*oneMinuseta; // N_2
shp[2][2][i] = 0.25*onePlusxi*onePluseta; // N_3
shp[2][3][i] = 0.25*oneMinusxi*onePluseta; // N_4
J[0][0] = 0.25 * (-nd1Crds(0)*oneMinuseta + nd2Crds(0)*oneMinuseta +
nd3Crds(0)*(onePluseta) - nd4Crds(0)*(onePluseta));
J[0][1] = 0.25 * (-nd1Crds(0)*oneMinusxi - nd2Crds(0)*onePlusxi +
nd3Crds(0)*onePlusxi + nd4Crds(0)*oneMinusxi);
J[1][0] = 0.25 * (-nd1Crds(1)*oneMinuseta + nd2Crds(1)*oneMinuseta +
nd3Crds(1)*onePluseta - nd4Crds(1)*onePluseta);
J[1][1] = 0.25 * (-nd1Crds(1)*oneMinusxi - nd2Crds(1)*onePlusxi +
nd3Crds(1)*onePlusxi + nd4Crds(1)*oneMinusxi);
detJ = J[0][0]*J[1][1] - J[0][1]*J[1][0];
oneOverdetJ = 1.0/detJ;
// L = inv(J)
L[0][0] = J[1][1]*oneOverdetJ;
L[1][0] = -J[0][1]*oneOverdetJ;
L[0][1] = -J[1][0]*oneOverdetJ;
L[1][1] = J[0][0]*oneOverdetJ;
L00 = 0.25*L[0][0];
L10 = 0.25*L[1][0];
L01 = 0.25*L[0][1];
L11 = 0.25*L[1][1];
L00oneMinuseta = L00*oneMinuseta;
L00onePluseta = L00*onePluseta;
L01oneMinusxi = L01*oneMinusxi;
L01onePlusxi = L01*onePlusxi;
L10oneMinuseta = L10*oneMinuseta;
L10onePluseta = L10*onePluseta;
L11oneMinusxi = L11*oneMinusxi;
L11onePlusxi = L11*onePlusxi;
// B: See Cook, Malkus, Plesha p. 169 for the derivation of these terms
shp[0][0][i] = -L00oneMinuseta - L01oneMinusxi; // N_1,1
shp[0][1][i] = L00oneMinuseta - L01onePlusxi; // N_2,1
shp[0][2][i] = L00onePluseta + L01onePlusxi; // N_3,1
shp[0][3][i] = -L00onePluseta + L01oneMinusxi; // N_4,1
shp[1][0][i] = -L10oneMinuseta - L11oneMinusxi; // N_1,2
shp[1][1][i] = L10oneMinuseta - L11onePlusxi; // N_2,2
shp[1][2][i] = L10onePluseta + L11onePlusxi; // N_3,2
shp[1][3][i] = -L10onePluseta + L11oneMinusxi; // N_4,2
dvol[i] = detJ * thickness * wts[i];
vol += dvol[i];
for (k=0; k<3; k++) {
for (l=0; l<4; l++) {
shpBar[k][l] += shp[k][l][i] * dvol[i];
}
}
}
for (k=0; k<3; k++) {
for (l=0; l<4; l++) {
shpBar[k][l] /= vol;
}
}
}
