const Matrix &
SSPquadUP::getDamp(void)
{
Matrix dampC(8,8);
// solid phase stiffness matrix
GetSolidStiffness();
// contribution of stiffness matrix for Rayleigh damping
if (betaK != 0.0) {
dampC.addMatrix(1.0, mSolidK, betaK);
} if (betaK0 != 0.0) {
dampC.addMatrix(1.0, mSolidK, betaK0);
} if (betaKc != 0.0) {
dampC.addMatrix(1.0, mSolidK, betaKc);
}
// contribution of mass matrix for Rayleigh damping
if (alphaM != 0.0) {
dampC.addMatrix(1.0, mSolidM, alphaM);
}
// assemble full element damping matrix [ C -Q ]
// comprised of C, Q, and H submatrices [ -Q' -H ]
mDamp.Zero();
for (int i = 0; i < 4; i++) {
int I = 2*i;
int Ip1 = 2*i+1;
int II = 3*i;
int IIp1 = 3*i+1;
int IIp2 = 3*i+2;
for (int j = 0; j < 4; j++) {
int J = 2*j;
int Jp1 = 2*j+1;
int JJ = 3*j;
int JJp1 = 3*j+1;
int JJp2 = 3*j+2;
// contribution of solid phase damping matrix
mDamp(II,JJ) = dampC(I,J);
mDamp(IIp1,JJ) = dampC(Ip1,J);
mDamp(IIp1,JJp1) = dampC(Ip1,Jp1);
mDamp(II,JJp1) = dampC(I,Jp1);
// contribution of solid-fluid coupling matrix
mDamp(JJp2,II) = -J0*mThickness*Mmem(0,I);
mDamp(JJp2,IIp1) = -J0*mThickness*Mmem(1,Ip1);
mDamp(II,JJp2) = -J0*mThickness*Mmem(0,I);
mDamp(IIp1,JJp2) = -J0*mThickness*Mmem(1,Ip1);
// contribution of permeability matrix
mDamp(IIp2,JJp2) = -mPerm(i,j);
}
}
return mDamp;
}
/* B := inv(A), both A and B are real matrices of rank x rank */
void Minv (int rank, double A[], double B[])
{
int info, *ipiv;
double *a, *work;
a = Mmem(rank*rank);
work = Vmem(rank);
ipiv = VImem(rank);
memcpy (a, A, (long)rank*rank*sizeof(double));
FORTRAN_SYMBOL(dgetrf) (&rank, &rank, a, &rank, ipiv, &info);
if ( info != 0 )
{
printf ("error: Minv: matrix of rank %d is singular.\n",
rank);
mump(rank,A);
exit(1);
}
FORTRAN_SYMBOL(dgetri) (&rank, a, &rank, ipiv, work, &rank, &info);
memcpy (B, a, (long)rank*rank*sizeof(double));
VFREE (ipiv);
Vfree (work);
Mfree (a);
return;
} /* end Minv() */
void
SSPquad::GetStab(void)
// this function computes the stabilization stiffness matrix for the element
{
Vector g1(SSPQ_NUM_DIM);
Vector g2(SSPQ_NUM_DIM);
Matrix I(SSPQ_NUM_DIM,SSPQ_NUM_DIM);
Matrix FCF(SSPQ_NUM_DIM,SSPQ_NUM_DIM);
Matrix Jmat(SSPQ_NUM_DIM,SSPQ_NUM_DIM);
Matrix Jinv(SSPQ_NUM_DIM,SSPQ_NUM_DIM);
Matrix dNloc(SSPQ_NUM_NODE,SSPQ_NUM_DIM);
Matrix dN(SSPQ_NUM_NODE,SSPQ_NUM_DIM);
Matrix Mben(2,SSPQ_NUM_DOF);
double Hss;
double Hst;
double Htt;
// shape function derivatives (local crd) at center
dNloc(0,0) = -0.25;
dNloc(1,0) = 0.25;
dNloc(2,0) = 0.25;
dNloc(3,0) = -0.25;
dNloc(0,1) = -0.25;
dNloc(1,1) = -0.25;
dNloc(2,1) = 0.25;
dNloc(3,1) = 0.25;
// jacobian matrix
Jmat = mNodeCrd*dNloc;
// inverse of the jacobian matrix
Jmat.Invert(Jinv);
// shape function derivatives (global crd)
dN = dNloc*Jinv;
// define hourglass stabilization vector gamma = 0.25*(h - (h^x)*bx - (h^y)*by);
double hx = mNodeCrd(0,0) - mNodeCrd(0,1) + mNodeCrd(0,2) - mNodeCrd(0,3);
double hy = mNodeCrd(1,0) - mNodeCrd(1,1) + mNodeCrd(1,2) - mNodeCrd(1,3);
double gamma[4];
gamma[0] = 0.25*( 1.0 - hx*dN(0,0) - hy*dN(0,1));
gamma[1] = 0.25*(-1.0 - hx*dN(1,0) - hy*dN(1,1));
gamma[2] = 0.25*( 1.0 - hx*dN(2,0) - hy*dN(2,1));
gamma[3] = 0.25*(-1.0 - hx*dN(3,0) - hy*dN(3,1));
// define mapping matrices
Mmem.Zero();
Mben.Zero();
for (int i = 0; i < 4; i++) {
Mmem(0,2*i) = dN(i,0);
Mmem(1,2*i+1) = dN(i,1);
Mmem(2,2*i) = dN(i,1);
Mmem(2,2*i+1) = dN(i,0);
Mben(0,2*i) = gamma[i];
Mben(1,2*i+1) = gamma[i];
}
// base vectors
g1(0) = Jmat(0,0);
g1(1) = Jmat(1,0);
g2(0) = Jmat(0,1);
g2(1) = Jmat(1,1);
// normalize base vectors
g1.Normalize();
g2.Normalize();
// compute second moment of area tensor
double fourThree = 4.0/3.0;
I = fourThree*mThickness*J0*(DyadicProd(g1,g1) + DyadicProd(g2,g2));
// stabilization terms
Hss = (I(0,0)*Jinv(1,0)*Jinv(1,0) + I(0,1)*Jinv(0,0)*Jinv(1,0) + I(1,1)*Jinv(0,0)*Jinv(0,0))*0.25;
Htt = (I(0,0)*Jinv(1,1)*Jinv(1,1) + I(0,1)*Jinv(0,1)*Jinv(1,1) + I(1,1)*Jinv(0,1)*Jinv(0,1))*0.25;
Hst = (I(0,0)*Jinv(1,1)*Jinv(1,0) + I(0,1)*(Jinv(1,0)*Jinv(0,1) + Jinv(1,1)*Jinv(0,0)) + I(1,1)*Jinv(0,1)*Jinv(0,0))*0.25;
// get material tangent
const Matrix &CmatI = theMaterial->getInitialTangent();
// compute stabilization matrix
FCF(0,0) = (CmatI(0,0) - (CmatI(0,1) + CmatI(1,0)) + CmatI(1,1))*Hss;
FCF(0,1) = (CmatI(0,1) - (CmatI(0,0) + CmatI(1,1)) + CmatI(1,0))*Hst;
FCF(1,0) = (CmatI(1,0) - (CmatI(0,0) + CmatI(1,1)) + CmatI(0,1))*Hst;
FCF(1,1) = (CmatI(1,1) - (CmatI(0,1) + CmatI(1,0)) + CmatI(0,0))*Htt;
// compute stiffness matrix for stabilization terms
Kstab.Zero();
Kstab.addMatrixTripleProduct(1.0, Mben, FCF, 1.0);
return;
}
