void
RotationMatrixTest::rotVtoU(RealVectorValue v, RealVectorValue u)
{
RealTensorValue ident(1,0,0, 0,1,0, 0,0,1);
RealVectorValue vhat = v/v.size();
RealVectorValue uhat = u/u.size();
RealTensorValue r = RotationMatrix::rotVec1ToVec2(v, u);
RealVectorValue rotated_v = r*vhat;
for (unsigned i = 0 ; i < LIBMESH_DIM ; ++i)
CPPUNIT_ASSERT_DOUBLES_EQUAL( rotated_v(i), uhat(i), 0.0001);
CPPUNIT_ASSERT( r*r.transpose() == ident);
CPPUNIT_ASSERT( r.transpose()*r == ident);
}
int main() {
const std::string Help =
"-------------------------------------------------------------------------\n"
"TestComputeAKV: \n"
"-------------------------------------------------------------------------\n"
"OPTIONS: \n"
"Nth=<int> theta resolution [default 3] \n"
"Nph=<int> phi resolution [default 4] \n"
"Radius=<double> radius of sphere. [default 1.0] \n"
"AKVGuess=MyVector<double> a guess for the values of THETA, thetap, phip \n"
" [default (0.,0.,0.)] \n"
"L_resid_tol=<double> tolerance for L residuals when finding approximate \n"
" Killing vectors. [default 1.e-12] \n"
"v_resid_tol=<double> tolerance for v residuals when finding approximate \n"
" Killing vectors. [default 1.e-12] \n"
"min_thetap = for values less than this, thetap is considered close to \n"
" zero. [default 1.e-5] \n"
"symmetry_tol=<double> abs(THETA) must be less than this value to be \n"
" considered an exact symmetry. [default 1.e-11] \n"
"ResidualSize=<double> determines the tolerance for residuals from the \n"
" multidimensional root finder. [default to 1.e-11] \n"
"Solver = <std::string> which gsl multidimensional root finding algorith \n"
" should be used. [default Newton] \n"
"Verbose=<bool> Print spectral coefficients and warnings if true \n"
" [default false] \n"
;
std::string Options = ReadFileIntoString("Test.input");
OptionParser op(Options,Help);
const int Nth = op.Get<int>("Nth", 3);
const int Nph = op.Get<int>("Nph", 4);
const double rad = op.Get<double>("Radius",1.0);
MyVector<double> AKVGuess =
op.Get<MyVector<double> >("AKVGuess",MyVector<double>(MV::Size(3),0.0));
//must be three-dimensional
REQUIRE(AKVGuess.Size()==3,"AKVGuess has Size " << AKVGuess.Size()
<< ", should be 3.");
const double L_resid_tol = op.Get<double>("L_resid_tol", 1.e-12);
const double v_resid_tol = op.Get<double>("L_resid_tol", 1.e-12);
const double residualSize = op.Get<double>("ResidualSize", 1.e-11);
const double min_thetap = op.Get<double>("min_theta",1.e-5);
const double symmetry_tol = op.Get<double>("symmetry_tol",1.e-11);
const std::string solver = op.Get<std::string>("Solver","Newton");
const bool verbose = op.Get<bool>("Verbose", false);
const MyVector<bool> printDiagnostic = MyVector<bool>(MV::Size(6), true);
//create skm
const StrahlkorperMesh skm(Nth, Nph);
//create surface basis
const SurfaceBasis sb(skm);
//get theta, phi
const DataMesh theta(skm.SurfaceCoords()(0));
const DataMesh phi(skm.SurfaceCoords()(1));
//set the initial guesses to be along particular axes
const int axes = 3; //the number of perpendicular axes
//create conformal factors for every rotation
const int syms = 5; //the number of axisymmetries we are testing
for(int s=4; s<5; s++) { //index over conformal factor symmetries
//for(int s=0; s<syms; s++){//index over conformal factor symmetries
//create conformal factor
const DataMesh Psi = ConstructConformalFactor(theta, phi, s);
//set the initial guesses
double THETA[3] = {AKVGuess[0],0.,0.};
double thetap[3] = {AKVGuess[1],0.,0.};
double phip[3] = {AKVGuess[2],0.,0.};
//save the v, xi solutions along particular axes
MyVector<DataMesh> v(MV::Size(3),DataMesh::Empty);
MyVector<DataMesh> rotated_v(MV::Size(3),DataMesh::Empty);
MyVector<Tensor<DataMesh> > xi(MV::Size(axes),Tensor<DataMesh>(2,"1",DataMesh::Empty));
//save the <v_i|v_j> inner product solutions
double v0v0 = 0.;
double v1v1 = 0.;
double v2v2 = 0.;
double v0v1 = 0.;
double v0v2 = 0.;
double v1v2 = 0.;
//int symmetries[3] = 0; //counts the number of symmetries
//compute some useful quantities
const DataMesh rp2 = rad * Psi * Psi;
const DataMesh r2p4 = rp2*rp2;
const DataMesh llncf = sb.ScalarLaplacian(log(Psi));
const DataMesh Ricci = 2.0 * (1.0-2.0*llncf) / r2p4;
const Tensor<DataMesh> GradRicci = sb.Gradient(Ricci);
for(int a=0; a<axes; a++) { //index over perpendicular axes to find AKV solutions
//if the diagnostics below decide that there is a bad solution for v[a]
//(usually a repeated solution), this flag will indicate that the
//solver should be run again
bool badAKVSolution = false;
//generate a guess for the next axis of symmetry based on prior solutions.
AxisInitialGuess(thetap, phip, a);
//create L
DataMesh L(DataMesh::Empty);
//setup struct with all necessary data
rparams p = {theta, phi, rp2, sb, llncf, GradRicci,
L, v[a], L_resid_tol, v_resid_tol, verbose, true
};
RunAKVsolvers(THETA[a], thetap[a], phip[a], min_thetap,
residualSize, verbose, &p, solver);
std::cout << "Solution found with : THETA[" << a << "] = " << THETA[a] << "\n"
<< " thetap[" << a << "] = " << (180.0/M_PI)*thetap[a] << "\n"
<< " phip[" << a << "] = " << (180.0/M_PI)*phip[a]
<< std::endl;
//check inner products
// <v_i|v_j> = Integral 0.5 * Ricci * Grad(v_i) \cdot Grad(v_j) dA
switch(a) {
case 0:
//compute inner product <v_0|v_0>
v0v0 = AKVInnerProduct(v[0],v[0],Ricci,sb)*sqrt(2.)*M_PI;
//if(v0v0<symmetry_tol) //symmetries++;
std::cout << "<v_0|v_0> = " << v0v0 << std::endl;
std::cout << "-THETA <v_0|v_0> = " << -THETA[a]*v0v0 << std::endl;
break;
case 1:
//compute inner products <v_1|v_1>, <v_0|v_1>
v1v1 = AKVInnerProduct(v[1],v[1],Ricci,sb)*sqrt(2.)*M_PI;
v0v1 = AKVInnerProduct(v[0],v[1],Ricci,sb)*sqrt(2.)*M_PI;
//if(v1v1<symmetry_tol) //symmetries++;
std::cout << "<v_1|v_1> = " << v1v1 << std::endl;
std::cout << "<v_0|v_1> = " << v0v1 << std::endl;
std::cout << "-THETA <v_1|v_1> = " << -THETA[a]*v1v1 << std::endl;
if(fabs(v0v0) == fabs(v1v2)) badAKVSolution = true;
break;
case 2:
//compute inner products <v_2|v_2>, <v_0|v_2>, <v_1|v_2>
v2v2 = AKVInnerProduct(v[2],v[2],Ricci,sb)*sqrt(2.)*M_PI;
v0v2 = AKVInnerProduct(v[0],v[2],Ricci,sb)*sqrt(2.)*M_PI;
v1v2 = AKVInnerProduct(v[1],v[2],Ricci,sb)*sqrt(2.)*M_PI;
//if(v2v2<symmetry_tol) //symmetries++;
std::cout << "<v_2|v_2> = " << v2v2 << std::endl;
std::cout << "<v_0|v_2> = " << v0v2 << std::endl;
std::cout << "<v_1|v_2> = " << v1v2 << std::endl;
std::cout << "-THETA <v_2|v_2> = " << -THETA[a]*v2v2 << std::endl;
if(fabs(v0v0) == fabs(v0v2)) badAKVSolution = true;
if(fabs(v1v1) == fabs(v1v2)) badAKVSolution = true;
break;
}
//Gram Schmidt orthogonalization
switch(a) {
case 1:
if(v0v0<symmetry_tol && v1v1<symmetry_tol) { //two symmetries, v2v2 should also be symmetric
GramSchmidtOrthogonalization(v[0], v0v0, v[1], v0v1);
}
break;
case 2:
if(v0v0<symmetry_tol) {
if(v1v1<symmetry_tol) {
REQUIRE(v2v2<symmetry_tol, "Three symmetries required, but only two found.");
GramSchmidtOrthogonalization(v[0], v0v0, v[2], v0v2);
GramSchmidtOrthogonalization(v[1], v1v1, v[2], v1v2);
} else if(v2v2<symmetry_tol) {
REQUIRE(false, "Three symmetries required, but only two found.");
} else {
GramSchmidtOrthogonalization(v[1], v1v1, v[2], v1v2);
}
} else if(v1v1<symmetry_tol) {
if(v2v2<symmetry_tol) {
REQUIRE(false, "Three symmetries required, but only two found.");
} else {
GramSchmidtOrthogonalization(v[0], v0v0, v[2], v0v2);
}
} else if(v2v2<symmetry_tol) {
GramSchmidtOrthogonalization(v[0], v0v0, v[1], v0v1);
}
break;
}
//create xi (1-form)
xi[a] = ComputeXi(v[a], sb);
//perform diagnostics
//Psi and xi are unscaled and unrotated
KillingDiagnostics(sb, L, Psi, xi[a], rad, printDiagnostic);
//rotate v, Psi for analysis
rotated_v[a] = RotateOnSphere(v[a],theta,phi,
sb,thetap[a],phip[a]);
DataMesh rotated_Psi = RotateOnSphere(Psi,theta,phi,
sb,thetap[a],phip[a]);
//compare scale factors
const double scaleAtEquator =
NormalizeAKVAtOnePoint(sb, rotated_Psi, rotated_v[a], rad, M_PI/2., 0.0);
PrintSurfaceNormalization(sb,rotated_Psi,theta,phi,rotated_v[a],scaleAtEquator,rad);
MyVector<double> scaleInnerProduct = InnerProductScaleFactors(v[a], v[a], Ricci, r2p4, sb);
PrintSurfaceNormalization(sb,rotated_Psi,theta,phi,rotated_v[a],scaleInnerProduct[0],rad);
PrintSurfaceNormalization(sb,rotated_Psi,theta,phi,rotated_v[a],scaleInnerProduct[1],rad);
PrintSurfaceNormalization(sb,rotated_Psi,theta,phi,rotated_v[a],scaleInnerProduct[2],rad);
OptimizeScaleFactor(rotated_v[a], rotated_Psi, rad, sb, theta,
phi, scaleAtEquator, scaleInnerProduct[0], scaleInnerProduct[1], scaleInnerProduct[2]);
//scale v
v[a] *= scaleAtEquator;
//recompute scaled xi (1-form)
xi[a] = ComputeXi(v[a], sb);
if(badAKVSolution) {
v[a] = 0.;
thetap[a] += M_PI/4.;
phip[a] += M_PI/4.;
a--;
std::cout << "This was a bad / repeated solution, and will be recomputed." << std::endl;
}
std::cout << std::endl;
}//end loop over perpendicular AKV axes
std::cout << "\n" << std::endl;
}
// Return 0 for success
return NumberOfTestsFailed;
}
