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Commit 349dcbe4 authored by Chris Cantwell's avatar Chris Cantwell
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Added Laplacian Op updates to TetExp.

parent f8ca5c6d
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library/LocalRegions/TetExp.cpp
View file @ 349dcbe4
......@@ -1797,12 +1797,100 @@ namespace Nektar
Array<OneD, NekDouble> &outarray,
Array<OneD, NekDouble> &wsp)
{
// This implementation is only valid when there are no
// coefficients associated to the Laplacian operator
if (m_metrics.count(MetricLaplacian00) == 0)
{
ComputeLaplacianMetric();
}
int nquad0 = m_base[0]->GetNumPoints();
int nquad1 = m_base[1]->GetNumPoints();
int nquad2 = m_base[2]->GetNumPoints();
int nqtot = nquad0*nquad1*nquad2;
int i, j;
ASSERTL1(wsp.num_elements() >= 6*nqtot,
"Insufficient workspace size.");
const Array<OneD, const NekDouble>& base0 = m_base[0]->GetBdata();
const Array<OneD, const NekDouble>& base1 = m_base[1]->GetBdata();
const Array<OneD, const NekDouble>& base2 = m_base[2]->GetBdata();
const Array<OneD, const NekDouble>& dbase0 = m_base[0]->GetDbdata();
const Array<OneD, const NekDouble>& dbase1 = m_base[1]->GetDbdata();
const Array<OneD, const NekDouble>& dbase2 = m_base[2]->GetDbdata();
const Array<OneD, const NekDouble>& metric00 = m_metrics[MetricLaplacian00];
const Array<OneD, const NekDouble>& metric01 = m_metrics[MetricLaplacian01];
const Array<OneD, const NekDouble>& metric02 = m_metrics[MetricLaplacian02];
const Array<OneD, const NekDouble>& metric11 = m_metrics[MetricLaplacian11];
const Array<OneD, const NekDouble>& metric12 = m_metrics[MetricLaplacian12];
const Array<OneD, const NekDouble>& metric22 = m_metrics[MetricLaplacian22];
// Allocate temporary storage
Array<OneD,NekDouble> wsp0 (wsp);
Array<OneD,NekDouble> wsp1 (wsp+1*nqtot );// TensorDeriv 1
Array<OneD,NekDouble> wsp2 (wsp+2*nqtot);// TensorDeriv 2
Array<OneD,NekDouble> wsp3 (wsp+3*nqtot);// TensorDeriv 3
Array<OneD,NekDouble> wsp4 (wsp+4*nqtot);// wsp4 == g1
Array<OneD,NekDouble> wsp5 (wsp+5*nqtot);// wsp5 == g2
// LAPLACIAN MATRIX OPERATION
// wsp1 = du_dxi1 = D_xi1 * inarray = D_xi1 * u
// wsp2 = du_dxi2 = D_xi2 * inarray = D_xi2 * u
// wsp2 = du_dxi3 = D_xi3 * inarray = D_xi3 * u
StdExpansion3D::PhysTensorDeriv(inarray,wsp0,wsp1,wsp2);
// wsp0 = k = g0 * wsp1 + g1 * wsp2 = g0 * du_dxi1 + g1 * du_dxi2
// wsp2 = l = g1 * wsp1 + g2 * wsp2 = g0 * du_dxi1 + g1 * du_dxi2
// where g0, g1 and g2 are the metric terms set up in the GeomFactors class
// especially for this purpose
Vmath::Vvtvvtp(nqtot,&metric00[0],1,&wsp0[0],1,&metric01[0],1,&wsp1[0],1,&wsp3[0],1);
Vmath::Vvtvp (nqtot,&metric02[0],1,&wsp2[0],1,&wsp3[0],1,&wsp3[0],1);
Vmath::Vvtvvtp(nqtot,&metric01[0],1,&wsp0[0],1,&metric11[0],1,&wsp1[0],1,&wsp4[0],1);
Vmath::Vvtvp (nqtot,&metric12[0],1,&wsp2[0],1,&wsp4[0],1,&wsp4[0],1);
Vmath::Vvtvvtp(nqtot,&metric02[0],1,&wsp0[0],1,&metric12[0],1,&wsp1[0],1,&wsp5[0],1);
Vmath::Vvtvp (nqtot,&metric22[0],1,&wsp2[0],1,&wsp5[0],1,&wsp5[0],1);
// outarray = m = (D_xi1 * B)^T * k
// wsp1 = n = (D_xi2 * B)^T * l
IProductWRTBase_SumFacKernel(dbase0,base1,base2,wsp3,outarray,wsp0,false,true,true);
IProductWRTBase_SumFacKernel(base0,dbase1,base2,wsp4,wsp1, wsp0,true,false,true);
IProductWRTBase_SumFacKernel(base0,base1,dbase2,wsp5,wsp2, wsp0,true,true,false);
// outarray = outarray + wsp1
// = L * u_hat
Vmath::Vadd(m_ncoeffs,wsp1.get(),1,outarray.get(),1,outarray.get(),1);
Vmath::Vadd(m_ncoeffs,wsp2.get(),1,outarray.get(),1,outarray.get(),1);
}
void TetExp::v_ComputeLaplacianMetric()
{
if (m_metrics.count(MetricQuadrature) == 0)
{
ComputeQuadratureMetric();
}
int i, j;
const SpatialDomains::GeomType type = m_metricinfo->GetGtype();
const unsigned int nqtot = GetTotPoints();
const unsigned int dim = 3;
const MetricType m[3][3] = { {MetricLaplacian00, MetricLaplacian01, MetricLaplacian02},
{MetricLaplacian01, MetricLaplacian11, MetricLaplacian12},
{MetricLaplacian02, MetricLaplacian12, MetricLaplacian22}
};
Array<OneD, NekDouble> dEta_dXi[2] = {Array<OneD, NekDouble>(nqtot,1.0),
Array<OneD, NekDouble>(nqtot,1.0)};
for (unsigned int i = 0; i < dim; ++i)
{
for (unsigned int j = i; j < dim; ++j)
{
m_metrics[m[i][j]] = Array<OneD, NekDouble>(nqtot);
}
}
// Allocate temporary storage
Array<OneD,NekDouble> alloc(13*nqtot,0.0);
Array<OneD,NekDouble> wsp1 (alloc );// TensorDeriv 1
......@@ -1825,27 +1913,15 @@ namespace Nektar
Array<OneD,NekDouble> wsp7 (alloc+ 9*nqtot);// wsp7 == h0
Array<OneD,NekDouble> wsp8 (alloc+10*nqtot);// wsp8 == h1
Array<OneD,NekDouble> wsp9 (alloc+11*nqtot);// wsp9 == h2
const Array<OneD, const NekDouble>& base0 = m_base[0]->GetBdata();
const Array<OneD, const NekDouble>& base1 = m_base[1]->GetBdata();
const Array<OneD, const NekDouble>& base2 = m_base[2]->GetBdata();
const Array<OneD, const NekDouble>& dbase0 = m_base[0]->GetDbdata();
const Array<OneD, const NekDouble>& dbase1 = m_base[1]->GetDbdata();
const Array<OneD, const NekDouble>& dbase2 = m_base[2]->GetDbdata();
// LAPLACIAN MATRIX OPERATION
// wsp1 = du_dxi1 = D_xi1 * inarray = D_xi1 * u
// wsp2 = du_dxi2 = D_xi2 * inarray = D_xi2 * u
// wsp2 = du_dxi3 = D_xi3 * inarray = D_xi3 * u
StdExpansion3D::PhysTensorDeriv(inarray,wsp1,wsp2,wsp3);
const Array<TwoD, const NekDouble>& gmat = m_metricinfo->GetGmat();
const Array<OneD, const NekDouble>& z0 = m_base[0]->GetZ();
const Array<OneD, const NekDouble>& z1 = m_base[1]->GetZ();
const Array<OneD, const NekDouble>& z2 = m_base[2]->GetZ();
// Step 2. Calculate the metric terms of the collapsed
// coordinate transformation (Spencer's book P152)
const Array<TwoD, const NekDouble>& df = m_metricinfo->GetDerivFactors();
const Array<OneD, const NekDouble>& z0 = m_base[0]->GetZ();
const Array<OneD, const NekDouble>& z1 = m_base[1]->GetZ();
const Array<OneD, const NekDouble>& z2 = m_base[2]->GetZ();
const unsigned int nquad0 = m_base[0]->GetNumPoints();
const unsigned int nquad1 = m_base[1]->GetNumPoints();
const unsigned int nquad2 = m_base[2]->GetNumPoints();
for(j = 0; j < nquad2; ++j)
{
for(i = 0; i < nquad1; ++i)
......@@ -1860,180 +1936,106 @@ namespace Nektar
{
Blas::Dscal(nquad1*nquad2, 1+z0[i], &h1[0]+i, nquad0);
}
// Step 3. Construct combined metric terms for physical space to
// collapsed coordinate system.
// Order of construction optimised to minimise temporary storage
if(m_metricinfo->GetGtype() == SpatialDomains::eDeformed)
{
// wsp4
Vmath::Vadd(nqtot, &gmat[1][0], 1, &gmat[2][0], 1, &wsp4[0], 1);
Vmath::Vvtvvtp(nqtot, &gmat[0][0], 1, &h0[0], 1, &wsp4[0], 1, &h1[0], 1, &wsp4[0], 1);
Vmath::Vadd(nqtot, &df[1][0], 1, &df[2][0], 1, &wsp4[0], 1);
Vmath::Vvtvvtp(nqtot, &df[0][0], 1, &h0[0], 1, &wsp4[0], 1, &h1[0], 1, &wsp4[0], 1);
// wsp5
Vmath::Vadd(nqtot, &gmat[4][0], 1, &gmat[5][0], 1, &wsp5[0], 1);
Vmath::Vvtvvtp(nqtot, &gmat[3][0], 1, &h0[0], 1, &wsp5[0], 1, &h1[0], 1, &wsp5[0], 1);
Vmath::Vadd(nqtot, &df[4][0], 1, &df[5][0], 1, &wsp5[0], 1);
Vmath::Vvtvvtp(nqtot, &df[3][0], 1, &h0[0], 1, &wsp5[0], 1, &h1[0], 1, &wsp5[0], 1);
// wsp6
Vmath::Vadd(nqtot, &gmat[7][0], 1, &gmat[8][0], 1, &wsp6[0], 1);
Vmath::Vvtvvtp(nqtot, &gmat[6][0], 1, &h0[0], 1, &wsp6[0], 1, &h1[0], 1, &wsp6[0], 1);
Vmath::Vadd(nqtot, &df[7][0], 1, &df[8][0], 1, &wsp6[0], 1);
Vmath::Vvtvvtp(nqtot, &df[6][0], 1, &h0[0], 1, &wsp6[0], 1, &h1[0], 1, &wsp6[0], 1);
// g0
Vmath::Vvtvvtp(nqtot, &wsp4[0], 1, &wsp4[0], 1, &wsp5[0], 1, &wsp5[0], 1, &g0[0], 1);
Vmath::Vvtvp (nqtot, &wsp6[0], 1, &wsp6[0], 1, &g0[0], 1, &g0[0], 1);
// g4
Vmath::Vvtvvtp(nqtot, &gmat[2][0], 1, &wsp4[0], 1, &gmat[5][0], 1, &wsp5[0], 1, &g4[0], 1);
Vmath::Vvtvp (nqtot, &gmat[8][0], 1, &wsp6[0], 1, &g4[0], 1, &g4[0], 1);
Vmath::Vvtvvtp(nqtot, &df[2][0], 1, &wsp4[0], 1, &df[5][0], 1, &wsp5[0], 1, &g4[0], 1);
Vmath::Vvtvp (nqtot, &df[8][0], 1, &wsp6[0], 1, &g4[0], 1, &g4[0], 1);
// overwrite h0, h1, h2
// wsp7 (h2f1 + h3f2)
Vmath::Vvtvvtp(nqtot, &gmat[1][0], 1, &h2[0], 1, &gmat[2][0], 1, &h3[0], 1, &wsp7[0], 1);
Vmath::Vvtvvtp(nqtot, &df[1][0], 1, &h2[0], 1, &df[2][0], 1, &h3[0], 1, &wsp7[0], 1);
// wsp8 (h2f4 + h3f5)
Vmath::Vvtvvtp(nqtot, &gmat[4][0], 1, &h2[0], 1, &gmat[5][0], 1, &h3[0], 1, &wsp8[0], 1);
Vmath::Vvtvvtp(nqtot, &df[4][0], 1, &h2[0], 1, &df[5][0], 1, &h3[0], 1, &wsp8[0], 1);
// wsp9 (h2f7 + h3f8)
Vmath::Vvtvvtp(nqtot, &gmat[7][0], 1, &h2[0], 1, &gmat[8][0], 1, &h3[0], 1, &wsp9[0], 1);
Vmath::Vvtvvtp(nqtot, &df[7][0], 1, &h2[0], 1, &df[8][0], 1, &h3[0], 1, &wsp9[0], 1);
// g3
Vmath::Vvtvvtp(nqtot, &wsp4[0], 1, &wsp7[0], 1, &wsp5[0], 1, &wsp8[0], 1, &g3[0], 1);
Vmath::Vvtvp (nqtot, &wsp6[0], 1, &wsp9[0], 1, &g3[0], 1, &g3[0], 1);
// overwrite wsp4, wsp5, wsp6
// g1
Vmath::Vvtvvtp(nqtot, &wsp7[0], 1, &wsp7[0], 1, &wsp8[0], 1, &wsp8[0], 1, &g1[0], 1);
Vmath::Vvtvp (nqtot, &wsp9[0], 1, &wsp9[0], 1, &g1[0], 1, &g1[0], 1);
// g5
Vmath::Vvtvvtp(nqtot, &gmat[2][0], 1, &wsp7[0], 1, &gmat[5][0], 1, &wsp8[0], 1, &g5[0], 1);
Vmath::Vvtvp (nqtot, &gmat[8][0], 1, &wsp9[0], 1, &g5[0], 1, &g5[0], 1);
Vmath::Vvtvvtp(nqtot, &df[2][0], 1, &wsp7[0], 1, &df[5][0], 1, &wsp8[0], 1, &g5[0], 1);
Vmath::Vvtvp (nqtot, &df[8][0], 1, &wsp9[0], 1, &g5[0], 1, &g5[0], 1);
// g2
Vmath::Vvtvvtp(nqtot, &gmat[2][0], 1, &gmat[2][0], 1, &gmat[5][0], 1, &gmat[5][0], 1, &g2[0], 1);
Vmath::Vvtvp (nqtot, &gmat[8][0], 1, &gmat[8][0], 1, &g2[0], 1, &g2[0], 1);
Vmath::Vvtvvtp(nqtot, &df[2][0], 1, &df[2][0], 1, &df[5][0], 1, &df[5][0], 1, &g2[0], 1);
Vmath::Vvtvp (nqtot, &df[8][0], 1, &df[8][0], 1, &g2[0], 1, &g2[0], 1);
}
else
{
// wsp4
Vmath::Svtsvtp(nqtot, gmat[0][0], &h0[0], 1, gmat[1][0] + gmat[2][0], &h1[0], 1, &wsp4[0], 1);
Vmath::Svtsvtp(nqtot, df[0][0], &h0[0], 1, df[1][0] + df[2][0], &h1[0], 1, &wsp4[0], 1);
// wsp5
Vmath::Svtsvtp(nqtot, gmat[3][0], &h0[0], 1, gmat[4][0] + gmat[5][0], &h1[0], 1, &wsp5[0], 1);
Vmath::Svtsvtp(nqtot, df[3][0], &h0[0], 1, df[4][0] + df[5][0], &h1[0], 1, &wsp5[0], 1);
// wsp6
Vmath::Svtsvtp(nqtot, gmat[6][0], &h0[0], 1, gmat[7][0] + gmat[8][0], &h1[0], 1, &wsp6[0], 1);
Vmath::Svtsvtp(nqtot, df[6][0], &h0[0], 1, df[7][0] + df[8][0], &h1[0], 1, &wsp6[0], 1);
// g0
Vmath::Vvtvvtp(nqtot, &wsp4[0], 1, &wsp4[0], 1, &wsp5[0], 1, &wsp5[0], 1, &g0[0], 1);
Vmath::Vvtvp (nqtot, &wsp6[0], 1, &wsp6[0], 1, &g0[0], 1, &g0[0], 1);
// g4
Vmath::Svtsvtp(nqtot, gmat[2][0], &wsp4[0], 1, gmat[5][0], &wsp5[0], 1, &g4[0], 1);
Vmath::Svtvp (nqtot, gmat[8][0], &wsp6[0], 1, &g4[0], 1, &g4[0], 1);
Vmath::Svtsvtp(nqtot, df[2][0], &wsp4[0], 1, df[5][0], &wsp5[0], 1, &g4[0], 1);
Vmath::Svtvp (nqtot, df[8][0], &wsp6[0], 1, &g4[0], 1, &g4[0], 1);
// overwrite h0, h1, h2
// wsp7 (h2f1 + h3f2)
Vmath::Svtsvtp(nqtot, gmat[1][0], &h2[0], 1, gmat[2][0], &h3[0], 1, &wsp7[0], 1);
Vmath::Svtsvtp(nqtot, df[1][0], &h2[0], 1, df[2][0], &h3[0], 1, &wsp7[0], 1);
// wsp8 (h2f4 + h3f5)
Vmath::Svtsvtp(nqtot, gmat[4][0], &h2[0], 1, gmat[5][0], &h3[0], 1, &wsp8[0], 1);
Vmath::Svtsvtp(nqtot, df[4][0], &h2[0], 1, df[5][0], &h3[0], 1, &wsp8[0], 1);
// wsp9 (h2f7 + h3f8)
Vmath::Svtsvtp(nqtot, gmat[7][0], &h2[0], 1, gmat[8][0], &h3[0], 1, &wsp9[0], 1);
Vmath::Svtsvtp(nqtot, df[7][0], &h2[0], 1, df[8][0], &h3[0], 1, &wsp9[0], 1);
// g3
Vmath::Vvtvvtp(nqtot, &wsp4[0], 1, &wsp7[0], 1, &wsp5[0], 1, &wsp8[0], 1, &g3[0], 1);
Vmath::Vvtvp (nqtot, &wsp6[0], 1, &wsp9[0], 1, &g3[0], 1, &g3[0], 1);
// overwrite wsp4, wsp5, wsp6
// g1
Vmath::Vvtvvtp(nqtot, &wsp7[0], 1, &wsp7[0], 1, &wsp8[0], 1, &wsp8[0], 1, &g1[0], 1);
Vmath::Vvtvp (nqtot, &wsp9[0], 1, &wsp9[0], 1, &g1[0], 1, &g1[0], 1);
// g5
Vmath::Svtsvtp(nqtot, gmat[2][0], &wsp7[0], 1, gmat[5][0], &wsp8[0], 1, &g5[0], 1);
Vmath::Svtvp (nqtot, gmat[8][0], &wsp9[0], 1, &g5[0], 1, &g5[0], 1);
// g2
Vmath::Fill(nqtot, gmat[2][0]*gmat[2][0] + gmat[5][0]*gmat[5][0] + gmat[8][0]*gmat[8][0], &g2[0], 1);
}
// Compute component derivatives into wsp7, 8, 9
Vmath::Vvtvvtp(nqtot,&g0[0],1,&wsp1[0],1,&g3[0],1,&wsp2[0],1,&wsp7[0],1);
Vmath::Vvtvp (nqtot,&g4[0],1,&wsp3[0],1,&wsp7[0],1,&wsp7[0],1);
Vmath::Vvtvvtp(nqtot,&g1[0],1,&wsp2[0],1,&g3[0],1,&wsp1[0],1,&wsp8[0],1);
Vmath::Vvtvp (nqtot,&g5[0],1,&wsp3[0],1,&wsp8[0],1,&wsp8[0],1);
Vmath::Vvtvvtp(nqtot,&g2[0],1,&wsp3[0],1,&g4[0],1,&wsp1[0],1,&wsp9[0],1);
Vmath::Vvtvp (nqtot,&g5[0],1,&wsp2[0],1,&wsp9[0],1,&wsp9[0],1);
// Step 4.
// Multiply by quadrature metric
MultiplyByQuadratureMetric(wsp7,wsp7);
MultiplyByQuadratureMetric(wsp8,wsp8);
MultiplyByQuadratureMetric(wsp9,wsp9);
IProductWRTBase_SumFacKernel(dbase0,base1,base2,wsp7,wsp1, wsp,false,true,true);
IProductWRTBase_SumFacKernel(base0,dbase1,base2,wsp8,wsp2, wsp,true,false,true);
IProductWRTBase_SumFacKernel(base0,base1,dbase2,wsp9,outarray,wsp,true,true,false);
// Step 5.
// Sum contributions from wsp1, wsp2 and outarray.
Vmath::Vadd(m_ncoeffs,wsp1.get(),1,outarray.get(),1,outarray.get(),1);
Vmath::Vadd(m_ncoeffs,wsp2.get(),1,outarray.get(),1,outarray.get(),1);
}
// g5
Vmath::Svtsvtp(nqtot, df[2][0], &wsp7[0], 1, df[5][0], &wsp8[0], 1, &g5[0], 1);
Vmath::Svtvp (nqtot, df[8][0], &wsp9[0], 1, &g5[0], 1, &g5[0], 1);
void TetExp::v_ComputeLaplacianMetric()
{
if (m_metrics.count(MetricQuadrature) == 0)
{
ComputeQuadratureMetric();
// g2
Vmath::Fill(nqtot, df[2][0]*df[2][0] + df[5][0]*df[5][0] + df[8][0]*df[8][0], &g2[0], 1);
}
int i, j;
const SpatialDomains::GeomType type = m_metricinfo->GetGtype();
const unsigned int nqtot = GetTotPoints();
const unsigned int dim = 2;
const MetricType m[3][3] = { {MetricLaplacian00, MetricLaplacian01, MetricLaplacian02},
{MetricLaplacian01, MetricLaplacian11, MetricLaplacian12},
{MetricLaplacian02, MetricLaplacian12, MetricLaplacian22}
};
Array<OneD, NekDouble> dEta_dXi[2] = {Array<OneD, NekDouble>(nqtot,1.0),
Array<OneD, NekDouble>(nqtot,1.0)};
for (unsigned int i = 0; i < dim; ++i)
{
for (unsigned int j = i; j < dim; ++j)
{
m_metrics[m[i][j]] = Array<OneD, NekDouble>(nqtot);
}
}
Array<OneD,NekDouble> alloc(13*nqtot,0.0);
Array<OneD,NekDouble> h0 (alloc+ 9*nqtot);// h0
Array<OneD,NekDouble> h1 (alloc+10*nqtot);// h1
Array<OneD,NekDouble> h2 (alloc+11*nqtot);// h2
Array<OneD,NekDouble> h3 (alloc+12*nqtot);// h3
MultiplyByQuadratureMetric(m_metrics[m[i][j]],
m_metrics[m[i][j]]);
const Array<TwoD, const NekDouble>& gmat = m_metricinfo->GetGmat();
const Array<OneD, const NekDouble>& z0 = m_base[0]->GetZ();
const Array<OneD, const NekDouble>& z1 = m_base[1]->GetZ();
const Array<OneD, const NekDouble>& z2 = m_base[2]->GetZ();
const unsigned int nquad0 = m_base[0]->GetNumPoints();
const unsigned int nquad1 = m_base[1]->GetNumPoints();
const unsigned int nquad2 = m_base[2]->GetNumPoints();
for(j = 0; j < nquad2; ++j)
{
for(i = 0; i < nquad1; ++i)
{
Vmath::Fill(nquad0, 4.0/(1.0-z1[i])/(1.0-z2[j]), &h0[0]+i*nquad0 + j*nquad0*nquad1,1);
Vmath::Fill(nquad0, 2.0/(1.0-z1[i])/(1.0-z2[j]), &h1[0]+i*nquad0 + j*nquad0*nquad1,1);
Vmath::Fill(nquad0, 2.0/(1.0-z2[j]), &h2[0]+i*nquad0 + j*nquad0*nquad1,1);
Vmath::Fill(nquad0, (1.0+z1[i])/(1.0-z2[j]), &h3[0]+i*nquad0 + j*nquad0*nquad1,1);
}
}
for(i = 0; i < nquad0; i++)
{
Blas::Dscal(nquad1*nquad2, 1+z0[i], &h1[0]+i, nquad0);
}
}
......
solvers/CompressibleFlowSolver/Tests/Couette_WeakDG_LDG_MODIFIED.tst
View file @ 349dcbe4
......@@ -11,12 +11,12 @@
<value variable="rho" tolerance="1e-12">0.0878644</value>
<value variable="rhou" tolerance="1e-12">60.3309</value>
<value variable="rhov" tolerance="1e-8">0.230213</value>
<value variable="E" tolerance="1e-12">4926.56</value>
<value variable="E" tolerance="1e-12">4926.55</value>
</metric>
<metric type="Linf" id="2">
<value variable="rho" tolerance="1e-12">0.0743376</value>
<value variable="rhou" tolerance="1e-12">60.8075</value>
<value variable="rhov" tolerance="1e-8">0.302439</value>
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<value variable="E" tolerance="1e-12">4448.92</value>
</metric>
</metrics>
......
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