2118 lines
70 KiB
C++
2118 lines
70 KiB
C++
// Copyright (c) 2010-2025, Lawrence Livermore National Security, LLC. Produced
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// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
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// LICENSE and NOTICE for details. LLNL-CODE-806117.
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//
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// This file is part of the MFEM library. For more information and source code
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// availability visit https://mfem.org.
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//
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// MFEM is free software; you can redistribute it and/or modify it under the
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// terms of the BSD-3 license. We welcome feedback and contributions, see file
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// CONTRIBUTING.md for details.
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// Implementation of IntegrationRule(s) classes
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// Acknowledgment: Some of the high-precision triangular and tetrahedral
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// quadrature rules below were obtained from the Encyclopaedia of Cubature
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// Formulas at http://nines.cs.kuleuven.be/research/ecf/ecf.html
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#include "fem.hpp"
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#include "../mesh/nurbs.hpp"
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#include <cmath>
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#ifdef MFEM_USE_MPFR
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#include <mpfr.h>
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#endif
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using namespace std;
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namespace mfem
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{
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IntegrationRule::IntegrationRule(IntegrationRule &irx, IntegrationRule &iry)
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{
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int i, j, nx, ny;
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nx = irx.GetNPoints();
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ny = iry.GetNPoints();
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SetSize(nx * ny);
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SetPointIndices();
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Order = std::min(irx.GetOrder(), iry.GetOrder());
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for (j = 0; j < ny; j++)
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{
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IntegrationPoint &ipy = iry.IntPoint(j);
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for (i = 0; i < nx; i++)
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{
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IntegrationPoint &ipx = irx.IntPoint(i);
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IntegrationPoint &ip = IntPoint(j*nx+i);
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ip.x = ipx.x;
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ip.y = ipy.x;
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ip.weight = ipx.weight * ipy.weight;
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}
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}
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}
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IntegrationRule::IntegrationRule(IntegrationRule &irx, IntegrationRule &iry,
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IntegrationRule &irz)
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{
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const int nx = irx.GetNPoints();
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const int ny = iry.GetNPoints();
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const int nz = irz.GetNPoints();
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SetSize(nx*ny*nz);
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SetPointIndices();
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Order = std::min({irx.GetOrder(), iry.GetOrder(), irz.GetOrder()});
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for (int iz = 0; iz < nz; ++iz)
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{
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IntegrationPoint &ipz = irz.IntPoint(iz);
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for (int iy = 0; iy < ny; ++iy)
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{
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IntegrationPoint &ipy = iry.IntPoint(iy);
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for (int ix = 0; ix < nx; ++ix)
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{
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IntegrationPoint &ipx = irx.IntPoint(ix);
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IntegrationPoint &ip = IntPoint(iz*nx*ny + iy*nx + ix);
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ip.x = ipx.x;
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ip.y = ipy.x;
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ip.z = ipz.x;
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ip.weight = ipx.weight*ipy.weight*ipz.weight;
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}
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}
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}
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}
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const Array<real_t> &IntegrationRule::GetWeights() const
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{
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if (weights.Size() != GetNPoints())
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{
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weights.SetSize(GetNPoints());
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for (int i = 0; i < GetNPoints(); i++)
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{
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weights[i] = IntPoint(i).weight;
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}
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}
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return weights;
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}
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void IntegrationRule::SetPointIndices()
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{
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for (int i = 0; i < Size(); i++)
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{
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IntPoint(i).index = i;
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}
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}
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void IntegrationRule::GrundmannMollerSimplexRule(int s, int n)
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{
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// for pow on older compilers
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using std::pow;
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const int d = 2*s + 1;
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Vector fact(d + n + 1);
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Array<int> beta(n), sums(n);
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fact(0) = 1.;
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for (int i = 1; i < fact.Size(); i++)
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{
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fact(i) = fact(i - 1)*i;
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}
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// number of points is \binom{n + s + 1}{n + 1}
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int np = 1, f = 1;
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for (int i = 0; i <= n; i++)
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{
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np *= (s + i + 1), f *= (i + 1);
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}
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np /= f;
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SetSize(np);
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SetPointIndices();
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Order = 2*s + 1;
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int pt = 0;
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for (int i = 0; i <= s; i++)
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{
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real_t weight;
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weight = pow(2., -2*s)*pow(static_cast<real_t>(d + n - 2*i),
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d)/fact(i)/fact(d + n - i);
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if (i%2)
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{
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weight = -weight;
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}
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// loop over all beta : beta_0 + ... + beta_{n-1} <= s - i
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int k = s - i;
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beta = 0;
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sums = 0;
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while (true)
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{
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IntegrationPoint &ip = IntPoint(pt++);
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ip.weight = weight;
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ip.x = real_t(2*beta[0] + 1)/(d + n - 2*i);
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ip.y = real_t(2*beta[1] + 1)/(d + n - 2*i);
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if (n == 3)
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{
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ip.z = real_t(2*beta[2] + 1)/(d + n - 2*i);
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}
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int j = 0;
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while (sums[j] == k)
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{
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beta[j++] = 0;
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if (j == n)
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{
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goto done_beta;
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}
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}
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beta[j]++;
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sums[j]++;
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for (j--; j >= 0; j--)
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{
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sums[j] = sums[j+1];
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}
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}
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done_beta:
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;
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}
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}
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IntegrationRule*
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IntegrationRule::ApplyToKnotIntervals(KnotVector const& kv) const
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{
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const int np = this->GetNPoints();
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const int ne = kv.GetNE();
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IntegrationRule *kvir = new IntegrationRule(ne * np);
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kvir->SetOrder(GetOrder());
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real_t x0 = kv[0];
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real_t x1 = x0;
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int id = 0;
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for (int e=0; e<ne; ++e)
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{
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x0 = x1;
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if (e == ne-1)
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{
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x1 = kv[kv.Size() - 1];
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}
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else
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{
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// Find the next unique knot
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while (id < kv.Size() - 1)
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{
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id++;
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if (kv[id] != x0)
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{
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x1 = kv[id];
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break;
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}
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}
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}
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const real_t s = x1 - x0;
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for (int j=0; j<this->GetNPoints(); ++j)
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{
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const real_t x = x0 + (s * (*this)[j].x);
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(*kvir)[(e * np) + j].Set1w(x, (*this)[j].weight);
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}
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}
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return kvir;
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}
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#ifdef MFEM_USE_MPFR
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// Class for computing hi-precision (HP) quadrature in 1D
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class HP_Quadrature1D
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{
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protected:
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mpfr_t pi, z, pp, p1, p2, p3, dz, w, rtol;
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public:
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static const mpfr_rnd_t rnd = GMP_RNDN;
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static const int default_prec = 128;
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// prec = MPFR precision in bits
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HP_Quadrature1D(const int prec = default_prec)
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{
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mpfr_inits2(prec, pi, z, pp, p1, p2, p3, dz, w, rtol, (mpfr_ptr) 0);
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mpfr_const_pi(pi, rnd);
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mpfr_set_si_2exp(rtol, 1, -32, rnd); // 2^(-32) < 2.33e-10
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}
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// set rtol = 2^exponent
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// this is a tolerance for the last correction of x_i in Newton's algorithm;
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// this gives roughly rtol^2 accuracy for the final x_i.
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void SetRelTol(const int exponent = -32)
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{
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mpfr_set_si_2exp(rtol, 1, exponent, rnd);
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}
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// n - number of quadrature points
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// k - index of the point to compute, 0 <= k < n
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// see also: QuadratureFunctions1D::GaussLegendre
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void ComputeGaussLegendrePoint(const int n, const int k)
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{
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MFEM_ASSERT(n > 0 && 0 <= k && k < n, "invalid n = " << n
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<< " and/or k = " << k);
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int i = (k < (n+1)/2) ? k+1 : n-k;
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// Initial guess for the x-coordinate:
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// set z = cos(pi * (i - 0.25) / (n + 0.5)) =
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// = sin(pi * ((n+1-2*i) / (2*n+1)))
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mpfr_set_si(z, n+1-2*i, rnd);
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mpfr_div_si(z, z, 2*n+1, rnd);
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mpfr_mul(z, z, pi, rnd);
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mpfr_sin(z, z, rnd);
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bool done = false;
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while (1)
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{
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mpfr_set_si(p2, 1, rnd);
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mpfr_set(p1, z, rnd);
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for (int j = 2; j <= n; j++)
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{
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mpfr_set(p3, p2, rnd);
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mpfr_set(p2, p1, rnd);
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// p1 = ((2 * j - 1) * z * p2 - (j - 1) * p3) / j;
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mpfr_mul_si(p1, z, 2*j-1, rnd);
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mpfr_mul_si(p3, p3, j-1, rnd);
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mpfr_fms(p1, p1, p2, p3, rnd);
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mpfr_div_si(p1, p1, j, rnd);
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}
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// p1 is Legendre polynomial
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// derivative of the Legendre polynomial:
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// pp = n * (z*p1-p2) / (z*z - 1);
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mpfr_fms(pp, z, p1, p2, rnd);
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mpfr_mul_si(pp, pp, n, rnd);
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mpfr_sqr(p2, z, rnd);
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mpfr_sub_si(p2, p2, 1, rnd);
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mpfr_div(pp, pp, p2, rnd);
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if (done) { break; }
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// set delta_z: dz = p1/pp;
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mpfr_div(dz, p1, pp, rnd);
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// compute absolute tolerance: atol = rtol*(1-z)
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mpfr_t &atol = w;
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mpfr_si_sub(atol, 1, z, rnd);
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mpfr_mul(atol, atol, rtol, rnd);
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if (mpfr_cmpabs(dz, atol) <= 0)
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{
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done = true;
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// continue the computation: get pp at the new point, then exit
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}
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// update z = z - dz
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mpfr_sub(z, z, dz, rnd);
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}
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// map z to (0,1): z = (1 - z)/2
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mpfr_si_sub(z, 1, z, rnd);
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mpfr_div_2si(z, z, 1, rnd);
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// weight: w = 1/(4*z*(1 - z)*pp*pp)
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mpfr_sqr(w, pp, rnd);
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mpfr_mul_2si(w, w, 2, rnd);
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mpfr_mul(w, w, z, rnd);
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mpfr_si_sub(p1, 1, z, rnd); // p1 = 1-z
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mpfr_mul(w, w, p1, rnd);
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mpfr_si_div(w, 1, w, rnd);
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if (k >= (n+1)/2) { mpfr_swap(z, p1); }
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}
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// n - number of quadrature points
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// k - index of the point to compute, 0 <= k < n
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// see also: QuadratureFunctions1D::GaussLobatto
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void ComputeGaussLobattoPoint(const int n, const int k)
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{
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MFEM_ASSERT(n > 1 && 0 <= k && k < n, "invalid n = " << n
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<< " and/or k = " << k);
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int i = (k < (n+1)/2) ? k : n-1-k;
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if (i == 0)
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{
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mpfr_set_si(z, 0, rnd);
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mpfr_set_si(p1, 1, rnd);
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mpfr_set_si(w, n*(n-1), rnd);
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mpfr_si_div(w, 1, w, rnd); // weight = 1/(n*(n-1))
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return;
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}
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// initial guess is the corresponding Chebyshev point, z:
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// z = -cos(pi * i/(n-1)) = sin(pi * (2*i-n+1)/(2*n-2))
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mpfr_set_si(z, 2*i-n+1, rnd);
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mpfr_div_si(z, z, 2*(n-1), rnd);
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mpfr_mul(z, pi, z, rnd);
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mpfr_sin(z, z, rnd);
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bool done = false;
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for (int iter = 0 ; true ; ++iter)
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{
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// build Legendre polynomials, up to P_{n}(z)
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mpfr_set_si(p1, 1, rnd);
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mpfr_set(p2, z, rnd);
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for (int l = 1 ; l < (n-1) ; ++l)
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{
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// P_{l+1}(x) = [ (2*l+1)*x*P_l(x) - l*P_{l-1}(x) ]/(l+1)
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mpfr_mul_si(p1, p1, l, rnd);
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mpfr_mul_si(p3, z, 2*l+1, rnd);
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mpfr_fms(p3, p3, p2, p1, rnd);
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mpfr_div_si(p3, p3, l+1, rnd);
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mpfr_set(p1, p2, rnd);
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mpfr_set(p2, p3, rnd);
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}
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if (done) { break; }
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// compute dz = resid/deriv = (z*p2 - p1) / (n*p2);
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mpfr_fms(dz, z, p2, p1, rnd);
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mpfr_mul_si(p3, p2, n, rnd);
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mpfr_div(dz, dz, p3, rnd);
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// update: z = z - dz
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mpfr_sub(z, z, dz, rnd);
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// compute absolute tolerance: atol = rtol*(1 + z)
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mpfr_t &atol = w;
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mpfr_add_si(atol, z, 1, rnd);
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mpfr_mul(atol, atol, rtol, rnd);
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// check for convergence
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if (mpfr_cmpabs(dz, atol) <= 0)
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{
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done = true;
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// continue the computation: get p2 at the new point, then exit
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}
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// If the iteration does not converge fast, something is wrong.
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MFEM_VERIFY(iter < 8, "n = " << n << ", i = " << i
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<< ", dz = " << mpfr_get_d(dz, rnd));
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}
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// Map to the interval [0,1] and scale the weights
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mpfr_add_si(z, z, 1, rnd);
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mpfr_div_2si(z, z, 1, rnd);
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// set the symmetric point
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mpfr_si_sub(p1, 1, z, rnd);
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// w = 1/[ n*(n-1)*[P_{n-1}(z)]^2 ]
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mpfr_sqr(w, p2, rnd);
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mpfr_mul_si(w, w, n*(n-1), rnd);
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mpfr_si_div(w, 1, w, rnd);
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if (k >= (n+1)/2) { mpfr_swap(z, p1); }
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}
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real_t GetPoint() const { return mpfr_get_d(z, rnd); }
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real_t GetSymmPoint() const { return mpfr_get_d(p1, rnd); }
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real_t GetWeight() const { return mpfr_get_d(w, rnd); }
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const mpfr_t &GetHPPoint() const { return z; }
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const mpfr_t &GetHPSymmPoint() const { return p1; }
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const mpfr_t &GetHPWeight() const { return w; }
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~HP_Quadrature1D()
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{
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mpfr_clears(pi, z, pp, p1, p2, p3, dz, w, rtol, (mpfr_ptr) 0);
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mpfr_free_cache();
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}
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};
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#endif // MFEM_USE_MPFR
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void QuadratureFunctions1D::GaussLegendre(const int np, IntegrationRule* ir)
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{
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ir->SetSize(np);
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ir->SetPointIndices();
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ir->SetOrder(2*np - 1);
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switch (np)
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{
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case 1:
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ir->IntPoint(0).Set1w(0.5, 1.0);
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return;
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case 2:
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ir->IntPoint(0).Set1w(0.21132486540518711775, 0.5);
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ir->IntPoint(1).Set1w(0.78867513459481288225, 0.5);
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return;
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case 3:
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ir->IntPoint(0).Set1w(0.11270166537925831148, 5./18.);
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ir->IntPoint(1).Set1w(0.5, 4./9.);
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ir->IntPoint(2).Set1w(0.88729833462074168852, 5./18.);
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return;
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}
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const int n = np;
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const int m = (n+1)/2;
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#ifndef MFEM_USE_MPFR
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for (int i = 1; i <= m; i++)
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{
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real_t z = cos(M_PI * (i - 0.25) / (n + 0.5));
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real_t pp, p1, dz, xi = 0.;
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bool done = false;
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while (1)
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{
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real_t p2 = 1;
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p1 = z;
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for (int j = 2; j <= n; j++)
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{
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real_t p3 = p2;
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p2 = p1;
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p1 = ((2 * j - 1) * z * p2 - (j - 1) * p3) / j;
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}
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// p1 is Legendre polynomial
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pp = n * (z*p1-p2) / (z*z - 1);
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if (done) { break; }
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dz = p1/pp;
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#ifdef MFEM_USE_SINGLE
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if (std::abs(dz) < 1e-7)
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#elif defined MFEM_USE_DOUBLE
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if (std::abs(dz) < 1e-16)
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#else
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MFEM_ABORT("Floating point type undefined");
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if (std::abs(dz) < 1e-16)
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#endif
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{
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done = true;
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// map the new point (z-dz) to (0,1):
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xi = ((1 - z) + dz)/2; // (1 - (z - dz))/2 has bad round-off
|
|
// continue the computation: get pp at the new point, then exit
|
|
}
|
|
// update: z = z - dz
|
|
z -= dz;
|
|
}
|
|
|
|
ir->IntPoint(i-1).x = xi;
|
|
ir->IntPoint(n-i).x = 1 - xi;
|
|
ir->IntPoint(i-1).weight =
|
|
ir->IntPoint(n-i).weight = 1./(4*xi*(1 - xi)*pp*pp);
|
|
}
|
|
|
|
#else // MFEM_USE_MPFR is defined
|
|
|
|
HP_Quadrature1D hp_quad;
|
|
for (int i = 1; i <= m; i++)
|
|
{
|
|
hp_quad.ComputeGaussLegendrePoint(n, i-1);
|
|
|
|
ir->IntPoint(i-1).x = hp_quad.GetPoint();
|
|
ir->IntPoint(n-i).x = hp_quad.GetSymmPoint();
|
|
ir->IntPoint(i-1).weight = ir->IntPoint(n-i).weight = hp_quad.GetWeight();
|
|
}
|
|
|
|
#endif // MFEM_USE_MPFR
|
|
|
|
}
|
|
|
|
void QuadratureFunctions1D::GaussLobatto(const int np, IntegrationRule* ir)
|
|
{
|
|
/* An np point Gauss-Lobatto quadrature has (np - 2) free abscissa the other
|
|
(2) abscissa are the interval endpoints.
|
|
|
|
The interior x_i are the zeros of P'_{np-1}(x). The weights of the
|
|
interior points on the interval [-1,1] are:
|
|
|
|
w_i = 2/(np*(np-1)*[P_{np-1}(x_i)]^2)
|
|
|
|
The end point weights (on [-1,1]) are: w_{end} = 2/(np*(np-1)).
|
|
|
|
The interior abscissa are found via a nonlinear solve, the initial guess
|
|
for each point is the corresponding Chebyshev point.
|
|
|
|
After we find all points on the interval [-1,1], we will map and scale the
|
|
points and weights to the MFEM natural interval [0,1].
|
|
|
|
References:
|
|
[1] E. E. Lewis and W. F. Millier, "Computational Methods of Neutron
|
|
Transport", Appendix A
|
|
[2] the QUADRULE software by John Burkardt,
|
|
https://people.sc.fsu.edu/~jburkardt/cpp_src/quadrule/quadrule.cpp
|
|
*/
|
|
|
|
ir->SetSize(np);
|
|
ir->SetPointIndices();
|
|
if ( np == 1 )
|
|
{
|
|
ir->IntPoint(0).Set1w(0.5, 1.0);
|
|
ir->SetOrder(1);
|
|
}
|
|
else
|
|
{
|
|
ir->SetOrder(2*np - 3);
|
|
|
|
#ifndef MFEM_USE_MPFR
|
|
|
|
// endpoints and respective weights
|
|
ir->IntPoint(0).x = 0.0;
|
|
ir->IntPoint(np-1).x = 1.0;
|
|
ir->IntPoint(0).weight = ir->IntPoint(np-1).weight = 1.0/(np*(np-1));
|
|
|
|
// interior points and weights
|
|
// use symmetry and compute just half of the points
|
|
for (int i = 1 ; i <= (np-1)/2 ; ++i)
|
|
{
|
|
// initial guess is the corresponding Chebyshev point, x_i:
|
|
// x_i = -cos(\pi * (i / (np-1)))
|
|
real_t x_i = std::sin(M_PI * ((real_t)(i)/(np-1) - 0.5));
|
|
real_t z_i = 0., p_l;
|
|
bool done = false;
|
|
for (int iter = 0 ; true ; ++iter)
|
|
{
|
|
// build Legendre polynomials, up to P_{np}(x_i)
|
|
real_t p_lm1 = 1.0;
|
|
p_l = x_i;
|
|
|
|
for (int l = 1 ; l < (np-1) ; ++l)
|
|
{
|
|
// The Legendre polynomials can be built by recursion:
|
|
// x * P_l(x) = 1/(2*l+1)*[ (l+1)*P_{l+1}(x) + l*P_{l-1} ], i.e.
|
|
// P_{l+1}(x) = [ (2*l+1)*x*P_l(x) - l*P_{l-1} ]/(l+1)
|
|
real_t p_lp1 = ( (2*l + 1)*x_i*p_l - l*p_lm1)/(l + 1);
|
|
|
|
p_lm1 = p_l;
|
|
p_l = p_lp1;
|
|
}
|
|
if (done) { break; }
|
|
// after this loop, p_l holds P_{np-1}(x_i)
|
|
// resid = (x^2-1)*P'_{np-1}(x_i)
|
|
// but use the recurrence relationship
|
|
// (x^2 -1)P'_l(x) = l*[ x*P_l(x) - P_{l-1}(x) ]
|
|
// thus, resid = (np-1) * (x_i*p_l - p_lm1)
|
|
|
|
// The derivative of the residual is:
|
|
// \frac{d}{d x} \left[ (x^2 -1)P'_l(x) ] \right] =
|
|
// l * (l+1) * P_l(x), with l = np-1,
|
|
// therefore, deriv = np * (np-1) * p_l;
|
|
|
|
// compute dx = resid/deriv
|
|
real_t dx = (x_i*p_l - p_lm1) / (np*p_l);
|
|
#ifdef MFEM_USE_SINGLE
|
|
if (std::abs(dx) < 1e-7)
|
|
#elif defined MFEM_USE_DOUBLE
|
|
if (std::abs(dx) < 1e-16)
|
|
#else
|
|
MFEM_ABORT("Floating point type undefined");
|
|
if (std::abs(dx) < 1e-16)
|
|
#endif
|
|
{
|
|
done = true;
|
|
// Map the point to the interval [0,1]
|
|
z_i = ((1.0 + x_i) - dx)/2;
|
|
// continue the computation: get p_l at the new point, then exit
|
|
}
|
|
// If the iteration does not converge fast, something is wrong.
|
|
MFEM_VERIFY(iter < 8, "np = " << np << ", i = " << i
|
|
<< ", dx = " << dx);
|
|
// update x_i:
|
|
x_i -= dx;
|
|
}
|
|
// Map to the interval [0,1] and scale the weights
|
|
IntegrationPoint &ip = ir->IntPoint(i);
|
|
ip.x = z_i;
|
|
// w_i = (2/[ n*(n-1)*[P_{n-1}(x_i)]^2 ]) / 2
|
|
ip.weight = (real_t)(1.0 / (np*(np-1)*p_l*p_l));
|
|
|
|
// set the symmetric point
|
|
IntegrationPoint &symm_ip = ir->IntPoint(np-1-i);
|
|
symm_ip.x = 1.0 - z_i;
|
|
symm_ip.weight = ip.weight;
|
|
}
|
|
|
|
#else // MFEM_USE_MPFR is defined
|
|
|
|
HP_Quadrature1D hp_quad;
|
|
// use symmetry and compute just half of the points
|
|
for (int i = 0 ; i <= (np-1)/2 ; ++i)
|
|
{
|
|
hp_quad.ComputeGaussLobattoPoint(np, i);
|
|
ir->IntPoint(i).x = hp_quad.GetPoint();
|
|
ir->IntPoint(np-1-i).x = hp_quad.GetSymmPoint();
|
|
ir->IntPoint(i).weight =
|
|
ir->IntPoint(np-1-i).weight = hp_quad.GetWeight();
|
|
}
|
|
|
|
#endif // MFEM_USE_MPFR
|
|
|
|
}
|
|
}
|
|
|
|
void QuadratureFunctions1D::OpenUniform(const int np, IntegrationRule* ir)
|
|
{
|
|
ir->SetSize(np);
|
|
ir->SetPointIndices();
|
|
ir->SetOrder(np - 1 + np%2);
|
|
|
|
// The Newton-Cotes quadrature is based on weights that integrate exactly the
|
|
// interpolatory polynomial through the equally spaced quadrature points.
|
|
for (int i = 0; i < np ; ++i)
|
|
{
|
|
ir->IntPoint(i).x = real_t(i+1) / real_t(np + 1);
|
|
}
|
|
|
|
CalculateUniformWeights(ir, Quadrature1D::OpenUniform);
|
|
}
|
|
|
|
void QuadratureFunctions1D::ClosedUniform(const int np,
|
|
IntegrationRule* ir)
|
|
{
|
|
ir->SetSize(np);
|
|
ir->SetPointIndices();
|
|
ir->SetOrder(np - 1 + np%2);
|
|
if ( np == 1 ) // allow this case as "closed"
|
|
{
|
|
ir->IntPoint(0).Set1w(0.5, 1.0);
|
|
return;
|
|
}
|
|
|
|
for (int i = 0; i < np ; ++i)
|
|
{
|
|
ir->IntPoint(i).x = real_t(i) / (np-1);
|
|
}
|
|
|
|
CalculateUniformWeights(ir, Quadrature1D::ClosedUniform);
|
|
}
|
|
|
|
void QuadratureFunctions1D::OpenHalfUniform(const int np, IntegrationRule* ir)
|
|
{
|
|
ir->SetSize(np);
|
|
ir->SetPointIndices();
|
|
ir->SetOrder(np - 1 + np%2);
|
|
|
|
// Open half points: the centers of np uniform intervals
|
|
for (int i = 0; i < np ; ++i)
|
|
{
|
|
ir->IntPoint(i).x = real_t(2*i+1) / (2*np);
|
|
}
|
|
|
|
CalculateUniformWeights(ir, Quadrature1D::OpenHalfUniform);
|
|
}
|
|
|
|
void QuadratureFunctions1D::ClosedGL(const int np, IntegrationRule* ir)
|
|
{
|
|
ir->SetSize(np);
|
|
ir->SetPointIndices();
|
|
ir->IntPoint(0).x = 0.0;
|
|
ir->IntPoint(np-1).x = 1.0;
|
|
ir->SetOrder(np - 1 + np%2); // Is this the correct order?
|
|
|
|
if ( np > 2 )
|
|
{
|
|
IntegrationRule gl_ir;
|
|
GaussLegendre(np-1, &gl_ir);
|
|
|
|
for (int i = 1; i < np-1; ++i)
|
|
{
|
|
ir->IntPoint(i).x = (gl_ir.IntPoint(i-1).x + gl_ir.IntPoint(i).x)/2;
|
|
}
|
|
}
|
|
|
|
CalculateUniformWeights(ir, Quadrature1D::ClosedGL);
|
|
}
|
|
|
|
void QuadratureFunctions1D::GivePolyPoints(const int np, real_t *pts,
|
|
const int type)
|
|
{
|
|
IntegrationRule ir(np);
|
|
|
|
switch (type)
|
|
{
|
|
case Quadrature1D::GaussLegendre:
|
|
{
|
|
GaussLegendre(np,&ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::GaussLobatto:
|
|
{
|
|
GaussLobatto(np, &ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::OpenUniform:
|
|
{
|
|
OpenUniform(np,&ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::ClosedUniform:
|
|
{
|
|
ClosedUniform(np,&ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::OpenHalfUniform:
|
|
{
|
|
OpenHalfUniform(np, &ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::ClosedGL:
|
|
{
|
|
ClosedGL(np, &ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::Invalid:
|
|
{
|
|
MFEM_ABORT("Asking for an unknown type of 1D Quadrature points, "
|
|
"type = " << type);
|
|
}
|
|
}
|
|
|
|
for (int i = 0 ; i < np ; ++i)
|
|
{
|
|
pts[i] = ir.IntPoint(i).x;
|
|
}
|
|
}
|
|
|
|
void QuadratureFunctions1D::CalculateUniformWeights(IntegrationRule *ir,
|
|
const int type)
|
|
{
|
|
/* The Lagrange polynomials are:
|
|
p_i = \prod_{j \neq i} {\frac{x - x_j }{x_i - x_j}}
|
|
|
|
The weight associated with each abscissa is the integral of p_i over
|
|
[0,1]. To calculate the integral of p_i, we use a Gauss-Legendre
|
|
quadrature rule. This approach does not suffer from bad round-off/
|
|
cancellation errors for large number of points.
|
|
*/
|
|
const int n = ir->Size();
|
|
switch (n)
|
|
{
|
|
case 1:
|
|
ir->IntPoint(0).weight = 1.;
|
|
return;
|
|
case 2:
|
|
ir->IntPoint(0).weight = .5;
|
|
ir->IntPoint(1).weight = .5;
|
|
return;
|
|
}
|
|
|
|
#ifndef MFEM_USE_MPFR
|
|
|
|
// This algorithm should work for any set of points, not just uniform
|
|
const IntegrationRule &glob_ir = IntRules.Get(Geometry::SEGMENT, n-1);
|
|
const int m = glob_ir.GetNPoints();
|
|
Vector xv(n);
|
|
for (int j = 0; j < n; j++)
|
|
{
|
|
xv(j) = ir->IntPoint(j).x;
|
|
}
|
|
Poly_1D::Basis basis(n-1, xv.GetData()); // nodal basis, with nodes at 'xv'
|
|
Vector w(n);
|
|
// Integrate all nodal basis functions using 'glob_ir':
|
|
w = 0.0;
|
|
for (int i = 0; i < m; i++)
|
|
{
|
|
const IntegrationPoint &ip = glob_ir.IntPoint(i);
|
|
basis.Eval(ip.x, xv);
|
|
w.Add(ip.weight, xv); // w += ip.weight * xv
|
|
}
|
|
for (int j = 0; j < n; j++)
|
|
{
|
|
ir->IntPoint(j).weight = w(j);
|
|
}
|
|
|
|
#else // MFEM_USE_MPFR is defined
|
|
|
|
static const mpfr_rnd_t rnd = HP_Quadrature1D::rnd;
|
|
HP_Quadrature1D hp_quad;
|
|
mpfr_t l, lk, w0, wi, tmp, *weights;
|
|
mpfr_inits2(hp_quad.default_prec, l, lk, w0, wi, tmp, (mpfr_ptr) 0);
|
|
weights = new mpfr_t[n];
|
|
for (int i = 0; i < n; i++)
|
|
{
|
|
mpfr_init2(weights[i], hp_quad.default_prec);
|
|
mpfr_set_si(weights[i], 0, rnd);
|
|
}
|
|
hp_quad.SetRelTol(-48); // rtol = 2^(-48) ~ 3.5e-15
|
|
const int p = n-1;
|
|
const int m = p/2+1; // number of points for Gauss-Legendre quadrature
|
|
int hinv = 0, ihoffset = 0; // x_i = (i+ihoffset/2)/hinv
|
|
switch (type)
|
|
{
|
|
case Quadrature1D::ClosedUniform:
|
|
// x_i = i/p, i=0,...,p
|
|
hinv = p;
|
|
ihoffset = 0;
|
|
break;
|
|
case Quadrature1D::OpenUniform:
|
|
// x_i = (i+1)/(p+2), i=0,...,p
|
|
hinv = p+2;
|
|
ihoffset = 2;
|
|
break;
|
|
case Quadrature1D::OpenHalfUniform:
|
|
// x_i = (i+1/2)/(p+1), i=0,...,p
|
|
hinv = p+1;
|
|
ihoffset = 1;
|
|
break;
|
|
case Quadrature1D::GaussLegendre:
|
|
case Quadrature1D::GaussLobatto:
|
|
case Quadrature1D::ClosedGL:
|
|
case Quadrature1D::Invalid:
|
|
MFEM_ABORT("invalid Quadrature1D type: " << type);
|
|
}
|
|
// set w0 = (-1)^p*(p!)/(hinv^p)
|
|
mpfr_fac_ui(w0, p, rnd);
|
|
mpfr_ui_pow_ui(tmp, hinv, p, rnd);
|
|
mpfr_div(w0, w0, tmp, rnd);
|
|
if (p%2) { mpfr_neg(w0, w0, rnd); }
|
|
|
|
for (int j = 0; j < m; j++)
|
|
{
|
|
hp_quad.ComputeGaussLegendrePoint(m, j);
|
|
|
|
// Compute l = \prod_{i=0}^p (x-x_i) and lk = l/(x-x_k), where
|
|
// x = hp_quad.GetHPPoint(), x_i = (i+ihoffset/2)/hinv, and x_k is the
|
|
// node closest to x, i.e. k = min(max(round(x*hinv-ihoffset/2),0),p)
|
|
mpfr_mul_si(tmp, hp_quad.GetHPPoint(), hinv, rnd);
|
|
mpfr_sub_d(tmp, tmp, 0.5*ihoffset, rnd);
|
|
mpfr_round(tmp, tmp);
|
|
int k = min(max((int)mpfr_get_si(tmp, rnd), 0), p);
|
|
mpfr_set_si(lk, 1, rnd);
|
|
for (int i = 0; i <= p; i++)
|
|
{
|
|
mpfr_set_si(tmp, 2*i+ihoffset, rnd);
|
|
mpfr_div_si(tmp, tmp, 2*hinv, rnd);
|
|
mpfr_sub(tmp, hp_quad.GetHPPoint(), tmp, rnd);
|
|
if (i != k)
|
|
{
|
|
mpfr_mul(lk, lk, tmp, rnd);
|
|
}
|
|
else
|
|
{
|
|
mpfr_set(l, tmp, rnd);
|
|
}
|
|
}
|
|
mpfr_mul(l, l, lk, rnd);
|
|
mpfr_set(wi, w0, rnd);
|
|
for (int i = 0; true; i++)
|
|
{
|
|
if (i != k)
|
|
{
|
|
// tmp = l/(wi*(x - x_i))
|
|
mpfr_set_si(tmp, 2*i+ihoffset, rnd);
|
|
mpfr_div_si(tmp, tmp, 2*hinv, rnd);
|
|
mpfr_sub(tmp, hp_quad.GetHPPoint(), tmp, rnd);
|
|
mpfr_mul(tmp, tmp, wi, rnd);
|
|
mpfr_div(tmp, l, tmp, rnd);
|
|
}
|
|
else
|
|
{
|
|
// tmp = lk/wi
|
|
mpfr_div(tmp, lk, wi, rnd);
|
|
}
|
|
// weights[i] += hp_quad.weight*tmp
|
|
mpfr_mul(tmp, tmp, hp_quad.GetHPWeight(), rnd);
|
|
mpfr_add(weights[i], weights[i], tmp, rnd);
|
|
|
|
if (i == p) { break; }
|
|
|
|
// update wi *= (i+1)/(i-p)
|
|
mpfr_mul_si(wi, wi, i+1, rnd);
|
|
mpfr_div_si(wi, wi, i-p, rnd);
|
|
}
|
|
}
|
|
for (int i = 0; i < n; i++)
|
|
{
|
|
ir->IntPoint(i).weight = mpfr_get_d(weights[i], rnd);
|
|
mpfr_clear(weights[i]);
|
|
}
|
|
delete [] weights;
|
|
mpfr_clears(l, lk, w0, wi, tmp, (mpfr_ptr) 0);
|
|
|
|
#endif // MFEM_USE_MPFR
|
|
|
|
}
|
|
|
|
|
|
int Quadrature1D::CheckClosed(int type)
|
|
{
|
|
switch (type)
|
|
{
|
|
case GaussLobatto:
|
|
case ClosedUniform:
|
|
case ClosedGL:
|
|
return type;
|
|
default:
|
|
return Invalid;
|
|
}
|
|
}
|
|
|
|
int Quadrature1D::CheckOpen(int type)
|
|
{
|
|
switch (type)
|
|
{
|
|
case GaussLegendre:
|
|
case GaussLobatto:
|
|
case OpenUniform:
|
|
case ClosedUniform:
|
|
case OpenHalfUniform:
|
|
case ClosedGL:
|
|
return type; // all types can work as open
|
|
default:
|
|
return Invalid;
|
|
}
|
|
}
|
|
|
|
|
|
IntegrationRules IntRules(0, Quadrature1D::GaussLegendre);
|
|
|
|
IntegrationRules RefinedIntRules(1, Quadrature1D::GaussLegendre);
|
|
|
|
IntegrationRules::IntegrationRules(int ref, int type)
|
|
: quad_type(type)
|
|
{
|
|
refined = ref;
|
|
|
|
if (refined < 0) { own_rules = 0; return; }
|
|
|
|
own_rules = 1;
|
|
|
|
const MemoryType h_mt = MemoryType::HOST;
|
|
PointIntRules.SetSize(2, h_mt);
|
|
PointIntRules = NULL;
|
|
|
|
SegmentIntRules.SetSize(32, h_mt);
|
|
SegmentIntRules = NULL;
|
|
|
|
// TriangleIntegrationRule() assumes that this size is >= 26
|
|
TriangleIntRules.SetSize(32, h_mt);
|
|
TriangleIntRules = NULL;
|
|
|
|
SquareIntRules.SetSize(32, h_mt);
|
|
SquareIntRules = NULL;
|
|
|
|
// TetrahedronIntegrationRule() assumes that this size is >= 10
|
|
TetrahedronIntRules.SetSize(32, h_mt);
|
|
TetrahedronIntRules = NULL;
|
|
|
|
PyramidIntRules.SetSize(32, h_mt);
|
|
PyramidIntRules = NULL;
|
|
|
|
PrismIntRules.SetSize(32, h_mt);
|
|
PrismIntRules = NULL;
|
|
|
|
CubeIntRules.SetSize(32, h_mt);
|
|
CubeIntRules = NULL;
|
|
|
|
#if defined(MFEM_THREAD_SAFE) && defined(MFEM_USE_OPENMP)
|
|
IntRuleLocks.SetSize(Geometry::NUM_GEOMETRIES, h_mt);
|
|
for (int i = 0; i < Geometry::NUM_GEOMETRIES; i++)
|
|
{
|
|
omp_init_lock(&IntRuleLocks[i]);
|
|
}
|
|
#endif
|
|
}
|
|
|
|
const IntegrationRule &IntegrationRules::Get(int GeomType, int Order)
|
|
{
|
|
Array<IntegrationRule *> *ir_array = NULL;
|
|
|
|
switch (GeomType)
|
|
{
|
|
case Geometry::POINT: ir_array = &PointIntRules; Order = 0; break;
|
|
case Geometry::SEGMENT: ir_array = &SegmentIntRules; break;
|
|
case Geometry::TRIANGLE: ir_array = &TriangleIntRules; break;
|
|
case Geometry::SQUARE: ir_array = &SquareIntRules; break;
|
|
case Geometry::TETRAHEDRON: ir_array = &TetrahedronIntRules; break;
|
|
case Geometry::CUBE: ir_array = &CubeIntRules; break;
|
|
case Geometry::PRISM: ir_array = &PrismIntRules; break;
|
|
case Geometry::PYRAMID: ir_array = &PyramidIntRules; break;
|
|
case Geometry::INVALID:
|
|
case Geometry::NUM_GEOMETRIES:
|
|
MFEM_ABORT("Unknown type of reference element!");
|
|
}
|
|
|
|
if (Order < 0)
|
|
{
|
|
Order = 0;
|
|
}
|
|
|
|
#if defined(MFEM_THREAD_SAFE) && defined(MFEM_USE_OPENMP)
|
|
omp_set_lock(&IntRuleLocks[GeomType]);
|
|
#endif
|
|
|
|
if (!HaveIntRule(*ir_array, Order))
|
|
{
|
|
IntegrationRule *ir = GenerateIntegrationRule(GeomType, Order);
|
|
#ifdef MFEM_DEBUG
|
|
int RealOrder = Order;
|
|
while (RealOrder+1 < ir_array->Size() && (*ir_array)[RealOrder+1] == ir)
|
|
{
|
|
RealOrder++;
|
|
}
|
|
MFEM_VERIFY(RealOrder == ir->GetOrder(), "internal error");
|
|
#else
|
|
MFEM_CONTRACT_VAR(ir);
|
|
#endif
|
|
}
|
|
|
|
#if defined(MFEM_THREAD_SAFE) && defined(MFEM_USE_OPENMP)
|
|
omp_unset_lock(&IntRuleLocks[GeomType]);
|
|
#endif
|
|
|
|
return *(*ir_array)[Order];
|
|
}
|
|
|
|
void IntegrationRules::Set(int GeomType, int Order, IntegrationRule &IntRule)
|
|
{
|
|
Array<IntegrationRule *> *ir_array = NULL;
|
|
|
|
switch (GeomType)
|
|
{
|
|
case Geometry::POINT: ir_array = &PointIntRules; break;
|
|
case Geometry::SEGMENT: ir_array = &SegmentIntRules; break;
|
|
case Geometry::TRIANGLE: ir_array = &TriangleIntRules; break;
|
|
case Geometry::SQUARE: ir_array = &SquareIntRules; break;
|
|
case Geometry::TETRAHEDRON: ir_array = &TetrahedronIntRules; break;
|
|
case Geometry::CUBE: ir_array = &CubeIntRules; break;
|
|
case Geometry::PRISM: ir_array = &PrismIntRules; break;
|
|
case Geometry::PYRAMID: ir_array = &PyramidIntRules; break;
|
|
case Geometry::INVALID:
|
|
case Geometry::NUM_GEOMETRIES:
|
|
MFEM_ABORT("Unknown type of reference element!");
|
|
}
|
|
|
|
#if defined(MFEM_THREAD_SAFE) && defined(MFEM_USE_OPENMP)
|
|
omp_set_lock(&IntRuleLocks[GeomType]);
|
|
#endif
|
|
|
|
if (HaveIntRule(*ir_array, Order))
|
|
{
|
|
MFEM_ABORT("Overwriting set rules is not supported!");
|
|
}
|
|
|
|
AllocIntRule(*ir_array, Order);
|
|
|
|
(*ir_array)[Order] = &IntRule;
|
|
|
|
#if defined(MFEM_THREAD_SAFE) && defined(MFEM_USE_OPENMP)
|
|
omp_unset_lock(&IntRuleLocks[GeomType]);
|
|
#endif
|
|
}
|
|
|
|
void IntegrationRules::DeleteIntRuleArray(
|
|
Array<IntegrationRule *> &ir_array) const
|
|
{
|
|
// Many of the intrules have multiple contiguous copies in the ir_array
|
|
// so we have to be careful to not delete them twice.
|
|
IntegrationRule *ir = NULL;
|
|
for (int i = 0; i < ir_array.Size(); i++)
|
|
{
|
|
if (ir_array[i] != NULL && ir_array[i] != ir)
|
|
{
|
|
ir = ir_array[i];
|
|
delete ir;
|
|
}
|
|
}
|
|
}
|
|
|
|
IntegrationRules::~IntegrationRules()
|
|
{
|
|
#if defined(MFEM_THREAD_SAFE) && defined(MFEM_USE_OPENMP)
|
|
for (int i = 0; i < Geometry::NUM_GEOMETRIES; i++)
|
|
{
|
|
omp_destroy_lock(&IntRuleLocks[i]);
|
|
}
|
|
#endif
|
|
|
|
if (!own_rules) { return; }
|
|
|
|
DeleteIntRuleArray(PointIntRules);
|
|
DeleteIntRuleArray(SegmentIntRules);
|
|
DeleteIntRuleArray(TriangleIntRules);
|
|
DeleteIntRuleArray(SquareIntRules);
|
|
DeleteIntRuleArray(TetrahedronIntRules);
|
|
DeleteIntRuleArray(CubeIntRules);
|
|
DeleteIntRuleArray(PrismIntRules);
|
|
DeleteIntRuleArray(PyramidIntRules);
|
|
}
|
|
|
|
|
|
IntegrationRule *IntegrationRules::GenerateIntegrationRule(int GeomType,
|
|
int Order)
|
|
{
|
|
switch (GeomType)
|
|
{
|
|
case Geometry::POINT:
|
|
return PointIntegrationRule(Order);
|
|
case Geometry::SEGMENT:
|
|
return SegmentIntegrationRule(Order);
|
|
case Geometry::TRIANGLE:
|
|
return TriangleIntegrationRule(Order);
|
|
case Geometry::SQUARE:
|
|
return SquareIntegrationRule(Order);
|
|
case Geometry::TETRAHEDRON:
|
|
return TetrahedronIntegrationRule(Order);
|
|
case Geometry::CUBE:
|
|
return CubeIntegrationRule(Order);
|
|
case Geometry::PRISM:
|
|
return PrismIntegrationRule(Order);
|
|
case Geometry::PYRAMID:
|
|
return PyramidIntegrationRule(Order);
|
|
case Geometry::INVALID:
|
|
case Geometry::NUM_GEOMETRIES:
|
|
MFEM_ABORT("Unknown type of reference element!");
|
|
}
|
|
return NULL;
|
|
}
|
|
|
|
|
|
// Integration rules for a point
|
|
IntegrationRule *IntegrationRules::PointIntegrationRule(int Order)
|
|
{
|
|
if (Order > 1)
|
|
{
|
|
MFEM_ABORT("Point Integration Rule of Order > 1 not defined");
|
|
return NULL;
|
|
}
|
|
|
|
IntegrationRule *ir = new IntegrationRule(1);
|
|
ir->IntPoint(0).x = .0;
|
|
ir->IntPoint(0).weight = 1.;
|
|
ir->SetOrder(1);
|
|
|
|
PointIntRules[1] = PointIntRules[0] = ir;
|
|
|
|
return ir;
|
|
}
|
|
|
|
// Integration rules for line segment [0,1]
|
|
IntegrationRule *IntegrationRules::SegmentIntegrationRule(int Order)
|
|
{
|
|
int RealOrder = GetSegmentRealOrder(Order); // RealOrder >= Order
|
|
// Order is one of {RealOrder-1,RealOrder}
|
|
AllocIntRule(SegmentIntRules, RealOrder);
|
|
|
|
IntegrationRule *ir = new IntegrationRule;
|
|
|
|
int n = 0;
|
|
// n is the number of points to achieve the exact integral of a
|
|
// degree Order polynomial
|
|
switch (quad_type)
|
|
{
|
|
case Quadrature1D::GaussLegendre:
|
|
{
|
|
// Gauss-Legendre is exact for 2*n-1
|
|
n = Order/2 + 1;
|
|
QuadratureFunctions1D::GaussLegendre(n, ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::GaussLobatto:
|
|
{
|
|
// Gauss-Lobatto is exact for 2*n-3
|
|
n = Order/2 + 2;
|
|
QuadratureFunctions1D::GaussLobatto(n, ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::OpenUniform:
|
|
{
|
|
// Open Newton Cotes is exact for n-(n+1)%2 = n-1+n%2
|
|
n = Order | 1; // n is always odd
|
|
QuadratureFunctions1D::OpenUniform(n, ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::ClosedUniform:
|
|
{
|
|
// Closed Newton Cotes is exact for n-(n+1)%2 = n-1+n%2
|
|
n = Order | 1; // n is always odd
|
|
QuadratureFunctions1D::ClosedUniform(n, ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::OpenHalfUniform:
|
|
{
|
|
// Open half Newton Cotes is exact for n-(n+1)%2 = n-1+n%2
|
|
n = Order | 1; // n is always odd
|
|
QuadratureFunctions1D::OpenHalfUniform(n, ir);
|
|
break;
|
|
}
|
|
case Quadrature1D::Invalid:
|
|
{
|
|
MFEM_ABORT("unknown Quadrature1D type: " << quad_type);
|
|
}
|
|
}
|
|
if (refined)
|
|
{
|
|
// Effectively passing memory management to SegmentIntegrationRules
|
|
IntegrationRule *refined_ir = new IntegrationRule(2*n);
|
|
refined_ir->SetOrder(ir->GetOrder());
|
|
for (int j = 0; j < n; j++)
|
|
{
|
|
refined_ir->IntPoint(j).x = ir->IntPoint(j).x/2.0;
|
|
refined_ir->IntPoint(j).weight = ir->IntPoint(j).weight/2.0;
|
|
refined_ir->IntPoint(j+n).x = 0.5 + ir->IntPoint(j).x/2.0;
|
|
refined_ir->IntPoint(j+n).weight = ir->IntPoint(j).weight/2.0;
|
|
}
|
|
delete ir;
|
|
ir = refined_ir;
|
|
}
|
|
SegmentIntRules[RealOrder-1] = SegmentIntRules[RealOrder] = ir;
|
|
return ir;
|
|
}
|
|
|
|
// Integration rules for reference triangle {[0,0],[1,0],[0,1]}
|
|
IntegrationRule *IntegrationRules::TriangleIntegrationRule(int Order)
|
|
{
|
|
IntegrationRule *ir = NULL;
|
|
// Note: Set TriangleIntRules[*] to ir only *after* ir is fully constructed.
|
|
// This is needed in multithreaded environment.
|
|
|
|
// assuming that orders <= 25 are pre-allocated
|
|
switch (Order)
|
|
{
|
|
case 0: // 1 point - degree 1
|
|
case 1:
|
|
ir = new IntegrationRule(1);
|
|
ir->AddTriMidPoint(0, 0.5);
|
|
ir->SetOrder(1);
|
|
TriangleIntRules[0] = TriangleIntRules[1] = ir;
|
|
return ir;
|
|
|
|
case 2: // 3 point - 2 degree
|
|
ir = new IntegrationRule(3);
|
|
ir->AddTriPoints3(0, 1./6., 1./6.);
|
|
ir->SetOrder(2);
|
|
TriangleIntRules[2] = ir;
|
|
// interior points
|
|
return ir;
|
|
|
|
case 3: // 4 point - 3 degree (has one negative weight)
|
|
ir = new IntegrationRule(4);
|
|
ir->AddTriMidPoint(0, -0.28125); // -9./32.
|
|
ir->AddTriPoints3(1, 0.2, 25./96.);
|
|
ir->SetOrder(3);
|
|
TriangleIntRules[3] = ir;
|
|
return ir;
|
|
|
|
case 4: // 6 point - 4 degree
|
|
ir = new IntegrationRule(6);
|
|
ir->AddTriPoints3(0, 0.091576213509770743460, 0.054975871827660933819);
|
|
ir->AddTriPoints3(3, 0.44594849091596488632, 0.11169079483900573285);
|
|
ir->SetOrder(4);
|
|
TriangleIntRules[4] = ir;
|
|
return ir;
|
|
|
|
case 5: // 7 point - 5 degree
|
|
ir = new IntegrationRule(7);
|
|
ir->AddTriMidPoint(0, 0.1125);
|
|
ir->AddTriPoints3(1, 0.10128650732345633880, 0.062969590272413576298);
|
|
ir->AddTriPoints3(4, 0.47014206410511508977, 0.066197076394253090369);
|
|
ir->SetOrder(5);
|
|
TriangleIntRules[5] = ir;
|
|
return ir;
|
|
|
|
case 6: // 12 point - 6 degree
|
|
ir = new IntegrationRule(12);
|
|
ir->AddTriPoints3(0, 0.063089014491502228340, 0.025422453185103408460);
|
|
ir->AddTriPoints3(3, 0.24928674517091042129, 0.058393137863189683013);
|
|
ir->AddTriPoints6(6, 0.053145049844816947353, 0.31035245103378440542,
|
|
0.041425537809186787597);
|
|
ir->SetOrder(6);
|
|
TriangleIntRules[6] = ir;
|
|
return ir;
|
|
|
|
case 7: // 12 point - degree 7
|
|
ir = new IntegrationRule(12);
|
|
ir->AddTriPoints3R(0, 0.062382265094402118174, 0.067517867073916085443,
|
|
0.026517028157436251429);
|
|
ir->AddTriPoints3R(3, 0.055225456656926611737, 0.32150249385198182267,
|
|
0.043881408714446055037);
|
|
// slightly better with explicit 3rd area coordinate
|
|
ir->AddTriPoints3R(6, 0.034324302945097146470, 0.66094919618673565761,
|
|
0.30472650086816719592, 0.028775042784981585738);
|
|
ir->AddTriPoints3R(9, 0.51584233435359177926, 0.27771616697639178257,
|
|
0.20644149867001643817, 0.067493187009802774463);
|
|
ir->SetOrder(7);
|
|
TriangleIntRules[7] = ir;
|
|
return ir;
|
|
|
|
case 8: // 16 point - 8 degree
|
|
ir = new IntegrationRule(16);
|
|
ir->AddTriMidPoint(0, 0.0721578038388935841255455552445323);
|
|
ir->AddTriPoints3(1, 0.170569307751760206622293501491464,
|
|
0.0516086852673591251408957751460645);
|
|
ir->AddTriPoints3(4, 0.0505472283170309754584235505965989,
|
|
0.0162292488115990401554629641708902);
|
|
ir->AddTriPoints3(7, 0.459292588292723156028815514494169,
|
|
0.0475458171336423123969480521942921);
|
|
ir->AddTriPoints6(10, 0.008394777409957605337213834539296,
|
|
0.263112829634638113421785786284643,
|
|
0.0136151570872174971324223450369544);
|
|
ir->SetOrder(8);
|
|
TriangleIntRules[8] = ir;
|
|
return ir;
|
|
|
|
case 9: // 19 point - 9 degree
|
|
ir = new IntegrationRule(19);
|
|
ir->AddTriMidPoint(0, 0.0485678981413994169096209912536443);
|
|
ir->AddTriPoints3b(1, 0.020634961602524744433,
|
|
0.0156673501135695352684274156436046);
|
|
ir->AddTriPoints3b(4, 0.12582081701412672546,
|
|
0.0389137705023871396583696781497019);
|
|
ir->AddTriPoints3(7, 0.188203535619032730240961280467335,
|
|
0.0398238694636051265164458871320226);
|
|
ir->AddTriPoints3(10, 0.0447295133944527098651065899662763,
|
|
0.0127888378293490156308393992794999);
|
|
ir->AddTriPoints6(13, 0.0368384120547362836348175987833851,
|
|
0.2219629891607656956751025276931919,
|
|
0.0216417696886446886446886446886446);
|
|
ir->SetOrder(9);
|
|
TriangleIntRules[9] = ir;
|
|
return ir;
|
|
|
|
case 10: // 25 point - 10 degree
|
|
ir = new IntegrationRule(25);
|
|
ir->AddTriMidPoint(0, 0.0454089951913767900476432975500142);
|
|
ir->AddTriPoints3b(1, 0.028844733232685245264984935583748,
|
|
0.0183629788782333523585030359456832);
|
|
ir->AddTriPoints3(4, 0.109481575485037054795458631340522,
|
|
0.0226605297177639673913028223692986);
|
|
ir->AddTriPoints6(7, 0.141707219414879954756683250476361,
|
|
0.307939838764120950165155022930631,
|
|
0.0363789584227100543021575883096803);
|
|
ir->AddTriPoints6(13, 0.025003534762686386073988481007746,
|
|
0.246672560639902693917276465411176,
|
|
0.0141636212655287424183685307910495);
|
|
ir->AddTriPoints6(19, 0.0095408154002994575801528096228873,
|
|
0.0668032510122002657735402127620247,
|
|
4.71083348186641172996373548344341E-03);
|
|
ir->SetOrder(10);
|
|
TriangleIntRules[10] = ir;
|
|
return ir;
|
|
|
|
case 11: // 28 point -- 11 degree
|
|
ir = new IntegrationRule(28);
|
|
ir->AddTriPoints6(0, 0.0,
|
|
0.141129718717363295960826061941652,
|
|
3.68119189165027713212944752369032E-03);
|
|
ir->AddTriMidPoint(6, 0.0439886505811161193990465846607278);
|
|
ir->AddTriPoints3(7, 0.0259891409282873952600324854988407,
|
|
4.37215577686801152475821439991262E-03);
|
|
ir->AddTriPoints3(10, 0.0942875026479224956305697762754049,
|
|
0.0190407859969674687575121697178070);
|
|
ir->AddTriPoints3b(13, 0.010726449965572372516734795387128,
|
|
9.42772402806564602923839129555767E-03);
|
|
ir->AddTriPoints3(16, 0.207343382614511333452934024112966,
|
|
0.0360798487723697630620149942932315);
|
|
ir->AddTriPoints3b(19, 0.122184388599015809877869236727746,
|
|
0.0346645693527679499208828254519072);
|
|
ir->AddTriPoints6(22, 0.0448416775891304433090523914688007,
|
|
0.2772206675282791551488214673424523,
|
|
0.0205281577146442833208261574536469);
|
|
ir->SetOrder(11);
|
|
TriangleIntRules[11] = ir;
|
|
return ir;
|
|
|
|
case 12: // 33 point - 12 degree
|
|
ir = new IntegrationRule(33);
|
|
ir->AddTriPoints3b(0, 2.35652204523900E-02, 1.28655332202275E-02);
|
|
ir->AddTriPoints3b(3, 1.20551215411079E-01, 2.18462722690190E-02);
|
|
ir->AddTriPoints3(6, 2.71210385012116E-01, 3.14291121089425E-02);
|
|
ir->AddTriPoints3(9, 1.27576145541586E-01, 1.73980564653545E-02);
|
|
ir->AddTriPoints3(12, 2.13173504532100E-02, 3.08313052577950E-03);
|
|
ir->AddTriPoints6(15, 1.15343494534698E-01, 2.75713269685514E-01,
|
|
2.01857788831905E-02);
|
|
ir->AddTriPoints6(21, 2.28383322222570E-02, 2.81325580989940E-01,
|
|
1.11783866011515E-02);
|
|
ir->AddTriPoints6(27, 2.57340505483300E-02, 1.16251915907597E-01,
|
|
8.65811555432950E-03);
|
|
ir->SetOrder(12);
|
|
TriangleIntRules[12] = ir;
|
|
return ir;
|
|
|
|
case 13: // 37 point - 13 degree
|
|
ir = new IntegrationRule(37);
|
|
ir->AddTriPoints3b(0, 0.0,
|
|
2.67845189554543044455908674650066E-03);
|
|
ir->AddTriMidPoint(3, 0.0293480398063595158995969648597808);
|
|
ir->AddTriPoints3(4, 0.0246071886432302181878499494124643,
|
|
3.92538414805004016372590903990464E-03);
|
|
ir->AddTriPoints3b(7, 0.159382493797610632566158925635800,
|
|
0.0253344765879434817105476355306468);
|
|
ir->AddTriPoints3(10, 0.227900255506160619646298948153592,
|
|
0.0250401630452545330803738542916538);
|
|
ir->AddTriPoints3(13, 0.116213058883517905247155321839271,
|
|
0.0158235572961491595176634480481793);
|
|
ir->AddTriPoints3b(16, 0.046794039901841694097491569577008,
|
|
0.0157462815379843978450278590138683);
|
|
ir->AddTriPoints6(19, 0.0227978945382486125477207592747430,
|
|
0.1254265183163409177176192369310890,
|
|
7.90126610763037567956187298486575E-03);
|
|
ir->AddTriPoints6(25, 0.0162757709910885409437036075960413,
|
|
0.2909269114422506044621801030055257,
|
|
7.99081889046420266145965132482933E-03);
|
|
ir->AddTriPoints6(31, 0.0897330604516053590796290561145196,
|
|
0.2723110556841851025078181617634414,
|
|
0.0182757511120486476280967518782978);
|
|
ir->SetOrder(13);
|
|
TriangleIntRules[13] = ir;
|
|
return ir;
|
|
|
|
case 14: // 42 point - 14 degree
|
|
ir = new IntegrationRule(42);
|
|
ir->AddTriPoints3b(0, 2.20721792756430E-02, 1.09417906847145E-02);
|
|
ir->AddTriPoints3b(3, 1.64710561319092E-01, 1.63941767720625E-02);
|
|
ir->AddTriPoints3(6, 2.73477528308839E-01, 2.58870522536460E-02);
|
|
ir->AddTriPoints3(9, 1.77205532412543E-01, 2.10812943684965E-02);
|
|
ir->AddTriPoints3(12, 6.17998830908730E-02, 7.21684983488850E-03);
|
|
ir->AddTriPoints3(15, 1.93909612487010E-02, 2.46170180120000E-03);
|
|
ir->AddTriPoints6(18, 5.71247574036480E-02, 1.72266687821356E-01,
|
|
1.23328766062820E-02);
|
|
ir->AddTriPoints6(24, 9.29162493569720E-02, 3.36861459796345E-01,
|
|
1.92857553935305E-02);
|
|
ir->AddTriPoints6(30, 1.46469500556540E-02, 2.98372882136258E-01,
|
|
7.21815405676700E-03);
|
|
ir->AddTriPoints6(36, 1.26833093287200E-03, 1.18974497696957E-01,
|
|
2.50511441925050E-03);
|
|
ir->SetOrder(14);
|
|
TriangleIntRules[14] = ir;
|
|
return ir;
|
|
|
|
case 15: // 54 point - 15 degree
|
|
ir = new IntegrationRule(54);
|
|
ir->AddTriPoints3b(0, 0.0834384072617499333, 0.016330909424402645);
|
|
ir->AddTriPoints3b(3, 0.192779070841738867, 0.01370640901568218);
|
|
ir->AddTriPoints3(6, 0.293197167913025367, 0.01325501829935165);
|
|
ir->AddTriPoints3(9, 0.146467786942772933, 0.014607981068243055);
|
|
ir->AddTriPoints3(12, 0.0563628676656034333, 0.005292304033121995);
|
|
ir->AddTriPoints3(15, 0.0165751268583703333, 0.0018073215320460175);
|
|
ir->AddTriPoints6(18, 0.0099122033092248, 0.239534554154794445,
|
|
0.004263874050854718);
|
|
ir->AddTriPoints6(24, 0.015803770630228, 0.404878807318339958,
|
|
0.006958088258345965);
|
|
ir->AddTriPoints6(30, 0.00514360881697066667, 0.0950021131130448885,
|
|
0.0021459664703674175);
|
|
ir->AddTriPoints6(36, 0.0489223257529888, 0.149753107322273969,
|
|
0.008117664640887445);
|
|
ir->AddTriPoints6(42, 0.0687687486325192, 0.286919612441334979,
|
|
0.012803670460631195);
|
|
ir->AddTriPoints6(48, 0.1684044181246992, 0.281835668099084562,
|
|
0.016544097765822835);
|
|
ir->SetOrder(15);
|
|
TriangleIntRules[15] = ir;
|
|
return ir;
|
|
|
|
case 16: // 61 point - 17 degree (used for 16 as well)
|
|
case 17:
|
|
ir = new IntegrationRule(61);
|
|
ir->AddTriMidPoint(0, 1.67185996454015E-02);
|
|
ir->AddTriPoints3b(1, 5.65891888645200E-03, 2.54670772025350E-03);
|
|
ir->AddTriPoints3b(4, 3.56473547507510E-02, 7.33543226381900E-03);
|
|
ir->AddTriPoints3b(7, 9.95200619584370E-02, 1.21754391768360E-02);
|
|
ir->AddTriPoints3b(10, 1.99467521245206E-01, 1.55537754344845E-02);
|
|
ir->AddTriPoints3 (13, 2.52141267970953E-01, 1.56285556093100E-02);
|
|
ir->AddTriPoints3 (16, 1.62047004658461E-01, 1.24078271698325E-02);
|
|
ir->AddTriPoints3 (19, 7.58758822607460E-02, 7.02803653527850E-03);
|
|
ir->AddTriPoints3 (22, 1.56547269678220E-02, 1.59733808688950E-03);
|
|
ir->AddTriPoints6 (25, 1.01869288269190E-02, 3.34319867363658E-01,
|
|
4.05982765949650E-03);
|
|
ir->AddTriPoints6 (31, 1.35440871671036E-01, 2.92221537796944E-01,
|
|
1.34028711415815E-02);
|
|
ir->AddTriPoints6 (37, 5.44239242905830E-02, 3.19574885423190E-01,
|
|
9.22999660541100E-03);
|
|
ir->AddTriPoints6 (43, 1.28685608336370E-02, 1.90704224192292E-01,
|
|
4.23843426716400E-03);
|
|
ir->AddTriPoints6 (49, 6.71657824135240E-02, 1.80483211648746E-01,
|
|
9.14639838501250E-03);
|
|
ir->AddTriPoints6 (55, 1.46631822248280E-02, 8.07113136795640E-02,
|
|
3.33281600208250E-03);
|
|
ir->SetOrder(17);
|
|
TriangleIntRules[16] = TriangleIntRules[17] = ir;
|
|
return ir;
|
|
|
|
case 18: // 73 point - 19 degree (used for 18 as well)
|
|
case 19:
|
|
ir = new IntegrationRule(73);
|
|
ir->AddTriMidPoint(0, 0.0164531656944595);
|
|
ir->AddTriPoints3b(1, 0.020780025853987, 0.005165365945636);
|
|
ir->AddTriPoints3b(4, 0.090926214604215, 0.011193623631508);
|
|
ir->AddTriPoints3b(7, 0.197166638701138, 0.015133062934734);
|
|
ir->AddTriPoints3 (10, 0.255551654403098, 0.015245483901099);
|
|
ir->AddTriPoints3 (13, 0.17707794215213, 0.0120796063708205);
|
|
ir->AddTriPoints3 (16, 0.110061053227952, 0.0080254017934005);
|
|
ir->AddTriPoints3 (19, 0.05552862425184, 0.004042290130892);
|
|
ir->AddTriPoints3 (22, 0.012621863777229, 0.0010396810137425);
|
|
ir->AddTriPoints6 (25, 0.003611417848412, 0.395754787356943,
|
|
0.0019424384524905);
|
|
ir->AddTriPoints6 (31, 0.13446675453078, 0.307929983880436,
|
|
0.012787080306011);
|
|
ir->AddTriPoints6 (37, 0.014446025776115, 0.26456694840652,
|
|
0.004440451786669);
|
|
ir->AddTriPoints6 (43, 0.046933578838178, 0.358539352205951,
|
|
0.0080622733808655);
|
|
ir->AddTriPoints6 (49, 0.002861120350567, 0.157807405968595,
|
|
0.0012459709087455);
|
|
ir->AddTriPoints6 (55, 0.075050596975911, 0.223861424097916,
|
|
0.0091214200594755);
|
|
ir->AddTriPoints6 (61, 0.03464707481676, 0.142421601113383,
|
|
0.0051292818680995);
|
|
ir->AddTriPoints6 (67, 0.065494628082938, 0.010161119296278,
|
|
0.001899964427651);
|
|
ir->SetOrder(19);
|
|
TriangleIntRules[18] = TriangleIntRules[19] = ir;
|
|
return ir;
|
|
|
|
case 20: // 85 point - 20 degree
|
|
ir = new IntegrationRule(85);
|
|
ir->AddTriMidPoint(0, 0.01380521349884976);
|
|
ir->AddTriPoints3b(1, 0.001500649324429, 0.00088951477366337);
|
|
ir->AddTriPoints3b(4, 0.0941397519389508667, 0.010056199056980585);
|
|
ir->AddTriPoints3b(7, 0.2044721240895264, 0.013408923629665785);
|
|
ir->AddTriPoints3(10, 0.264500202532787333, 0.012261566900751005);
|
|
ir->AddTriPoints3(13, 0.211018964092076767, 0.008197289205347695);
|
|
ir->AddTriPoints3(16, 0.107735607171271333, 0.0073979536993248);
|
|
ir->AddTriPoints3(19, 0.0390690878378026667, 0.0022896411388521255);
|
|
ir->AddTriPoints3(22, 0.0111743797293296333, 0.0008259132577881085);
|
|
ir->AddTriPoints6(25, 0.00534961818733726667, 0.0635496659083522206,
|
|
0.001174585454287792);
|
|
ir->AddTriPoints6(31, 0.00795481706619893333, 0.157106918940706982,
|
|
0.0022329628770908965);
|
|
ir->AddTriPoints6(37, 0.0104223982812638, 0.395642114364374018,
|
|
0.003049783403953986);
|
|
ir->AddTriPoints6(43, 0.0109644147961233333, 0.273167570712910522,
|
|
0.0034455406635941015);
|
|
ir->AddTriPoints6(49, 0.0385667120854623333, 0.101785382485017108,
|
|
0.0039987375362390815);
|
|
ir->AddTriPoints6(55, 0.0355805078172182, 0.446658549176413815,
|
|
0.003693067142668012);
|
|
ir->AddTriPoints6(61, 0.0496708163627641333, 0.199010794149503095,
|
|
0.00639966593932413);
|
|
ir->AddTriPoints6(67, 0.0585197250843317333, 0.3242611836922827,
|
|
0.008629035587848275);
|
|
ir->AddTriPoints6(73, 0.121497787004394267, 0.208531363210132855,
|
|
0.009336472951467735);
|
|
ir->AddTriPoints6(79, 0.140710844943938733, 0.323170566536257485,
|
|
0.01140911202919763);
|
|
ir->SetOrder(20);
|
|
TriangleIntRules[20] = ir;
|
|
return ir;
|
|
|
|
case 21: // 126 point - 25 degree (used also for degrees from 21 to 24)
|
|
case 22:
|
|
case 23:
|
|
case 24:
|
|
case 25:
|
|
ir = new IntegrationRule(126);
|
|
ir->AddTriPoints3b(0, 0.0279464830731742, 0.0040027909400102085);
|
|
ir->AddTriPoints3b(3, 0.131178601327651467, 0.00797353841619525);
|
|
ir->AddTriPoints3b(6, 0.220221729512072267, 0.006554570615397765);
|
|
ir->AddTriPoints3 (9, 0.298443234019804467, 0.00979150048281781);
|
|
ir->AddTriPoints3(12, 0.2340441723373718, 0.008235442720768635);
|
|
ir->AddTriPoints3(15, 0.151468334609017567, 0.00427363953704605);
|
|
ir->AddTriPoints3(18, 0.112733893545993667, 0.004080942928613246);
|
|
ir->AddTriPoints3(21, 0.0777156920915263, 0.0030605732699918895);
|
|
ir->AddTriPoints3(24, 0.034893093614297, 0.0014542491324683325);
|
|
ir->AddTriPoints3(27, 0.00725818462093236667, 0.00034613762283099815);
|
|
ir->AddTriPoints6(30, 0.0012923527044422, 0.227214452153364077,
|
|
0.0006241445996386985);
|
|
ir->AddTriPoints6(36, 0.0053997012721162, 0.435010554853571706,
|
|
0.001702376454401511);
|
|
ir->AddTriPoints6(42, 0.006384003033975, 0.320309599272204437,
|
|
0.0016798271630320255);
|
|
ir->AddTriPoints6(48, 0.00502821150199306667, 0.0917503222800051889,
|
|
0.000858078269748377);
|
|
ir->AddTriPoints6(54, 0.00682675862178186667, 0.0380108358587243835,
|
|
0.000740428158357803);
|
|
ir->AddTriPoints6(60, 0.0100161996399295333, 0.157425218485311668,
|
|
0.0017556563053643425);
|
|
ir->AddTriPoints6(66, 0.02575781317339, 0.239889659778533193,
|
|
0.003696775074853242);
|
|
ir->AddTriPoints6(72, 0.0302278981199158, 0.361943118126060531,
|
|
0.003991543738688279);
|
|
ir->AddTriPoints6(78, 0.0305049901071620667, 0.0835519609548285602,
|
|
0.0021779813065790205);
|
|
ir->AddTriPoints6(84, 0.0459565473625693333, 0.148443220732418205,
|
|
0.003682528350708916);
|
|
ir->AddTriPoints6(90, 0.0674428005402775333, 0.283739708727534955,
|
|
0.005481786423209775);
|
|
ir->AddTriPoints6(96, 0.0700450914159106, 0.406899375118787573,
|
|
0.00587498087177056);
|
|
ir->AddTriPoints6(102, 0.0839115246401166, 0.194113987024892542,
|
|
0.005007800356899285);
|
|
ir->AddTriPoints6(108, 0.120375535677152667, 0.32413434700070316,
|
|
0.00665482039381434);
|
|
ir->AddTriPoints6(114, 0.148066899157366667, 0.229277483555980969,
|
|
0.00707722325261307);
|
|
ir->AddTriPoints6(120, 0.191771865867325067, 0.325618122595983752,
|
|
0.007440689780584005);
|
|
ir->SetOrder(25);
|
|
TriangleIntRules[21] =
|
|
TriangleIntRules[22] =
|
|
TriangleIntRules[23] =
|
|
TriangleIntRules[24] =
|
|
TriangleIntRules[25] = ir;
|
|
return ir;
|
|
|
|
default:
|
|
// Grundmann-Moller rules
|
|
int i = (Order / 2) * 2 + 1; // Get closest odd # >= Order
|
|
AllocIntRule(TriangleIntRules, i);
|
|
ir = new IntegrationRule;
|
|
ir->GrundmannMollerSimplexRule(i/2,2);
|
|
TriangleIntRules[i-1] = TriangleIntRules[i] = ir;
|
|
return ir;
|
|
}
|
|
}
|
|
|
|
// Integration rules for unit square
|
|
IntegrationRule *IntegrationRules::SquareIntegrationRule(int Order)
|
|
{
|
|
int RealOrder = GetSegmentRealOrder(Order);
|
|
// Order is one of {RealOrder-1,RealOrder}
|
|
if (!HaveIntRule(SegmentIntRules, RealOrder))
|
|
{
|
|
SegmentIntegrationRule(RealOrder);
|
|
}
|
|
AllocIntRule(SquareIntRules, RealOrder); // RealOrder >= Order
|
|
SquareIntRules[RealOrder-1] =
|
|
SquareIntRules[RealOrder] =
|
|
new IntegrationRule(*SegmentIntRules[RealOrder],
|
|
*SegmentIntRules[RealOrder]);
|
|
return SquareIntRules[Order];
|
|
}
|
|
|
|
/** Integration rules for reference tetrahedron
|
|
{[0,0,0],[1,0,0],[0,1,0],[0,0,1]} */
|
|
IntegrationRule *IntegrationRules::TetrahedronIntegrationRule(int Order)
|
|
{
|
|
IntegrationRule *ir;
|
|
// Note: Set TetrahedronIntRules[*] to ir only *after* ir is fully
|
|
// constructed. This is needed in multithreaded environment.
|
|
|
|
// assuming that orders <= 9 are pre-allocated
|
|
switch (Order)
|
|
{
|
|
case 0: // 1 point - degree 1
|
|
case 1:
|
|
ir = new IntegrationRule(1);
|
|
ir->AddTetMidPoint(0, 1./6.);
|
|
ir->SetOrder(1);
|
|
TetrahedronIntRules[0] = TetrahedronIntRules[1] = ir;
|
|
return ir;
|
|
|
|
case 2: // 4 points - degree 2
|
|
ir = new IntegrationRule(4);
|
|
// ir->AddTetPoints4(0, 0.13819660112501051518, 1./24.);
|
|
ir->AddTetPoints4b(0, 0.58541019662496845446, 1./24.);
|
|
ir->SetOrder(2);
|
|
TetrahedronIntRules[2] = ir;
|
|
return ir;
|
|
|
|
case 3: // 5 points - degree 3 (negative weight)
|
|
ir = new IntegrationRule(5);
|
|
ir->AddTetMidPoint(0, -2./15.);
|
|
ir->AddTetPoints4b(1, 0.5, 0.075);
|
|
ir->SetOrder(3);
|
|
TetrahedronIntRules[3] = ir;
|
|
return ir;
|
|
|
|
case 4: // 11 points - degree 4 (negative weight)
|
|
ir = new IntegrationRule(11);
|
|
ir->AddTetPoints4(0, 1./14., 343./45000.);
|
|
ir->AddTetMidPoint(4, -74./5625.);
|
|
ir->AddTetPoints6(5, 0.10059642383320079500, 28./1125.);
|
|
ir->SetOrder(4);
|
|
TetrahedronIntRules[4] = ir;
|
|
return ir;
|
|
|
|
case 5: // 14 points - degree 5
|
|
ir = new IntegrationRule(14);
|
|
ir->AddTetPoints6(0, 0.045503704125649649492,
|
|
7.0910034628469110730E-03);
|
|
ir->AddTetPoints4(6, 0.092735250310891226402, 0.012248840519393658257);
|
|
ir->AddTetPoints4b(10, 0.067342242210098170608,
|
|
0.018781320953002641800);
|
|
ir->SetOrder(5);
|
|
TetrahedronIntRules[5] = ir;
|
|
return ir;
|
|
|
|
case 6: // 24 points - degree 6
|
|
ir = new IntegrationRule(24);
|
|
ir->AddTetPoints4(0, 0.21460287125915202929,
|
|
6.6537917096945820166E-03);
|
|
ir->AddTetPoints4(4, 0.040673958534611353116,
|
|
1.6795351758867738247E-03);
|
|
ir->AddTetPoints4b(8, 0.032986329573173468968,
|
|
9.2261969239424536825E-03);
|
|
ir->AddTetPoints12(12, 0.063661001875017525299, 0.26967233145831580803,
|
|
8.0357142857142857143E-03);
|
|
ir->SetOrder(6);
|
|
TetrahedronIntRules[6] = ir;
|
|
return ir;
|
|
|
|
case 7: // 31 points - degree 7 (negative weight)
|
|
ir = new IntegrationRule(31);
|
|
ir->AddTetPoints6(0, 0.0, 9.7001763668430335097E-04);
|
|
ir->AddTetMidPoint(6, 0.018264223466108820291);
|
|
ir->AddTetPoints4(7, 0.078213192330318064374, 0.010599941524413686916);
|
|
ir->AddTetPoints4(11, 0.12184321666390517465,
|
|
-0.062517740114331851691);
|
|
ir->AddTetPoints4b(15, 2.3825066607381275412E-03,
|
|
4.8914252630734993858E-03);
|
|
ir->AddTetPoints12(19, 0.1, 0.2, 0.027557319223985890653);
|
|
ir->SetOrder(7);
|
|
TetrahedronIntRules[7] = ir;
|
|
return ir;
|
|
|
|
case 8: // 43 points - degree 8 (negative weight)
|
|
ir = new IntegrationRule(43);
|
|
ir->AddTetPoints4(0, 5.7819505051979972532E-03,
|
|
1.6983410909288737984E-04);
|
|
ir->AddTetPoints4(4, 0.082103588310546723091,
|
|
1.9670333131339009876E-03);
|
|
ir->AddTetPoints12(8, 0.036607749553197423679, 0.19048604193463345570,
|
|
2.1405191411620925965E-03);
|
|
ir->AddTetPoints6(20, 0.050532740018894224426,
|
|
4.5796838244672818007E-03);
|
|
ir->AddTetPoints12(26, 0.22906653611681113960, 0.035639582788534043717,
|
|
5.7044858086819185068E-03);
|
|
ir->AddTetPoints4(38, 0.20682993161067320408, 0.014250305822866901248);
|
|
ir->AddTetMidPoint(42, -0.020500188658639915841);
|
|
ir->SetOrder(8);
|
|
TetrahedronIntRules[8] = ir;
|
|
return ir;
|
|
|
|
case 9: // orders 9 and higher -- Grundmann-Moller rules
|
|
ir = new IntegrationRule;
|
|
ir->GrundmannMollerSimplexRule(4,3);
|
|
TetrahedronIntRules[9] = ir;
|
|
return ir;
|
|
|
|
default: // Grundmann-Moller rules
|
|
int i = (Order / 2) * 2 + 1; // Get closest odd # >= Order
|
|
AllocIntRule(TetrahedronIntRules, i);
|
|
ir = new IntegrationRule;
|
|
ir->GrundmannMollerSimplexRule(i/2,3);
|
|
TetrahedronIntRules[i-1] = TetrahedronIntRules[i] = ir;
|
|
return ir;
|
|
}
|
|
}
|
|
|
|
// Integration rules for reference pyramid
|
|
IntegrationRule *IntegrationRules::PyramidIntegrationRule(int Order)
|
|
{
|
|
// This is a simple integration rule adapted from an integration
|
|
// rule for a cube which seems to be adequate for now. We should continue
|
|
// to search for a more appropriate integration rule designed specifically
|
|
// for pyramid elements.
|
|
const IntegrationRule &irc = Get(Geometry::CUBE, Order);
|
|
int npts = irc.GetNPoints();
|
|
AllocIntRule(PyramidIntRules, Order);
|
|
PyramidIntRules[Order] = new IntegrationRule(npts);
|
|
PyramidIntRules[Order]->SetOrder(Order);
|
|
|
|
if (npts == 1)
|
|
{
|
|
// We handle this as a special case because with only one integration
|
|
// point we cannot accurately integrate the quadratic factor
|
|
// pow(1.0 - ipc.z, 2) and the resulting weight does not match the volume
|
|
// of the reference element.
|
|
IntegrationPoint &ipp = PyramidIntRules[Order]->IntPoint(0);
|
|
ipp.x = 0.375;
|
|
ipp.y = 0.375;
|
|
ipp.z = 0.25;
|
|
ipp.weight = 1.0 / 3.0;
|
|
}
|
|
else
|
|
{
|
|
for (int k=0; k<npts; k++)
|
|
{
|
|
const IntegrationPoint &ipc = irc.IntPoint(k);
|
|
IntegrationPoint &ipp = PyramidIntRules[Order]->IntPoint(k);
|
|
ipp.x = ipc.x * (1.0 - ipc.z);
|
|
ipp.y = ipc.y * (1.0 - ipc.z);
|
|
ipp.z = ipc.z;
|
|
ipp.weight = ipc.weight * pow(1.0 - ipc.z, 2);
|
|
}
|
|
}
|
|
return PyramidIntRules[Order];
|
|
}
|
|
|
|
// Integration rules for reference prism
|
|
IntegrationRule *IntegrationRules::PrismIntegrationRule(int Order)
|
|
{
|
|
const IntegrationRule &irt = Get(Geometry::TRIANGLE, Order);
|
|
const IntegrationRule &irs = Get(Geometry::SEGMENT, Order);
|
|
int nt = irt.GetNPoints();
|
|
int ns = irs.GetNPoints();
|
|
AllocIntRule(PrismIntRules, Order);
|
|
PrismIntRules[Order] = new IntegrationRule(nt * ns);
|
|
PrismIntRules[Order]->SetOrder(std::min(irt.GetOrder(), irs.GetOrder()));
|
|
while (Order < std::min(irt.GetOrder(), irs.GetOrder()))
|
|
{
|
|
AllocIntRule(PrismIntRules, ++Order);
|
|
PrismIntRules[Order] = PrismIntRules[Order-1];
|
|
}
|
|
|
|
for (int ks=0; ks<ns; ks++)
|
|
{
|
|
const IntegrationPoint &ips = irs.IntPoint(ks);
|
|
for (int kt=0; kt<nt; kt++)
|
|
{
|
|
int kp = ks * nt + kt;
|
|
const IntegrationPoint &ipt = irt.IntPoint(kt);
|
|
IntegrationPoint &ipp = PrismIntRules[Order]->IntPoint(kp);
|
|
ipp.x = ipt.x;
|
|
ipp.y = ipt.y;
|
|
ipp.z = ips.x;
|
|
ipp.weight = ipt.weight * ips.weight;
|
|
}
|
|
}
|
|
return PrismIntRules[Order];
|
|
}
|
|
|
|
// Integration rules for reference cube
|
|
IntegrationRule *IntegrationRules::CubeIntegrationRule(int Order)
|
|
{
|
|
int RealOrder = GetSegmentRealOrder(Order);
|
|
if (!HaveIntRule(SegmentIntRules, RealOrder))
|
|
{
|
|
SegmentIntegrationRule(RealOrder);
|
|
}
|
|
AllocIntRule(CubeIntRules, RealOrder);
|
|
CubeIntRules[RealOrder-1] =
|
|
CubeIntRules[RealOrder] =
|
|
new IntegrationRule(*SegmentIntRules[RealOrder],
|
|
*SegmentIntRules[RealOrder],
|
|
*SegmentIntRules[RealOrder]);
|
|
return CubeIntRules[Order];
|
|
}
|
|
|
|
IntegrationRule& NURBSMeshRules::GetElementRule(const int elem,
|
|
const int patch, const int *ijk,
|
|
Array<const KnotVector*> const& kv) const
|
|
{
|
|
// First check whether a rule has been assigned to element index elem.
|
|
auto search = elementToRule.find(elem);
|
|
if (search != elementToRule.end())
|
|
{
|
|
return *elementRule[search->second];
|
|
}
|
|
|
|
#ifndef MFEM_THREAD_SAFE
|
|
// If no prescribed rule is given for the current element, a temporary one is
|
|
// formed by restricting a tensor-product of 1D rules to the element. The
|
|
// ownership model for this temporary rule is not thread-safe.
|
|
|
|
MFEM_VERIFY(patchRules1D.NumRows(),
|
|
"Undefined rule in NURBSMeshRules::GetElementRule");
|
|
|
|
// Use a tensor product of rules on the patch.
|
|
MFEM_VERIFY(kv.Size() == dim, "");
|
|
|
|
int np = 1;
|
|
std::vector<std::vector<real_t>> el(dim);
|
|
|
|
std::vector<int> npd;
|
|
npd.assign(3, 0);
|
|
|
|
for (int d=0; d<dim; ++d)
|
|
{
|
|
const int order = kv[d]->GetOrder();
|
|
|
|
const real_t kv0 = (*kv[d])[order + ijk[d]];
|
|
const real_t kv1 = (*kv[d])[order + ijk[d] + 1];
|
|
|
|
const bool rightEnd = (order + ijk[d] + 1) == (kv[d]->Size() - 1);
|
|
|
|
for (int i=0; i<patchRules1D(patch,d)->Size(); ++i)
|
|
{
|
|
const IntegrationPoint& ip = (*patchRules1D(patch,d))[i];
|
|
if (kv0 <= ip.x && (ip.x < kv1 || rightEnd))
|
|
{
|
|
const real_t x = (ip.x - kv0) / (kv1 - kv0);
|
|
el[d].push_back(x);
|
|
el[d].push_back(ip.weight);
|
|
}
|
|
}
|
|
|
|
npd[d] = static_cast<int>(el[d].size() / 2);
|
|
np *= npd[d];
|
|
}
|
|
|
|
temporaryElementRule.SetSize(np);
|
|
|
|
// Set temporaryElementRule[i + j*npd[0] + k*npd[0]*npd[1]] =
|
|
// (el[0][2*i], el[1][2*j], el[2][2*k])
|
|
|
|
MFEM_VERIFY(npd[0] > 0 && npd[1] > 0, "Assuming 2D or 3D");
|
|
|
|
for (int i = 0; i < npd[0]; ++i)
|
|
{
|
|
for (int j = 0; j < npd[1]; ++j)
|
|
{
|
|
for (int k = 0; k < std::max(npd[2], 1); ++k)
|
|
{
|
|
const int id = i + j*npd[0] + k*npd[0]*npd[1];
|
|
temporaryElementRule[id].x = el[0][2*i];
|
|
temporaryElementRule[id].y = el[1][2*j];
|
|
|
|
temporaryElementRule[id].weight = el[0][(2*i)+1];
|
|
temporaryElementRule[id].weight *= el[1][(2*j)+1];
|
|
|
|
if (npd[2] > 0)
|
|
{
|
|
temporaryElementRule[id].z = el[2][2*k];
|
|
temporaryElementRule[id].weight *= el[2][(2*k)+1];
|
|
}
|
|
}
|
|
}
|
|
}
|
|
|
|
return temporaryElementRule;
|
|
#else
|
|
MFEM_ABORT("Temporary integration rules on NURBS elements "
|
|
"are not thread-safe.");
|
|
#endif
|
|
}
|
|
|
|
void NURBSMeshRules::GetIntegrationPointFrom1D(const int patch, int i, int j,
|
|
int k, IntegrationPoint & ip)
|
|
{
|
|
MFEM_VERIFY(patchRules1D.NumRows() > 0,
|
|
"Assuming patchRules1D is set.");
|
|
|
|
ip.weight = (*patchRules1D(patch,0))[i].weight;
|
|
ip.x = (*patchRules1D(patch,0))[i].x;
|
|
|
|
if (dim > 1)
|
|
{
|
|
ip.weight *= (*patchRules1D(patch,1))[j].weight;
|
|
ip.y = (*patchRules1D(patch,1))[j].x; // 1D rule only has x
|
|
}
|
|
|
|
if (dim > 2)
|
|
{
|
|
ip.weight *= (*patchRules1D(patch,2))[k].weight;
|
|
ip.z = (*patchRules1D(patch,2))[k].x; // 1D rule only has x
|
|
}
|
|
}
|
|
|
|
void NURBSMeshRules::Finalize(Mesh const& mesh)
|
|
{
|
|
if ((int) pointToElem.size() == npatches) { return; } // Already set
|
|
|
|
MFEM_VERIFY(elementToRule.empty() && patchRules1D.NumRows() > 0
|
|
&& npatches > 0, "Assuming patchRules1D is set.");
|
|
MFEM_VERIFY(mesh.NURBSext, "");
|
|
MFEM_VERIFY(mesh.Dimension() == dim, "");
|
|
|
|
pointToElem.resize(npatches);
|
|
patchRules1D_KnotSpan.resize(npatches);
|
|
|
|
// First, find all the elements in each patch.
|
|
std::vector<std::vector<int>> patchElements(npatches);
|
|
|
|
for (int e=0; e<mesh.GetNE(); ++e)
|
|
{
|
|
patchElements[mesh.NURBSext->GetElementPatch(e)].push_back(e);
|
|
}
|
|
|
|
Array<int> ijk(3);
|
|
Array<int> maxijk(3);
|
|
Array<int> np(3); // Number of points in each dimension
|
|
ijk = 0;
|
|
|
|
Array<const KnotVector*> pkv;
|
|
|
|
for (int p=0; p<npatches; ++p)
|
|
{
|
|
patchRules1D_KnotSpan[p].resize(dim);
|
|
|
|
// For each patch, get the range of ijk.
|
|
mesh.NURBSext->GetPatchKnotVectors(p, pkv);
|
|
MFEM_VERIFY((int) pkv.Size() == dim, "");
|
|
|
|
maxijk = 1;
|
|
np = 1;
|
|
for (int d=0; d<dim; ++d)
|
|
{
|
|
maxijk[d] = pkv[d]->GetNKS();
|
|
np[d] = patchRules1D(p,d)->Size();
|
|
}
|
|
|
|
// For each patch, set a map from ijk to element index.
|
|
Array3D<int> ijk2elem(maxijk[0], maxijk[1], maxijk[2]);
|
|
ijk2elem = -1;
|
|
|
|
for (auto elem : patchElements[p])
|
|
{
|
|
mesh.NURBSext->GetElementIJK(elem, ijk);
|
|
MFEM_VERIFY(ijk2elem(ijk[0], ijk[1], ijk[2]) == -1, "");
|
|
ijk2elem(ijk[0], ijk[1], ijk[2]) = elem;
|
|
}
|
|
|
|
// For each point, find its ijk and from that its element index.
|
|
// It is assumed here that the NURBSFiniteElement kv the same as the
|
|
// patch kv.
|
|
|
|
for (int d=0; d<dim; ++d)
|
|
{
|
|
patchRules1D_KnotSpan[p][d].SetSize(patchRules1D(p,d)->Size());
|
|
|
|
for (int r=0; r<patchRules1D(p,d)->Size(); ++r)
|
|
{
|
|
const IntegrationPoint& ip = (*patchRules1D(p,d))[r];
|
|
|
|
const int order = pkv[d]->GetOrder();
|
|
|
|
// Find ijk_d such that ip.x is in the corresponding knot-span.
|
|
int ijk_d = 0;
|
|
bool found = false;
|
|
while (!found)
|
|
{
|
|
const real_t kv0 = (*pkv[d])[order + ijk_d];
|
|
const real_t kv1 = (*pkv[d])[order + ijk_d + 1];
|
|
|
|
const bool rightEnd = (order + ijk_d + 1) == (pkv[d]->Size() - 1);
|
|
|
|
if (kv0 <= ip.x && (ip.x < kv1 || rightEnd))
|
|
{
|
|
found = true;
|
|
}
|
|
else
|
|
{
|
|
ijk_d++;
|
|
}
|
|
}
|
|
|
|
patchRules1D_KnotSpan[p][d][r] = ijk_d;
|
|
}
|
|
}
|
|
|
|
pointToElem[p].SetSize(np[0], np[1], np[2]);
|
|
for (int i=0; i<np[0]; ++i)
|
|
for (int j=0; j<np[1]; ++j)
|
|
for (int k=0; k<np[2]; ++k)
|
|
{
|
|
const int elem = ijk2elem(patchRules1D_KnotSpan[p][0][i],
|
|
patchRules1D_KnotSpan[p][1][j],
|
|
patchRules1D_KnotSpan[p][2][k]);
|
|
MFEM_VERIFY(elem >= 0, "");
|
|
pointToElem[p](i,j,k) = elem;
|
|
}
|
|
} // Loop (p) over patches
|
|
}
|
|
|
|
void NURBSMeshRules::SetPatchRules1D(const int patch,
|
|
std::vector<const IntegrationRule*> & ir1D)
|
|
{
|
|
MFEM_VERIFY((int) ir1D.size() == dim, "Wrong dimension");
|
|
|
|
for (int i=0; i<dim; ++i)
|
|
{
|
|
patchRules1D(patch,i) = ir1D[i];
|
|
}
|
|
}
|
|
|
|
NURBSMeshRules::~NURBSMeshRules()
|
|
{
|
|
for (int i=0; i<patchRules1D.NumRows(); ++i)
|
|
for (int j=0; j<patchRules1D.NumCols(); ++j)
|
|
{
|
|
delete patchRules1D(i, j);
|
|
}
|
|
}
|
|
|
|
}
|