bool Target2DUntangle::evaluate_with_hess( const MsqMatrix<2,2>& A,
const MsqMatrix<2,2>& W,
double& result,
MsqMatrix<2,2>& deriv_wrt_A,
MsqMatrix<2,2> second_wrt_A[3],
MsqError& err )
{
const MsqMatrix<2,2> Winv = inverse(W);
const MsqMatrix<2,2> T = A * Winv;
double tau = det(T);
if (tau < mGamma) {
double d = tau - mGamma;
result = 16 * d*d*d*d;
const MsqMatrix<2,2> adjt = transpose_adj(T);
deriv_wrt_A = 64 * d*d*d * adjt;
deriv_wrt_A = deriv_wrt_A * transpose(Winv);
set_scaled_outer_product( second_wrt_A, 192*d*d, adjt );
pluseq_scaled_2nd_deriv_of_det( second_wrt_A, 64*d*d*d );
second_deriv_wrt_product_factor( second_wrt_A, Winv );
}
else {
result = 0.0;
second_wrt_A[0] = second_wrt_A[1] = second_wrt_A[2] = deriv_wrt_A = MsqMatrix<2,2>(0.0);
}
return true;
}
bool Target3DShape::evaluate_with_hess( const MsqMatrix<3,3>& A,
const MsqMatrix<3,3>& W,
double& result,
MsqMatrix<3,3>& deriv_wrt_A,
MsqMatrix<3,3> second_wrt_A[6],
MsqError& err )
{
MsqMatrix<3,3> Winv = inverse(W);
MsqMatrix<3,3> T = A * Winv;
double f = Frobenius(T);
double d = det(T);
result = f*f*f - 3*MSQ_SQRT_THREE*d;
deriv_wrt_A = T;
deriv_wrt_A *= f;
deriv_wrt_A -= MSQ_SQRT_THREE*transpose_adj(T);
deriv_wrt_A *= 3;
deriv_wrt_A = deriv_wrt_A * transpose(Winv);
set_scaled_2nd_deriv_of_det( second_wrt_A, -3 * MSQ_SQRT_THREE, T );
pluseq_scaled_outer_product( second_wrt_A, 3.0/f, T );
pluseq_scaled_I( second_wrt_A, 3.0*f );
second_deriv_wrt_product_factor( second_wrt_A, Winv );
return true;
}
/** \f$ \frac{\partial^2 \mu}{\partial T^2}
= \frac{1}{\tau} I_4
- \frac{1}{\tau^2} \left( T \otimes \frac{\partial \tau}{\partial T}
+ \frac{\partial \tau}{\partial T} \otimes T \right)
+ \frac{|T|^2}{\tau^3} \left( \frac{\partial \tau}{\partial T} \otimes
\frac{\partial \tau}{\partial T} \right)
- \frac{|T|^2}{2 \tau^3} \frac{\partial^2 \tau}{\partial T^2} \f$
*/
bool Target2DShapeBarrier::evaluate_with_hess( const MsqMatrix<2,2>& A,
const MsqMatrix<2,2>& W,
double& result,
MsqMatrix<2,2>& deriv_wrt_A,
MsqMatrix<2,2> second_wrt_A[3],
MsqError& err )
{
const MsqMatrix<2,2> Winv = inverse(W);
const MsqMatrix<2,2> T = A * Winv;
const double d = det(T);
if (invalid_determinant(d)) { // barrier
result = 0.0;
return false;
}
double inv_d = 1.0/d;
double f1 = sqr_Frobenius(T) * inv_d;
result = 0.5 * f1;
const MsqMatrix<2,2> adjt = transpose_adj(T);
deriv_wrt_A = T;
deriv_wrt_A -= result * adjt;
deriv_wrt_A *= inv_d;
deriv_wrt_A = deriv_wrt_A * transpose(Winv);
set_scaled_outer_product( second_wrt_A, f1 * inv_d * inv_d, adjt );
pluseq_scaled_sum_outer_product( second_wrt_A, -inv_d*inv_d, T, adjt );
pluseq_scaled_I( second_wrt_A, inv_d );
pluseq_scaled_2nd_deriv_of_det( second_wrt_A, -result * inv_d );
second_deriv_wrt_product_factor( second_wrt_A, Winv );
result -= 1.0;
return true;
}
bool Target3DShapeOrientBarrierAlt1::evaluate_with_Hess( const MsqMatrix<3,3>& A,
const MsqMatrix<3,3>& W,
double& result,
MsqMatrix<3,3>& deriv_wrt_A,
MsqMatrix<3,3> second_wrt_A[6],
MsqError& err )
{
MsqMatrix<3,3> Winv = inverse(W);
MsqMatrix<3,3> T = A * Winv;
double tau = det(T);
if (invalid_determinant(tau)) {
result = 0.0;
return false;
}
const double b = 0.5/tau;
// calculate non-barrier value (ShapeOrientAlt1)
const double tr = trace(T);
const double f = MSQ_ONE_THIRD * fabs(tr);
result = sqr_Frobenius( T ) - f * tr;
// calculate non-barrier first derivatives
deriv_wrt_A = T;
deriv_wrt_A(0,0) -= f;
deriv_wrt_A(1,1) -= f;
deriv_wrt_A(2,2) -= f;
deriv_wrt_A *= 2;
// calculate barrier second derivs
const MsqMatrix<3,3> adjt = transpose_adj(T);
set_scaled_sum_outer_product( second_wrt_A, -b/tau, deriv_wrt_A, adjt );
pluseq_scaled_outer_product( second_wrt_A, result/(tau*tau*tau), adjt );
pluseq_scaled_2nd_deriv_of_det( second_wrt_A, -result * b / tau, T );
// calculate non-barrier barrier portion of second derivs
pluseq_scaled_I( second_wrt_A, 1/tau );
pluseq_scaled_outer_product_I_I( second_wrt_A, MSQ_ONE_THIRD/tau * (tr < 0 ? 1 : -1) );
// calculate barrier derivs from non-barrier
deriv_wrt_A *= tau;
deriv_wrt_A -= result * adjt;
deriv_wrt_A *= b/tau;
// barrier value from non-barrier
result *= b;
// convert from derivs wrt T to derivs wrt A
deriv_wrt_A = deriv_wrt_A * transpose(Winv);
second_deriv_wrt_product_factor( second_wrt_A, Winv );
return true;
}
bool Target3DShapeSizeBarrierAlt1::evaluate_with_hess( const MsqMatrix<3,3>& A,
const MsqMatrix<3,3>& W,
double& result,
MsqMatrix<3,3>& wrt_A,
MsqMatrix<3,3> second[6],
MsqError& err )
{
const MsqMatrix<3,3> Winv = inverse(W);
const MsqMatrix<3,3> T = A * Winv;
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
result = 0.0;
return false;
}
const double f = sqr_Frobenius(T);
const double g = sqr_Frobenius(adj(T));
result = (f + g)/(6 * tau);
MsqMatrix<3,3> dtau = transpose_adj(T);
MsqMatrix<3,3> dg = -transpose(T) * T;
dg(0,0) += f;
dg(1,1) += f;
dg(2,2) += f;
dg = T * dg;
dg *= 2;
wrt_A = T;
wrt_A += 0.5*dg;
wrt_A *= 1.0/3.0;
wrt_A -= result * dtau;
wrt_A *= 1.0/tau;
wrt_A = wrt_A * transpose(Winv);
set_scaled_2nd_deriv_norm_sqr_adj( second, 1.0/6.0, T );
pluseq_scaled_I( second, 1.0/3.0 );
pluseq_scaled_sum_outer_product( second, -1./3./tau, T, dtau );
pluseq_scaled_sum_outer_product( second, -1./6./tau, dg, dtau );
pluseq_scaled_outer_product( second, 2*result/tau, dtau );
pluseq_scaled_2nd_deriv_of_det( second, -result, T );
hess_scale( second, 1.0/tau );
second_deriv_wrt_product_factor( second, Winv );
result -= 1.0;
return true;
}
bool Target2DShapeOrientBarrier::evaluate_with_hess( const MsqMatrix<2,2>& A,
const MsqMatrix<2,2>& W,
double& result,
MsqMatrix<2,2>& deriv,
MsqMatrix<2,2> second[3],
MsqError& err )
{
const MsqMatrix<2,2> Winv = inverse(W);
const MsqMatrix<2,2> T = A * Winv;
const double norm = Frobenius(T);
const double invroot = 1.0/MSQ_SQRT_TWO;
const double tau = det(T);
if (invalid_determinant(tau)) { // barrier
result = 0.0;
return false;
}
const double inv_tau = 1.0/tau;
const double invnorm = 1.0/norm;
const double f = norm - invroot * trace(T);
result = 0.5 * inv_tau * f;
const MsqMatrix<2,2> adjt = transpose_adj(T);
deriv = invnorm * T;
deriv(0,0) -= invroot;
deriv(1,1) -= invroot;
deriv *= 0.5;
deriv -= result * adjt;
deriv *= inv_tau;
deriv = deriv * transpose(Winv);
const double a = 0.5 * inv_tau * invnorm;
set_scaled_outer_product( second, -a*invnorm*invnorm, T );
pluseq_scaled_I( second, a );
pluseq_scaled_outer_product( second, f*inv_tau*inv_tau*inv_tau, adjt );
pluseq_scaled_2nd_deriv_of_det( second, -0.5*f*inv_tau*inv_tau );
pluseq_scaled_sum_outer_product( second, -0.5*inv_tau*inv_tau*invnorm, T, adjt );
pluseq_scaled_sum_outer_product_I( second, 0.5*inv_tau*inv_tau*invroot, adjt );
second_deriv_wrt_product_factor( second, Winv );
return true;
}
bool TQualityMetric::evaluate_with_Hessian_diagonal(
PatchData& pd,
size_t handle,
double& value,
std::vector<size_t>& indices,
std::vector<Vector3D>& grad,
std::vector<SymMatrix3D>& diagonal,
MsqError& err )
{
const Sample s = ElemSampleQM::sample( handle );
const size_t e = ElemSampleQM:: elem( handle );
MsqMeshEntity& elem = pd.element_by_index( e );
EntityTopology type = elem.get_element_type();
unsigned edim = TopologyInfo::dimension( type );
size_t num_idx = 0;
const NodeSet bits = pd.non_slave_node_set( e );
bool rval;
if (edim == 3) { // 3x3 or 3x2 targets ?
const MappingFunction3D* mf = pd.get_mapping_function_3D( type );
if (!mf) {
MSQ_SETERR(err)( "No mapping function for element type", MsqError::UNSUPPORTED_ELEMENT );
return false;
}
MsqMatrix<3,3> A, W, dmdT, d2mdT2[6];
mf->jacobian( pd, e, bits, s, mIndices, mDerivs3D, num_idx, A, err );
MSQ_ERRZERO(err);
targetCalc->get_3D_target( pd, e, s, W, err ); MSQ_ERRZERO(err);
const MsqMatrix<3,3> Winv = inverse(W);
const MsqMatrix<3,3> T = A*Winv;
rval = targetMetric->evaluate_with_hess( T, value, dmdT, d2mdT2, err );
MSQ_ERRZERO(err);
gradient<3>( num_idx, mDerivs3D, dmdT * transpose(Winv), grad );
second_deriv_wrt_product_factor( d2mdT2, Winv );
diagonal.resize( num_idx );
hessian_diagonal<3>(num_idx, mDerivs3D, d2mdT2, arrptr(diagonal) );
#ifdef PRINT_INFO
print_info<3>( e, s, A, W, A * inverse(W) );
#endif
}
else if (edim == 2) {
#ifdef NUMERICAL_2D_HESSIAN
// use finite diference approximation for now
return QualityMetric::evaluate_with_Hessian_diagonal( pd, handle,
value, indices, grad, diagonal,
err );
#else
MsqMatrix<2,2> W, A, dmdT, d2mdT2[3];
MsqMatrix<3,2> M;
rval = evaluate_surface_common( pd, s, e, bits, mIndices, num_idx,
mDerivs2D, W, A, M, err );
if (MSQ_CHKERR(err) || !rval)
return false;
const MsqMatrix<2,2> Winv = inverse(W);
const MsqMatrix<2,2> T = A*Winv;
rval = targetMetric->evaluate_with_hess( T, value, dmdT, d2mdT2, err );
MSQ_ERRZERO(err);
gradient<2>( num_idx, mDerivs2D, M * dmdT * transpose(Winv), grad );
second_deriv_wrt_product_factor( d2mdT2, Winv );
diagonal.resize( num_idx );
for (size_t i = 0; i < num_idx; ++i) {
MsqMatrix<2,2> block2d;
block2d(0,0) = transpose(mDerivs2D[i]) * d2mdT2[0] * mDerivs2D[i];
block2d(0,1) = transpose(mDerivs2D[i]) * d2mdT2[1] * mDerivs2D[i];
block2d(1,0) = block2d(0,1);
block2d(1,1) = transpose(mDerivs2D[i]) * d2mdT2[2] * mDerivs2D[i];
MsqMatrix<3,2> p = M * block2d;
SymMatrix3D& H = diagonal[i];
H[0] = p.row(0) * transpose(M.row(0));
H[1] = p.row(0) * transpose(M.row(1));
H[2] = p.row(0) * transpose(M.row(2));
H[3] = p.row(1) * transpose(M.row(1));
H[4] = p.row(1) * transpose(M.row(2));
H[5] = p.row(2) * transpose(M.row(2));
}
#ifdef PRINT_INFO
print_info<2>( e, s, J, Wp, A * inverse(W) );
#endif
#endif
}
else {
assert(0);
return false;
}
// pass back index list
indices.resize( num_idx );
std::copy( mIndices, mIndices+num_idx, indices.begin() );
// apply target weight to value
if (!num_idx)
weight( pd, s, e, num_idx, value, 0, 0, 0, err );
else
weight( pd, s, e, num_idx, value, arrptr(grad), arrptr(diagonal), 0, err );
MSQ_ERRZERO(err);
return rval;
}
bool TQualityMetric::evaluate_with_Hessian( PatchData& pd,
size_t handle,
double& value,
std::vector<size_t>& indices,
std::vector<Vector3D>& grad,
std::vector<Matrix3D>& Hessian,
MsqError& err )
{
const Sample s = ElemSampleQM::sample( handle );
const size_t e = ElemSampleQM:: elem( handle );
MsqMeshEntity& elem = pd.element_by_index( e );
EntityTopology type = elem.get_element_type();
unsigned edim = TopologyInfo::dimension( type );
size_t num_idx = 0;
const NodeSet bits = pd.non_slave_node_set( e );
bool rval;
if (edim == 3) { // 3x3 or 3x2 targets ?
const MappingFunction3D* mf = pd.get_mapping_function_3D( type );
if (!mf) {
MSQ_SETERR(err)( "No mapping function for element type", MsqError::UNSUPPORTED_ELEMENT );
return false;
}
MsqMatrix<3,3> A, W, dmdT, d2mdT2[6];
mf->jacobian( pd, e, bits, s, mIndices, mDerivs3D, num_idx, A, err );
MSQ_ERRZERO(err);
targetCalc->get_3D_target( pd, e, s, W, err ); MSQ_ERRZERO(err);
const MsqMatrix<3,3> Winv = inverse(W);
const MsqMatrix<3,3> T = A*Winv;
rval = targetMetric->evaluate_with_hess( T, value, dmdT, d2mdT2, err );
MSQ_ERRZERO(err);
gradient<3>( num_idx, mDerivs3D, dmdT*transpose(Winv), grad );
second_deriv_wrt_product_factor( d2mdT2, Winv );
Hessian.resize( num_idx*(num_idx+1)/2 );
if (num_idx)
hessian<3>( num_idx, mDerivs3D, d2mdT2, arrptr(Hessian) );
#ifdef PRINT_INFO
print_info<3>( e, s, A, W, A * inverse(W) );
#endif
}
else if (edim == 2) {
#ifdef NUMERICAL_2D_HESSIAN
// return finite difference approximation for now
return QualityMetric::evaluate_with_Hessian( pd, handle,
value, indices, grad, Hessian,
err );
#else
MsqMatrix<2,2> W, A, dmdT, d2mdT2[3];
MsqMatrix<3,2> M;
rval = evaluate_surface_common( pd, s, e, bits, mIndices, num_idx,
mDerivs2D, W, A, M, err );
if (MSQ_CHKERR(err) || !rval)
return false;
const MsqMatrix<2,2> Winv = inverse(W);
const MsqMatrix<2,2> T = A*Winv;
rval = targetMetric->evaluate_with_hess( T, value, dmdT, d2mdT2, err );
MSQ_ERRZERO(err);
gradient<2>( num_idx, mDerivs2D, M * dmdT * transpose(Winv), grad );
// calculate 2D hessian
second_deriv_wrt_product_factor( d2mdT2, Winv );
const size_t n = num_idx*(num_idx+1)/2;
hess2d.resize(n);
if (n)
hessian<2>( num_idx, mDerivs2D, d2mdT2, arrptr(hess2d) );
// calculate surface hessian as transform of 2D hessian
Hessian.resize(n);
for (size_t i = 0; i < n; ++i)
Hessian[i] = Matrix3D( (M * hess2d[i] * transpose(M)).data() );
#ifdef PRINT_INFO
print_info<2>( e, s, J, Wp, A * inverse(W) );
#endif
#endif
}
else {
assert(0);
return false;
}
// pass back index list
indices.resize( num_idx );
std::copy( mIndices, mIndices+num_idx, indices.begin() );
// apply target weight to value
if (!num_idx)
weight( pd, s, e, num_idx, value, 0, 0, 0, err );
else
weight( pd, s, e, num_idx, value, arrptr(grad), 0, arrptr(Hessian), err );
MSQ_ERRZERO(err);
return rval;
}
