void Tet21Stokes :: computeStiffnessMatrix(FloatMatrix &answer, MatResponseMode mode, TimeStep *tStep)
{
FluidDynamicMaterial *mat = static_cast< FluidCrossSection * >( this->giveCrossSection() )->giveFluidMaterial();
FloatMatrix B(6, 30), EdB, K, G, Dp, DvT, C, Ed, dN;
FloatArray dN_V(30), Nlin, Ep, Cd, tmpA, tmpB;
double Cp;
B.zero();
for ( GaussPoint *gp: *this->integrationRulesArray [ 0 ] ) {
// Compute Gauss point and determinant at current element
const FloatArray &lcoords = gp->giveNaturalCoordinates();
double detJ = fabs( this->interpolation_quad.evaldNdx( dN, lcoords, FEIElementGeometryWrapper(this) ) );
double dV = detJ * gp->giveWeight();
this->interpolation_lin.evalN( Nlin, lcoords, FEIElementGeometryWrapper(this) );
for ( int j = 0, k = 0; j < dN.giveNumberOfRows(); j++, k += 3 ) {
dN_V(k + 0) = B(0, k + 0) = B(3, k + 1) = B(4, k + 2) = dN(j, 0);
dN_V(k + 1) = B(1, k + 1) = B(3, k + 0) = B(5, k + 2) = dN(j, 1);
dN_V(k + 2) = B(2, k + 2) = B(4, k + 0) = B(5, k + 1) = dN(j, 2);
}
// Computing the internal forces should have been done first.
// dsigma_dev/deps_dev dsigma_dev/dp deps_vol/deps_dev deps_vol/dp
mat->giveStiffnessMatrices(Ed, Ep, Cd, Cp, mode, gp, tStep);
EdB.beProductOf(Ed, B);
K.plusProductSymmUpper(B, EdB, dV);
G.plusDyadUnsym(dN_V, Nlin, -dV);
C.plusDyadSymmUpper(Nlin, Cp * dV);
tmpA.beTProductOf(B, Ep);
Dp.plusDyadUnsym(tmpA, Nlin, dV);
tmpB.beTProductOf(B, Cd);
DvT.plusDyadUnsym(Nlin, tmpB, dV);
}
K.symmetrized();
C.symmetrized();
FloatMatrix GTDvT, GDp;
GTDvT.beTranspositionOf(G);
GTDvT.add(DvT);
GDp = G;
GDp.add(Dp);
answer.resize(34, 34);
answer.zero();
answer.assemble(K, this->momentum_ordering);
answer.assemble(GDp, this->momentum_ordering, this->conservation_ordering);
answer.assemble(GTDvT, this->conservation_ordering, this->momentum_ordering);
answer.assemble(C, this->conservation_ordering);
// K.printYourself();
// GDp.printYourself();
// GTDvT.printYourself();
// C.printYourself();
}
void Tr21Stokes :: computeInternalForcesVector(FloatArray &answer, TimeStep *tStep)
{
IntegrationRule *iRule = integrationRulesArray [ 0 ];
FluidDynamicMaterial *mat = ( FluidDynamicMaterial * ) this->domain->giveMaterial(this->material);
FloatArray a_pressure, a_velocity, devStress, epsp, BTs, Nh, dNv(12);
double r_vol, pressure;
FloatMatrix dN, B(3, 12);
B.zero();
this->computeVectorOf(EID_MomentumBalance, VM_Total, tStep, a_velocity);
this->computeVectorOf(EID_ConservationEquation, VM_Total, tStep, a_pressure);
FloatArray momentum(12), conservation(3);
momentum.zero();
conservation.zero();
GaussPoint *gp;
for ( int i = 0; i < iRule->getNumberOfIntegrationPoints(); i++ ) {
gp = iRule->getIntegrationPoint(i);
FloatArray *lcoords = gp->giveCoordinates();
double detJ = fabs(this->interpolation_quad.giveTransformationJacobian(* lcoords, FEIElementGeometryWrapper(this)));
this->interpolation_quad.evaldNdx(dN, * lcoords, FEIElementGeometryWrapper(this));
this->interpolation_lin.evalN(Nh, * lcoords, FEIElementGeometryWrapper(this));
double dA = detJ * gp->giveWeight();
for ( int j = 0, k = 0; j < 6; j++, k += 2 ) {
dNv(k) = B(0, k) = B(2, k + 1) = dN(j, 0);
dNv(k + 1) = B(1, k + 1) = B(2, k) = dN(j, 1);
}
pressure = Nh.dotProduct(a_pressure);
epsp.beProductOf(B, a_velocity);
mat->computeDeviatoricStressVector(devStress, r_vol, gp, epsp, pressure, tStep);
BTs.beTProductOf(B, devStress);
momentum.add(dA, BTs);
momentum.add(-pressure*dA, dNv);
conservation.add(r_vol*dA, Nh);
}
FloatArray temp(15);
temp.zero();
temp.addSubVector(momentum, 1);
temp.addSubVector(conservation, 13);
answer.resize(15);
answer.zero();
answer.assemble(temp, this->ordering);
}
const Matrix &
SSPquadUP::getMass(void)
{
mMass.Zero();
// compute compressibility matrix term
double oneOverQ = -0.25*J0*mThickness*mPorosity/fBulk;
// get mass density from the material
double density = theMaterial->getRho();
// transpose the shape function derivative array
Matrix dNp(2,4);
dNp(0,0) = dN(0,0); dNp(0,1) = dN(1,0); dNp(0,2) = dN(2,0); dNp(0,3) = dN(3,0);
dNp(1,0) = dN(0,1); dNp(1,1) = dN(1,1); dNp(1,2) = dN(2,1); dNp(1,3) = dN(3,1);
// compute stabilization matrix for incompressible problems
Matrix Kp(4,4);
Kp = -4.0*mAlpha*J0*mThickness*dN*dNp;
// return zero matrix if density is zero
if (density == 0.0) {
return mMass;
}
// full mass matrix for the element [ M 0 ]
// includes M and S submatrices [ 0 -S ]
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;
mMass(II,JJ) = mSolidM(I,J);
mMass(IIp1,JJ) = mSolidM(Ip1,J);
mMass(IIp1,JJp1) = mSolidM(Ip1,Jp1);
mMass(II,JJp1) = mSolidM(I,Jp1);
// contribution of compressibility matrix
mMass(IIp2,JJp2) = Kp(i,j) + oneOverQ;
}
}
return mMass;
}
ZSparseMatrix Assembler2D::getLaplacianMatrix() {
typedef Eigen::Triplet<double> T;
std::vector<T> triplets;
for (size_t i=0; i<m_mesh->getNbrElements(); ++i)
{
VectorI idx = m_mesh->getElement(i);
assert(idx.size() == 3);
Eigen::MatrixXd& dN = m_DN[i];
// Small strain-displacement matrix
//
Eigen::MatrixXd B(2,3);
B << dN(0,0), dN(1,0), dN(2,0),
dN(0,1), dN(1,1), dN(2,1);
Eigen::MatrixXd k_el = B.transpose() * B * m_mesh->getElementVolume(i);
for (size_t j=0; j<3; ++j)
for (size_t k=0; k<3; ++k)
triplets.push_back(T(idx[j], idx[k], k_el(j,k)));
}
Eigen::SparseMatrix<double> L = Eigen::SparseMatrix<double>(m_mesh->getNbrNodes(), m_mesh->getNbrNodes());
L.setFromTriplets(triplets.begin(), triplets.end());
return ZSparseMatrix(L);
}
void Tr21Stokes :: computeInternalForcesVector(FloatArray &answer, TimeStep *tStep)
{
FluidDynamicMaterial *mat = static_cast< FluidCrossSection * >( this->giveCrossSection() )->giveFluidMaterial();
FloatArray a_pressure, a_velocity, devStress, epsp, Nh, dNv(12);
double r_vol, pressure;
FloatMatrix dN, B(3, 12);
B.zero();
this->computeVectorOfVelocities(VM_Total, tStep, a_velocity);
this->computeVectorOfPressures(VM_Total, tStep, a_pressure);
FloatArray momentum, conservation;
for ( GaussPoint *gp: *integrationRulesArray [ 0 ] ) {
const FloatArray &lcoords = gp->giveNaturalCoordinates();
double detJ = fabs( this->interpolation_quad.evaldNdx( dN, lcoords, FEIElementGeometryWrapper(this) ) );
this->interpolation_lin.evalN( Nh, lcoords, FEIElementGeometryWrapper(this) );
double dA = detJ * gp->giveWeight();
for ( int j = 0, k = 0; j < dN.giveNumberOfRows(); j++, k += 2 ) {
dNv(k) = B(0, k) = B(2, k + 1) = dN(j, 0);
dNv(k + 1) = B(1, k + 1) = B(2, k) = dN(j, 1);
}
pressure = Nh.dotProduct(a_pressure);
epsp.beProductOf(B, a_velocity);
mat->computeDeviatoricStressVector(devStress, r_vol, gp, epsp, pressure, tStep);
momentum.plusProduct(B, devStress, dA);
momentum.add(-pressure * dA, dNv);
conservation.add(r_vol * dA, Nh);
}
answer.resize(15);
answer.zero();
answer.assemble(momentum, this->momentum_ordering);
answer.assemble(conservation, this->conservation_ordering);
}
Matrix QuadTriSF::dNds(point pt){
Matrix dN(6,2);
double r = pt.x;
double s = pt.y;
dN[0][0]=4.*r-1.;
dN[0][1]=0.;
dN[1][0]=0.;
dN[1][1]=4.*s-1.;
dN[2][0]=-3.+4.*s+4.*r;
dN[2][1]=-3.+4.*r+4.*s;
dN[3][0]=4.*s;
dN[3][1]=4.*r;
dN[4][0]=-4.*s;
dN[4][1]=4.-4.*r-8.*s;
dN[5][0]=4.-4.*s-8.*r;
dN[5][1]=-4.*r;
return dN;
};
void
SSPquadUP::GetPermeabilityMatrix(void)
// this function computes the permeability matrix for the element
{
mPerm.Zero();
Matrix k(2,2);
Matrix dNp(2,4);
// permeability tensor
k(0,0) = perm[0];
k(1,1) = perm[1];
// transpose the shape function derivative array
dNp(0,0) = dN(0,0); dNp(0,1) = dN(1,0); dNp(0,2) = dN(2,0); dNp(0,3) = dN(3,0);
dNp(1,0) = dN(0,1); dNp(1,1) = dN(1,1); dNp(1,2) = dN(2,1); dNp(1,3) = dN(3,1);
// compute permeability matrix
mPerm.addMatrixTripleProduct(1.0, dNp, k, 4.0*J0*mThickness);
return;
}
ZSparseMatrix Assembler2D::getStiffnessMatrix() {
// Elastic modulii
//
Eigen::MatrixXd& D = m_D;
Eigen::MatrixXd& C = m_C;
typedef Eigen::Triplet<double> T;
std::vector<T> triplets;
for (size_t i=0; i<m_mesh->getNbrElements(); ++i)
{
VectorI idx = m_mesh->getElement(i);
assert(idx.size() == 3);
Eigen::MatrixXd& dN = m_DN[i];
// Small strain-displacement matrix
//
Eigen::MatrixXd B(3,6);
B << dN(0,0), 0.0,dN(1,0), 0.0,dN(2,0), 0.0,
0.0 ,dN(0,1), 0.0,dN(1,1), 0.0,dN(2,1),
0.5*dN(0,1),0.5*dN(0,0),
0.5*dN(1,1),0.5*dN(1,0),
0.5*dN(2,1),0.5*dN(2,0);
Eigen::MatrixXd k_el = B.transpose() * D * C * B * m_mesh->getElementVolume(i);
for (size_t j=0; j<3; ++j)
for (size_t k=0; k<3; ++k)
for (size_t l=0; l<2; ++l)
for (size_t m=0; m<2; ++m)
triplets.push_back(T(2*idx[j]+l, 2*idx[k]+m, k_el(2*j+l, 2*k+m)));
}
Eigen::SparseMatrix<double> K = Eigen::SparseMatrix<double>(2*m_mesh->getNbrNodes(), 2*m_mesh->getNbrNodes());
K.setFromTriplets(triplets.begin(), triplets.end());
ZSparseMatrix tmp = ZSparseMatrix(K);
return tmp;
}
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;
}
int R() {
int SG;
int tu;
int b;
int cbnn;
int fJA;
int Ywh;
int Sd3i;
int U;
int FTe;
;
if (AgC9 > + (Y0V(UnE, 791438648 + cK(517618502, + N_ != Ed() != - (Yk(((883538282)), nBa)), sdDM(kC, (1604897559), ! ! 528836364 / (1815076270) / 1477268768), n), psl, - 126855483))) while (om(AZ)) {
int xF;
int yd;
int xEmV;
int vc;
int m9;
while (2044928990) while (- 644557172 + 1573824677 + 1727264726 < TvAP(1980609521, Jy47, C, 953854953, - WyD(RlhU(LY0), + ipA(G, (Vg((1882502790), (1790689388) == 711715734 / (_) != + ! (134548752)))), nk1(((2078180300)) - + - 1230367323 - gD13, pK38 / 1377998756, m4G(1127784867) > u, (QSs()), 1389625403), 929895424, 895835811 < dW) * - 1339857856) / 2045295101) while (pLb) while (+ 1321345616) return ! 708468611;
(J);
return eUY;
continue;
(bRvz = 1838176344);
r(F + ! 196992386, Gg);
if (1419330106) ! (- (uG)) <= ! 996971325;
else T((- ! 2092297653 <= AA >= ! 1494541558 == o((! + v7nD))), 1818433226);
}
else break;
;
;
if (! (! cQ63(1853496451, i(DW9, 988295904 > _Gv, Wlbr, b) / ! ((O(xl))) - - O(+ a_O > 318640956, 1005354893, 167517361) / If, GDD, u, B) * B) + 443946612 + (dN(BFD3(mg <= 1303591176 + hFH2, x) % hch > 1053610009 / RaZ0(ZC7O((720576768)), 124325761, K(! C, 1901412912 * 1202992483, + (+ + e * 389490648), + 1244843051 < (sm = dH), W9(1583501857)), - ! sIs, 1297610228)))) lYAi(1443174874, U8R, + 925787186);
else continue;
continue;
if (+ ! 329152016 + K2()) continue;
else ;
continue;
continue;
}
void Tr21Stokes :: computeStiffnessMatrix(FloatMatrix &answer, TimeStep *tStep)
{
// Note: Working with the components; [K, G+Dp; G^T+Dv^T, C] . [v,p]
FluidDynamicMaterial *mat = ( FluidDynamicMaterial * ) this->domain->giveMaterial(this->material);
IntegrationRule *iRule = this->integrationRulesArray [ 0 ];
GaussPoint *gp;
FloatMatrix B(3, 12), EdB, K(12,12), G, Dp, DvT, C, Ed, dN;
FloatArray *lcoords, dN_V(12), Nlin, Ep, Cd, tmpA, tmpB;
double Cp;
K.zero();
G.zero();
for ( int i = 0; i < iRule->getNumberOfIntegrationPoints(); i++ ) {
// Compute Gauss point and determinant at current element
gp = iRule->getIntegrationPoint(i);
lcoords = gp->giveCoordinates();
double detJ = fabs(this->interpolation_quad.giveTransformationJacobian(* lcoords, FEIElementGeometryWrapper(this)));
double dA = detJ * gp->giveWeight();
this->interpolation_quad.evaldNdx(dN, * lcoords, FEIElementGeometryWrapper(this));
this->interpolation_lin.evalN(Nlin, * lcoords, FEIElementGeometryWrapper(this));
for ( int j = 0, k = 0; j < 6; j++, k += 2 ) {
dN_V(k) = B(0, k) = B(2, k + 1) = dN(j, 0);
dN_V(k + 1) = B(1, k + 1) = B(2, k) = dN(j, 1);
}
// Computing the internal forces should have been done first.
mat->giveDeviatoricStiffnessMatrix(Ed, TangentStiffness, gp, tStep); // dsigma_dev/deps_dev
mat->giveDeviatoricPressureStiffness(Ep, TangentStiffness, gp, tStep); // dsigma_dev/dp
mat->giveVolumetricDeviatoricStiffness(Cd, TangentStiffness, gp, tStep); // deps_vol/deps_dev
mat->giveVolumetricPressureStiffness(Cp, TangentStiffness, gp, tStep); // deps_vol/dp
EdB.beProductOf(Ed,B);
K.plusProductSymmUpper(B, EdB, dA);
G.plusDyadUnsym(dN_V, Nlin, -dA);
C.plusDyadSymmUpper(Nlin, Nlin, Cp*dA);
tmpA.beTProductOf(B, Ep);
Dp.plusDyadUnsym(tmpA, Nlin, dA);
tmpB.beTProductOf(B, Cd);
DvT.plusDyadUnsym(Nlin, tmpB, dA);
}
K.symmetrized();
C.symmetrized();
FloatMatrix GTDvT, GDp;
GTDvT.beTranspositionOf(G);
GTDvT.add(DvT);
GDp = G;
GDp.add(Dp);
FloatMatrix temp(15, 15);
temp.setSubMatrix(K, 1, 1);
temp.setSubMatrix(GTDvT, 13, 1);
temp.setSubMatrix(GDp, 1, 13);
temp.setSubMatrix(C, 13, 13);
answer.resize(15, 15);
answer.zero();
answer.assemble(temp, this->ordering);
}
